MEAN-SQUARE APPROXIMATION OF ITERATED ITO AND STRATONOVICH STOCHASTIC INTEGRALS: METHOD OF GENERALIZED MULTIPLE FOURIER SERIES. APPLICATION TO NUMERICAL INTEGRATION OF ITO SDES AND SEMILINEAR SPDES (THIRD EDITION)

Kuznetsov Dmitriy Feliksovich

r

dx

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DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N 1, 2023 Electronic Journal, reg. N&C77-39410 at 15.04.2010 ISSN 1817-2172

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http://diffjournal. spbu. ru /

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Stochastic differential equations Numerical methods Computer modeling in dynamical and control systems

Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple

Fourier Series. Application to Numerical Integration of Ito SDEs and Semilinear SPDEs

Peter the Great Saint-Petersburg Polytechnic University e-mail: sde_kuznetsov@inbox.ru

(Third Edition)

Dmitriy F. Kuznetsov

Dedicated to My Family

The first and second editions of this monograph is published in the Journal

" Differencialnie Uravnenia i Protsesy Upravlenia"

(Differential Equations and Control Processes),

no. 4 (2020), A.1-A.606 and no. 4 (2021), A.1-A.788

Available at:

http:/ / diffj ournal. spbu.ru ZEN; 1 numbers / '2020.4/ 'article.l.S.html

http:/ /diffj ournal. spbu.ru/ ZEN; 1 numbers / '2021.4/ 'article.l.Q.html

Author's Comments

on the Third Edition of the Book

The main difference between this (third) edition and the second edition of the book is that the third edition includes original material (Sect. 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. The above results allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. Moreover, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. In addition, the book includes new material presented in Sect. 1.11-1.13, 2.1.4, 2.27, 5.7. Among these results, we would like to especially note the results of Sect. 1.11-1.13 on the series expansion of iterated Ito stochastic integrals. The material of these sections generalizes the results obtained earlier by the author in the indicated direction and is closely related to the multiple Wiener stochastic integral introduced by Ito in 1951.

The list of references has been supplemented. Some inaccuracies have been corrected.

Preface

The book is devoted to the problem of strong (mean-square) approximation of iterated Ito and Stratonovich stochastic integrals in the context of numerical integration of Ito stochastic differential equations (SDEs) and non-commutative semilinear stochastic partial differential equations (SPDEs) with nonlinear multiplicative trace class noise. The presented monograph opens up a new direction in researching of iterated stochastic integrals and summarizes the author's research on the mentioned problem carried out in the period 1994-2023.

The basis of this book composes on the monographs [1]-[16] and recent author's results [17]-[66].

This monograph (also see books [6]-[11], [14]-[16]) is the first monograph

where the problem of strong (mean-square) approximation of iterated Ito and Stratonovich stochastic integrals is systematically analyzed in application to the numerical solution of SDEs.

For the first time we successfully use the generalized multiple Fourier series converging in the sense of norm in Hilbert space L2([t, T]k) for the expansion and strong approximation of iterated Ito stochastic integrals of arbitrary multiplicity k, k e N (Chapter 1).

This result has been adapted for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of continuously differentiable weight functions) for the Legendre polynomial system and the system of trigonometric functions (Chapter 2) as well as for some other types of iterated stochastic integrals (Chapter 1). The mentioned adaptation has also been carried out for iterated Stratonovich stochastic integrals of multiplicity k (k e N) under the condition of convergence of trace series (Chapter 2).

Two theorems on expansions of iterated Stratonovich stochastic integrals of multiplicity k (k e N) based on generalized iterated Fourier series with the pointwise convergence are formulated and proved (Chapter 2).

The integration order replacement technique for the class of iterated Ito stochastic integrals has been introduced (Chapter 3). This result is generalized for the class of iterated stochastic integrals with respect to martingales.

Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k, k £ N (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4) using the system of Legendre polynomials and the system of trigonometric functions.

The methods formulated in this book have been compared with some existing methods of strong approximation of iterated Ito and Stratonovich stochastic integrals (Chapter 6).

The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation WM of the infinite-dimensional Q-Wiener process Wt (for integrals of arbitrary multiplicity k, k £ N) and to the approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process Wt (for integrals of multiplicities 1 to 3).

This book will be interesting for specialists dealing with the theory of stochastic processes, applied and computational mathematics as well as senior students and postgraduates of technical institutes and universities.

Exact solutions of Ito SDEs and semilinear SPDEs are known in rather rare cases. Therefore, the need arises to construct numerical procedures for solving these equations.

The importance of the problem of numerical integration of Ito SDEs and semilinear SPDEs is explained by a wide range of their applications related to the construction of adequate mathematical models of dynamic systems of various physical nature under random disturbances and to the application of these equations for solving various mathematical problems, among which we mention signal filtering in the background of random noise, stochastic optimal control, stochastic stability, evaluating the parameters of stochastic systems, etc.

It is well known that one of the effective and perspective approaches to the numerical integration of Ito SDEs and semilinear SPDEs is an approach based on the stochastic analogues of the Taylor formula for solutions of these equations. This approach uses the finite discretization of temporal variable and performs numerical modeling of solutions of Ito SDEs and semilinear SPDEs in

discrete moments of time using stochastic analogues of the Taylor formula.

Speaking about Ito SDEs, note that the most important feature of the mentioned stochastic analogues of the Taylor formula for solutions of Ito SDEs is a presence in them of the so-called iterated Ito and Stratonovich stochastic integrals which are the functionals of a complex structure with respect to components of the multidimensional Wiener process. These iterated Ito and Stratonovich stochastic integrals are subject for study in this book and are defined by the following formulas T t

tk (tk)... i t (i1)dwt(;i)... dwt(!fc) (Ito integrals),

-.T

(tk).. .J ^i(ti)dwt(|l)... (Stratonovich integrals),

t

where ^1(r),...,(t) : [t,T] ^ R are nonrandom functions (as a rule, in the applications they are identically equal to 1 or have a binomial form (see Chapter 4)), wT is a random vector with an m + 1 components: wT^ = f-^ for i = 1,..., m and wT0) = t, f-j^ (i = 1,..., m) are independent standard Wiener processes, i1,..., ik = 0,1,..., m.

Apparently, one of the first who began the study of such stochastic integrals (the case k = 2, m = 2, ^1(t),^2(t) = 1, i1 = 1, i2 = 2) was Levy, who introduced the concept of the so-called Levy stochastic area and studied its properties.

The above iterated stochastic integrals are the specific objects in the theory of stochastic processes. From the one side, nonrandomness of weight functions ^(t) (/ = 1,..., k) is the factor simplifying their structure. From the other side, nonscalarity of the Wiener process fT with independent components fT^ (i = 1,..., m) and the fact that the functions ^(t) (/ = 1,..., k) are different for various I (/ = 1,..., k) are essential complicating factors of the structure of iterated stochastic integrals. Taking into account features mentioned above, we suppose that the systems of iterated Ito and Stratonovich stochastic integrals play the extraordinary and perhaps the key role for solving the problem of numerical integration of Ito SDEs.

A natural question arises: is it possible to construct a numerical scheme for Ito SDE that includes only increments of the Wiener processes fT^ (i = 1,..., m), but has a higher order of convergence than the Euler method? It

is known that this is impossible for m > 1 in the general case. This fact is called the "Clark-Cameron paradox" [67] and explains the need to use iterated stochastic integrals for constructing high-order numerical methods for Ito SDEs.

We want to mention in short that there are two main criteria of numerical methods convergence for Ito SDEs: a strong or mean-square criterion and a weak criterion where the subject of approximation is not the solution of Ito SDE, simply stated, but the distribution of Ito SDE solution. Both mentioned criteria are independent, i.e. in general it is impossible to state that from the execution of strong criterion follows the execution of weak criterion and vice versa. Each of two convergence criteria is oriented on the solution of specific classes of mathematical problems connected with Ito SDEs.

Numerical integration of Ito SDEs based on the strong convergence criterion of approximation is widely used for the numerical simulation of sample trajectories of solutions to Ito SDEs (which is required for constructing new mathematical models based on such equations and for the numerical solution of different mathematical problems connected with Ito SDEs). Among these problems, we note the following: signal filtering under influence of random noises in various statements (linear Kalman-Bucy filtering, nonlinear optimal filtering, filtering of continuous time Markov chains with a finite space of states, etc.), optimal stochastic control (including incomplete data control), testing estimation procedures of parameters of stochastic systems, stochastic stability and bifurcations analysis.

The problem of effective jointly numerical modeling (with respect to the mean-square convergence criterion) of iterated Ito or Stratonovich stochastic integrals is very important and difficult from theoretical and computing point of view.

Seems that iterated stochastic integrals may be approximated by multiple integral sums. However, this approach implies the partitioning of the interval of integration [t, T] for iterated stochastic integrals. The length T — t of this interval is already fairly small (because it is a step of integration of numerical methods for Ito SDEs) and does not need to be partitioned. Computational experiments show that the application of numerical simulation for iterated stochastic integrals (in which the interval of integration is partitioned) leads to unacceptably high computational cost and accumulation of computation errors.

The problem of effective decreasing of the mentioned cost (in several times or even in several orders) is very difficult and requires new complex investigations.

The only exception is connected with a narrow particular case, when i1 = ... = ik = 0 and (t),... (t) = ^(t). This case allows the investigation with using of the Ito formula. In the more general case, when not all numbers i1,...,ik are equal, the mentioned problem turns out to be more complex (it cannot be solved using the Ito formula and requires more deep and complex investigation). Note that even for the case i1 = ... = ik = 0, but for different functions ^1(t),... (t) the mentioned difficulties persist and simple sets of iterated Ito and Stratonovich stochastic integrals, which can be often met in the applications, cannot be expressed effectively in a finite form (with respect to the mean-square approximation) using the system of standard Gaussian random variables. The Ito formula is also useless in this case and as a result we need to use more complex but effective expansions.

Why the problem of the mean-square approximation of iterated stochastic integrals is so complex?

Firstly, the mentioned stochastic integrals (in the case of fixed limits of integration) are the random variables, whose density functions are unknown in the general case. The exception is connected with the narrow particular case which is the simplest iterated Ito stochastic integral with multiplicity 2 and ^(t), ^2(t) = 1; i1,i2 = 1,..., m. Nevertheless, the knowledge of this density function not gives a simple way for approximation of iterated Ito stochastic integral of multiplicity 2.

Secondly, we need to approximate not only one stochastic integral, but several iterated stochastic integrals that are complexly dependent in a probabilistic sense.

Often, the problem of combined mean-square approximation of iterated Ito and Stratonovich stochastic integrals occurs even in cases when the exact solution of Ito SDE is known. It means that even if you know the solution of Ito SDE exactly, you cannot model it numerically without the combined numerical modeling of iterated stochastic integrals.

Note that for a number of special types of Ito SDEs the problem of approximation of iterated stochastic integrals may be simplified but cannot be solved. Equations with additive vector noise, with non-additive scalar noise, with additive scalar noise, with a small parameter are related to such types of equations. In these cases, simplifications are connected to the fact that some members from stochastic Taylor expansions are equal to zero or we may neglect some members from these expansions due to the presence of a small parameter.

Furthermore, the problem of combined numerical modeling (with respect to the mean-square convergence criterion) of iterated Ito and Stratonovich stochastic integrals is rather new.

One of the main and unexpected achievements of this book is the successful usage of functional analysis methods (more concretely, we mean generalized multiple and iterated Fourier series (convergence in L2([t,T]k) and pointwise correspondently) through various systems of basis functions) in this scientific field.

The problem of combined numerical modeling (with respect to the mean-square convergence criterion) of systems of iterated Ito and Stratonovich stochastic integrals was analyzed in the context of the problem of numerical integration of Ito SDEs in the following monographs:

[I] Milstein G.N. Numerical Integration of Stochastic Differential Equations. Kluwer Academic Publishers. Dordrecht. 1995 (Russian Ed. 1988).

[II] Kloeden P.E., Platen E. Numerical Solution of Stochastic Differential Equations. Springer-Verlag. Berlin. 1992 (2nd Ed. 1995, 3rd Ed. 1999).

[III] Milstein G.N., Tretyakov M. V. Stochastic Numerics for Mathematical Physics. Springer-Verlag. Berlin. 2004.

[IV] Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. Polytechnical University Publ. St.-Petersburg. 2007 [2] (2nd Ed. 2007 [3], 3rd Ed. 2009 [4], 4th Ed. 2010 [5], 5th Ed. 2017 [12], 6th Ed. 2018 [13]).

Note that the initial version of the book [IV] has been published in 2006 [1]. Also we mention the books [6] (2010), [7] (2011), [8] (2011), [9] (2012), [10" (2013), [11] (2017) and [14] (2020), [15] (2021), [16] (2022).

The books [I] and [III] analyze the problem of the mean-square approximation of iterated stochastic integrals only for two simplest iterated Ito stochastic integrals of 1st and 2nd multiplicities (k = 1 and 2, ^1(t) and ^2(t) = 1) for the multidimensional case: i1, i2 = 0, 1, . . . , m. In addition, the main idea is based on the expansion of the so-called Brownian bridge process into the trigonometric Fourier series (version of the so-called Karhunen-Loeve expansion). This method is called in [I] and [III] as the Fourier method1.

In [II] using the Fourier method [I], the attempt was made to obtain the

1To date, there is confusion in the literature about who first proposed the Fourier method [I], [III]. As far as the author of this book knows, the mentioned method first appeared in the Russian edition of the monograph by G.N. Milstein [77] (pp. 121-135), which was published in 1988.

mean-square approximation of elementary iterated Stratonovich stochastic integrals of multiplicities 1 to 3 (k = 1,...,3, (t),...,^3(t) = 1) for the multidimensional case: i1, . . . , i3 = 0, 1, . . . , m. However, as we can see in the presented book, the results of the monograph [II], related to the mean-square approximation of iterated Stratonovich stochastic integrals of 3rd multiplicity, cause a number of critical remarks (see discussions in Sect. 2.20, 2.21, 6.2).

The main purpose of this book is to construct and develop newer and more effective methods (than presented in the books [I]—[III]) of combined mean-square approximation of iterated Ito and Stratonovich stochastic integrals.

Talking about the history of solving the problem of combined mean-square approximation of iterated stochastic integrals, the idea to find a basis of random variables using which we may represent iterated stochastic integrals turned out to be useful. This idea was transformed several times during last decades.

Attempts to approximate the iterated stochastic integrals using various integral sums were made until 1980s and later, i.e. the interval of integration [t, T] of the stochastic integral was divided into n parts and the iterated stochastic integral was represented approximately by the multiple integral sum, which included the system of independent standard Gaussian random variables, whose numerical modeling is not a problem.

However, as we noted above, it is obvious that the length T — t of integration interval [t, T] of the iterated stochastic integrals is a step of integration of numerical methods for Ito SDEs, which is already a rather small value even without the additional splitting. Numerical experiments demonstrate that such approach results in drastic increasing of computational costs accompanied by the growth of multiplicity of the stochastic integrals (beginning from 2nd and 3rd multiplicity) that is necessary for construction of high-order strong numerical methods for Ito SDEs or in the case of decrease of integration step of numerical methods, and thereby it is almost useless for practice.

The new step for solution of the problem of combined mean-square approximation of iterated stochastic integrals was made by Milstein G.N. in his monograph [I] (1988). For the expansion of iterated stochastic integrals, he proposed to use the trigonometric Fourier expansion of the Brownian bridge process (version of the so-called Karhunen-Loeve expansion). Using this method, expansions of two simplest iterated Ito stochastic integrals of multiplicities 1 and 2 are obtained and their mean-square convergence is proved.

As we noted above, the attempt to develop this idea together with the

Wong-Zakai approximation [68]-[70] was made in the monograph [II] (1992), where the expansions of simplest iterated Stratonovich stochastic integrals of multiplicities 1 to 3 were obtained. However, due to a number of limitations and technical difficulties which are typical for the method [I], in [II] and following publications this problem was not solved more completely. In addition, the author has reasonable doubts about application of the Wong-Zakai results 68]-[70] to approximation of iterated Stratonovich stochastic integrals of 3rd multiplicity in the monograph [II] (see discussions in Sect. 2.20, 2.21, 6.2).

It is necessary to note that the computational cost for the method [I] is significantly less than for the method of multiple integral sums.

Regardless of the method [I] positive features, the number of its limitations are also outlined. Among them let us mention the following.

1. The absence of explicit formula for calculation of expansion coefficients for iterated stochastic integrals.

2. The practical impossibility of exact calculation of the mean-square approximation error of iterated stochastic integrals with the exception of simplest integrals of 1st and 2nd multiplicity (as a result, it is necessary to consider redundant terms of expansions and it results to the growth of computational cost and complication of the numerical methods for Ito SDEs).

3. There is a hard limitation on the system of basis functions — it may be only the trigonometric functions.

4. There are some technical problems if we use this method for iterated stochastic integrals whose multiplicity is greater than 2nd.

Nevertheless, it should be noted that the analyzed method is a concrete step forward in this scientific field.

The author thinks that the method presented by him in [IV] (for the first time this method is appeared in the final form in [1] (2006)) and in this book (hereafter this method is reffered to as the method of generalized multiple Fourier series) is a breakthrough in solution of the problem of combined mean-square approximation of iterated Ito stochastic integrals.

The idea of this method is as follows: the iterated Ito stochastic integral of multiplicity k (k £ N) is represented as the multiple stochastic integral from the certain nonrandom discontinuous function of k variables defined on the hypercube [t,T]k, where [t,T] is the interval of integration of the iterated Ito stochastic integral. Then, the mentioned nonrandom function of k variables is expanded in the hypercube [t,T]k into the generalized multiple Fourier series

converging in the mean-square sense in the space L2([t,T]k). After a number of nontrivial transformations we come to the mean-square converging expansion of the iterated Ito stochastic integral into the multiple series of products of standard Gaussian random variables. The coefficients of this series are the coefficients of generalized multiple Fourier series for the mentioned nonrandom function of k variables, which can be calculated using the explicit formula regardless of the multiplicity k of the iterated Ito stochastic integral.

As a result, we obtain the following new possibilities and advantages in comparison with the Fourier method [I].

1. There is an explicit formula for calculation of expansion coefficients of iterated Ito stochastic integral with any fixed multiplicity k. In other words, we can calculate (without any preliminary and additional work) the expansion coefficient with any fixed number in the expansion of iterated Ito stochastic integral of the preset fixed multiplicity. At that, we do not need any knowledge about coefficients with other numbers or about other iterated Ito stochastic integrals included in the considered set.

2. We have new possibilies for obtainment the exact and approximate expressions for the mean-square approximation errors of iterated Ito stochastic integrals. These possibilities are realized by the exact and estimate formulas for the mentioned mean-square approximation errors. As a result, we would not need to consider redundant terms of expansions that may complicate approximations of iterated Ito stochastic integrals.

3. Since the used multiple Fourier series is a generalized in the sense that it is built using various complete orthonormal systems of functions in the space L2([t,T]k), we have new possibilities for approximation — we can use not only the trigonometric functions as in [I] but the Legendre polynomials as well as the systems of Haar and Rademacher-Walsh functions.

4. As it turned out, it is more convenient to work with Legendre polynomials for approximation of iterated Ito stochastic integrals. The approximations themselves are simpler than their analogues based on the system of trigonometric functions. Probably for the systems of Haar and Rademacher-Walsh functions the expansions of iterated stochastic integrals become more complex and less effective for practice [IV]. Expansions based on Haar functions for k = 2 were also considered in [82], [90], [201]. Note that the multiple Fourier-Walsh

and Fourier-Haar series (k G N) were applied to the mean-square approximation of multiple Stratonovich stochastic integrals (defined as in [128], [129]) in 200]. The convergence of these approximations was proved with respect to the

special subsequence nm = 2m (m ^ to) [200 .

5. The question about what kind of functions (polynomial or trigonometric) is more convenient in the context of computational costs for approximation turns out to be nontrivial, since it is necessary to compare approximations not for one stochastic integral but for several stochastic integrals at the same time. At that there is a possibility that computational costs for some integrals will be smaller for the system of Legendre polynomials and for others — for the system of trigonometric functions. The author proved [20] (also see Sect. 5.3 in this book) that the computational costs are significantly less for the system of Legendre polynomials at least in the case of approximation of the special set of iterated Ito stochastic integrals, which are necessary for the implementation of strong numerical methods for Ito SDEs with the order of convergence 7 = 1.5. In addition, the author supposes that this effect will be more impressive when analyzing more complex sets of iterated Ito stochastic integrals (7 = 2.0, 2.5, 3.0, ...). This supposition is based on the fact that the polynomial system of functions has a significant advantage (in comparison with the trigonometric system of functions) in the mean-square approximation of iterated Ito stochastic integrals for which not all weight functions are equal to 1.

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6. The Milstein approach [I] for approximation of iterated Ito stochastic integrals leads to iterated applicaton of the operation of limit transition (in contrast with the method of generalized multiple Fourier series, for which the operation of limit transition is implemented only once) starting at least from the second or third multiplicity of iterated Ito stochastic integrals (we mean at least double or triple integration with respect to components of the multidimensional Wiener process). Multiple series are more preferential for approximation than the iterated ones, since the partial sums of multiple series converge for any possible case of joint converging to infinity of their upper limits of summation (let us denote them as p1,... ,pk). For example, when p1 = ... = pk = p ^ to. For iterated series, the condition p1 = ... = pk = p ^ to obviously does not guarantee the convergence of this series. However, in [II] the authors use (without rigorous proof) the condition p1 = p2 = p3 = p ^ to within the frames of the Milstein approach [I] together with the Wong-Zakai approximation [68 -70] (see discussions in Sect. 2.20, 2.21, 6.2).

7. The convergence in the mean of degree 2n (n £ N) as well as the convergence with probability 1 of approximations from the method of generalized multiple Fourier series are proved. The convergence rate for these two types of convergence is estimated.

8. The method of generalized multiple Fourier series has been applied for some other types of iterated stochastic integrals (iterated stochastic integrals with respect to martingale Poisson random measures and iterated stochastic integrals with respect to martingales) as well as for approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process.

9. Another modification of the method of generalized multiple Fourier series is connected with the application of complete orthonormal with weight r(ti)... r(tk) > 0 systems of functions in the space L2([t,T]k).

10. As it turned out, the method of generalized multiple Fourier series can be adapted for iterated Stratonovich stochastic integrals. This adaptation is carried out in Chapter 2 for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of continuously differentiable weight functions) and multiplicity k (the case of convergence of trace series). At the same time, for the approximation of the mentioned iterated Stratonovich stochastic integrals of multiplicities 1 to 6, we used the complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]). The rate of mean-square convergence of approximations of iterated Stratonovich stochastic integrals is found (Sect. 2.8, 2.15, 2.16).

11. The method of generalized multiple Fourier series is reformulated using Hermite polynomials in Sect. 1.10 and generalized to the case of an arbitrary complete orthonormal systems of functions in the space L2([t,T]) and ^i(r),...,^k(t) G L2([t,T]) in Sect. 1.11, 1.12. At that, in Sect. 1.11, 1.12 we use the multiple Wiener stochastic integral with respect to the components of a multidimensional Wiener process.

12. The results of Chapter 1 (Theorems 1.1, 1.2, 1.14, 1.16) and Chapter 2 (Theorems 2.1-2.10, 2.14, 2.17, 2.30, 2.32-2.35, 2.40) can be considered from the point of view of the Wong-Zakai approximation [68]-[70] for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions (see discussions in Sect. 2.20, 2.21, 6.2). These results overcome a number of difficulties that were noted above and relate to the Fourier method [I].

The theory presented in this book was realized [52], [53] in the form of a

software package in the Python programming language. The mentioned software package implements the strong numerical methods with convergence orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with multidimensional non-commutative noise based on the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). At that for the numerical simulation of iterated Ito

and Stratonovich stochastic integrals of multiplicities 1 to 6 we applied the formulas based on multiple Fourier-Legendre series (Chapter 5). Moreover, we used [52], [53] the database with 270,000 exactly calculated Fourier-Legendre

coefficients.

Throughout the book, special attention is paid to two systems of basis functions in the space L2([t, T]). Namely, we mainly use the complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]). This is due to two reasons. The first of these is that the trigonometric basis system has already been used to approximate iterated stochastic integrals in the 1980s-1990s (see above), and the author needed to compare his results with the results of other authors. The second reason is that the system of Legendre polynomials is optimal (see Sect. 5.3) for the implementation of strong numerical methods with convergence order 1.5 and higher for Ito SDEs with multidimensional non-commutative noise. The system of Legendre polynomials was first applied to the approximation of iterated stochastic integrals in the author's work [71] in 1997 (also see [72]-[74]). According to the author's

opinion, other complete orthonormal systems of functions in the space L2([t, T]) (for example, systems of Haar and Rademacher-Walsh functions) turn out to be less efficient for the mean-square approximation of iterated Ito and Stratonovich stochastic integrals.

The attentive reader will notice that Chapters 1 and 2 of this book can be somewhat shortened since Theorem 1.16 is a generalization of Theorems 1.1, 1.2 and Theorems 2.32, 2.33, 2.40 are generalizations of Theorems 2.4-2.9. However, the author did not make the appropriate changes in Chapters 1, 2 for a number of reasons. In particular, the application of the multiple Wiener stochastic integral with respect to the components of a multidimensional Wiener process to the expansion of iterated Ito stochastic integrals (Theorem 1.16) and a new approach to the expansion of iterated Stratonovich stochastic integrals (Theorems 2.30-2.35, 2.40) were obtained by the author recently (in 20212022), while Theorems 1.1, 1.2, 2.4-2.9 were obtained by the author in the period from 2005 to 2013. In addition, the proof of each of the mentioned theorems contains some original ideas that the author would like to keep in Chapters 1 and 2. Thus, the results of Chapters 1, 2 are presented primarily in the order in which they were obtained by the author.

Dmitriy F. Kuznetsov March, 2023

Acknowledgements

I would like to thank the Deputy Editor of the Journal "Differencialnie Uravnenia i Protsesy Upravlenia" Dr. Nataly B. Ampilova for her timely administrative support and encouragement and Dr. Konstantin A. Rybakov for useful discussion of some presented results.

Basic Notations

N

R, R1 Rn

(ai,... ,an)

|«i,... ,an}

n!

(2n - 1)!!

def

rim Cn

0

1a

x G X X U Y X x Y lim

n^œ

lim

n^œ

x C y

[x]

set of natural numbers

set of real numbers

n-dimensional Euclidean space

ordered set with elements a1,..., an

unordered set with elements a1,..., an

1 ■ 2 ■ ... ■ n for n G N (0! = 1) 1 ■ 3 ■ ... ■ (2n - 1) for n G N

equal by definition

identically equal to

binomial coefficient n!/(m!(n — m)!)

empty set

indicator of the set A

x is an element of the set X

union of sets X and Y

Cartesian product of sets X and Y

lim sup

n^œ

lim inf

x much less than y

largest integer number not exceeding x

|x|

F : X^Y

absolute value of the real number x

function F from X into Y

A(ij)

Ai

x(i) O(x)

E

(il,-..,» k )

Mie}

Mie |F}

e - N(m, a2)

l.i.m.

B(X ) ft ft

w. p. 1 Wt

dF

dx(i)

<92F

dx(i) 3x(j)

T

ijth element of the matrix A

ith colomn of the matrix A

ith component of the vector x G Rn

expression being divided by x remains bounded as x ^ 0 sum with respect to all possible permutations (¿i,..., ik)

expectation of e

conditional expectation of £ with respect to F

Gaussian random variable £ with expectation m and variance a2

limit in the mean-square sense

a-algebra of Borel subsets of X

scalar standard Wiener process

vector standard Wiener process with independent components ft(i), i = 1,..., m

with probability 1

vector with components w(i), i = 0,1,...,m and property

(i) (i)

(0)

t

wf = ft( ) for i = 1,..., m and wt

partial derivative of F : Rn ^ R

2nd order partial derivative of F : Rn ^ R

dwTi)

Ito stochastic integral

J

... dw«

Stratonovich stochastic integral

T

...

dw«

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t

Wt

J№(fc)kt, /¡fcS

(¿1 ...Zfc)T,t

I *(il...ifc)

t' J(¿i...¿fc)T,t

j*[^(fc)]T,t, /;(ii-ifc)

JS [^(fc)](i1-ik)

JT,t

J [^(fc)]

P1, — ,Pk T(i l-^fc )p T,t

(Zi...Zfc )T,t

j*[«,t,

JM(t, J[^]Ti>1t...ifc) j'[$]Tfc], J'[$grifc)

Pn(x)

H„(x), hn(x) L2(D)

h"nl2(d) tr A

ii'iih

(u, V)h Lhs (U,H)

Stratonovich stochastic integral

Q-Wiener process

iterated Ito stochastic integrals

iterated Stratonovich stochastic integrals

iterated Stratonovich stochastic integral

multiple Stratonovich stochastic integral

approximations of iterated Ito stochastic integrals

approximations of iterated Stratonovich stochastic integrals

multiple Stratonovich stochastic integrals

multiple Wiener stochastic integrals

Legendre polynomials

Hermite polynomials

polynomials related to the Hermite polynomials

Hilbert space of square integrable functions on D norm in the Hilbert space L2(D)

trace of the operator A

norm in the Hilbert space H

scalar product in the Hilbert space H

space of Hilbert-Schmidt operators from U to H

operator norm in the space of Hilbert-Schmidt operators from U to H

stochastic integral with respect to the Q-Wiener process

H'IlLffs (U,H ) T

J... dWr

t

Contents

Preface

Acknowledgements

Basic Notations

17

18

1 Method of Expansion and Mean-Square Approximation of Iterated Ito Sto-

1.1.2

1.1.3

1.1.4

1.1.5

1.1.6

1.1.7

1.1.8

Expansion of Iterated Ito Stochastic Integrals of Multiplicity k (k G N) Based on Theorem 1.1...........................

Comparison of Theorem 1.2 with the Representations of Iterated Ito Stochastic Integrals Based on Hermite Polynomials...........

On Usage of Discontinuous Complete Orthonormal Systems of Functions in Theorem 1.1 ...............................

Remark on Usage of Complete Orthonormal Systems of Functions in Theorem 1.1 .................................

1.2

30

31

chastic Integrals Based on Generalized Multiple Fourier Series

1.1 Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean..........

1.1.1 Introduction........... ....................... 31

Ito Stochastic Integral............................ 32

Theorem on Expansion of Iterated Ito Stochastic Integrals of Multiplicity k (k G N)................................... 34

Expansions of Iterated Itô Stochastic Integrals with Multiplicities 1 to 7 Based on Theorem 1.1 ...........................

49

60

66

72

1.1.9 Convergence in the Mean of Degree 2n (n G N) of Expansions of Iterated

Ito Stochastic Integrals from Theorem 1.1 ................. 73

1.1.10 Conclusions.................................. 78

Exact Calculation of the Mean-Square Error in the Method of Approximation of Iterated Ito Stochastic integrals Based on Generalized Multiple Fourier Series .

1.2.1 Introduction .................................

1.2.2 Theorem on Exact Calculation of the Mean-Square Approximation Error

for Iterated Itô Stochastic integrals..................... 81

80 80

1.2.3 Exact Calculation of the Mean-Square Approximation Errors for the

Cases k = 1, . . . , 5 ............................... 89

1.2.4 Estimate for the Mean-Square Approximation Error of Iterated Ito

Stochastic Integrals Based on Theorem 1.1.................105

1.3 Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series. The Case of Complete Orthonormal with Weight r(ti)... r(tk) Systems of Functions in the Space L2([t,T]k)....................109

1.4 Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson

Measures Based on Generalized Multiple Fourier Series .............. 11

1.4.1 Stochastic Integral with Respect to Martingale Poisson Measure.....115

1.4.2 Expansion of Iterated Stochastic Integrals with Respect to Martingale

Poisson Measures...............................118

1.5 Expansion of Iterated Stochastic Integrals with Respect to Martingales Based

on Generalized Multiple Fourier Series ....................... 12

1.5.1 Stochastic Integral with Respect to Martingale...............12!

1.5.2 Expansion of Iterated Stochastic Integrals with Respect to Martingales . 12

1.6 One Modification of Theorems 1.5 and 1.8.....................134

1.6.1 Expansion of Iterated Stochastic Integrals with Respect to Martingales Based on Generalized Multiple Fourier Series. The Case p(x)/r(x) < œ . 134

1.6.2 Example on Application of Theorem 1.9 and the System of Bessel Functions 136

1.7 Convergence with Probability 1 of Expansions of Iterated Itoô Stochastic Integrals

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in Theorem 1.1 .................................... 139

1.7.1 Convergence with Probability 1 of Expansions of Iterated Itôo Stochastic Integrals of Multiplicities 1 and 2......................139

1.7.2 Convergence with Probability 1 of Expansions of Iterated Itôo Stochastic Integrals of Multiplicity k (k G N) .....................[144

1.7.3 Rate of Convergence with Probability 1 of Expansions of Iterated Itoô Stochastic Integrals of Multiplicity k (k G N) ...............163

1.8 Modification of Theorem 1.1 for the Case of Integration Interval [t, s] (s G (t, T])

of Iterated Itôo Stochastic Integrals .......... ................164

1.8.1 Formulation and Proof of Theorem 1.1 Modification ............ 164

1.8.2 Expansions of Iterated Itôo Stochastic Integrals with Multiplicities 1 to 5

and Miltiplicity k Based on Theorem 1.11 ................. 17

1.9 Expansion of Multiple Wiener Stochastic Integral Based on Generalized Multiple

Fourier Series.....................................175

1.10 Reformulation of Theorems 1.1, 1.2, and 1.13 Using Hermite Polynomials .... 179

1.11 Generalization of Theorems 1.1, 1.2, 1.14, and 1.15 to the Case of an Arbitrary Complete Orthonormal System of Functions in the Space L2([t,T]) and (t),

^k(t) G L([t,T]), $(ti,...,tfc) G L([t,T]k)..................19

1.12 Generalization of Theorems 1.3, 1.4 to the Case of an Arbitrary Complete Orthonormal System of Functions in the Space L2([t, T]) and ^1(t),..., (t) G L([t,T])205

1.13 Generalization of Theorems 1.5, 1.6 to the Case of an Arbitrary Complete Orthonormal with Weight r(x) > 0 System of Functions in the Space L2([i,T]) and

^(аО^/ф), (z)VT(X) G L([t,T]) .....................208

Expansions of Iterated Stratonovich Stochastic Integrals Based on Generalized Multiple and Iterated Fourier Series

2.1 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicity 2 Based

210

on Theorem 1.1. The case pi ,p2 ^ ^ and Smooth Weight Functions......210

2.1.1 Approach Based on Theorem 1.1 and Integration by Parts........210

2.1.2 Approach Based on Theorem 1.1 and Double Fourier-Legendre Series

Summarized by Pringsheim Method.....................221

2.1.3 Approach Based on Generalized Double Multiple and Iterated Fourier Series ....................................

248

2.2

2.1.4 Approach Based on Arbitrary Complete Orthonormal Systems of Functions in the Space L2([t, T]) .........................253

Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Theorem 1.1....................................

2.2.1 The Case рьр2,р3 ^ то and Constant Weight Functions (The Case of

25

2.2.2

2.2.3

2.2.4

Legendre Polynomials)............................258

The Case p^p2,p3 ^ œ, Binomial Weight Functions, and Additional Restrictive Conditions (The Case of Legendre Polynomials).......

The Case p1 ,p2,p3 ^ œ and Constant Weight Functions (The Case of

27;

Trigonometric Functions)...........................290

The Case p1 = p2 = p3 ^ œ, Smooth Weight Functions, and Additional Restrictive Conditions (The Cases of Legendre Polynomials and

2.2.5

2.3

2.4

Trigonometric Functions)...........................299

The Case p1 = p2 = p3 ^ то, Smooth Weight Functions, and without Additional Restrictive Conditions (The Cases of Legendre Polynomials and Trigonometric Functions).......................

Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4 Based on Theorem 1.1. The Case p1 = ... = p4 ^ то (Cases of Legendre Polynomials

30

and Trigonometric Functions) ............................ 316

Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k G N) Based on Generalized Iterated Fourier Series Converging Pointwise . . . '.....34

2.4.1 Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of

Multiplicity k (k G N)............................34

2.4.2

2.4.3

Further Remarks...............................378

Refinement of Theorems 2.10 and 2.14 for Iterated Stratonovich Stochastic Integrals of Multiplicities 2 and 3 (i1,i2,i3 = 1,... ,m). The Case of

2.5

Mean-Square Convergence..........................390

The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of

Multiplicity k (k G N) Based on Theorem 1.1 ...................398

2

2.6 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 3 and 4. Combained Approach Based on Generalized Multiple and Iterated Fourier

series. Another Proof of Theorems 2.8 and 2.9...................402

2.6.1 Another Proof of Theorem 2.8........................40

2.6.2 Another Proof of Theorem 2.9........................408

2.7 Modification of Theorems 2.2, 2.8, and 2.9 for the Case of Integration Interval [t, s] (s G (t,T]) of Iterated Stratonovich Stochastic Integrals of Multiplicities 2

to 4 and Wong-Zakai Type Theorems........................1421

2.7.1 Modification of Theorem 2.2 for the Case of Integration Interval [t, s]

(s G (t,T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 2 421

2.7.2 Modification of Theorem 2.8 for the Case of Integration Interval [t, s]

(s G (t,T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 42

2.7.3 Modification of Theorem 2.9 for the Case of Integration Interval [t, s]

(s G (t,T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 4 434

2.8 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich

Stochastic Integrals of Multiplicities 2 to 4 in Theorems 2.2, 2.8, and 2.9 .... 448

2.8.1 Rate of the Mean-Square Convergence of Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2................448

2.8.2 Rate of the Mean-Square Convergence of Expansion of Iterated Strato-

novich Stochastic Integrals of Multiplicity 3................450

2.8.3 Rate of the Mean-Square Convergence of Expansion of Iterated Strato-

novich Stochastic Integrals of Multiplicity 4 ................ 454

2.9 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 4 in Theorems 2.18, 2.20, and 2.22

(The Case of Integration Interval [t, s] (s G (t,T])).................466

2.10 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity k (k G N). Proof of Hypothesis 2.2 Under the Condition of Convergence of Trace Series..........................................471

2.11 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3. The Case p1 = p2 = p3 ^ œ and Continuously Differentiable Weight Functions "01 (t), "2 (t), "3 (t) (The Cases of Legendre Polynomials and Trigonometric Func-

tions) .......................................... 509

2.12 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4. The Case p1 = ... = p4 ^ œ and Continuously Differentiable Weight Functions "1(t), ..., "4(t) (The Cases of Legendre Polynomials and Trigonometric Func-

tions) .......................................... 515

2.13 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 5. The Case p1 = ... = p5 ^ œ and Continuously Differentiable Weight Functions "1(t), ..., "5(t) (The Cases of Legendre Polynomials and Trigonometric Func-

tions) .......................................... 528

2.14 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 6. The Case p1 = ... = p6 — œ and ^1(r),... , ^б(т) = 1 (The Cases of Legendre Polynomials and Trigonometric Functions) ..................... 54

2.15 Estimates for the Mean-Square Approximation Error of Iterated Stratonovich

Stochastic Integrals of Multiplicity k in Theorems 2.30, 2.31 ........... 578

2.16 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich

Stochastic Integrals of Multiplicities 3-5 in Theorems 2.32-2.34 .........1581

2.17 Generalization of Theorems 2.4-2.8. The Case pi, p2, p3 — to and Continuously Differetiable Weight Functions (The Cases of Legendre Polynomials and

Trigonometric Functions). Proof of Hypothesis 2.3 for the Case k = 3......|58

2.18 Generalization of Theorem 2.30 for Complete Orthonormal Systems of Functions in L2([t,T]) and ^1(r),...,^fc(t) G L2([t,T]) such that Condition 3 of

Theorem 2.30 is Satisfied...............................589

2.19 Algorithm of the Proof of Hypothesis 2.2 ...................... 59

2.20 Theorems 2.1-2.9, 2.32-2.35, 2.40 on Expansion of Iterated Stratonovich

Stochastic Integrals from Point of View of the Wong-Zakai Approximation . . .

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2.21 Wong-Zakai Type Theorems for Iterated Stratonovich Stochastic Integrals. The Case of Approximation of the Multidimensional Wiener Process Based on its

Series Expansion Using Legendre Polynomials and Trigonometric Functions . . . 612 2.22 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k. The

Case i1 = ... = = 0 and Different Weight Functions ^1(r),..., (т).....1622

2.23 Comparison of Theorems 2.2 and 2.7 with the Representations of Iterated Stratonovich Stochastic Integrals With Respect to the Scalar Standard Wiener

Process ......................................... 628

2.24 One Result on the Expansion of Multiple Stratonovich Stochastic Integrals of Multiplicity k. The Case i1 = ... = = 1,..., m .................1631

2.25 A Different Look at Hypotheses 2.1-2.3 on the Expansion of Iterated Stratono-

vich Stochastic Integrals of Multiplicity k......................635

2.26 Invariance of Expansions of Iterated Ito and Stratonovich Stochastic Integrals

from Theorems 1.1 and 2.58 ............................. 639

2.27 Expansion of Multiple Stratonovich Stochastic Integrals of Arbitrary Multiplicity k. The case of a multidimensional Wiener process and a smooth function Ф(*1,...,4 )......................................645

3 Integration Order Replacement Technique for Iterated Ito Stochastic Inte^

grals and Iterated Stochastic Integrals with Respect to Martingales 648

3.1 Introduction ...................................... 648

3.2 Formulation of the Theorem on Integration Order Replacement for Iterated Ito Stochastic Integrals of Multiplicity k (k £ N) ...................65;

3.3 Proof of Theorem 3.1 for the Case of Iterated Ito Stochastic Integrals of Multi-

plicity 2........................................654

3.4 Proof of Theorem 3.1 for the Case of Iterated Ito Stochastic Integrals of Multi-

plicity k (k G N) ...................................659

3.5 Corollaries and Generalizations of Theorem 3.1 ..................66i

3.6 Examples of Integration Order Replacement Technique for the Concrete Iterated

Itô Stochastic Integrals................................66'

3.7 Integration Order Replacement Technique for Iterated Stochastic Integrals with

Respect to Martingale ................................671

4 Four New Forms of the Taylor—Ito and Taylor—Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochas-

tic Differential Equations 678

4.1 Introduction......................................678

4.2 Auxiliary Lemmas .................................. 684

4.3 The Taylor-Ito Expansion..............................1688

4.4 The First Form of the Unified Taylor-Ito Expansion................1691

4.5 The Second Form of the Unified Taylor-Ito Expansion ..............1694

4.6 The Taylor-Stratonovich Expansion.........................1696

4.7 The First Form of the Unified Taylor-Stratonovich Expansion..........1699

4.8 The Second Form of the Unified Taylor-Stratonovich Expansion.........170

4.9 Comparison of the Unified Taylor-Itô and Taylor-Stratonovich Expansions with

the Classical Taylor-Ito and Taylor-Stratonovich Expansions...........L70i

4.10 Application of First Form of the Unified Taylor-Ito Expansion to the High-Order

Strong Numerical Methods for Ito SDEs......................710

4.11 Application of First Form of the Unified Taylor-Stratonovich Expansion to the

High-Order Strong Numerical Methods for Itô SDEs................716

5 Mean-Square Approximation of Specific Iterated Ito and Stratonovich Sto^ chastic Integrals of Multiplicities 1 to 6 from the Taylor—Ito and Taylor—Stra^

tonovich Expansions Based on Theorems From Chapters 1 and 2 723

5.1 Mean-Square Approximation of Specific Iterated Ito and Stratonovich Stochastic

Integrals of multiplicities 1 to 6 Based on Legendre Polynomials.........72;

5.2 Mean-Square Approximation of Specific Iterated Stratonovich Stochastic Inte-

grals of multiplicities 1 to 3 Based on Trigonometric System of Functions .... 766 5.3 A Comparative Analysis of Efficiency of Using the Legendre Polynomials and

Trigonometric Functions for the Numerical Solution of Itô SDEs.........776

5.3.1 A Comparative Analysis of Efficiency of Using the Legendre Polynomials

and Trigonometric Functions for the Integral Jü)^ t............1781

5.3.2 A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Integrals t, t, JolyT t,

t(îi°) т(пг2гз) 7oq J(10)T, t' J(111)T, t................................783

5.3.3 A Comparative Analysis of Efficiency of Using the Legendre Polynomials

and Trigonometric Functions for the Integral jOlljyt...........1791

5.3.4 Conclusions..................................793

5.4 Optimization of the Mean-Square Approximation Procedures for Iterated Ito

Stochastic Integrals Based on Theorem 1.1 and Multiple Fourier-Legendre Series 1794

5.5 Exact Calculation of the Mean-Square Approximation Errors for Iterated Stratonovich Stochastic Integrals , I*!)^, 1(oo)t t> 1(ooo)t3 ............

5.6 Exact Calculation of the Mean-Square Approximation Error for Iterated Stratonovich Stochastic Integral 1(o(3oo)iz3, i/4).........................181

5.7 Optimization of the Mean-Square Approximation Procedures for Iterated Stratonovich Stochastic Integrals Based on Theorems 2.2, 2.8 and Multiple Fourier-

Legendre Series....................................830

6 Other Methods of Approximation of Specific Iterated Ito and Stratonovich

Stochastic Integrals of Multiplicities 1 to 4 835

6.1 New Simple Method for Obtainment an Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on the Wiener Process Expansion Using Legendre Polynomials and Trigonometric Functions ....... "..............835

6.2 Milstein method of Expansion of Iterated Ito and Stratonovich Stochastic Integrals841

6.3 Usage of Integral Sums for Approximation of Iterated Ito Stochastic Integrals . . 849

6.4 Iterated Ito Stochastic Integrals as Solutions of Systems of Linear Ito SDEs . . . 854

6.5 Combined Method of the Mean-Square Approximation of Iterated Ito Stochastic

Integrals ........................................ 855

6.6 Representation of Iterated Ito Stochastic Integrals of Multiplicity k with Respect to the Scalar Standard Wiener Process Based on Hermite Polynomials.....

6.7 Representation of Iterated Stratonovich Stochastic Integrals of Multiplicity k with Respect to the Scalar Standard Wiener Process ..............

6.8 Weak Approximation of Iterated Ito Stochastic Integrals of Multiplicity 1 to 4

7 Approximation of Iterated Stochastic Integrals with Respect to the Q-Wiener Process. Application to the High-Order Strong Numerical Methods for Non-Commutative Semilinear SPDEs with Nonliear Multiplicative Trace Class

Noise 881

7.1 Introduction......................................881

7.2 Exponential Milstein and Wagner-Platen Numerical Schemes for Non-Commutative Semilinear SPDEs...............................888

7.3 Approximation of Iterated Stochastic Integrals of Multiplicity k (k G N) with Respect to the Finite-Dimensional Approximation W^ of the Q-Wiener Process 892

7.3.1 Theorem on the Mean-Square Approximation of Iterated Stochastic Integrals of Multiplicity k (k G N) with Respect to the Finite-Dimensional

Approximation of the Q-Wiener Process...............892

7.3.2 Approximation of Some Iterated Stochastic Integrals of Miltiplicities 2 and 3 with Respect to the Finite-Dimensional Approximation W^ of

the Q-Wiener Process ............................901

7.3.3 Approximation of Some Iterated Stochastic Integrals of Miltiplicities 3 and 4 with Respect to the Finite-Dimensional Approximation W^ of

the Q-Wiener Process ............................904

7.4 Approximation of Iterated Stochastic Integrals of Miltiplicities 1 to 3 with Respect to the Infinite-Dimensional Q-Wiener Process................91

7.4.1 Formulas for the Numerical Modeling of Iterated Stochastic Integrals of Miltiplicities 1 to 3 with Respect to the Infinite-Dimensional Q-Wiener Process Based on Theorem 1.1 and Legendre Polynomials.........915

7.4.2 Theorem on the Mean-Square Approximation of Iterated Stochastic Integrals of Multiplicities 2 and 3 with Respect to the Ininite-Dimensional

Q-Wiener Process...............................919

Epilogue

931

Bibliography

932

Chapter 1

Method of Expansion and Mean-Square Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series

This chapter is devoted to the expansions of iterated Ito stochastic integrals with respect to components of the multidimensional Wiener process based on generalized multiple Fourier series converging in the sense of norm in the space L2([t,T]k), k € N. The method of generalized multiple Fourier series for expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity k (k € N) is proposed and developed. The obtained expansions contain only one operation of the limit transition in contrast to existing analogues. In this chapter it is also obtained the generalization of the proposed method for the case of an arbitrary complete orthonormal systems of functions in the space L2([t,T]k), k € N as well as for the case of complete orthonormal with weight r(ti)... r(tk) > 0 systems of functions in the space L2([t,T]k), k € N. It is shown that in the case of scalar Wiener process the proposed method leads to the well known expansion of iterated Ito stochastic integrals based on the Ito formula and Hermite polynomials. The convergence in the mean of degree 2n (n € N) as well as the convergence with probability 1 of the proposed method are proved. The exact and approximate expressions for the mean-square approximation error of iterated Ito stochastic integrals of multiplicity k (k € N) have been derived. The considered method has been applied for other types of iterated stochastic integrals (iterated stochastic integrals with respect to martingale Poisson random measures and iterated stochastic integrals with respect to martingales).

1.1 Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean

1.1.1 Introduction

The idea of representing the iterated Ito and Stratonovich stochastic integrals in the form of multiple stochastic integrals from specific discontinuous nonrandom functions of several variables and following expansion of these functions using multiple and iterated Fourier series in order to get effective mean-square approximations of the mentioned stochastic integrals was proposed and developed in a lot of author's publications [1]-[65] (also see early publications [71] (1997),

72] (1998), [73] (2000), [74] (2001), [75] (1994), [76] (1996)). Note that another

approaches to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals can be found in [66], [77]-[94 .

Specifically, the approach [1]-[65] appeared for the first time in [75], In these works the mentioned idea is formulated more likely at the level of guess (without any satisfactory grounding), and as a result the papers [75], [76

contain rather fuzzy formulations and a number of incorrect conclusions. Note that in [75], [76] we used the trigonometric multiple Fourier series converging in the sense of norm in the space L2([t,T]k), k = 1, 2,3. It should be noted that the results of [75], [76] are correct for a sufficiently narrow particular case when

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the numbers i1,..., ik are pairwise different, i1,..., ik = 1,..., m (see Theorem 1.1 below).

Usage of Fourier series with respect to the system of Legendre polynomials for approximation of iterated stochastic integrals took place for the first time in the publications of the author [71]-[74] (also see [1]-[66]).

The question about what integrals (Ito or Stratonovich) are more suitable for expansions within the frames of distinguished direction of researches has turned out to be rather interesting and difficult.

On the one side, the results of Chapter 1 (see Theorems 1.1, 1.2, 1.16) conclusively demonstrate that the structure of iterated Ito stochastic integrals is rather convenient for expansions into multiple series with respect to the system of standard Gaussian random variables regardless of the multiplicity k of the iterated Ito stochastic integral.

On the other side, the results of Chapter 2 [6]-[22], [25], [27], [29], [31 -

[41],

[44]-[46], [51],

[71]-[74] convincingly testify that

there is a doubtless relation between multiplier factor 1/2, which is typical for Stratonovich stochastic integral and included into the sum connecting Stratonovich and Ito stochastic integrals, and the fact that in the point of finite discontinuity of piecewise smooth function f (x) its Fourier series converges to the value (f (x — 0) + f (x + 0))/2. In addition, as it is demonstrated in Chapter [27], [29], [31]-[38], [41], [42], [44]-[46], [51], [63], [64], the final

2

formulas for expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of continuously differentiable weight functions) and multiplicity k (the case of convergence of trace series) are more compact than their analogues for iterated Ito stochastic integrals.

1.1.2 Ito Stochastic Integral

Let (Q, F, p) be a complete probability space and let f (t, w) : [0, T] x Q ^ R be the standard Wiener process defined on the probability space (Q, F, p). Further,

we will use the following notation: f (t,w) == ft.

Let us consider the right-continous family of a-algebras {Ft, t £ [0,T]} defined on the probability space (Q, F, p) and connected with the Wiener process ft in such a way that

1. Fs C Ft C F for s < t.

2. The Wiener process ft is Ft-measurable for all t £ [0,T].

3. The process ft+A — ft for all t > 0, A > 0 is independent with the events of a-algebra Ft.

Let us introduce the class M2([0,T]) of functions £ : [0,T] x Q ^ R, which satisfy the conditions:

1. The function £(t,w) is measurable with respect to the pair of variables (t,w).

2. The function £(t,w) is Ft-measurable for all t £ [0,T] and £(t, w) is independent with increments ft+A — ft for t > t, A > 0.

3. The following relation is fulfilled

T

j m {(£(t,w))2} dt < oo. 0

4. m {(£(t,w))^ < o for all t £ [0,T].

For any partition t(n\ j = 0,1,..., N of the interval [0, T] such that

0 = t«n> < t<n> < ... < TNN> = T, max

0 1 N 0<j<N-1

T(N) _ T(N) Tj+1 Tj

— 0 if N —y to (1.1)

we will define the sequence of step functions

€(N >(t , w) = €j (w) w. p. 1 for t €

T(N T(N Tj , Tj+1

where €(N) (t,w) € M2 ([0,T]), j = 0,1,...,N -1, N = 1, 2,... Here and further, w. p. 1 means with probability 1.

Let us define the Ito stochastic integral for €(t,w) € M2([0,T]) as the following mean-square limit [95], [96] (also see [79])

N-1

ni-e €(n' j u)(/(t<+>) -/j >w

T

d=f i €tdf, (1.2)

j=0

where €(N)(t,w) is any step function from the class M2([0,T]), which converges to the function €(t, w) in the following sense

T

lim / m

n—to J 0

€(N)(t,w) - €(t,w)

dt = 0.

(1.3)

Further, we will denote €(t, w) as €T.

It is well known [95] that the Ito stochastic integral exists as the limit (1.2) and it does not depend on the selection of sequence €(N)(t,w). Furthermore, the Ito stochastic integral satisfies w. p. 1 to the following properties [95

m < I €tdft 0

F0 = 0,

m

T

0

F

0

Fo > = m < I €t dt

T T

(a€t + = a €t dft + P #ft,

0

2

where &, ^ e M2([0,T]), a, £ e R1.

Let us define the stochastic integral for e M2([0,T]) as the following mean-square limit

N —1 T

14m. £ i(N) j,") j — j') =7 ^t, (1.4)

j =0 0

where &(N)(t,w) is any step function from the class M2([0,T]), which converges in the sense (1.3) to the function &(t,w).

1.1.3 Theorem on Expansion of Iterated Ito Stochastic Integrals of Multiplicity k (k e N)

Let F, p) be a complete probability space, let {Ft,t e [0,T]} be a non-decreasing right-continuous family of a-algebras of F, and let ft be a standard m-dimensional Wiener stochastic process, which is Ft-measurable for any t e [0,T]. We assume that the components ft(i) (i = 1,... ,m) of this process are independent.

Let us consider the following iterated Ito stochastic integrals

t t2

J [^k)]T,t = / (tk)... f ^i(ti)dwt(;i).. .dw£k), (1.5)

where every ^(t) (l = 1,..., k) is a nonrandom function on [t,T], wT*' = f-^' for i = 1,..., m and wT0) = t, i1,..., = 0,1,..., m.

Let us consider the approach to expansion of the iterated Ito stochastic integrals (1.5) [1]-[65] (the so-called method of generalized multiple Fourier

series). The idea of this method is as follows: the iterated Ito stochastic integral (1.5) of multiplicity k (k e N) is represented as the multiple stochastic integral from the certain discontinuous nonrandom function of k variables defined on the hypercube [t,T]k. Here [t,T] is the interval of integration of the iterated Ito stochastic integral (1.5). Then, the mentioned nonrandom function of k variables is expanded in the hypercube [t,T]k into the generalized multiple Fourier series converging in the mean-square sense in the space L2([t, T]k). After a number of nontrivial transformations we come to the mean-square converging expansion of the iterated Ito stochastic integral (1.5) into the multiple series of products of standard Gaussian random variables. The coefficients of this

series are the coefficients of generalized multiple Fourier series for the mentioned nonrandom function of k variables, which can be calculated using the explicit formula regardless of the multiplicity k of the iterated Ito stochastic integral (1.5).

Suppose that every ^ (t) (l = 1,..., k) is a continuous nonrandom function on [t, T] (we will also consider the case ^i(t),..., (t) £ L2([t, T]) in Sect. 1.11, 1.12). Define the following function on the hypercube [t,T]k

(ti ) ...^k (tk ), ti <...<tk

K (ti,...,tk ) =

k k-i n^i(ti) n i <ti+1},

0,

otherwise

i=i

(1.6)

where txG [t,T] (k > 2) and K(ti) = ^(ti) for tx G [t,T]. Here 1A denotes the indicator of the set A.

Suppose that (x)}°=0 is a complete orthonormal system of functions in the space L2([t, T]).

The function K(ti,... ,tk) is piecewise continuous in the hypercube [t,T]k. At this situation it is well known that the generalized multiple Fourier series of K(ti,..., tk) G L2([t, T]k) is converging to K(ti,..., tk) in the hypercube [t, T]k in the mean-square sense, i.e.

lim

Plv,Pk ^^

Pi Pk

K (ti,...,tk ) ..^Cjk ..¿n (ti )

ji=0 jk=0 i=i

= 0, (1.7)

L2([t,T ]k )

where

C =

Cjk •••ji =

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[t,T ]k

is the Fourier coefficient, and

/k

K (ti,...,tk 0ji (ti )dti ...dtk

.m,k i=i

(1.8)

m

L2([t,T ]k)

/

\

i/2

f 2(ti,...,tk )dti ...dtk

y

Consider the partition {Tj}N=0 of [t,T] such that

t = t0 < ... < tn = T, AN = max At, ^ 0 if N ^ to, At,

0<j<N-i

Tj+i - T

j"

(1.9)

k

Theorem 1.12 [1] (2006) (also see [2]-[65]). Suppose that every ^(т) (/ = 1,..., k) is a continuous nonrandom function on [t, T] and {ф (x)}°=0 is a complete orthonormal system of continuous functions in the space L2([t,T]). Then

Pi Pk /к

t = i.i.m Ej

pi,-",pk^^ z—' z—' \ ji=0 jk=0 \/=1

— l.i.m. Y, j(Til)AwT;;'.. j(Tik)AwT;fck) , (1.10)

(l;,...,lk)GGk 1 k )

where

Gk = Hk\Lk, Hk = {(/i,...,/k) : 1i,...,1k = 0, 1,...,N — 1}, Lk = {(1i,...,1k) : 1i,...,1k = 0, 1,...,N—1; lg = ¿r (g = r); g,r = 1,...,k}, l.i.m. is a limit in the mean-square sense, i1,... ,ik = 0,1,... ,m,

T

j = J to (s)dw<" (1.11)

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjk...j; is the Fourier coefficient (11.81). AwTj = wTj++; — wTj (i = 0,1,... ,m), {Tj}=0 is a partition of [t,T], which satisfies the condition (1.9).

Proof. At first, let us prove preparatory lemmas.

Lemma 1.1. Suppose that every ^ (t) (l = 1,..., k) is a continuous non-random function on [t,T]. Then

N—1 j2 1 k

J[^(k)]T,t = l.i.m. £ ..^n^i(Tj;)AwT;.;) w. p. 1, (1.12)

where AwTj = wTj+; — wTj (i = 0,1,..., m), {Tj}N=0 is a partition of the interval [t,T] satisfying the condition (1.9).

Proof. It is easy to notice that using the property of stochastic integrals additivity, we can write

N—1 j2 — 1 k

J[^(k)]T,t = £ . . . £ n J+ ^ w. p. 1, (1.13)

jk =0 j;=0 l=1

2 Theorem 1.1 will be generalized to the case of an arbitrary complete orthonormal system of functions

{ф(x)}~0 in the space L2([t, T]) and ^i(r), ..., (т) G L2([t, T]) in Sect. 1.11 (see Theorem 1.16).

where

s

J]s,e = j ^ (t)dwTil) e

and

N—1 Tjfc+1 s

= ^ / ^k(s) / ^k-x(T)J[^(k-2)]T,tdwTik-l)dwiik) + jk=0

Tjk Tjk

k—3

+ E G[^<-r+1]N X

r=1

jfc-r+1-1 """ r

1 Tjk-r + 1 S

X ^ J ^k-r(s) J ^k-r-i(T)J[^(k-r-2)]T,tdwTik-r-l)dwSik-r) +

jk-r =0 _ _

j3-1

J[^Vij

¿2=0

where

N —1 jk-1 jm+l-1 k

G[^mk)]N = £ £ .. ■ £ n jW'l]

jk=0 jk-i=0 jm=0 l=m (^m,^m+1, . . . ) =f (^1, . . . ,^k) = ^(k) =

Using the standard estimates (1.26), (1.27) (see below) for the moments of stochastic integrals, we obtain w. p. 1

l.i.m. £n = 0. (1.14)

N ^TO

Comparing (1.13) and (1.14), we get

N — 1 j2 1 k

J[^(k)]T,t = l.i.m. ^ ... £ n J[^kw. p. 1. (1.15)

jk=0 ji=0 l=1

Let us rewrite J[^l]Tji+i;Tji in the form

J]rJi+i,rji = ^l(Tji)AwTj.;) + / (Mt) — «j))dwTil)

and substitute it into (1.15). Then, due to the moment properties of stochastic integrals and continuity (which means uniform continuity) of the functions ^ (s) (l = 1,..., k) it is easy to see that the prelimit expression on the right-hand side of (1.15) is a sum of the prelimit expression on the right-hand side of (1.12) and the value which tends to zero in the mean-square sense if N ^ to. Lemma 1.1 is proved.

Remark 1.1. It is easy to see that if Awj in (11.121) for some I G {1,..., k} is replaced with ^Awj^ (p = 2, i/ = 0), then the differential dw(;i) in the integral J[^(k)]T,t will be replaced with dt/. If p = 3,4,..., then the right-hand side of the formula (1.12) will become zero w. p. 1. If we replace Awj ^ in (1.12) for some I G {1,..., k} with (At^)p (p = 2,3,...), then the right-hand side of the formula (1.12) also will be equal to zero w. p. 1.

Let us define the following multiple stochastic integral

N-1 k

l.i.m. £ $(t31 ,...,TA)nAw<j> = J[<4ki, (1.16)

JlvJfc =0 / = 1

where $(ti,...,tk): [t,T]k ^ R1

is a nonrandom function (the properties of this function will be specified further).

Denote

Dk = {(ti,...,tk): t < ti <...<tk < T}. (1.17)

We will use the same symbol Dk to denote the open and closed domains corresponding to the domain Dk defined by (1.17). However, we always specify what domain we consider (open or closed).

Also we will write $(t1,...,tk) G C(Dk) if $(t1,...,tk) is a continuous nonrandom function of k variables in the closed domain Dk.

Let us consider the iterated Ito stochastic integral

t t2

/[*]$ d=f / ...f $(i1,...,tk)dw«;i> ...dw«:k), (1.18)

where ..., tk) G C(Dk).

Using the arguments which similar to the arguments used in the proof of Lemma 1.1 it is easy to demonstrate that if $(t1,... ,tk) G C(Dk), then the

following equality is fulfilled

N-i j2 i k

I (k)

I [$]Tk) = l.i.m. £ ...£ ,...,Tjk ^Awj w.p. 1. (1.19)

jfc=0 ji=0 1=i

In order to explain this, let us check the correctness of the equality (1.19) when k = 3. For definiteness we will suppose that i1, i2, i3 = 1,..., m. We have

T is t2

IKi = // |$(ii,i2,t3)dw<;i)dwii2)dwiis) =

i i i N-i j i2

= I /^(ti,t2,Tjs^w^W^Awj

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œ js=0

ii

N-i js-i j+1 '<2

= l.i.m. ££ / /$(ti,t2,Tjs )dwi;i)dw(;2)Awii.s) =

N^ js=0 j2=0 j i 3

N-i js-i Tj '2 \

=l„Liœ-E£/ J +j $(ti,t2,Tjs)dw<:i'dw<:2>Aw<;S>

V ' Tj /

N-i js-i j2 i j+i j+1

N^œ . .

js=0 j2=0

/ / $(ti,t2,Tjs)dwi:i)dw(:2)Awi:s)+

N^œ . J J

js =0 j2=0 ji=0 T,

N-i js-i j+i '2

+ Li.m. ££ / /$(ti,t2,rjs rfdw^Aw^. (1.20)

Njs=0 j2=0 j j s

Let us demonstrate that the second limit on the right-hand side of (1.20) equals to zero.

Actually, for the second moment of its prelimit expression we get

N-i js-i j+i '2 N-i js-i 1

EE I J ^I-'-'-^sAr,3 < .1/- E E 2 Ar,3 > 0

js=0 j2=0 js=0 j2=0

j2 j2

when N ^ œ. Here M is a constant, which restricts the module of the function ^(ti, t2, t3) due to its continuity, Arj = rj+i - Tj.

Considering the obtained conclusions, we have

T is t2

i Ki = // y$(ii,i2,is)dw((;i)dwi;2)dwi;s) =

i i i N-l js-i j2 i j+1 j+1

= l.i.m. £££ / / $(ii,t2,Tjs^w^W^Awj

N^œ . „ . „ . „ J J s

js=0 j2=0 ji=0 N-l js-l j2 l j + 1 TjV+1

ïts^ ^ _ n-.Wri™ dw

i1 i2 js

= l.i.m. £££ / / ($(ii,i2,Tjs) - $(ii,Tj2, Tjs ) ) dw<;1)dw(:2)Aw<'S) +

N^œ . . . „ „ js=0 j2=0 j1=0 T,

j2 j1

N-l js-l j2-1 j+1 +l.i.m. £££ / / ($(il,ri2,Tjs) - $(Tj1 ,Tj2,Tjs))dw(;'W<;2)Awj +

N^œ . . . „

js=0 j2=0 j1=0 T, j2 j1

N-1 j3-1 j2-1

+ l.i.m. £ £ £ , Tj2, j)Aw(;;)Awi;2)Awi;3). (1.21)

N^to . . .

j3=0 j2=0 j;=0

In order to get the sought result, we just have to demonstrate that the first two limits on the right-hand side of (1.21) equal to zero. Let us prove that the first one of them equals to zero (proof for the second limit is similar).

The second moment of prelimit expression of the first limit on the right-hand side of (1.21) equals to the following expression

N-1 j3-1 j2 1 Ty; Ty;

EEE / I W1,t2,Tj3) - $(i1,Tj-2,Tj3))2ATj3. (1.22)

j3=0 j2 =0 j;=0

j2 j1

Since the function $(t1,t2,t3) is continuous in the closed bounded domain D3, then it is uniformly continuous in this domain. Therefore, if the distance between two points of the domain D3 is less than (£(e) > 0 exists for any £ > 0 and it does not depend on mentioned points), then the corresponding oscillation of the function $(t1,t2,t3) for these two points of the domain D3 is less than

If we assume that At,- < (j = 0,1,...,N - 1), then the distance between points (t1,t2,Tj3), (t1,Tj2,Tj3) is obviously less than £(e). In this case

|$(t1,t2,Tj3 ) - $(t1,Tj2 , Tjg )| <£.

Consequently, when Ar^ < (j = 0, 1,..., N — 1) the expression (1.22) is estimated by the following value

N — 1 j3 — 1 j2 — 1 (T-t)3

£2 E E E AtjAtJ2ATj, <

j3=0 j2=0 ji=0

Therefore, the first limit on the right-hand side of (1.21) equals to zero. Similarly, we can prove that the second limit on the right-hand side of (1.21) equals to zero.

Consequently, the equality (1.19) is proved for k = 3. The cases k = 2 and k > 3 are analyzed absolutely similarly.

It is necessary to note that the proof of correctness of (1.19) is similar when the nonrandom function $(t1,... , tk) is continuous in the open domain Dk and bounded at its boundary.

Let us consider the following multiple stochastic integral

N —1 k

l.i.m. £ «(j ,...,rjt) H Awj =f J'[$]<k>, (1.23)

J1vjfc=0 1=1

jq =jr; q=r; q,r=1,...,fc

where «(t1,... ,tk) : [t,T]k ^ R1 is the same function as in (1.16). According to (1.19), we get the following equality

T t2

J'[<4k] = /...J E («(t1,...,tk)dwt;i)...dw

tfc

w. p. 1, (1.24)

where

£

(ti,...,tfe)

means the sum with respect to all possible permutations (t1,..., tk). At the same time permutations (t1,... ,tk) when summing are performed in (1.24) only in the expression, which is enclosed in parentheses. Moreover, the nonrandom function «(t1,... ,tk) is assumed to be continuous in the corresponding closed domains of integration. The case when the nonrandom function «(t1,..., tk) is continuous in the open domains of integration and bounded at their boundaries is also possible.

It is not difficult to see that (1.24) can be rewritten in the form

t t2

J'№Tk] = E /•••/.....tk)d

w(il)... dw

(ifc) ifc

w. p. 1, (1.25)

(ti,"-,tfc) t

where permutations (tl5..., tk) when summing are performed only in the values dwt(|l)... dwt(ik). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,..., tk), then ir swapped with in the permutation (i1?..., ik).

Lemma 1.2. Suppose that $(ti;... , tk) G C(Dk) or $(ti;... ,tk) is a continuous nonrandom function in the open domain Dk and bounded at its boundary. Then

T t2

m

I [*]

(k) T,t

< Ck y ...y ^(ti,...,tk )dti ...dtk, Ck < oo,

t t

where I [$]Tk] is defined by the formula (11.18).

Proof. Using standard properties and estimates of stochastic integrals for

£T G M2([t,T]), we have [96]

2'

m

T

£t d/r

T

= / M{|er|2}dT,

t

(1.26)

m

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T

eT dr

Let us denote

em

T

< (t - t)J m{ieT i2}dr.

U + 1 t2

$(ti,...,tk)dwt(;i)

. . dw

(il)

(1.27)

where l = 1,..., k - 1 and £[0..,tfc,t = $(ti,... ,tk). By induction it is easy to demonstrate that

£[*](+ i,--.,tk,t G M2([t,T])

2

with respect to the variable t/+1. Further, using the estimates (1.26), (1.27) repeatedly we obtain the statement of Lemma 1.2.

It is not difficult to see that in the case i1,..., = 1,..., m from the proof of Lemma 1.2 we obtain

m

I [*]

(k) T,t

T t2

(1.28)

t t

Lemma 1.3. Suppose that every ^(s) (l = 1,..., k) is a continuous non-random function on [t, T]. Then

II J[^l]T,t = J[$]Tkt w. p. 1,

where

1=1

T

J[w]T,t = W(s)dw(il), $(ti,...,tk) = HW(ti),

1=1

and the integral Jis defined by the equality (11.161). Proof. Let at first i/ = 0, I = 1,..., k. Let us denote

n-1

Jb/]n = £ (Tj)Aw|i|). j=0

Since

IP[Wi]n -ft J[w]

T,t =

1=1

k /1- 1

¿(n J bg]T,t) J [Wl]N - J bfo) n J 1=1 \g=1 / ^ ' \g=1+1

(1.29)

(1.30)

then due to the Minkowski inequality and the inequality of Cauchy-Bunyakov-sky we obtain

m

IP [W1]N -]! J M

T,t

l=1

1/2

<

2

k

2

J [p/]n - J [pfo

< c*^ ( m '

/=1

where Ck is a constant. Note that

J[P/]N - J[P/]T>t = X] J[A^]Tj+i,Tj ,

j=0

rj+i

J[Ap /]WJ = / (p/ (Tj) - p/(s)) dw(il).

n- 1

(1.31)

Since J[Ap/]T-+1>r- are independent for various j, then [97

m

N-1

E J[A^/] Wj j=0

N-1

m

j=0

J [Ap/]

Tj+i>

+

N-1

+ 6^; m j=0

J [Ap/]

Tj+i>

2ï j-1

m

q=0

J [Ap/]

Tq+1>

(1.32)

Moreover, since J[Ap/]T >T. is a Gaussian random variable, we have

m

J [Ap/]

rj+i>rj

rj+i

= / (P/(Tj) - P/(s))2ds

m

4

J [AP/]Tj+i>Tj -3

/ rj+i

\

(P /(Tj) - P /(s))2ds

V-j )

Using these relations and continuity (which means uniform continuity) of the functions p /(s), we get

I N -1 4

m

N-1

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E J[AP/]Tj+i>Tj j=0

\ / N-1 N-1 j-1 \

< ^(Atj )2 + 6 £ At, £ At, <

J V j=0 j=0 ,=0 J

< 3e4 (5(e)(T - t) + (T - t)2) ,

4

2

where Arj < 6(e), j = 0,1,..., N - 1 (6(e) > 0 exists for any £ > 0 and it does not depend on points of the interval [t,T]). Then the right-hand side of the formula (1.32) tends to zero when N ^ to. Considering this fact as well as (1.31), we obtain (1.29).

If w(i^ = t/ for some I G {1,..., k}, then the proof of Lemma 1.3 becomes obviously simpler and it is performed similarly. Lemma 1.3 is proved.

Remark 1.2. It is easy to see that if Awj in (11.29) for some I G {1,..., k}

is replaced with ^Awlj^ (p = 2, i/ = 0), then the differential dw(il) in the

integral J[$(k)]Tjt will be replaced with dt/. If p = 3, 4,..., then the right-hand side of the formula (1.29) will become zero w. p. 1.

Let us consider the case p = 2 in detail. Let AwTj^ in (11.29) for some I G {1,..., k} is replaced with ^AwTj^ (i/ = 0) and

N1

T

J[w]n = e W(Tj) (Awj>) , J[w]T,t d=f / W/(s)ds. j=0 {

We have

/

m

V

m <

V

j=0

m

N1

J[^/]n - JW]T,t

1/4

T

¿>(Tj)(Awij^ - [ w(s)ds j=0 {

4 1/4

/

rj+;

N-1 f 2

^ W/(Tj)(Aw(i1 >) - (w/(s) - w/(Tj) + W/(Tj)) ds

V

m

N-1 / 2

£>/(Tj )((Aw£ >) - ATj

j=0

1/4

+

\ 4 \

>

/ J /

1/4

<

+

N-1

X] / (W/(Tj) - W/(s)) ds

j=0 T

(1.33)

4

From the relation, which is similar to (1.32), we obtain

m

N-1 , 2

(Tj) (Aw^) - At;

N-i f /

(pi(Tj))4 mH (Awi;

j=0 Ni

«•'>1 - ArJ \ +

N -1 / 2

+6^(pi(Tj))2 m (AwJi1 ^ - ATj

j=0 I v

X

j-i

X > (Pi (Tq q=0

N-i j-i

22

£ (pi(tq))2 m ((Aw£>)S - At,

N-1

60 (Tj))4 (ATj)4 +

j=0

+ 24 £ (pi(Tj))2 (ATj)2 £ (pi(Tq))2 (ATq)2 < C (An)2 ^ 0 (1.34)

j=0 q=0

if N ^ o, where constant C does not depend on N.

The second term on the right-hand side of (1.33) tends to zero if N ^ o due to continuity (which means uniform continuity) of the function pi (s) at the interval [t,T]. Then, taking into account (1.30), (1.31), (1.33), (1.34), we come to the affirmation of Remark 1.2.

Let us prove Theorem 1.1. According to Lemma 1.1, we have

N i i2 i

J[^(k)]T,t = l.i.m. £ ... £ ^i(Tii)... ^(Tik)Aw[;;)... Awl;

r(;fc) = 'k

ik=0 ii=0 N-1 i2-1

l.i.m^... £ K (Til,..., Tik )Aw(;;)... Awi;;) =

N

ik =0 i1=0 N i N i

u.m. £... £ K (Ti1,..., Tik )Aw<;;>... aw<;;> =

N

ik=0 i1=0

Ni

= l.i.m.

N

£ k (Ti1,..., Tik )Aw(;;)... AwT;; ) = (1.35)

i1,...,ik=0

= lr; q=r; q,r=1,...,k

4

2

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T t2

= / -j E (K(ii,...,ik)dw(;i)>) w.p.i, (1.36)

t t )

where permutations (tl5... , tk) when summing are performed only in the expression enclosed in parentheses.

It is easy to see that (1.36) can be rewritten in the form

T t2

J[^(k)]T,t = £ J.. ^K(ti,...,tk)dwill) ...dw(::) w.p.i, (1.37)

ti,

(ti,...,t:) t t

where permutations (tl5..., tk) when summing are performed only in the values dwt(il)... dwt(::). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,..., ), then ir swapped with in the permutation (i1,..., ik).

Since integration of a bounded function with respect to the set with measure zero for Riemann or Lebesgue integrals gives zero result, then the following formula is correct for these integrals

T t2

J |G(ti,...,tk)|dti ...dtk = £ /.../|G(ti,...,tk)|dti ...dtk,

[t,T]: (ti,...,t:) t t

(1.38)

where permutations (£i;... , tk) when summing are performed only in the values dti.. . dtk. At the same time the indices near upper limits of integration in the iterated integrals are changed correspondently and |G(ti,... ,tk)| is the integrable function on the hypercube [t,Tjk.

According to Lemmas 1.1, 1.3 and (1.24), (1.25), (1.36), (1.37), we get the following representation

Pi P: n n

. • / E (j(ti)(tk^wi;1'...dw(::>) +

; _n ;__n J J ti. J. ^

J 'jT,t =

T t2

j'i=0 j:=0 t t (ti,...,t:)

T?pi,...,p: _

+ RT,t =

P1 p; N-1

E . . . E Cjk---j1 l.i.m. £ j (Ti1)... j (Tik )Aw[;;)... Awt;;)+

j1=0 jk=0 11 '...?'k=0

=lr; q=r; q,r=1,...,k

+ RT1t---'Pk = (1.39)

P1 Pk / N-1

£ ... £ Cjk---j1 l.i.m. £ j (Ti1)... j (Tik )Aw[;;)... Awl-)-

—n —n \ N^o 1 k

j1=0 jk=0 \ i1,---,ik=0

l.i.m. £ j (Ti1 )Aw[;;)... j (Tik )Awt;; M +

N^o ^^ 1 k '

(i1,---,ik )GGk

+RTT1t---'Pk

P1 p;

E . . . E Cjk-j1 X j1=0 jk=0

k

XI I "X Zj(i) - l.i.m. £ j (Ti1 )AwT;;)... j (Tik )AwT;;: i = 1 (i1,---,ik )GG;

+ RT1,t---'Pk w. p. 1, (1.40)

where

T ^ / P1 p; k \

£ /.../ K(ti,...,tk)...£Cj-k-j-1n^(ti) 1,---,tk) t i V j1=0 jk=0 i=i /

R?t---'Pk = V /.../IK (ti.....tk) -> \..VCj,j| Uj, (ti )| X

(t1,---,tk) t t ^ j1=0 jk=0

X dwt(i1) ...dw(;k), (1.41)

where permutations (t1,..., tk) when summing are performed only in the values dwt(i1)... dwt(;k). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped

with tq in the permutation (ti,... , tk), then ir swapped with in the permutation (¿1,..., ik).

Let us estimate the remainder RrV ^ of the series. According to Lemma 1.2 and (1.38), we have

m^ (RTrN ^ <

T ^ / pi Pk k \ "

< Ck£ ... K(ti,...,tk) n^j M dti ...dtk

(t 1 */ V j i=0 jk=0 1=1 /

(t i,...,tk) t t v j 1=0 jk=0

2

// P 1 Pk k \

K(ti,..., tk) - £ ... £ Cjk...j 1 n (ti) dti... dtk ^ 0

[tT]k V j 1=0 jk=0 i=i /

(1.42)

if pi,... ,pk ^ oo, where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral J[^(k)]Tjt. Theorem 1.1 is proved.

Note that from (1.39) and (1.42) it follows that

p i Pk

J[^(k)]T,t = l.i.m. £ ... £ Cjk...j i JU- i... j]Ti;t-ik) w. p. 1, (1.43)

j =0 jk=0

where J'[0ji... jis defined by (1.23).

It is not difficult to see that for the case of pairwise different numbers ii,...,ik = 0,1,..., m from Theorem 1.1 we obtain

P i Pk

i c (i i > z )

P i,...,Pk

J i^(k)]T,< = l.i.nm. £ ...J>V..j i Cj; i' ...Cik''. (1.44)

p.....Pi„ f ■« ' * J i Jk

ji =0 jk=0

1.1.4 Expansions of Iterated Ito Stochastic Integrals with Multiplicities 1 to 7 Based on Theorem 1.1

In order to evaluate the significance of Theorem 1.1 for practice we will demonstrate its transformed particular cases (see Remark 1.2) for k = 1,..., 7 [1]-[62]

P i

Ji^(i>]T,( = l.i.m. ]T CjiCj; i>, (1.45)

?ii —^no ' * J1

p i j i =0

Pi P2

J=^ E E ji (Cji°zJ' - 1(!1=,2=0!1{J1=J2}), (1.46)

jl=0 j2=0

Pi P2 P3

j ^=pi^. e E E j <jij z<:2> j« -

jl=0 J2=0 J3=0

1{i1=i2=0!1{j1=j2}C7(33) - 1{i2=i3=0!1{j2=j3}Cjil) - 1{i1 = i3=0!1{j1=j3}C]22) I , (1'47)

Pi

P4

Pi,...,P4^-ro z-' z-' I -1--1- -

(¿l)

ji=0 j4=0

J=1

1 1 /-(i3)/-(i4) ^ 1 A(«2^(«4)

-1{ii=i2=0!1{ji=j2!Zj3 j - 1{ii=i3=0!1{ji=j3!Zj2 j -

1 I /-(i2)/-(i3) ^ -I A(«iV(i4)

-1{i1 = i4=0!1{j1=j4!Zj2 Zj3 - 1{i2=i3=0!1{j2=j3!Zji ZJ4 ~

i i Aii)^(i3) n i /-(nM^) ,

-1{i2=i4=0!1{j2=j4!Zji ZJ3 - 1{»3 = i4=0!1{j3=j4!Zji Zj2 +

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+ 1{i1 = i2=0!1{ji=j2!1{i3=i4=0}1{j3=j4! + 1{i1 = i3=0!1{ji=j3!1{i2 = i4=0}1{j2=j4! +

+ 1{i1=i4=0!1{j1=j4!1{i2 = i3=0!1{j2=j3! (1-48)

pi

P5

J [^(5)]T,t = l.i.m. V

ji=0

+ 1{i1=i2=0} + 1{i1=i2=0} + 1{ii=«3=0}

{i 1 = ^2=0} {ii = i4=0} {¿2 = «3=0} {i2 = i5=0} {i3=i5=0}

Z(i3)Z(i4) Z(i5 {jl=j2} ZJ3 Zj ZJ5

Z (i2) Z (i3)Z (i5 { jl=j4} Z J2 ZJ3 ZJ5

Z (i1)Z (i4) Z (i5

{ j2=j3} Z ji Zj Zj

Z (i1)Z (i3)Z (i4

{j2=j5} Zj Zj Zj

Z (i1)Z (i2 ) Z (i4

{ j3=j5} Zji Zj ZJ4

{j 1 = J2 } 1 { ¿3 = ¿4=0 } 1 {J3 =J4 } Z

(¿5

J5

{j 1 = J2 } 1 { ¿4 = ¿5=0} 1 {J4 = J5 } Z { j 1 =J3 } 1 { «2 = ¿5=0} 1 {J2 = J5 } C

(¿3 j3

(¿4 j4

■EC,,

J5=0

Ji

n Zj;

j=I

)- {¿1= ¿3=0}

)- {¿1= =¿5=0} 1

)- {¿2 = ¿4=0}

)- {¿3 = ¿4=0}1

)- {¿4 = = ¿5=0} 1

) + {¿1 = ¿2=0}1

) + {¿1 = = ¿3=0} 1

) + {¿1 = =¿3=0}1

j3 } ZJ2

(¿2

1 = j5 } ZJ2

J4 } Z J i

Z(ii j4} Z ji

¿;)

Z(i4)Z(i5

Sj Z

J4 J

Z(i3)Z(i4 ZJ3 Z J4

Z(i3)Z(i5 ZJ3 ZJ5

Z(i2)Z(i5 Z j Z

1{ji= 1{ji= 1 { J2 = 1{j3 =

I {J4 = 1{ji =

II 1 Z (¿5)1 1 {j 1 = j3 } 1 { ¿2 = ¿4 =0 } 1 {j2 = j4 } Z j5 +

1{ji =

J2 J

Z^z(¿2)z(;3) ,

4=j5} Zji ZJ2 ZJ3 +

1 1 Z ^^

J2! 1{«3 = «5=0! 1{j3=j5!Zj4 +

J3} 1 {M = *5=0} 1 {J4=j5 } j ) +

4

+ l{i1 = i4=0}l{ji=j4}1{i2=i3=0}l{j2=j3}C]5i5) + 1{i1=i4=0}l{ji=j4}1{i2=i5=0}l{j2=j5}C]<i3) + + l{i1=i4=0}l{ji=j4}1{i3 = i5=0}l{j3=j5}Cj2i2) + 1{i1 = i5=0}1{ji=j5}1{i2 = i3=0}1{j2=j3}Cj4i4) + + 1{i1=i5=0}1{ji=j5}1{i2 = i4=0}1{j2=j4}C]<i3) + 1{i1 = i5=0}1{ji=j5}1{i3 = i4=0}1{j3=j4}Cj22) + + 1{i2=i3=0}1{j2=j3}1{i4 = i5=0}1{j4=j5}C] ^ + 1{«2 = «4=0} 1{j2=j4} 1{«3 = «5=0} 1{j3=j5}Cj ^^

+ 1{i2 = i5=0}1{j2=j5}1{i3=M=0}1{j3=j4}C

(i1)

¿1

(1.49)

P1

P6

J (= l.i.m ]T ..^C^.j^n Z

(«1 )

¿1=0 ¿6=0

J=1

1{i1= =«6=0} {¿1= Z («2 =¿6 } Z¿2 Z («3)Z («4) Z («5) _ Zi3 Z ¿4 Z.?5 {¿2 = =«6=0} { ¿ 2 Z(«1 =¿6^1 Z(«3)Z(«4) Zi3 Z ¿4 Z(«5) Zi5

A{«3 = =«6=0} {j3 = Z («1 =¿6^1 Z («2)Z («4) Z («5) _ Z ¿2 Z ¿4 Zi5 {¿4 = =«6=0} {¿4 Z(«1 =¿6^1 Z(«2)Z(«3) Zi2 Zi3 Z(«5) ZJ5 -

^{«5 = =«6=0} {¿5 = =¿6^1 Z(«2)Z(«3)Z(«4) _ Z ¿2 Zi3 Zi4 {«1 = =«2=0} {¿1 = ■ \Z(«3 =¿2 } Z¿3 Z(«4)Z(«5 Zi4 Z ¿5 Z («6)_ Zi6

1{«1 = =«3=0} {¿1 = Z («2 =¿3 } Z¿2 Z(«4) Z(«5)Z(«6) _ Z¿4 Z¿5 Zi6 {«1 = =«4=0} {¿1 = Z(«2 =¿4^2 Z («3)Z («5 Zi3 Z ¿5 Z («6)_ Zi6

1{«1 = =«5=0} {¿1 = =¿5 } Z¿2 Z («3)Z («4) z («6) _ Zi3 Z¿4 Zi6 {«2 = =«3=0} { ¿ 2 ■ xZ(«1 =¿3^1 Z(«4)Z(«5 Zi4 Z ¿5 Z («6)_ Zi6

1{«2 = =«4=0} {¿2 = =¿4^1 Z(«3)Z(«5)Z(«6) _ Zi3 Z ¿5 Zi6 {«2 = =«5=0} { ¿ 2 ■ xZ(«1 =¿5 } Z^ 1 Z(«3)Z(«4) Zi3 Z ¿4 Z («6)_ Zi6

A{«3 = =«4=0} {j3 = Z («1 =¿4^1 Z(«2)Z(«5)Z(«6) _ Z ¿2 Z ¿5 Zi6 {«3 = =«5=0} { ¿ 3 Z(«1 =¿5^1 Z(«2)Z(«4) Zi2 Z ¿4 Z («6)_ Zi6

1 1 Z(i1)Z(i2)Z(i3)z(i6) I

1{i4 = i5=0}1{j4=j5}Zj1 Z j2 Zj3 Zj6 +

+1 {«1= «2=0} {¿1= =i2} {«3 «4=0} {¿3 Z(«5)Z(«6) =¿4^5 Zi6 +

+1 {«1= «2=0} {¿1= =i2} {«3 «5=0} {¿3 Z(«4)Z(«6) =¿5^4 ^6 +

+1 {«1= «2=0} {¿1= =i2} {«4 «5=0} {¿4 Z(«3)Z(«6) =¿5 } Z¿3 ^6 +

+1 {«1= =«3=0} {¿1= =i3} {«2 «4=0} {¿2 Z(«5)Z(«6) =¿4 } ^5 Zi6 +

+1 {«1= =«3=0} {¿1= =i3} {«2 «5=0} {¿2 Z(«4)Z(«6) =¿5^4 ^6 +

+1 {«1= =«3=0} {¿1= =i3} {«4 «5=0} {¿4 Z(«2)Z(«6) =¿5^2 Zi6 +

+1 {«1= = «4=0} {¿1= =¿4} {«2 «3=0} {¿2 Z(«5)Z(«6) =¿3 } Z¿5 ^6 +

+1 {«1= = «4=0} {¿1= =¿4} {«2 «5=0} {¿2 Z(«3)Z(«6) =¿5 } Z¿3 Zi6 +

6

+ 1{H = =¿4=0} j =j4} {¿3 = = ¿5=0} {j3 Z (¿2) z (¿6) 1 =j5 } Z j2 Zj6 +

+ 1{H = = ¿5=0} j =j5} {¿2 =¿3=0} {j2 Z(¿4)Z(¿6) 1 =j3}Zj4 Zj6 +

+ 1{H = = ¿5=0} j =j5} {¿2 =¿4=0} {j2 Z (¿3)Z ^6) 1 =j4}Zj3 Zj6 +

+ 1{H = = ¿5=0} j =j5} {¿3 = =¿4=0} {j3 Z(¿2)Z(¿6) 1 =j4}Zj2 Zj6 +

+ I{i2 = =¿3=0} {j2 = =j3} {¿4 =¿5=0} {j4 Z (¿1)Z ^6) 1 =j5}Zji Zj6 +

+ I{i2 = = ¿4=0} {j2 = =j4} {¿3 = =¿5=0} {j3 Z ^V (¿6)+ =j5} Zji Zj6 +

+ I{i2 = = ¿5=0} {j2 = =j5} {¿3 = =¿4=0} {j3 Z (¿l)z ^6) + =j4}Zji Zj6 +

+!{ie= =ii=0} {j6= =ji} {¿3 = =¿4=0} {j3 Z(¿2)Z(¿5) 1 =j4}Zj2 Z j5 +

+ 1{ie= =ii=0} {j6= =ji} {¿3 = =¿5=0} {j3 Z(¿2)Z(¿4) 1 =j5}Zj2 Z j4 +

+ !{ie= =ii=0} {j6= =ji} {¿2 =¿5=0} {j2 Z (¿3)Z ^4^) 1 =j5 } Z j3 Z j4 +

+ 1{ie= =ii=0} {j6= =ji} {¿2 =¿4=0} {j2 Z (¿3)Z (¿5) 1 =j4}Zj3 Z j5 +

+ 1{ie= =ii=0} {j6= =ji} {¿4 =¿5=0} {j4 Z(¿2)Z(¿3)+ =j5 } Zj2 Zj3 +

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+ 1{ie= =ii=0} {j6 =ji} {¿2 =¿3=0} {j2 Z^4) Z(¿5) 1 =j3 } Z j'4 Z j5 +

+ 1{ie= = ¿2=0} {j6 =j2} {¿3 = =¿5=0} {j3 Z ^V (¿4) 1 =j5} Zji Z j4 +

+ 1{ie= = ¿2=0} {j6 =j2} {¿4 =¿5=0} {j4 Z (¿l)z (¿3) + =j5}Zji Zj3 +

+ 1{ie= = ¿2=0} {j6 =j2} {¿3 = =¿4=0} {j3 Z ^V (¿5) 1 =j4}Zji Z j5 +

+ 1{ie= = ¿2=0} {j6 =j2} {¿i =¿5=0} j Z (¿3)Z (¿4) 1 =j5}Zj3 Z j4 +

+ 1{ie= = ¿2=0} {j6 =j2} {¿i =¿4=0} j Z (¿3)Z (¿5) 1 =j4}Zj3 Z j5 +

+ 1{ie= = ¿2=0} {j6 =j2} {¿i =¿3=0} j Z^4) Z(¿5) 1 =j3 } Z j'4 Z j5 +

+ 1{ie= =¿3=0} {j6 =j3} {¿2 =¿5=0} {j2 Z ^V (¿4) 1 =j5} Zji Z j4 +

+ 1{ie= =¿3=0} {j6 =j3} {¿4 =¿5=0} j Z ^V (¿2) 1 =j5}Zji Z j2 +

+ 1{ie= =¿3=0} {j6 =j3} {¿2 =¿4=0} {j2 = Z (¿1)Z (¿5) 1 =j4}Zji Z j5 +

+ 1{ie= =¿3=0} {j6 =j3} {¿i =¿5=0} j Z^.¿2 ) Z(¿4) 1 =j5}Zj2 Z j4 +

+ 1{ie= =¿3=0} {j6 =j3} {¿i =¿4=0} j Z(^¿2 ) Z(¿5) 1 =j4}Zj2 Z j5 +

+ 1{ie= =¿3=0} {j6 =j3} {¿i =¿2=0} j Z(¿4) Z(¿5) 1 =j2 } Z j'4 Z j5 +

+1{i6= =¿4=0} {j6 = =j4} {¿3 = = ¿5=0} {j3 = Z (¿1)z (¿2) + =j5} Z j1 Zj2 +

+1{i6= = ¿4=0} {j6 = =j4} {¿2 = =¿5=0} {j2 = Z (¿1)Z =j5}Zj1 Zj3 +

+1{i6= = ¿4=0} {j6 = =j4} {¿2 = =¿3=0} {j2 = Z (¿1)Z ^ + =j3}Zj1 Z j5 +

+1{i6= = ¿4=0} {j6 = =j4} {¿1 = =¿5=0} {j1 = Z(^¿2 ) Z^ + =j5}Zj2 Zj3 +

+1{i6= = ¿4=0} {j6 = =j4} {¿1 = =¿3=0} {j1 = Z(¿2)Z^ + =j3 } Z j2 Zj5 +

+1{i6= = ¿4=0} {j6 = =j4} {¿1 = =¿2=0} j Z ^V ^ + =j2 } Z j3 Z j5 +

+1{i6= = ¿5=0} {j6 = =j5} {¿3 =¿4=0} {j3 = Z ^V ^ + =j4}Zj1 Z j2 +

+1{i6= = ¿5=0} {j6 = =j5} {¿2 =¿4=0} {j2 = Z ^V ^ + =j4}Zj1 Zj3 +

+1{i6= = ¿5=0} {j6 = =j5} {¿2 =¿3=0} {j2 = Z ^V ^ + =j3}Zj1 Z j4 +

+1{i6= = ¿5=0} {j6 = =j5} {¿1 =¿4=0} j Z(^¿2 ) z^ + =j4}Zj2 Zj3 +

+1{i6= = ¿5=0} {j6 = =j5} {¿1 =¿3=0} j Z^2 ) ZMi =j3 } Z j2 Zj4 +

+1{i6= =¿5=0} 1 {j6 = =j5} {¿1 = ¿2=0} 1 j Z ^V (¿4) =j2 } Z j3 Zj4 n

^¿6 = =¿1=0} {j6= =j1} {¿2 = ¿5=0} {j2 =j5} {¿3 =¿4=0} {j3 = =j4}-

^¿6 = =¿1=0} {j6= =j1} {¿2 = ¿4=0} {j2 =j4} {¿3 =¿5=0} {j3 = =j5} —

^¿6 = =¿1=0} {j6= =j1} {¿2 = ¿3=0} {j2 =j3} {¿4 =¿5=0} {j4 = =j5} —

^¿6 = = ¿2=0} {j6= =j2} {¿1 = ¿5=0} {j1 =j5} {¿3 =¿4=0} {j3 = =j4}-

^¿6 = = ¿2=0} {j6= =j2} {¿1 = ¿4=0} {j1 =j4} {¿3 =¿5=0} {j3 = =j5} —

^¿6 = = ¿2=0} {j6= =j2} {¿1 = ¿3=0} {j1 =j3} {¿4 =¿5=0} {j4 = =j5} —

^¿6 = =¿3=0} {j6= =j3} {¿1 = ¿5=0} {j1 =j5} {¿2 =¿4=0} {j2 = =j4}-

^¿6 = =¿3=0} {j6= =j3} {¿1 = ¿4=0} {j1 =j4} {¿2 =¿5=0} {j2 = =j5} —

^¿3 = =¿6=0} {j3 = =j6} {¿1 = ¿2=0} {j1 =j2} {¿4 =¿5=0} {j4 = =j5} —

^¿6= = ¿4=0} {j6= =j4} {¿1 = ¿5=0} {j1 =j5} {¿2 =¿3=0} {j2 = =j3} —

^¿6= = ¿4=0} {j6= =j4} {¿1 = ¿3=0} {j1 =j3} {¿2 =¿5=0} {j2 = =j5} —

^¿6= = ¿4=0} {j6= =j4} {¿1 = ¿2=0} {j1 =j2} {¿3 =¿5=0} {j3 = =j5} —

^¿6= = ¿5=0} {j6= =j5} {¿1 = ¿4=0} {j1 =j4} {¿2 =¿3=0} {j2 = =j3} —

^¿6= = ¿5=0} {j6= =j5} {¿1 = ¿2=0} {j1 =j2} {¿3 =¿4=0} {j3 = =j4 } \

^¿6= = ¿5=0} {j6= =j5 } {¿1 = =¿3=0} {j1 = =j3} {¿2 = = ¿4=0} {j2 = j4}

(1.50)

P1 P7 /7

j[^ = u.m E-E^..¿,11

¿1=0 ¿7=0 \1=1

777

— 1{«l=«6=0,¿l=¿6} n — n ^ — 1{«3=«6=0,¿3=¿6} H —

¡=1 ¡=1 ¡=1 ¡=1,6 1=2,6 1=3,6

777

— -^^^¿^¿^ n ^ — 1{«5=«6=0,¿5=¿6} H C^ — 1{«1=«2=0,¿1=¿2} C?'; —

1=1 ;=1 ;=1 ¡=4,6 ¡=5,6 ¡=1,2

777

1{«1 = «3=0,¿1=¿з} n 4«1) - 1{«1 = «4=0,¿1=¿4} n ) - 1{«1=«5=0,¿1=¿5} n -

¡=1 ¡ = 1 ¡ = 1 ¡=1,3 ¡=1,4 ¡=1,5

777

H C^ — 1{«2 = «4=0,¿2=¿4} H Z¿'¡ — 1{«2=«5=0,i2=i5} H Z¿'¡ —

¡=1 ¡=1 ¡ = 1 ¡=2,3 ¡=2,4 ¡=2,5

777

H (¿^ — 1{«з = «5=0,¿'з=¿5} H Z¿•¡ — 1{«4=«5=0,¿4=¿5} H Z¿'¡

¡=1 ¡=1 ¡ = 1 ¡=3,4 ¡=3,5 ¡=4,5

-1{«7 = «1=0,¿7=Jl} II (¿r - 1{«7 = «2=0,¿7=¿2} II Z¿¡«¡ - 1{«7=«3=0,¿7=¿3^ II Z "

-( «¡) i , , TT ^ ( «¡) i , , TT Z( «¡)

¡=1 ¡=1 ¡=1 ¡=1,7 ¡=2,7 ¡=3,7

777

^ («¡)

1 , TT An) TT

-1{«7=«4=0,¿7=¿4} II Z¿¡«¡ - 1{«7=«5=0,¿7=¿5^ II C^ - 1{«7 = «6=0,¿7=¿6} il Z¿¡«¡ +

¡ = 1 ¡ = 1 ¡=1 ¡=4,7 ¡=7,5 ¡=7,6

+ 1{«1=«2=0,J1=¿2,«3=«4=0,J3=¿4} H C^ + 1{«1=«2=0,J1=¿2,«3 = «5=0,J3=¿5} H C^ +

1=5,6,7 1=4,6,7

^^^^¿'^¿^^^¿^M H C^ + 1{«1=«3=0,J1=¿3,«2 = «4=0,J2=¿4} C^ +

1=3,6,7 1=5,6,7

+ 1{«1=«3=0,J1=¿3,«2=«5=0,J2=¿5} H Cft + 1{«1=«3=0,J1=¿3,«4 = «5=0,J4=¿5} H Cft +

^^^^¿^¿¿l^^^^M H Cft + 1{«1=«4=0,J1=¿4,«2 = «5=0,J2=¿5} H Cft +

{«1=«4=0,j1=j4,«2=«5=0,j2=j5}

1=5,6,7 1=3,6,7

+ 1{«1=«4=0,J1=¿4,«3=«5=0,J3=¿5} H C^ + 1{«1=«5=0,J1=¿5,«2 = «3=0,J2=¿3} C^ +

1=2,6,7 1=4,6,7

7

+ 1{H = =;5=0J1 = =j5,i2 = =«4=0,j2 = =j4} n zi^ /=3,6,7 + 1{n= =j5,;3 = = «4=0,i3 = =i4} n zir /=2,6,7

+ I{i2 = =«3=0,j2 = =j3,i4 = =«5=0,j4 = =J5} n zf /=1,6,7 + 1{r2 = =«4=0,i2 = =j4,;3 = = «5=0,i3 = =i5} n 4'°+ /=1,6,7

+ I{i2 = =«5=0,j2 = =j5,i3 = =«4=0,j3 = =j4} n i1 /=1,6,7 + 1{r6= =n=0,i6= = «4=0,i3 = =i4} n cf+ /=2,5,7

+!{ie= =«1=0,j6 = =j1,i3 = =«5=0,j3 = =j5} n i1 /=2,4,7 + 1{r6= =n=0,i6= =J1,;2 = = «5=0,i2 = =i5} n i+ /=3,4,7

+ !{ie= =«1=0,j6 = =j1,i2 = = «4=0,j2 = =j4} n zir1 /=3,5,7 + 1{r6= =n=0,i6= =i1 ,r4 = =«5=0,i4 = =i5} n zir1+ /=2,3,7

+1{ie= =ii=0,j6= =j1,i2 = = «3=0,j2 = =j3} n zir1 /=4,5,7 + 1{r6= =;2=0j6= =«5=0,i3 = =i5} n zir1+ /=1,4,7

+1{ie= = «2=0,j6 = =j2,i4 = = «5=0,j4 = =j5} n zir' /=1,3,7 + 1{r6 = = ¿2=0,^6 = =«4=0,i3 = =i4} n zir1+ /=1,5,7

+1{ie= = «2=0,j6 = =j2,i1 = = «5=0,j1 = =j5} n zir' /=3,4,7 + 1{r6 = = ¿2=0,^6 = =j2,;1 = =«4=0,i1 = =i4} n zir1+ /=3,5,7

+1{ie= = «2=0,j6 = =j2,i1 = =«3=0,j1 = =j3} n zir1 /=4,5,7 + 1{r6 = = ;3=0,J6 = =j3,;2= =¿5=0,^2 = =i5} n zir1+ /=1,4,7

+1{ie= = «3=0,j6 = =j3,i4 = = «5=0,j4 = =j5} n zir1 /=1,2,7 + 1 { r6 =;3=0,j6= =j3,;2= =¿4=0,^2 = =i4} n zir1+ /=1,5,7

+1{ie= =«3=0,j6= =j3,H = = «5=0,j1 = =j5} n zir1 /=2,4,7 + 1 { r6 =;3=0j6= =«4=0,i1 = =i4} n zir1+ /=2,5,7

+1{ie= =«3=0,j6= =j3,i1 = = «2=0,j1 = =j2} n zir1 /=4,5,7 + 1 { r6 = ¿4=0,^6 = =j4,;3 = =;5=0,j3= =i5} n zir 1+ /=1,2,7

+1{ie= = «4=0,j6 = =j4,i2 = = «5=0,j2 = =j5} n zir1 /=1,3,7 + 1 { r6 = ¿4=0,^6 = =j4,;2 = =;3=0,j2= =i3} n zir1+ /=1,5,7

+1{ie= = «4=0,j6 = =j4,i1 = = «5=0,j1 = =j5} n zir1 /=2,3,7 + 1 { r6 = ¿4=0,^6 = =j4,;1 = =«3=0,i1= =i3} n zir1+ /=2,5,7

+1{ie= = «4=0,j6 = =j4,i1 = = «2=0,j1 = =j2} n zir1 /=3,5,7 + 1 { r6 = ¿5=0,^6 = =i5 ,r3 = =;4=0,j3= =i4} n zir1+ /=1,2,7

+1{ie= = «5=0,j6 = =j5,i2 = = «4=0,j2 = =j4} n zir1 /=1,3,7 + 1 { r6 = ¿5=0,^6 = =i5 ,r2 = =;3=0j2= =i3} n zir1+ /=1,4,7

+1{i6= =«5=0,j6 = =j5,H = =«4=0,j'l = =j4} n zir > /=2,3,7 + 1{r6= =i5,«l = =i3} n zir 1+ /=2,4,7

+1{i6= =«5=0,j6 = =j5,i1 = =«2=0,j'l = =i2} n zir' /=3,4,7 + 1{r7 = =H=0J7 = =i3} n zir >+ /=4,5,6

+ 1{ir= =il=0,j7 = =j'l,i2 = =«4=0,j2 = =j4} n zir 1 /=3,5,6 + 1{^7 = =H=0J7 = =j'l,i2 = =i5} n z|' >+ /=3,4,6

+ 1{ir= =il=0,j7 = =j'l,i2 = =«6=0,j2 = =j6} n zir1 /=3,4,5 + 1{r7 = = H=0J7 = = ¿4=0^3= =i4} n zir 1+ /=2,5,6

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

+ 1{ir= =i1=0,j7 = =j'l,i3 = = «5=0,j3 = =j5} n zji'1 /=2,4,6 + 1{r7 = = H=0J7 = =i6} n zir 1+ /=2,4,5

+ 1{ir= = i1=0,j7 = =j1,i4 = = «5=0,j4 = =j5} n zji'1 /=2,3,6 + 1{r7 = = «l=0,i7 = =i'l ,r4 = =i6} n zir 1+ /=2,3,5

+1{i1= = «2=0,j7 = =j'l,i7 = = il=0,j5 = =j6} n zji'1 /=2,3,4 + 1{r7 = = «2=0,i7 = =r3=0,il= =i3} n zir 1+ /=4,5,6

+ 1{ir= = «2=0,j7 = =j2,il = = «4=0,j'l = =j4} n zir1 /=3,5,6 + 1{r7 = = «2=0,i7 = =i5=0,j'l = =i5} n zir 1+ /=3,4,6

+ 1{ir= = «2=0,j7 = =j2,il = = «6=0,j'l = =j6} n zir1 /=3,4,5 + 1{r7 = = «2=0,i7 = =M=0j3 = =i4} n zir 1+ /=1,5,6

+ 1{ir= = «2=0,j7 = =j2,«3 = = «5=0,j3 = =j5} n zir1 /=1,4,6 + 1 { r7 = «2=0,i7 = =r6=0,i3= =i6} n zir 1+ /=1,4,5

+ 1{ir= = «2=0,j7 = =j2,i4 = = «5=0,j4 = =j5} n zir1 /=1,3,6 + 1 { r7 = «2=0,i7 = =i2 ,r4 = =r6=0,i4 = =i6} n zir 1+ /=1,3,5

+ 1{ir= = «2=0,j7 = =j2,i5 = = «6=0,j5 = =j6} n zir1 /=1,3,4 + 1 { r7 = «3=0,i7 = =r2=0,il = =i2} n zir1+ /=4,5,6

+ 1{ir= = «3=0,j7 = =j3,il = = «4=0,j'l = =j4} n zir1 /=2,3,5 + 1 { r7 = «3=0,i7 = =i5=0,j'l = =i5} n zir1+ /=2,4,6

+ 1{ir= = «3=0,j7 = =j3,il = = «6=0,j'l = =j6} n zir1 /=4,2,5 + 1 { r7 = «3=0,i7 = =r4=0,i2 = =i4} n zir1+ /=3,5,6

+ 1{ir= = «3=0,j7 = =j3,«2 = = «5=0,j2 = =j5} n zir1 /=1,4,6 + 1 { r7 = «3=0,i7 = =r6=0,i2= =i6} n zir1+ /=1,4,5

+ 1{ir= = «3=0,j7 = =j3,i4 = = «5=0,j4 = =j5} n zir1 /=1,2,6 + 1 { r7 = «3=0,i7 = =j'з,i4 = =r6=0,i4 = =i6} n zir1+ /=1,2,5

+ 1{i7=i3=0J7=i3,i5=i6=0J5=i6} H Zj; + ^^M^^J^H^^J'^.M H Zj +

/=1,2,4 /=3,5,6

+ 1{i7=i4=0j7=i4,i1=i3=0j1=i3} H Zj + ^^M^^J^H^^J'^M H Zj +

/=2,5,6 /=2,3,6

+ 1{i7=i4=0,i7=i4,i1=i6=0,i1=i6^ H Zj + ^^M^,.^^,^^^,.^^} H Zj +

/=2,3,5 /=1,5,6

) ii, , tt

+ 1{;7=i4=0j7=i4,i2=i5=0J2=i5} H Zj + ^^^^^ H Zj +

/=1,3,6 /=1,3,5

+ 1{;7=i4=0j7=i4,i3 = i5=0J3=i5} Zj + 1{;7 = i4=0,i7=i4,i3=i6=0,i3=i6} H Zj +

/=1,2,6 /=1,2,5

+ 1{i7 = i4=0j7=i4,i5 = i6=0J5=i6} H Zj + ^^^^ J^M H Zj +

/=1,2,3 /=3,4,6

+ 1{i7 = i5=0j7=i5,i1 = i3=0j1=i3} H Zj + ^^^^ .1=^ H Zj +

/=2,4,6 /=2,3,6

(r ) + 1 (r )

/=2,3,4 /=1,4,6

(r ) (r )

+ 1{i7 = i5=0,i7=i5,i1 = i6=0,i1=i6} H Zj + H Zj +

/=2,3,4 /=1,4,6

+ 1{i7 = i5=0j7=i5,i2 = i4=0J2=i4} H Zj + ^^^^ ^^^^^ .2=.^ H Zj +

/=1,3,6 /=1,3,5

+ 1{;7 = ;5=0,i7=i5 ,;3 = ;4=0,i3=i4} H Zj + ^^^^ ^^^^^ Zj +

/=1,2,6 /=1,2,4

+ 1{г7=г5=0,i7=i5,г4=г6=0,i4=i6} H Zj + ^^^^ jf^^l^^ jl=j2} H Zj +

/=1,2,3 /=3,4,5

+ 1{i7 = i6=0j7=i6,i1 = i3=0j1=i3} H Zj + H Zj +

/=2,4,5 /=2,3,5

+ 1{г7=г6=0,i7=i6,гl=г5=0,il=i5} H Zj + ^^^^ J2=J3} H Zj +

/=2,3,4 /=1,4,5

+ 1{г7=г6=0,i7=i6,г2=г4=0,i2=i4} H Zj + ^^^^ J2=J5} H Zj +

/=1,3,5 /=1,3,4

+ 1{i7 = i6=0j7=i6,i3 = i5=0J3=i5} H Zj + ^^^^ ^^^^^ H Zj +

/=1,2,4 /=1,2,3

+ 1{г7=г6=0,i7=i6,гз=г4=0,iз=i4} H C

(r 1 i

/=1,2,5

^ 1{i2=«3=0,j'2=j'з,«4=«5=0,j'4=j'5,«6=«7=0,j'6=j'7} + 1{«2=«3=0,j'2=j'з,«4=«6=0,j'4=j'6,«5=«7=0,j'5=j'7} +

+ 1{«2=«з=0,j'2=j'з,«4=«7=0,j'4=j'7,«5=«6=0,j'5=j'6} + +

+ 1{«2=«4=0,j'2=j'4,«з=«6=0,j'з=j'6,«5=«7=0,j'5=j'7} + +

+ 1{«2=«5=0,j'2=j'5,«з=«4=0,j'з=j'4,«6=«7=0,j'6=j'7} + +

+ 1{«2=«5=0,j'2=j'5,«з=«7=0,j'з=j'7,«4=«6=0,j'4=j'6} + +

+ 1{«2=«6=0,j'2=j'6,«з=«5=0,j'з=j'5,«4=«7=0,j'4=j'7} + +

+ 1{«2=«7=0,j'2=j'7,«з=«4=0,j'з=j'4,«5=«6=0,j'5=j'6} + +

+ 1 v (¿i)_

+ 1{«2 = «7=0,j'2=j'7,«3 = «6=0,j'з=j'6,«4=«5=0,j'4=j'5^ I V^

" ^1{¿l=¿з=0,j'l=j'з,i4=«7=0,j'4=j'7,«5=«6=0,j'5=j'6} + ^¿I^^ ^^^^^ ^=7} +

+ 1{¿l=¿з=0,j'l=j'з,¿4=«6=0,j'4=j'6,«5=«7=0,j'5=j'7} + 1{н=г4=0,j\=j4,гз=г5=0,jз=j5,г6=г7=0,j6=j7} +

+ 1{¿l=¿4=0,j'l=j'4,iз=«6=0,j'з=j'6,«5=«7=0,j'5=j'7} + 1{н=г4=0,j\=j4,гз=г7=0,jз=j7,г5=г6=0,j5=j6} +

+ 1{¿1 = ¿5=0,j'l=j'5,iз = «4=0,j'з=j'4,«6 = «7=0,j'6=j'7} + ^H^^ Jl^,^^^^^,^^^^^} +

+ 1{¿l=¿5=0,j'l=j'5,iз=«7=0,j'з=j'7,«4=«6=0,j'4=j'6} + 1{н=г6=0,j\=j6,гз=г4=0,jз=j4,г5=г7=0,j5=j7} +

+1 {«6=«i =0,i6=il ,iз=«5 =0,i3=i5 ,«4=¿7=0,j4=j7 } + 1{г6=гl=0,j6=j^гз=г7=0,jз=j7,г4=г5=0,j4=j5} +

+ 1{¿l=¿7=0,j'l=j'7,iз=«4=0,j'з=j'4,«5=«6=0,j'5=j'6} + ^H^^,^^,^^^^^,^^^^^} +

+ 1 lz (r2)_

+ 1{¿1 = ¿7=0,j'l=j'7,i3 = «6=0,j'з=j'6,«4=«5=0,j'4=j'5^ I J

" + 1{«1=«2=0,j'l=j'2,i4 = «6=0,j'4=j'6,«5 = «7=0,j'5=j'7} +

+ 1{¿1 = ¿2=0,j'l=j'2,i4 = «7=0,j'4=j'7,«5 = «6=0,j'5=j'6} + 1{н = г4=0,j\=j4,г2=г5=0,j2=j5,г6=г7=0,j6=j7} +

+ 1{¿1 = ¿4=0,j'l=j'4,i2 = «6=0,j'2=j'6,«5 = «7=0,j'5=j'7} + 1{н = г4=0,j\=j4,г2=г7=0,j2=j7,г5=г6=0,j5=j6} +

+ 1{¿1 = ¿5=0,jl=j5,¿2 = «4=0,j2=j4,«6 = «7=0,j6=j7} + ^H^^,^^,^^^^^,^^^^^} +

+ 1{¿1 = ¿5=0,j'l=j'5,i2 = «7=0,j'2=j'7,«4 = «6=0,j'4=j'6} + 1{г6=гl=0,j6=j^г2=г4=0,j2=j4,г5=г7=0,j5=j7} +

+ 1{«6 = «1=0,j'6=j'l,i2 = «5=0,j'2=j'5,«4 = «7=0,j'4=j'7} + 1{г6=гl=0,j6=j^г2=г7=0,j2=j7,г4=г5=0,j4=j5} +

+ 1{¿1 = ¿7=0,j'l=j'7,i2 = «4=0,j'2=j'4,«5 = «6=0,j'5=j'6} + ^H^^,^^,^^^^^,^^^^^} +

+ 1 iz (r3 )_

+ 1{¿1 = ¿7=0,j'l=j'7,i2 = «6=0,j'2=j'6,«4=«5=0,j'4=j'5} I i

^1 {«l=«2=0.1=j2,;з=;5=o,jз=j5,;6=;7=o,j6=j7}+1{«l=«2=0 +

+ 1{«l=«2=0 ,jl=j2,«3=«7=0 J3=j7,«5=«6=0 J5=j6} + 1{гl=гз=0,jl=jз,г2=г5=0,j2=j5,г6=г7=0,j6=j7} +

+ 1{;i=«3=0 jl^,^^^ ^^^^^ J5=j7} + 1{гl=гз=0,il=iз,г2=г7=0,i2=i7,г5=г6=0,i5=i6} + + 1{гl=г5=0,il=i5,г2=гз=0,i2=iз,г6=г7=0,i6=i7} + 1{гl=г5=0,jl=j5,г2=г6=0,j2=j6,гз=г7=0,jз=j7} + + 1{гl=г5=0,il=i5,г2=г7=0,i2=i7,гз=г6=0,iз=i6} + 1{г6=гl=0,j6=jl,г2=гз=0,j2=jз,г5=г7=0,j5=j7} + + 1{г6=гl=0,i6=il,г2=г5=0,i2=i5,гз=г7=0,iз=i7} + 1{г6=гl=0,j6=jl,г2=г7=0,j2=j7,гз=г5=0,jз=j5} +

+ 1{«7=«I=0 j7=jl,г2=гз=0 j2=jз,г5=г6=0 J5=j6} + 1{г7=гl=0,j7=jl,г2=г5=0,j2=j5,гз=г6=0,jз=j6} +

+ 1 1Z (;4)_

+ 1{i7 = i1=0,i7=i1,i2 = i6=0,i2=i6,i3=i5=0,i3=i5^ I j

1{гl=г2=0,jl=j2,гз=г4=0,jз=j4,г6=г7=0,j6=j7} + 1{гl=г2=0,jl=j2,гз=г6=0,jз=j6,г4=г7=0,j4=j7} +

+ 1{гl=г2=0,il=i2,гз=г7=0,iз=i7,г4=г6=0,i4=i6} + 1{гl=гз=0,jl=jз,г2=г4=0,j2=j4,г6=г7=0,j6=j7} + + 1{гl=гз=0,jl=jз,г2=г6=0,j2=j6,г4=г7=0,j4=j7} + 1{гl=гз=0,jl=jз,г2=г7=0,j2=j7,г4=г6=0,j4=j6} + + 1{гl=г4=0,jl=j4,г2=гз=0,j2=jз,г6=г7=0,j6=j7} + 1{гl=г4=0,jl=j4,г2=г6=0,j2=j6,гз=г7=0,jз=j7} + + 1{гl=г4=0,jl=j4,г2=г7=0,j2=j7,гз=г6=0,jз=j6} + 1{г6=гl=0,j6=jl,г2=гз=0,j2=jз,г4=г7=0,j4=j7} + + 1{г6=гl=0,j6=jl,г2=г4=0,j2=j4,гз=г7=0,jз=j7} + 1{г6=гl=0,j6=jl,г2=г7=0,j2=j7,гз=г4=0,jз=j4} + + 1{гl=г7=0,jl=j7,г2=гз=0,j2=jз,г4=г6=0,j4=j6} + 1{гl=г7=0,jl=j7,г2=г4=0,j2=j4,гз=г6=0,jз=j6} +

+ 1 1Z (;5)_

+ 1{i7 = i1=0,j7=j1,i2 = i6=0,j2=j6,i3=i4=0,j3=j4^ I j

1{гl=г2=0,jl=j2,гз=г4=0,jз=j4,г5=г7=0,j5=j^^

+ 1{гl=г2=0,jl=j2,гз=г7=0,jз=j7,г4=г5=0,j4=j5} + 1{гl=гз=0,jl=jз,г2=г4=0,j2=j4,г5=г7=0,j5=j7} + + 1{гl=гз=0,jl=jз,г2=г5=0,j2=j5,г4=г7=0,j4=j7} + 1{гl=гз=0,jl=jз,г2=г7=0,j2=j7,г4=г5=0,j4=j5} + + 1{гl=г4=0,jl=j4,г2=гз=0,j2=jз,г5=г7=0,j5=j7} + 1{гl=г4=0,jl=j4,г2=г5=0,j2=j5,гз=г7=0,jз=j7} + + 1{гl=г4=0,jl=j4,г2=г7=0,j2=j7,гз=г5=0,jз=j5} + 1{гl=г5=0,jl=j5,г2=гз=0,j2=jз,г4=г7=0,j4=j7} + + 1{гl=г5=0,jl=j5,г2=г4=0,j2=j4,гз=г7=0,jз=j7} + 1{гl=г5=0,jl=j5,г2=г7=0,j2=j7,гз=г4=0,jз=j4} + + 1{г7=гl=0,j7=jl,г2=гз=0,j2=jз,г4=г5=0,j4=j5} + 1{г7=гl=0,j7=jl,г2=г4=0,j2=j4,гз=г5=0,jз=j5} +

+ 1 IZ (;6)_

+ 1{i7 = i1=0,j7=j1,i2 = i5=0,j2=j5,i3=i4=0,j3=j4} I Cj6

1{il=i2=0,j1=j2,i3=i4=0,j3=j4,i5=i6=0,j5=j6} + 1{i1=i2=0,j1=j2,i3=i5=0,j3=j5,i4=i6=0,j4=j6} +

+ l{il = «2=0 ji=j2,«3 = «6=0 j3=j6,M = «5=0 J4=j5} + I{ii = i3=0,j1=j3,i2=i4=0,j2=j4,i5=i6=0,j5=j6} +

+ I{il = i3=0,jl=j3,i2 = i5=0,j2=j5,i4 = i6=0,j4=je} + 1{il = i3=0,jl=j3,i2=i6=0,j2=j6,i4=i5=0,j4=j5} +

+ I{i4 = il=0,j4=ji,i2 = i3=0,j2=j3,i5 = i6=0,j5=j6} + 1 { i4 = i l=0,j4 = jl ,i2=i5=0,j2 = j5 ,i3=i6=0,j3= j6 } +

+1 { i4 = i l =0 ,j4 = j l ,i 2 = i6 =0 ,j2 = j6 ,i3 = i5 =0 ,j3 = j5 } + 1 { i5 = i l=0,j5 = jl ,i2=i3=0,j2 = j3 ,i4=i6=0,j4= j6 } +

+ 1{i5 = il=0,j5=jl,i2 = i4=0,j2=j4,i3 = i6=0,j3=j6} + 1 { i5 = i l=0,j5 = jl ,i2=i6=0,j2 = j6 ,i3=i4=0,j3= j4 } +

+ 1{i6 = il=0,j6=jl,i2 = i3=0,j2 = j3 ,i4 = i5 =0 ,j4 = j5 } + 1 { i6 = i l=0,j6 = jl ,i2=i4=0,j2 = j4 ,i3=i5=0,j3= j5 } +

+ 1{i6=il=0,j6=jl,i2 = i5=0,j2=j5,i3 = i4=0,j3=j4^ j ^ , (1'51)

where 1A is the indicator of the set A.

1.1.5 Expansion of Iterated Ito Stochastic Integrals of Multiplicity k (k E N) Based on Theorem 1.1

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Consider a generalization of the formulas (11.45I)-(TT5T1) for the case of arbitrary multiplicity k for J)]T,t. In order to do this, let us consider the unordered set {1, 2,..., k} and separate it into two parts: the first part consists of r unordered pairs (sequence order of these pairs is also unimportant) and the second one consists of the remaining k — 2r numbers. So, we have

({jffi,92}, • • •, {92,-1,92r}}, {<Zi, • • •, qk-2r)), (1-52)

part 1 part 2

where

{gi,g2,... ,g2r-i,g2r,qi,... ,qk-2r} = {1,2,... },

braces mean an unordered set, and parentheses mean an ordered set.

We will say that (1.52) is a partition and consider the sum with respect to all possible partitions

y ^ g2,---,g2r-1g2r,9i---9fc-2r . (1.53)

({{31,32 },---,{32r-1>32r }},{qi ■ ■■■>9fc-2r}) {31,32 >'">32r-1>32r>91 >'">9fe-2r } = {1>2>'">fc}

Below there are several examples of sums in the form (1.53)

E ag1g2 = a12,

({31 j32}) {31,32} = {1,2}

E

a3ig233g4 — a1234 + a1324 + a2314,

({{31>32}>{33>34}})

{S1,S2,S3,S4} = {1,2,3,4}

E

a

3ig2,qiq2

({31,32}>{91>92})

{S1,S2,91,92} = {1,2,3,4}

— a12,34 + a13,24 + a14,23 + a23,14 + a24,13 + a34,12,

E

a

â,1â,2,<M2q3

({31>S2}>{91>92>93}) {31>S2>91>92 >93} = {1>2>3>4>5}

— a12,345 + a13,245 + a14,235 + a15,234 + a23,145 + a24,135 + +a25,134 + a34,125 + a35,124 + a45,123,

E

a

31g2,33g4,q1

({{31,32}>{33 >34}}>{91}) {31,32>33>34>91} = {1>2>3>4>5}

— a12,34,5 + a13,24,5 + a14,23,5 + a12,35,4 + a13,25,4 + a15,23,4 + +a12,54,3 + a15,24,3 + a14,25,3 + a15,34,2 + a13,54,2 + a14,53,2 + +a52,34,1 + a53,24,1 + a54,23,1-

Now we can formulate Theorem 1.1 (see (1.10)) using alternative form.

Theorem 1.23 [4] (2009) (also see [5]-[16], [23], [28], [38], [47], [48]). Under the conditions of Theorem 1.1 the following expansion

J

l.i.m.

P1

e

j1=0

Pk

k [k/2]

■•EC^III j ' + E(-D

jk =0

-i)rx

,1=1

r=1

X

E

ire

({{31,32}>--->{32r-1>32r }},{91>--->9fc-2r }) S = 1 {S1,S2>"'>S2r-1>S2r>91>'">9fc-2r } = {1,2>'">k}

f2s-1 = ig2s =0} 1{jg2s-1 = jg2s }

k-2r

IT

1=1

)

(1.54)

r

converging in the mean-square sense is valid, where [x] is an integer part of a real number x.

In particular, from (1.54) for k = 5 we obtain

3The connection of formulas (11.45l)-(1.5111. (11.541) with Hermite polynomials is studied in Sect. 1.10, 1.11 (see

Theorems 1.14-1.17).

P1 P5 /5

J [^ = l.i.m £ ...£C,5..Jn j'

j1=0 j5=0 \/=1

3

)

iis1 = ¿s2 =0}1{js1 = } H j

({S1,S2}>{91>92>93}) 1=1

{31,32>91>92>93} = {1>2>3>4>5}

E 1{ifl1 = ^ =0} j = jfl2 }Yl j') +

+ E 1{is1 = ¿s2 =0}1ijs1 = js2 }1{ig3 = ¿s4 =0} 1{jg3 = js4 }Zjg1'

(i91 )

({{S1>S2},{S3>S4}}>{91}) {S1,S2>S3>34>91} = {1>2>3>4>5}

The last equality obviously agrees with (1.49).

1.1.6 Comparison of Theorem 1.2 with the Representations of Iterated Ito Stochastic Integrals Based on Hermite Polynomials

Note that the correctness of the formulas (I! .45I)-(ITT5T1) can be verified in the following way. If ii = ... = i7 = i = T,..., m and ^i(s),..., ^7(s) = ^(s), then we can derive from (T.45I) (TT5T)) [2]-[T6], [28] the well known equalities

T,t = J; sT,t,

JVF]t, = ± (4, " AT,), ■Wi3)kt = i (4, - 3^AT,) ,

= I - 64,AT, + 3A|,) ,

J[^]T, = i (4, - 104,iAT)i + ,

■Wi6)kt = i " 15AT, + ^ölAh ~ 15AT,) , J^V = i (4, - 214,AT, + 1054,A|, - 105^,A^)

w. p. 1, where

T T

¿t, = ^(sf AT,t =

r2'

t t

which can be independently obtained using the Ito formula and Hermite polynomials [103 .

When k = 1 everything is evident. Let us consider the cases k = 2 and k = 3 in detail. When k = 2 and p1 = p2 = p we have (see (1.46)) [2]-[16], [28

(¿Mi)

J[^ = l.i,n. ( £ 6- c'M1 -E C

jiji

ji 72=0

ji=0

( p jl-1 / \ p / ■is EE 7 + )zjij+ E6iji (

Vi=0 72=0 v 7 ji=0 v

/ p ji-1 1 p /

ee^^E^:1

Vi=0 72 =0 ji=0 v

cf) -1

1=

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= l.i.m.

pis

ji=0 j2=0 1

7i=0 p

1 p 1 p /

o. o„-n n v

2

l Ji>J2~ \ ji=j2

1

ji=0 2

J

= £5- UfE^l-5EC?

1

ji=0

2

2 ^ ~ji 7i=0

^ - w. p. i.

(1.55)

Let us explain the last step in (1.55). For the Ito stochastic integral the following estimate [98] is valid

q/2

m

T

< m

I£t |2dT

(1.56)

where q > 0 is a fixed number, fT is a scalar standard Wiener process, eT £ M2([t,T]), is a constant depending only on q,

T

J |£r|2dr < s w. p. 1,

t

T \ q/2'

m

ieT i2dr

<.

p

2

Since

T

*T,t - 6ij = / (W) - 6jij (s)) dfs(i)

ji =0 *t v 7i=0

then applying the estimate (1.56) to the right-hand side of this expression and considering that

T p 2

J (V(S) -£ 6ij (S)) ds i 0

,s) - 6j(SM ds i

t ji=0

if p i s, we obtain

T

p

s)dfv = q - l.i.m. x 6 z(i)

i ^(s)dfs(i) = q - l.i.m. ^ Cj j, q > 0. (1.57)

J pis „

pis . „ t ji=0

Here q - l.i.m. is a limit in the mean of degree q. Hence, if q = 4, then it is

pis

easy to conclude that w. p. 1

^ (gCjC«)2 = 5T,t■

This equality as well as Parseval's equality were used in the last step of the formula (1.55).

When k = 3 and p1 = p2 = p3 = p we obtain (see (1.47)) [2]-[16], [28

J [#>]T,( = Urn. E Cj Cji^zj

ji ,j2 ,j3=0

p

(i)C (i)C (i) 73

ppp

E6 C(¿) _ V^ 6 C(¿) _ V^ 6 C(¿)

6j3ji ji Cj / v 67272 ji Cj / v 6ji72ji Cj

7i,73=0 ji,j2=0 ji,j2=0

= ^ ^ 6j3j2ji- £ iCj3jiji + Cj'ijij3 + C3i333ij CjijM =

\ji,j2,j3=0 ji,j3 =0 ^ /

/ p ji-1 j2 1 ✓ \ = ^ X): ( 6j3j2ji + Cj3jij2 + Cj2jij3 + Cj2j3ji + Cj'ij2 j3 + Cj'ij3j2 j X

p s Vi=0 j2=0 j3=0 7

> (i)C (i)C (i) XCji Cj2 Cj3 +

P ji-1 / \

+ y ^ ( Cj3j'ij3 + Cjij3j3 + Cj3j3j'i j f

— n ---n V /

z(:A z(:)+

Cj3 / Cji +

ji =0 j3=0 P ji-1

P ji- / \ 2

+E E (j j,+Cj,jij3+j (cj:0 zj3)+

ji =0 j3=0 J

+E Cj

P o P

'jijiji I Cji

ji =0 ji,j3 =0

y ^ (Cj3jiji + Cjijij3 + Cjij3ji) C

j3

/ P ji-1 j2 1

^ EEECj.Cj2o,Cj:)cj

\ji=0 j2=0 j3=0

P j i - 1

ji=0 j3=0

P ji 1 2

w'lVw^i

ji=0 j3=0

1 p x 3 1 p

T / u(:)\ 3 T

,3 M*) g i ji=0

2

E Cji Cj3 C

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(:) j3

ji,j3=°

l.i.m.

6

E

(:) (:) (:) Cji Cj2 Cj3 j Cj2 Cj3 +

ji j2 j3 = 0 ji=j2 j2=j3 Ji=j3

P ji 1 2 P ji 1 2

Ai) < 1st sr r2 r /7WwW+ K V^3 J v ^ 2 Z^ Z^ ./'• v / / ^3 ^

ji=0 j3=0 ji=0 j3=0

1 P X 3 1 P

- Vr3

g Z^ ii v ^

ji=0

E Cji Cj3 j

(:) j3

ji,j3 =0

ll E

ji,j2,j3=0

P j i - 1

a ( 3 S S ((^ ) (If + 3 S ^ V Sji

«V C(:) +

Cj3 +

ji =0 j3=0

+C

ji=0

3 /C(:) ji \Cji

ji=0 j3=0 3

+

1 p ji-1 2 -, p ji-1 2

(dTci:1 + JEE^ (d1)^

ji=0 j3=0 ji=0 j3=0

1 p 3 1 p

6 / v ^i y 2 7 v ji j j3 ji=0 ji,j3=0

p \ 3 p p

ji=0 / ji=0 j3=0

= ^ - ZSrATt) W. p. 1. (1.58)

The last step in (1.58) follows from Parseval's equality, Theorem 1.1 for k = 1, and the equality

lpi.m. Cjij') = w.p.l,

which can be obtained easily when q = 8 (see (1.57)).

In addition, we used the following relations between Fourier coefficients for the considered case

Cjlj2 + Cj2jl = Cjl Cj2 , 2Cjljl = , (1.59) Cjlj2j3 + Cjlj3j2 + Cj2j3jl + Cj2 jlj3 + Cj3j2jl + Cj3jl j2 = Cjl Cj2 Cj3 , (1.60) 2 (Cjljlj3 + Cjlj3jl + Cj3jl jl ) = Cjl Cj3 , (1.61)

6Cjljljl = Cjl. (1.62)

1.1.7 On Usage of Discontinuous Complete Orthonormal Systems of Functions in Theorem 1.1

Analyzing the proof of Theorem 1.1, we can ask the question: can we weaken the continuity condition for the functions (x), j = 1, 2,.. .?4

We will say that the function f (x) : [t,T] ^ R satisfies the condition (*), if it is continuous at the interval [t, T] except may be for the finite number of points of the finite discontinuity as well as it is right-continuous at the interval

[t,T ].

4 The results of this section will be generalized to the case of an arbitrary complete orthonormal system of functions j^j(x)}°=0 in the space L2([t, T]) and ^l(r), ..., ) € L2([t, T]) in Sect. 1.11 (see Theorem 1.16).

Furthermore, let us suppose that {0j(x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function (x) of which for j < oo satisfies the condition (*).

It is easy to see that continuity of the functions (x) was used substantially for the proof of Theorem 1.1 in two places. More precisely, we mean Lemma 1.3 and the formula (1.19). It is clear that without the loss of generality the partition {Tj}N=0 of the interval [t,T] in Lemma 1.3 and (1.19) can be taken so "dense" that among the points Tj of this partition there will be all points of jumps of the functions ^i(t) = j (t), ..., (t) = j (t) (ji,..., jk < o) and among the points (j,..., Tjk) for which 0 < j < ... < jk < N — 1 there will be all points of jumps of the function $(t1,..., ).

Let us demonstrate how to modify the proofs of Lemma 1.3 and the formula (1.19) in the case when {0j(x)}o=0 is a complete orthonormal system of functions in the space L2([t,T]), each function (x) of which for j < o satisfies the condition (*).

At first, appeal to Lemma 1.3. From the proof of this lemma it follows that

m

n-1

E J ^

rj+i,rj

j=0

N-1

m

J [AW ]

rj+1,rj

+

N-1

+ 6^ m

j=0

J [A^i]

tj+I>

2Ï j-1

m

q=0

rj+i

J [A^z]

Tq+1>

m

{|J [A^Z]rJ+i,Tj = J (^z (Tj ) - ^ (s))2ds,

m

{|J [A^]Wj|4}

/ rj+i

\

= 3

(^z(Tj) - ^z(s))2ds

\Tj

y

(1.63)

Suppose that the functions (s) (l = 1,...,k) satisfy the condition (*) and the partition {Tj}NL0 includes all points of jumps of the functions (s) (l = 1,..., k). It means that for the integral

rj+i

(^z(Tj) - ^z(s))2ds

4

2

the integrand function is continuous at the interval [Tj , Tj+1], except possibly the point Tj+1 of finite discontinuity.

Let ^ E (0, ATj) be fixed. Due to continuity (which means uniform continuity) of the functions p/(s) (/ = 1,... ) at the interval [Tj,rj+1 — p] we have

rj+i

(p/ (Tj) — p/ (s))2ds =

Tj+i—M "j+i

I I (, ^ .(,-,- \ , . f r->\\2 d ^ ^ r-2 I

= J (p/(Tj) — p/ (s))2ds + J (p/ (Tj) — p/(s))2ds < e2(Ar, — + M2

Tj Tj+1 - M

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(1.64)

When obtaining the inequality (1.64) we supposed that ATj < 5(e) for all j = 0,1,..., N — 1 (here 5(e) > 0 exists for any £ > 0 and it does not depend on s),

p(Tj) — P/(s)| <£

for s E [Tj ,Tj+1 — p.] (due to uniform continuity of the functions p^s), I = 1,..., k),

|p/(Tj) — p/(s)| <M

for s E [Tj+1 — p., Tj+1], M is a constant (potential discontinuity point of the function p/ (s) is the point Tj+1).

Performing the passage to the limit in the inequality (1.64) when ^ ^ +0, we get

rj+i

(p/(Tj) — p/(s))2ds < £2ATj.

(1.65)

Using (1.65) to estimate the right-hand side of (1.63), we obtain

( N—1

m >

\

E J [Ap/] j=0

Tj+i>

N1

N—1 j —1

< e4( ^(ATj)2 + 6 E ATj E AtJ <

j=0

j=0 q=0

< 3e4 (5(e)(T — t) + (T — t)2) .

This implies that

m

N1

E J [AP/]

Tj+i>

j=0

0

(1.66)

4

when N —to and Lemma 1.3 remains correct.

Now, let us present explanations concerning the correctness of (1.19), when

(x)}°=0 is a complete orthonormal system of functions in the space L2([t, T]), each function ^ (x) of which for j < to satisfies the condition (*).

Consider the case k = 3 and the representation (1.21). Let us demonstrate that in the studied case the first limit on the right-hand side of (1.21) equals to zero (similarly, we can demonstrate that the second limit on the right-hand side of (1.21) equals to zero; proof of the second limit equality to zero on the right-hand side of the formula (1.20) is the same as for the case of continuous functions (x), j = 0,1,...).

The second moment of the prelimit expression of first limit on the right-hand side of (1.21) looks as follows

N-1 ¿3-1 j2-1 j + 1 j+1

EEE J J (*(*!, t2 ,Tj3 ) - *(t1,Tj2 ,Tj3 ))2 «^¿3.

¿3=0 ¿2=0 ji =0

Further, for the fixed p £ (0, Arj2) and p £ (0, At^) we have

22 + 1 TJ1 + 1

J J ($(t1,t2,^¿3) - $(i1,Tj2 ,Tj3))2 «2 =

Tj2 Tj1

/ rj2 + 1-^ 22 + 1 \ / TJ1 + 1-P 21 + 1 \

,J + , .

\ j rJ2 + l-M /

J+J,

V j -ji+i-p /

(^(¿1,^2, Tj3 ) - T, , Tj3))2 dt1dt2 =

/ Tj2 + i-Mrji + 1-p Tj2 + i-M 2i + i 2'2 + i Tjl + 1-p TJ2 + i 2i + i \

/ hi hi I + / /.

\ j j j 2 i + i-P TJ2 + i-M j Tj2 + i-pTJ'i + i-P /

X

X ($(Î1,Î2,Tj3) - $(Î1,Tj2 ,T-3)) dM*2 < < e2 (At, - p) (At-i - p) + M2p (At, - p) + M2p (At-, i - p) + M2pp, (1.67)

where M is a constant, At- < Ô(e) for j = 0,1,..., N - 1 (£(e) > 0 exists for any e > 0 and it does not depend on points (t1,t2,Tj3), (t1 ,t-2, t-3)). We suppose here that the partition {t-}N=0 contains all discontinuity points of the

function $(t1,i2,i3) as points Tj (for each variable with fixed remaining two variables). When obtaining the inequality (1.67) we also supposed that potential discontinuity points of this function (for each variable with fixed remaining two variables) are contained among the points Tj1+1, Tj2+1, Tj3+1.

Let us explain in detail how we obtained the inequality (1.67). Since the function $(t1;t2,t3) is continuous at the closed bounded set

Q3 = ^ (t1, t2 ,t3) : ¿1 E [Tji ,Tji+1 — p],t2 E [Tj2 ,Tj2+1 — M],t3 E [Tj3 ,Tj3+1 — v ] j ,

where p, m, v are fixed small positive numbers such that

v E (0, Aj), m E (0, ATj2), p E (0, ATji),

then this function is also uniformly continous at this set. Moreover, the function ^(t1, t2, t3) is supposed to be bounded at the closed set D3 (see the proof of Theorem 1.1).

Since the distance between points (t1,t2,Tj3), (t1 ,Tj2,Tj3) E Q3 is obviously less than 5(e) (At, < 5(e) for j = 0,1,..., N — 1), then

|$(t1,t2 ,Tj3 ) — $(i1,Tj-2 ,Tj3 )| < e.

This inequality was used to estimate the first double integral in (1.67). Estimating the three remaining double integrals in (1.67) we used the boundedness property for the function $(t1,t2,t3) in the form of inequality

|$(*1,i2 ,Tj3 ) — $(i1,Tj2 ,Tj3 )| <M.

Performing the passage to the limit in the inequality (1.67) when m, p ^ +0, we obtain the estimate

TJ2 + i TJi + i

J ($(*!, *2,Tj3) — $(i1,Tj-2 ,Tj3))2 dt1dt2 < e2ATj2 ATji. Tj2 Tji This estimate provides

N — 1 j3 — 1 j2 1 Tj2 + i Tji/+i

EEE J J W1,t2,Tj3 ) — $(i1,Tj-2 ,Tj3 ))2 dt^Aj <

j3 =0 j2=0 ji=0 T,

N — 1 j3 — 1 j2 1 (T _ t)3

E E E A^J2Am < £

j3=0 j2=0 ji=0

The last inequality means that in the considered case the first limit on the right-hand side of (1.21) equals to zero (similarly, we can demonstrate that the second limit on the right-hand side of (1.21) equals to zero).

Consequently, the formula (1.19) is correct when k = 3 in the studied case. Similarly, we can perform the argumentation for the cases k = 2 and k > 3.

Therefore, in Theorem 1.1 we can use complete orthonormal systems of functions {^(x)}°=0 in the space L2([t,T]), each function ^(x) of which for j < to satisfies the condition (*).

One of the examples of such systems of functions is a complete orthonormal system of Haar functions in the space L2([t,T])

where n = 0,1,..., j = 1, 2,..., 2n, and the functions ^ (x) are defined as

n = 0,1,..., j = 1,2,..., 2n (we choose the values of Haar functions in the points of discontinuity in such a way that these functions will be right-continuous).

The other example of similar system of functions is a complete orthonormal system of Rademacher-Walsh functions in the space L2([t, T])

' 2n/2, x G [(j - 1)/2n, (j - 1)/2n + 1/2n+1)

Pnj(x) = < - 2n/2, x G [(j - 1)/2n + 1/2n+1, j/2n)

0,

otherwise

V

x — t

where 0 < m1 < ... < mk, m1,..., mk = 1, 2,..., k = 1, 2,.

1.1.8 Remark on Usage of Complete Orthonormal Systems of Functions in Theorem 1.1

Note that actually the functions (s) from the complete orthonormal system of functions (s)}°=0 in the space L2([t,T]) depend not only on s, but on t and T.

For example, the complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]) have the following form

2j + 1 // T + t\ 2

<t>Äs,t,T) = \HF—rPj[

n ' ' 7 V t -1 j VV" 2 y T - ty

where Pj(y) (j = 0,1, 2,...) is the Legendre polynomial,

1, j = 0

\/2sin (27rr(s - i)/(T -t)), j = 2r - 1, (1.68) \/2cos (27rr(s -t)/(T-t)), j = 2r

where r = 1, 2,...

Note that the specified systems of functions are assumed to be used in the context of implementation of numerical methods for Ito SDEs (see Chapter 4) for the sequences of time intervals

[T0,TJ, [T1,T2], [T2,T3], ...

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and Hilbert spaces

L2QT0, T1]), L2([T1,T2]), L2 ([T2, T3]) , ...

We can explain that the dependence of functions (s,t,T) on t and T (hereinafter these constants will mean fixed moments of time) will not affect on the main properties of independence of random variables

T

jt = i k(s,t,T)dw«

where i = 1,..., m and j = 0,1, 2,...

Indeed, for fixed t and T due to orthonormality of the mentioned systems of functions we have

m {= 1{»=r}1{j =g},

where i,r = 1,...,m, j,g = 0,1, 2,...

This means that C(j)T t and C^t t are independent for j = g or i = r (since these random variables are Gaussian).

From the other side, the random variables

Tl T2

CjjU = / h(s,t1,T1)dwii), C((;)t2,2 = / k(s,t2,T2)dwii) tl t2

are independent if [t1, T1] R [t2,T2] = 0 (the case T1 = t2 is possible) according to the properties of the Ito stochastic integral.

Therefore, the important properties of random variables C(j)Tt, which are the basic motive of their usage, are saved.

1.1.9 Convergence in the Mean of Degree 2n (n G N) of Expansions of Iterated Ito Stochastic Integrals from Theorem 1.1

Constructing the expansions of iterated Ito stochastic integrals from Theorem 1.1 we saved all information about these integrals. That is why it is natural to expect that the mentioned expansions will converge not only in the mean-square sense but in the stronger probabilistic senses.

We will obtain the general estimate which proves convergence in the mean of degree 2n (n G N) of expansions from Theorem 1.1.

According to the notations of Theorem 1.1 (see (1.41)), we have

RTt-= J[^(k)]T,t - J[^Tf"^ = T t2

= E /.../Rpi...pk(t1,...,tkf ...f;fc), (1.69)

(t1 r-'^fc ) t t

where

Pi Pk k

Rpi...pfc (t1,... ,tk) d=f k (t1,... ,tk) — e.. ^ Cjk...ji j] (ti),

j'i=0 jk=0 1=1

J [^(k)] T,t is the stochastic integral (O), J[^(k)]Tit''Pk is the expression on the right-hand side of (1.10) before passing to the limit l.i.m. .

Note that for definiteness we consider in this section the case i1,..., ik = 1,..., m. Another notations from this section are the same as in the formulation and proof of Theorem 1.1.

When proving Theorem 1.1 we obtained the following estimate (see (1.42))

m|(RTr^)2} < Ck J (t1,...,tk)dt1 ...dtk,

[t,T ]k

where Ck is a constant. Assume that

nt'—11 = J ■ ■ ! Rpi-«(t1, • •., tk)dfi:i)... df(1!;i). l = 2, 3,..., k + 1,

t t

nTk) = J... J Rpi...pk(t1,..., tk)<">... df'r), n!k+„( d=f nTkt.

tt

Using the Ito formula it is easy to demonstrate that

2A t r / s \ 2n—2

M ^d/T | > = n(2n — 1) J MM J £ud/u | £ > ds.

to I \to

Using the Holder inequality (under the integral sign on the right-hand side of the last equality) for p = n/(n — 1), q = n (n > 1) and using the increasing of the value

( t \ 2n' m < | j ^ d/

to

with the growth of t, we get

2n'

2n^ \ (n-1)/n

m

£td/T > < n(2n - 1)

VtQ

m

£t d/r

x

V

Vto

/

x / (m{if})1/nds.

to

After raising to power n the obtained inequality and dividing the result by

n— 1

/

m

t 2n

i ir d/

V

we get the following estimate

to

/

2n

m

iTd/J ^ < (n(2n - 1))n / (m {is2n})1/nds . (1.70)

to

Using the estimate (1.70) repeatedly, we have

T

m

(k)

2n

< (n(2n - 1))'

m

n ('-1)

2n 1/n

dt J <

< (n(2n - 1))nx

x

/ T / / tk

/ (n(2n - 1))n tt

m {(d?)

2n

1/n

dtk- 1

dtk

/

= (n(2n - 1))

2n

t tfc

tt

2n 1/n

m (nt(k-120 1 dtk-1dt j < ...

< (n(2n - 1))n(k-1)

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T tfc ts

t t t

m { (n(

(1) 2,t

2n

1/n

dt3... dtk- 1dtk =

t

n

t

n

T t2

(n(2n - 1))"(k-1)(2n - 1)!!

R2 (t1,..., tk)dt1... dtk I <

,t t

< (n(2n - 1))"(k-1)(2n - 1)!!x

x

/

V[t,T ]k

\

RPL...Pfc (t1, . . . , tk)dt1 . . . dtk

/

The penultimate step was obtained using the formula

m{ (nsf

t2

= (2n - 1)!! I I R2 (t1 ,...,tk )dt1

which follows from Gaussianity of

t2

ni2,i = I RPi...Pk (t1;

.,tk )dft(1l).

Similarly, we estimate each summand on the right-hand side of (1.69). Then, from (1.69) using the Minkowski inequality, we finally get

Ml (RTF**)2"j <

<

k!

(n(2n - 1))"(k-1)(2n - 1)!!

\[t.T ]*

\ " \ 1/2"\

2"

R 1 ...Pk (t1, • • • , tk)dt1 . . . dtk

y y

/

= (k!)2"(n(2n - 1))"(k-1)(2n - 1)!!x

x

/

\[t,T ]*

\

RP21...Pk (t1, . . . , tk)dt1 . . . dtk

(1.71)

/

"

Using the orthonormality of the functions ^(s) (j = 0,1, 2,...), we obtain

J RL-Pk(t1,...,tk)dt1 ...dtk =

[t,T ]k

// Pi Pk k \ 2

K(t1,..., tk) - ... ^ Cjk -ji n hji (ti) dt1... dtk =

™ V ji=0 jk=0 1=1 /

[t,T ]

= J K2(t1,...,tk)dt1 ...dtk-

[t,T ]k

Pi Pk k

1,...,6k ) ... V Cjk ---j'i T1 hji (t )dt1

[t,T v

k2

+ 'I Z^ . . . Z^ Cjk---ji l! j (ti ) I dt1

k ji=0 jk=0 1=1

/Pi Pk k

K(t1,..., tk) ^ ... ^ Cjk -ji n hji (ti)dt1... dtk+

!k ji=0 jk =0 1=1

p / Pi Pk k \ 2

J ( E .. . E Cjk - ji n hji (ti ) dt1... dtk

[tT]k Vi=0 jk=0 1 = 1 '

= J K2(t1,...,tk)dt1 ...dtk-

[t,T ]k

Pi Pk r. k

-2E..^Cjk -ji J K(t1,...,tk^hji(ti)dt1 ...dtk+

ji=0 jk=0 [t,T]k i=1

T

Pi Pi Pk Pk k „

+E E... E E Cjk---ji Cjk -ji n / hji (ti (ti )dti =

_A :! r\ _A :! r\ 7 1«-'

i

ji =0 ji=0 jk =0 jk=0 i=1

Pi Pk Pi Pk

= I K2(i1,..., tk)dt1... dtk - 2 £... £ cl,ji + £... £ cl-ji =

[t t]k ji=0 jk=0 ji =0 jk =0

^ Pi Pk

= J K 2(t1,...,tk )dt1 ...dtk ..^C2k---ji. (1.72)

[t,T]k ji=0 jk =0

Let us substitute (1.72) into (1.71)

X

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m{ J- jw^Tr^)2 Ï <

< (k!)2n(n(2n - 1))n(k-1)(2n - 1)!! x

/ r. P1 Pk ^

I K 2(ii,...,ik )dti ...rftk

V[t,T ]k j 1=0 j=0 /

n

Due to Parseval's equality

J RP21...pk (t1,---,tk )dt1 =

[t,T ]k

Pi Pk

22

(1.73)

/i'i ik

K2(ti,...,tk )dti ...dtk-E --^C2 ^ 0 (1.74)

[t,T ]k j1=0 jk=0

if pl5... ^ œ. Therefore, the inequality (1.71) (or (1.73)) means that the expansions of iterated ItO stochastic integrals obtained using Theorem 1.1 converge in the mean of degree 2n (n G N) to the appropriate iterated Ito stochastic integrals.

1.1.10 Conclusions

Thus, we obtain the following useful possibilities and modifications of the approach based on Theorem 1.1.5

1. There is an explicit formula (see (1.8)) for calculation of expansion coefficients of the iterated Ito stochastic integral (1.5) with any fixed multiplicity k

(k e N).

2. We have possibilities for exact calculation of the mean-square approximation error of the iterated Ito stochastic integral (1.5) [14]-[17], [30] (see Sect. 1.2).

3. Since the used multiple Fourier series is a generalized in the sense that it is built using various complete orthonormal systems of functions in the space L2([t,T]), then we have new possibilities for approximation — we can use not only the trigonometric functions as in [77]-[80], [87], [88], [91], [92], but the Legendre polynomials.

5 Theorem 1.1 will be generalized to the case of an arbitrary complete orthonormal system of functions (x)}~0 in the space L2([t, T]) and ^(t), ..., (t) e L2([t,T]) in Sect. 1.11 (see Theorem 1.16).

4. As it turned out [1]-[62], it is more convenient to work with Legendre polynomials for approximation of the iterated Ito stochastic integrals (1.5) (see Chapter 5). Approximations based on Legendre polynomials essentially simpler than their analogues based on trigonometric functions [1]-[62]. Another advantages of the application of Legendre polynomials in the framework of the

mentioned problem are considered in [20], [39] (see Sect. 5.3).

5. The Milstein approach [77] (see Sect. 6.2 in this book) to expansion of iterated stochastic integrals based on the Karhunen-Loeve expansion of the Brownian bridge process (also see [78]-[80], [87], [88], [91], [92]) leads to iterated application of the operation of limit transition (the operation of limit transition is implemented only once in Theorem 1.1) starting from the second or third multiplicity of the iterated Ito stochastic integral (1.5). Multiple series (the operation of limit transition is implemented only once) are more convenient for approximation than the iterated ones (iterated application of the operation of limit transition), since partial sums of multiple series converge for any possible case of convergence to infinity of their upper limits of summation (let us denote them as p1,... ,pk). For example, when p1 = ... = pk = p ^ to. For iterated series, the condition p1 = ... = pk = p ^ to obviously does not guarantee the convergence of this series. However, in [78]-[80], [88] the authors use (without rigorous proof) the condition p1 = p2 = p3 = p ^ to within the frames of the Milstein approach [77] together with the Wong-Zakai approximation [68]-[70] (see discussions in Sect. 2.20, 2.21, 6.2).

6. As we mentioned above, constructing the expansions of iterated Ito stochastic integrals from Theorem 1.1 we saved all information about these integrals. That is why it is natural to expect that the mentioned expansions will converge with probability 1. The convergence with probability 1 in Theorem 1.1 has been proved for some particular cases in [3]-[16], [31] (see Sect. 1.7.1) and for the general case of iterated Ito stochastic integrals of multiplicity k (k G N) in [14]-[16], [26],

30], [31] (see Sect. 1.7.2).

7. The generalizations of Theorem 1.1 for an arbitrary complete orthonormal systems of functions in L2([t,T]k) [28] and complete orthonormal with weight r(ti).. .r(tk) > 0 systems of functions in L2([t,T]k) [12]-[16], [40] as well as for iterated stochastic integrals with respect to martingale Poisson measures and iterated stochastic integrals with respect to martingales [1]-[16], [40] are presented in Sect. 1.3-1.6, 1.11.

8. The adaptation of Theorem 1.1 for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 was carried out in [6]-[22], [27], [29], [31]-

[44]-[46], [49], [51], [63], [64] (see Chapter 2).

9. Application of Theorem 1.1 for the mean-square approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener

process can be found in [14]-[16], [23], [24], [47], [48] (see Chapter 7).

1.2 Exact Calculation of the Mean-Square Error in the Method of Approximation of Iterated Ito Stochastic integrals Based on Generalized Multiple Fourier Series

This section is devoted to the obtainment of exact and approximate expressions for the mean-square approximation error in Theorem 1.1 for iterated Ito stochastic integrals of arbitrary multiplicity k (k e N). As a result, we do not need to use redundant terms of expansions of iterated Ito stochastic integrals.

1.2.1 Introduction

Recall that we called the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on Theorem 1.1 as the method of generalized multiple Fourier series. The question about how estimate or even calculate exactly the mean-square approximation error of iterated Ito stochastic integrals for the method of generalized multiple Fourier series composes the subject of Sect. 1.2. From the one side the mentioned question is essentially difficult in the case of a multidimensional Wiener process, because of we need to take into account all possible combinations of the components of a multidimensional Wiener process. From the other side an effective solution of the mentioned problem allows to construct more simple expansions of iterated Ito stochastic integrals than in [77]-[82], [87]-[89], [91], [92].

Sect. 1.2.2 is devoted to the formulation and proof of Theorem 1.3, which allows to calculate exacly the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (k e N) for the method of generalized multiple Fourier series. The particular cases (k = 1,..., 5) of Theorem 1.3 are considered in detail in Sect. 1.2.3. In Sect. 1.2.4 we prove an effective estimate for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (k e N) for the method of generalized multiple Fourier series.

1.2.2 Theorem on Exact Calculation of the Mean-Square Approximation Error for Iterated Ito Stochastic integrals

Theorem 1.36 [12]-[17], [30]. Suppose that every ^ (t) (l = 1,..., k) is a continuous nonrandom function on [t,T] and {fy(x)}TO=0 is a complete orthonormal system of functions in the space L2([t, T]), each function fy (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then

m{ J [^(k)]T,t - J [^(k)]^) 2} = J K 2(t1,...,tk )dt1 ...dtk -

[t,T ]k

T t2

¿...¿Cj.., M J [^(k)]T,^ /j (tk) .../j (tf ..f

_n _n I / \ J J

(1.75)

j1=0 jfc=0 ^ (jl,-,jfc) t t where

T t2

J [^(k)]T,t = J ^k (tk) ..J ^1(t1)dfi;i)... fk),

i i p p / k \

j= E ... E Cjk.ji n j" - . (1.76)

ji=0 jfc=0 \/=1 /

jj =1^. E j (TJi f (Tik ^ (1.77)

(/l,...,/fc )GGfc

the Fourier coefficient Cjk...j has the form (11.81),

T

j = f to (s)f (1.78)

are independent standard Gaussian random variables for various i or j (i = 1,...,m),

E

(ji,...,jfc)

6 Theorem 1.3 will be generalized to the case of an arbitrary complete orthonormal system of functions

{^(x)}~0 in the space L2([t,T]) and ^i(r), ..., (t) G L2([t,T]) in Sect. 1.12 (see Theorem 1.18).

means the sum with respect to all possible permutations (ji,..., jk). At the same time if jr swapped with in the permutation j..., jk), then ir swapped with iq in the permutation (ii,..., ik) (see (1.75)); another notations are the same as in Theorem 1.1.

Remark 1.3. Note that

{T t2

J^kt/ j (tk) ..J j (ti)dfi;i) ..f

r T t2 T t2

= m J (tk) ... | ^i(ii)df«i;i)... <f;; > / j (tk) ..J j (tf... df;; >

Lit t t

T t2

= J ^k (tk )j (tk) ..J Wil) j (ti)dti ...dtk = Cj; ...ji. (1.79)

tt

Therefore, in the case of pairwise different numbers ii,..., ik from Theorem 1.3 we obtain

m{ (J[^(k)]T,t - J = J K2(ti,...,tk )dti ...dtk -EC? ...ji. (1.80)

[t,T ]; ji=0 j;=0

Moreover, if ii = ... = ik, then from Theorem 1.3 we get

m<! (J [#k)]T,t - J [#k)]Tt

^ p p / J K2 (ti , . . . , tk )dti ...dtk ..^Cj;...jJ E Cj; ...ji

l rpi; ji=0 j;=0 V(ji ,...,j;)

[t,T]; ji=0 j;=0 \(ji,...,j;)

where

£

(ji,...,j;)

2

means the sum with respect to all possible permutations (ji,..., jk). For example, for the case k = 3 we have

T is t2

m{ (j[^(3)]t,î - J2} = J J ^(¿2) | ^?(ii)dtidt2dis-

t t t

CjSj2jl i CjSj2jl + Cjsjlj2 + Cj2jsjl + Cj2jljs + Cjlj2jS + Cjljsj2 j '

jl,j2 ,js=0

Proof. Using Theorem 1.1 for the case i1,..., ik = 1,..., m and p1 = ... = pk = p, we obtain

p p

J

T,t =

-ECk.jlin zj

SjlvJk

(1.81)

jl=0 jk =0

,1=i

For n > p we can write

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p n

J

n

T,t

E+E

jl=0 jl=p+b

(E + E K^II zf - jr)

jk=0 jk =p+1.

jk

J=i

=J

T,t

+

(1.82)

-Y(;l---;fc )

Let us prove that due to the special structure of random variables Sj jk (see (11.45l)-([TTïïT1). (11.54). (11.77)) the following relations are correct

M II z

(^ _ s(;l-;k^ =0 jl j-.-jjfc 1

i=i

(1.83)

m

n

J=i

A;i) c(;l---;fc )

jl,...,jk

n

J=i

z (;*) — s (il-;* ) ji j l,—k

= 0,

(1.84)

where

and

(ji,...,jk) g Kp, (ji) e Kn\Kp

Kn = {(ji,.. ., jk) : 0 < ji,.. ., jk < n}

k

p

n

p

= {(ji,. . . , jk) : 0 < ji,. . . , jk < P} .

For the case i1,..., ik = 1,..., m and p1 = .. (see the proof of Theorem 1.1) we obtain

= pk = p from (1.39), (1.40)

n

1=1

N-1

Ail) Cf(«1---«fc) i

l.i.m.

N->oo

E j (Til) ••• j (Tik )^fTii1l)... Afik) =

ll,-,lfc=0

iq = lr ; q=r; q,r=1,...,fc

T

t2

£ / j(tk)...ij(i1)dfi:i)..f' w.p. 1

?(ifc)

(1.85)

(ji,"-,jfc) t

where

£

means the sum with respect to all possible permutations (j,..., jk). At the same time if jr swapped with in the permutation (j15..., jk), then ir swapped with in the permutation (i1,... , ik); another notations are the same as in Theorem 1.1.

So, we obtain (1.83) from (1.85) due to the moment property of the Ito stochastic integral.

Let us prove (1.84). From (1.85) we have

0

m

n4:,) - e.i'h ik' - j.::

/(il-.-ifc) ,j k

(il)

^(i1 ...ifc ) ,j k

J=1

,1=1

( T t2

m\ E E / j (tk) . J j (t1)dft

U"!t(1:1)... dft(kik )x

(j1,---,jfc) (j1,---,jk) t T t2

x /j(tk)... j(t1)dft(1:1)..f

<

tt

T T

< E / j (tk ) j (tk )dtk ...J j (t1)0j1 (t1)dt1 (j1j-"jjk) t t

k

= E 1{ji=ji}... 1{jk=jk}, (1.86)

(jiv"Jfc)

where 1A is the indicator of the set A. From (1.86) we obtain (1.84).

Let us explain (1.86) for the cases k = 2 and k = 3 in detail. We have

( T t2 T t2

m e e J j fe)/ j (t1)dfi;i)df(^| j (t2^ j (tofif [(ji,j2) (ji,j2) t t t t

T T

= J j (s)j j (s)j(s)ds+

tt

T T

+1{i i=i2^ jj(s)j(s)ds = tt

= 1{j i =j'}1{j2=j2} + 1{i i = i2} • 1 {j2 =ji}1{j i =j'2}, (1.87) T t3 t2

m E E / j(*)/ jfe)/j«ofr>f:2)<f:3)x

(j i,j2,j3) (j i,j2,j3) t t t

T t3 t2

*/ j №>)/ w/ to ¡(tof: ■ )f:,)<:3)

ttt T T T

ttt

T T T

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+ 1{: i=:2^ j (s)j (s)dsy i (s)j toj2 i(s)ds + ttt T T T

+ 1{:2=:3}

/ toji(s)toj;(s)dW toj2(s)toj3(s)dW toj3(s)to'2(s)ds+ ttt T T T

+ 1{: i=:3^ i (s)j j (s)j j i(s)ds +

ttt

T T T

+ I{:1=:2=i3^ j(s) jj (s) j (S^y j(s) j (5)d5+ ttt T T T

=i2 = i3}

/ j (s)j(s)dW j(s) j(s)dW j(s) j (s)ds = ttt = 1{j3=j3}1{j2=j'2}1{j1=j1} + 1{i1=i2} • 1 {j3 =j'}1{j1=j2}1{j2=j1} + + I{i2 = :3} • 1 {j 1 = j 1} 1 {j2 = j3} 1 {j3 = j2} + 1{i1=i3} ^ 1{j1=j3} 1{j2=j2} 1{j3=j1} +

+ 1{i1=i2=i3} ^ 1 {j2 = j3} 1 {j 1 = j2} 1{j3=j1} +

+ 1{i1=i2=i3} ^ 1{j1=j3}1{j3=j2}1{j2 =j1}. (1.88)

From (1.87) and (1.88) we get

m

[(j1,j2) (j1,j2) t

T t2

j (t2) y j (t1)dft(1:1)dft(2i2)x t

xy j fe)/ j (tofif

tt

< 1 {j 1 = j 1} 1 {j2 = j'2 } + 1 {j2 = j 1}1 {j 1 = j'2 }

<

E 1{j1=j1}1 {j2=j2 } ' (j1,j2)

( T t3 t2

m{ E £ J j (MJ j (u-)j j (iOf^ff'f:3'

x

(j1,j2,j3) (j1,j2,j3) t

T

t3

t

t2

x / j (t3W j (t2W j fafdft(2i2)dft(3i3)

<

< 1 {j3 = j3} 1 {j2 = j2} 1 {j1 = j1} + 1 {j3 = j3} 1 {j1 = j2}1 {j2=j 1} + +1{j1=j1}1{j2=j3} 1{j3=j2} + 1{j1=j3}1 {j'2 = j2}1 {j3 = j 1} +

+1 {j2 = j3} 1 {j1 = j2}1 {j3 = j1} + 1 {j1 = j3}1 {j3=j2}1 {j2=j1} =

E X{j l =j 1} 1 {j2 = j'2}1 {js = js} ' (j i ,j2,jS)

where we used the relation

T

(s)0q(s)ds = 1{g=q}, = 0,1, 2

Note that the formula (1.84) (in the light of the results of Sect. 1.10, 1.11) can be interpreted as a consequence of the orthogonality of two random variables that are Hermite polynomials of vector random arguments.

From (1.84) we obtain

m

{j [^(k)]T,i£[^(k)]T+i'n}

= 0.

Due to (1.76), (1.81), and (1.82) we can write

£= j[^(k)]n,t - j

p

T,t>

l.i.m. £[^(k)]T+ii'n = J[^(k)]T,t - J

n

def

T,t

We have

0

m

{(£[^T? - £№'n + £^(k)Cn) J[^(k)]T,t}

<

m

{(£№ - £№(k)cn) j[^(%}| + | m {£[^ty

JT,t J[^(k)]T,i}

m{ (j[^(k)]T,t - J[^(k)]n,t) J

<

< ^/m j[^(k)]T,t - J[^(k)]nt

M J

p

2

< y m| (j[^(k)]T,t - J[^(k)]T,0 }x

<

< K^M | - ^ o if ?wog, (1.89)

where K is a constant.

From (1.89) it follows that

m{ e [^(k)]T+t1J №(%} = o

or

M { (J№(t)]T,( - JIT,) J№'%} = 0.

The last equality means that

m { JW.<k>]T,t J№<%} = m { (j. (1.90)

Taking into account (1.90), we obtain

m { (J[^(k)]T,t - Jfo>(%)2) = m { (Ji^(k)]T,)2} +

+m { J 2j - 2m { J [^(k)]T,tJ ty>(%} = m { (J [^(k) ]T,t) ^ -

-m { J [^(k)]T,tJ №(%} = = J K2(t1,...,tk)dt1 ...dtk - m{J[^(k)]T,tJ[^(k)]T,t} . (1.91)

[t,T ]k

Let us consider the value

m { j j №-'%}.

The relations (1.76) and (1.85) imply that

T t2

f f t> t> j= E... ECjk..ji E i to*(tk)... itoji(t1)f:11...f;;>.

j i=0 j';=0 (j i,...,jfc) t t

(1.92)

After substituting (1.92) into (1.91), we finally get

m{ J [^(k)]T,t - J [^(%) ^ = / K2(t1,...,tk )dt1 ...dtk -

[t,T ]k

p p ( T t2

-E• • . ECj; jm j j[#>]« E J to*(t»)■■■[toji(t1)df(i»...f

ji=0 jk=0 [ (j i,...,jfc) t t

Theorem 1.3 is proved.

1.2.3 Exact Calculation of the Mean-Square Approximation Errors for the Cases k = 1, . . . , 5

Let us denote

m{ (j[v>№)]t,( - J№<%)*} = Ep,

lKllL2([t,T]k) = J K2(t1, . . . , tk)dt1 . . . dtk ==

[t,T ]k

The case k = 1

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In this case from Theorem 1.3 we obtain

ep = -E c2 .

j i=0

p

The case k = 2

In this case from Theorem 1.3 we have

(I). ii = «2:

EP = /2 j, (1.93)

j1 j2=0

(II). ii = «2 :

p

EP = /2 - £ Cj2ji - £ jj. (1.94)

jlj2 =0 j1,j2=0

Note that from (1.76), (1.85), (1.87), (1.90), and (1.91) we obtain

pp

Ep = /2 - E Cj2j1 - 1{i1=i2} E Cj2j1 Cj1j2 • (1.95) jl ,j2=0 jl,j2 =0

Obviously, the relation (1.95) is consistent with (1.93) and (1.94).

Example 1.1. Let us consider the following iterated Ito stochastic integral

T t2

= // fM" (1.96)

t t

where i1, i2 = 1,..., m.

Approximation of the iterated Ito stochastic integral (1.96) based on the expansion (1.10) (Theorem 1.1, the case of Legendre polynomials) has the following form

t_f / ^ 1

jinh )p ' •-(i-i^ Jin) X > J-

An)Aii) , v^ (A^A^ — A^A^A _ i 1

So SO _ ] Sj Si —1 y } I •

(1.97)

(00)T,t- 2 \S0 So ^ z^ /4i2 _ !

=i -

Note that (1.97) has been derived for the first time in [71] (1997) (also see 72]- [74]) with using the another approach. This approach will be considered in

Sect. 2.4. Later (1.97) was obtained [1] (2006), [2]-[62] on the base of Theorem 1.1.

p

Using (1.93), we get

m I

(; ;2)

-I,

(: i:2)p

- (00)T,t J (00)T,t

(T - t)2 1

2-

1

:=1

4i2 1

(1.98)

where i1 = i2.

It should also be noted that the formula (1.98) has been obtained for the first time in [71] (1997) by direct calculation.

The case k = 3 In this case from Theorem 1.3 we obtain

(I). i1 = i2,i1 = i3,¿2 = i3 :

Ep

E3

= I3

e c

j i,j2,j3=0

2, j3j2j ,

(1.99)

(II). i1 = i2 = i3 :

p

Ep = I3 - E C

33323 i I Cj3j2:/ i

j i ,j2 ,j3=0 V(j i,j2 ,j3)

(1.100)

(III).1. i1 = i2 = i3 :

pp

_ i2 '3 S ■ Cj3 j2 j i

ji,j2 ,j3=0 j i ,j2 ,j3=0

Ep = 13 - E Cj3 j2j i - E ji j2 C

j3j2j

(1.101)

(III).2. i1 = i2 = i3 :

pp 2

j3j2j

ji,j2 ,j3=0 j i ,j2 ,j3=0

Ep = 13 - E Cj3 j2j i - E j3 j i C

j3j2j

(1.102)

(III).3. i1 = i3 = i2 :

pp

_ i2 '3 S ■ Cj3 j2j i

ji,j2 ,j3=0 j i ,j2 ,j3=0

Ep = 13 - E Cj3 j2j i - E Cj3j2 j i C

jij2j3 *

(1.103)

p

2

It is not difficult to see that from (1.76), (1.85), (1.88), (1.90), and (1.91) we obtain

p

Ep = '3 — Cinn —

jl j2 j3=0 p

— 1{il = »2| y v Cj3j2 jl Cj3jlj2 —

j1,j2,j3=0 p

r C. . . C

j2j3jl

"1{»2 = »3| Cj3j2 j1C

jl j2 j3=0

p

"1{»1 = »3| ^ ^ Cj3j2 jl Cj1j2j3" jl j2 j3=0

p

1{»1=»2=»3| ^ y Cj3j2jl (Cj2jlj3 + Cjlj3j2) • (1.104)

jl j2 j3=0

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Obviously, the relation (1.104) is consistent with (1.99)-(1.103).

Note that the cases k = 2 and k = 3 (excepting the formula (1.100)) were investigated for the first time in [2] (2007) using the direct calculation.

Example 1.2. Let us consider the following iterated Ito stochastic integral

T t3 t2

'S = // /ff <*>, (1.105)

i i i

where ii, i3 = 1,..., m.

Approximation of the iterated Ito stochastic integral (1.105) based on Theorem 1.1 (the case of Legendre polynomials and p1 = p2 = p3 = p) has the following form [1] (2006), [2]-[62]

jl

p

EC | z (il)Z (i2 )z(i

Cj3j2jl I zjl zj2 j

j2 ,j3 =0 \

'(ili2i3)p = V"^ C I z(il)Z(i2)z_ 1 1 z(i3)_

'(000)T,t Z^ Cj3j2jl I zjl z j2 zj3 1{il = i2} 1{jl =j2}zj3

— 1{i2=i3} 1 {j2 =j3} C^ — 1{il = i3} 1{jl=i3}zi22 ) ) , (1.106)

where

V(2jl + 1)(2r1)(2j3 + 1)(T-^TO., (1.107)

C . . =

WJ3J2J1 g

1 z y

i = j Pj3 (Z) j j (y) j Pj i (x)dxdydZ -1 -1 -1

where P:(x) is the Legendre polynomial (i = 0,1, 2,...). For example, using (1.101) and (1.102), we obtain

l\/l J I T(:i:2:3) t(:i:2:3)p\

2| (T - t)3 ^2 v- ^ ^

IVI | ^ (000)T,i J(000)T,t J \ 6 A' ./:;./: / Z^ ( ./':t./'\/Vt ./:;./: ./ •

jl,j2,js=0 jl,j2,Js=0

where ii = i2 = i3,

■V /■ I ( t(;li2;s) T(;li2;s)p \

lvl | ^J(000)T,i J(000)T,i j [" g Z^ ./:;./:/ Z^ * ./:./:■,/ ( ./:■,/:./•

j l j2,j3=0 jl,j2,Js=0

where ii = i2 = i3.

The exact values of Fourier-Legendre coefficients Cjsj2j l can be calculated for example using computer algebra system Derive [1]-[16], [31] (see Sect. 5.1,

Tables 5.4-5.36). For more details on calculating of Cj3j2ji using Python programming language see [52], [53].

For the case i1 = i2 = i3 it is convenient to use the following well known formula

'm ^ = ^-i)3/2((^1,)3-3C«")) w.p.i.

The case k = 4 In this case from Theorem 1.3 we have (I). i1,...,i4 are pairwise different:

p

Ep = J4 - E Cj4...ji, j i,...,j4=°

(II). ¿1 = ¿2 = ¿3 = ¿4:

Ep = '4 — Cj4...jl I E Cj4...jl

jl,-j4=0 \(jl,-,j4)

(III).1. ¿1 = ¿2 = ¿3, ¿4; ¿3 = ¿4 :

Ep

'4 — Cj4...jl ( E Cj4...

jlv"j4=0 V(jl,j2)

(1.108)

(III).2. ¿1 = ¿3 = ¿2, ¿4; ¿2 = ¿4 :

Ep

E4

'4 — ^ Cj4...jl ( E Cj4...

jl,...j4=0 V(jl,j3)

(1.109)

(III).3. ¿1 = ¿4 = ¿2, ¿3; ¿2 = ¿3 :

E4p

'4 — ^ Cj4..jl ( E Cj4...

jlv"j4=0 V(jl,j4)

jl

(1.110)

(III).4. ¿2 = ¿3 = ¿1,¿4; ¿1 = ¿4 :

Ep E4

'4 — E Cj4..jl ( E Cj4...

jlv"j4=0 V(j2,j3)

jl

(1.111)

(III).5. ¿2 = ¿4 = ¿1, ¿3; ¿1 = ¿3 :

E4p

'4 — E Cj4...jl ( E Cj4...

jl,...,j4=0 V(j2,j4)

(1.112)

(III).6. ¿3 = ¿4 = ¿1,¿2; ¿1 = ¿2 :

Ep E4

'4 — E Cj4..jl ( E Cj4...jl jl,...,j4=0 V(j3,j4)

(1.113)

p

(IV).1. i1 = i2 = i3 = i4:

Ep = I4 - E Cj4...ji E Cj4...ji ), (1.114)

ji,-,j4=0 V(ji,j2,j3)

(IV).2. i2 = i3 = i4 = i1:

Ep = I4 - E Cj4...ji E Cj4...ji ), (1.115)

ji,-,j4=0 V(j2 jj

(IV).3. i1 = i2 = i4 = i3:

Ep = I4 - E Cj4...ji E Cj4...ji ), (1.116)

ji,...,j4=0 V(ji,j2,j4)

(IV).4. i1 = i3 = i4 = i2:

Ep = I4 - E Cj4...ji E Cj4...ji ), (1.117)

ji,...,j4=0 V(ji,j3,j4)

(V).1. i1 = i2 = i3 = i4:

Ep = I4 - E Cj4...ji E E Cj4..ji ) ), (1.118)

j i,...,j4=0 \(j i,j2A(j3,j4)

(V).2. i1 = i3 = i2 = i4:

Ep = I4 - E Cj4...ji E E Cj4.ji ) ), (1.119)

j i,...,j4=0 \(j i,j3A(j2,j4)

(V).3. i1 = i4 = i2 = i3:

Ep = I4 - E Cj4...ji E E Cj4..ji ) ). (1.120)

j i,...,j4=0 \(j i,j4A(j2,j3)

p

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p

The case k = 5

In this case from Theorem 1.3 we obtain

(I). ¿1,..., ¿5 are pairwise different:

p

EP = '5 — E CLj,,

jl,...,j5=0

(II). ¿1 = ¿2 = ¿3 = ¿4 = ¿5:

p

Ep = '5 — E Cj5...jl ( E Cj5..J 1

jl,...j5=0 V(j'l,...,j5)

(III). 1. ¿1 = ¿2 = ¿3, ¿4, ¿5 (¿3, ¿4, ¿5 are pairwise different):

p

Ep = '5 — E Cj5...jl ( E Cj5...jl

jlv"j5=° V(j'l,j2)

■>(jlj2 )

(III).2. ¿1 = ¿3 = ¿2, ¿4, ¿5 (¿2, ¿4, ¿5 are pairwise different):

p

Ep = '5 — E Cj5...jl ( E Cj5...jl

jl,...,j5=0 V(jl,j3)

(III).3. ¿1 = ¿4 = ¿2, ¿3, ¿5 (¿2, ¿3, ¿5 are pairwise different):

Ep = '5 — E Cj5...jl ( E Cj5...

jlv"J5=° V(jl,j4)

(III).4. ¿1 = ¿5 = ¿2, ¿3, ¿4 (¿2, ¿3, ¿4 are pairwise different):

p

Ep = '5 — E Cj5. jl ( E Cj5...jl

jl,...,j5=0 V(jl,j5)

(III).5. ¿2 = ¿3 = ¿1, ¿4, ¿5 (¿1, ¿4, ¿5 are pairwise different):

Ep = '5 — E Cj5...jl ( E Cj5...

jlv"j5=0 V(j2,j3)

p

(III).6. i2 = i4 = i1, i3, i5 (i1, i3, i5 are pairwise different):

Ep = - E Cj5...j i ( E Cjs...-

ji,...,j5=0 V(j2,j4)

(III).7. i2 = i5 = i1, i3, i4 (i1, i3, i4 are pairwise different):

p

Ep = I5 - E Cj5...ji ( E Cj5..^ i

jiv",j5=0 V(j2,j5)

(III).8. i3 = i4 = i1, i2, i5 (i1, i2, i5 are pairwise different):

E5p = I5 - Cj5... j Cj5...

ji,...,j5=0 V(j3,j4)

(III).9. i3 = i5 = i1, i2, i4 (i1, i2, i4 are pairwise different):

p

E5p = I5 - Cj5... j Cj5... j

ji,...,j5=0 V(j3,j5)

(III).10. i4 = i5 = i1, i2, i3 (i1, i2, i3 are pairwise different):

E5p = I5 - Cj5... j Cj5...

ji,...,j5=0 V(j4,j5)

(IV).1. i1 = i2 = i3 = i4,i5 (i4 = i5):

p

E5p = I5 - Cj5... j Cj5... j

ji,...,j5 =0 V(ji,j2,j3)

(IV).2. i1 = i2 = i4 = i3,i5 (i3 = i5):

p

E5p = I5 - Cj5... j Cj5... j

ji,...,j5 =0 V(ji,j2,j4)

p

(IV).3. ¿1 = ¿2 = ¿5 = ¿3,¿4 (¿3 = ¿4):

p

Ep = '5 — E Cj5...jl ( E Cj5...j^ ,

(IV).4. ¿2 = ¿3 = ¿4 = ¿1, ¿5 (¿1 = ¿5):

p

Ep = '5 — E Cj5..jl ( E Cj5...j^ ,

jl,...,j5 =0 V(j2 J3J4) '

(IV).5. ¿2 = ¿3 = ¿5 = ¿1, ¿4 (¿1 = ¿4):

p

Ep = '5 — E Cj5..jl ( E Cj5..jl ) ,

jl,...,j5 =0 V(j2 ,j3,j5) '

(IV).6. ¿2 = ¿4 = ¿5 = ¿1, ¿3 (¿1 = ¿3):

p

Ep = '5 — E Cj5..jl ( E Cj5...jl ) ,

j'l,...j5 =0 \(j2 J4J5) /

(IV).7. ¿3 = ¿4 = ¿5 = ¿1, ¿2 (¿1 = ¿2):

p

Ep = '5 — E Cj5...jl ( E Cj5...jl ) , jl,...,j5 =0 V(j3 J4J5) /

(IV).8. ¿1 = ¿3 = ¿5 = ¿2, ¿4 (¿2 = ¿4):

p

Ep = '5 — E Cj5..jl ( E Cj5...jl ) ,

j'l,...j5 =0 \(jl J3J5) /

(IV).9. ¿1 = ¿3 = ¿4 = ¿2, ¿5 (¿2 = ¿5):

p

Ep = '5 — E Cj5...jl ( E Cj5...jl ) ,

jl,...j5 =0 \(jl J3J4) '

http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal. http://diffjournal.spbu.ru/ A.98

(IV).10. i1 = i4 = i5 = i2, i3 (i2 = i3):

E5p = I5 - Cj5... j Cj5...

j i,.-.,j5 =0 V(ji,j4,j5)

(V).1. i1 = i2 = i3 = i4 = i5:

E5p = I5 - Cj5... j Cj5... j

ji,...,j5=0 Mji,j2,j3 ,j4)

(V).2. i1 = i2 = i3 = i5 = i4:

E5p = I5 - Cj5... j Cj5...

ji,...,j5=0 \(ji,j2,j3 ,j5)

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(V).3. i1 = i2 = i4 = i5 = i3:

E5p = I5 - Cj5... j Cj5... j

ji,...,j5=0 \(ji,j2,j4 ,j5)

(V).4. i1 = i3 = i4 = i5 = i2:

E5p = I5 - Cj5... j Cj5...

ji,...,j5=0 \(ji,j3,j4 ,j5)

(V).5. i2 = i3 = i4 = i5 = i1:

E5p = I5 - Cj5... j Cj5... j

j i,...,j5=0 \(j2,j3,j4 ,j5)

(VI).1. i5 = i1 = i2 = i3 = i4 = i5:

ep = 15 - y, Cj5..ji e e j

j i,...,j5=0 \(j i,j2A(j3,j4)

p

(VI).2. i5 = i1 = i3 = i2 = i4 = i5:

ep=/5 - E cj . . j i E (Ec... .

j i,".,j5=0 V(j i,j3A(j2,j4)

(VI).3. i5 = i1 = i4 = i2 = i3 = i5:

E5p = I5 - Cj5... j Cj5... j

j i,...,j5=0 \(j i,j4A(j2,j3)

(VI).4. i4 = i1 = i2 = i3 = i5 = i4:

Ep = /5 - £ i PT Ecj5...

j i,...,j5=0 \(j i,j2A(j3,j5)

(VI).5. i4 = i1 = i5 = i2 = i3 = i4:

E5p = I5 - Cj5... j Cj5... j

j i,...,j5=0 \(j i,j5A(j2,j3)

(VI).6. i4 = i2 = i5 = i1 = i3 = i4:

ep=/5 - E Ci.j E (ECj...,

j i,...,j5=0 V(j2 ,j5)V(j i,j's)

(VI).7. i3 = i2 = i5 = i1 = i4 = i3:

E5p = I5 - Cj5... j Cj5... j

j i,...,j5=0 V(j2 ,j5A(j i )

(VI).8. i3 = i1 = i2 = i4 = i5 = i3:

ep = /5 - y, Cj5..ji E (ECfc-.

j i,...,j5=0 \(j i,j2A(j4,j5)

p

(VI).9. ¿3 = ¿2 = ¿4 = ¿1 = ¿5 = ¿3:

Ep E5

'5 — E Cj5.jl E E Cj5...jl

jlv"j5=0

^(j2 ,j4) \(jl,j5)

(VI).10. ¿2 = ¿1 = ¿4 = ¿3 = ¿5 = ¿2:

Ep = '5 — E Cj5..Jl E E Cj5...jl

jlv"j5=0

^(j'l ,j4^(j3,j5)

(VI).11. ¿2 = ¿1 = ¿3 = ¿4 = ¿5 = ¿2:

Ep E5

'5 — E Cj5. jl E E Cj5..jl

jl,...,j5=0

(VI).12. ¿2 = ¿1 = ¿5 = ¿3 = ¿4 = ¿2:

Ep = '5 — E Cj5..Jl E E Cj5...jl

jl,".j5=°

(VI).13. ¿1 = ¿2 = ¿3 = ¿4 = ¿5 = ¿1:

Ep = '5 — E Cj5..Jl E E Cj5...jl

jl,".j5=°

^(j2 ,j3) \(j4,j5)

(VI).14. ¿1 = ¿2 = ¿4 = ¿3 = ¿5 = ¿1:

Ep = '5 — E Cj5..Jl E E Cj5...jl

jl,".j5=0

^(j2 ,j4) \(j3,j5)

(VI).15. ¿1 = ¿2 = ¿5 = ¿3 = ¿4 = ¿1:

E5p

'5 — E Cj5.jl E E Cj5...jl

jl,...,j5=0

^(j2 ,j5^(j3,j4)

p

(VII).1. i1 = i2 = i3 = i4 = i5:

ep = /5 - e C, . . j i e e c, . . .

j i,...,j5 =0 V(j4,j5A(j i,j2 ,j3)

(VII).2. i1 = i2 = i4 = i3 = i5:

Ep = /5 - E Cj5...j i ( E ( E Cj5...j i

j i,...,j5 =0 V(j3,j5A(j ij2 ,j4)

(VII).3. i1 = i2 = i5 = i3 = i4:

Ep = 1 - E Cj5 .j i I E I E Cj5. j i

j i,...,j5=0 V(j3,j4A(j i,j2,j5)

(VII).4. i2 = i3 = i4 = i1 = i5:

Ep = /5 - E Cj5...j i ( E ( E Cj5...j i

j iv",j5 =0 V(j i,j5)\(j2,j3 ,j4)

(VII).5. i2 = i3 = i5 = i1 = i4:

Ep = /5 - E i E E j.

j i,...,j5 =0 V(j i,j4A(j2,j3 j5 )

(VII).6. i2 = i4 = i5 = i1 = i3:

Ep = /5 - E Cj5...j i ( E l E Cj5...j i

j iv..,j5 =0 \(j i,j3A(j2,j4 ,j5)

(VII).7. i3 = i4 = i5 = i1 = i2:

Ep = /5 - E CjejMr £ j

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j i,...,j5 =0 V(j i,j2A(j3,j4 ,j5)

p

(VII).8. ¿1 = ¿3 = ¿5 = ¿2 = ¿4:

Ep = '5 — E Cj5..Jl ( E l E Cj5...jl

j'lv"7 =0 \(72,74) \(7l ,73 ,75 )

(VII).9. ¿1 = ¿3 = ¿4 = ¿2 = ¿5:

Ep = '5 — E Cj5..Jl ( E l E Cj5...jl

7l,.",75 =0 \(72,75) \(7l,73 j'4)

(VII).10. ¿1 = ¿4 = ¿5 = ¿2 = ¿3:

p

E5 = '5 — E Cj5..Jl ( E l E Cj5...jl

...,j5 =0 M 72,73) M 71,74,75)

jl,...,j5 =0 V(72,73) \(7'i,74 ,75 )

Let us make a remark about Theorem 1.3. It is easy to see that the right-hand side of the formula (1.75) consists of two parts. The first part tends to zero when p ^ to by Parseval's equality. At the same time the second part also tends to zero when p ^ to, but due to the generalized Parseval equality. Let us explain the above reasoning in more detail for the case k = 3.

For the case k = 3 we have (see (1.104))

p

Ep = '3 — E c73727I —

7I ,72,73=0

p

"1{»l=»2} ^ ^ C737271 C737l72

7l,72,73=0 p

CC

727371

"1{»2=»3} ^ ^ C737271C 7l,72,73=0

p

"1{»l=»3} ^ ^ C737271 C7l7273" 7l,72,73=0

p

1{»l=»2=»3} ^ ^ C73727l (C727l73 + C7l7372) . (1.121)

7'l ,72,73=0

Applying the Parseval equality, we obtain

= 0. (1.122)

lim (/3 - V Cj2|

p—TO \ 3 Z^ i I

\ j U2 ,j3=0 /

The generalized Parseval equality gives

pp

lim Cj3j2j i Cj i j2 j3 = 0, lim Cj3j2j i Cj3j i j2 = 0, (1.123)

p—TO z—» p—TO z—»

j U2,j3 =0 j i,j2,j3 =0

pp

lim Zv Cj3j2j i Cj i j3j2 = 0, lim Cj3j2j i Cj2j i j3 = 0, (1.124)

p—TO ' * p—>-to '

j i,j2,j3 =0 j i,j2,j3 =0

p

lim Zv Cj3j2j i Cj2j3j i = 0' (1.125)

p—TO '

j i,j2,j3=0

Let us explain in more detail the first equality in (1.123). Using the generalized Parseval equality, we have

p

lim / v Cj i j2j3 Cj3j2j i rt—Vm ' *

p—TO

j i,j2,j3 =0

p T t3 t2

pl—i—TO E /^3 (t3)0j i (*)/ ^2 (¿2) j (t2^ j (t1)dtidt2 d^x

j i,j2,j3=0 t t t

T t3 t2

x y ^(tj) j (t3^ ^(¿2) j (t2^ ^fa) j (t1)dMt2dt3 =

ttt

p T T T

pl——To E J (t$) j (,)/ ^2(¿2) j (t2^ ^3(t1)0ji (t1)dtidt2dt3X

j i,j2,j3=0 t t3 t2

T t3 t2

x / Wtj) j(t$) / ^(¿2) j (t2M ^fa) j (t1)dt1dt2dt3 =

p » 3

Hm E / 1{i3<i2<ii}^3(ti)^2(t2)^i(t3 )JJ $3i (ti )dtidt2dt3X

3 3 3 =0 J l = 1

3i'32'33=0 [t,T]3

3

X J 1{ti<t2<t3}^1 (t 1 )^2(t2)^3(t3) H $31 (ti)dtidt2dt3 [t,T ]3

l=1

2

1{i3<i2<ii}1{ii<i2<i3}^3(t1)^1(t1) (^2^2)) ^3(t3)^1(t3)dt1dt2dt3 = 0.

[t,T ]3

Applying (1.121), (1.122), and (1.123)

case k = 3)

lim Ep = 0.

(1.126)

25), we get (see (1.75) for the

1.2.4 Estimate for the Mean-Square Approximation Error of Iterated Ito Stochastic Integrals Based on Theorem 1.1

In this section, we prove the useful estimate for the mean-square approximation error in Theorem 1.1.

Theorem 1.4 [12]-[16], [30]. Suppose that every ^(t) (l = 1,...,k) is a continuous nonrandom function on [t,T] and {$3(x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function $3(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the estimate

m

J

- J

Tf'pfc 1 <

/

< k!

n p i pk

J K 2(t1,...,tk )dt1 ...dtk ..^C3k...3- i

(1.127)

\[t,T ]k

3 i =0 3k =0

)

is valid for the following cases:

1. = 1,...,m and 0 <T — t< to,

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2. i1,...,ik = 0,1,..., m, if + ... + if > 0, and 0 <T — t< 1,

where Jis the iterated Ito stochastic integral (11.5), J[^(k)]j!t".'pfc is the expression on the right-hand side of (1.10) before passing to the limit l.i.m. ;

another notations are the same as in Theorem 1.1.

2

Proof. In the proof of Theorem 1.1 we obtained w. p. 1 the following representation (see (1.40))

j = j + RT!r,Pfc,

where J' "'Pfc is the expression on the right-hand side of (11.101) before passing to the limit l.i.m. and

T t2

T J / Pi Pk k \

= ^ /••7k(ti.....tk)-£(tinx

1 ,...,tk) i t V j i=0 jk=0 1=1 /

(t 1,...,tk) t t \ j 1 =0 jk=0

x dwt(; 1) ...dwt(ik), (1.128)

where

E

(t i,...,tk)

means the sum with respect to all possible permutations (t1,..., tk), which are performed only in the values dwt(| i)... dwt(ik). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,..., tk), then ir swapped with in the permutation (i1,..., ik).

The stochastic integrals on the right-hand side of (1.128) will be dependent in a stochastic sense (i1,..., ik = 1,..., m, k £ N). Let us estimate the second moment of

j[^(k)]T,t - j[^(k)]T;r'Pk.

Using (126). (1381). (ITTT281). the orthonormality of the system (x)}°=0 (see the relation (1.72)), and the elementary inequality

(«1 + «2 + ... + ap )2 < + + ... + ap) , P £ N, (1.129)

we obtain the following estimate

iTt-^rrTTt

T ^ / p i Pk k \2

< k! £ ..j K (t1,...,tk).. ^ Cj-k...j-i n^j. (tin dt1 ...dtk (t i,...,tk) t t V j i=0 jk=0 1=1 /

J[#k)lTt- J[#k)lPi r'Pk ^ <

// p1 Pk k \ 2

K(ti,..., tk) — ^ .. ^ Cjk...j 1 n j(ti) dti... dtk

u^ik V j 1=0 jk=0 1=1 /

[t,T

= k!

/

/p i Pk

k 2(ti.....tk № ...dik — £ i

~ _A ' _A

(1.130)

\[t,T

j =0 jk=0

/

where T — t £ (0, to) and il5..., ik = 1,..., m.

From (1.26), (1.27), (1.38), (1.128), (1.129), and the orthonormality of the system (x)}°=0 we obtain

M^ J

T,t

- J

T t2

<

p

T1f"'PM <

pk

K (ti

(t 1,---,tk) t

tk) — E.. ^ Cjk...j 1 n (ti)) dti... dtk

ji=0 jk=0

1=1

// P l Pk k \ 2

K(ti,..., tk) — ^ ... £ Cjk...j 1 n (ti) dti... dtk

u^k V j 1=0 jk=0 1=1 /

[t,T

= Ck

/P 1 Pk

K 2(t1.....tk )dt1 ...dtk —E .--ECL» 1

~ _A ~ _n

\[t,T

j =0 jk=0

/

where i1,... ,ik = 0,1,... ,m, if + ... + ik > 0, and Ck is a constant.

It is not difficult to see that the constant Ck depends on k (k is the multiplicity of the iterated Ito stochastic integral) and T — t (T — t is the length of integration interval of the iterated Ito stochastic integral). Moreover, Ck has the following form

Ck = k! • max{(T — t)a1, (T — t)a2, ..., (T — t)"k!},

where af,a2,...,ak! = 0, 1,..., k — 1.

However, T — t is an integration step of numerical procedures for Ito SDEs (see Chapter 4), which is a rather small value. For example, 0 < T — t < 1. Then Ck < k!

2

k

It means that for the case i1,... , ik = 0,1,... ,m, i2 + ... + ik > 0, and 0 < T — t < 1 we get (1.127). Theorem 1.4 is proved.

Example 1.3. The particular case of the estimate (1.127) for the iterated Ito stochastic integral 1((000)T)t (see (1.105)) has the following form

21 zr / (T — t)3 V- ^2 IVI j y (000)T,i J(000)T,t. J j - ° I g Z^ ( .k/y.r

j ,j2,j3=0

where i1, i2, i3 = 1, . . .

, m and Cj3j2j i is defined by the formula (1.107).

Let us consider the case of pairwise different i1,..., ik = 1,..., m and prove the following equality

m ( (J[^(k)]T,t — J[^(k)]?t...'Pk

/1 I I k

K2(ti,..., tk)dti... dtk - £... £ C2 ...j I, (1.131)

Ifc j I=0 jfc=0

[t,T

where notations are the same as in Theorem 1.4.

The stochastic integrals on the right-hand side of (1.128) are uncorrelated for the case of pairwise different i1,..., ik = 1,..., m. Moreover, these integrals have zero expectations. Then

m | (J[^(k)]T,t — J[^(k)]?r'Pk

T / pi Pk k \

E •• K(ti,...,tk) ...£Ck..ji n^ji (ti ) x

v(iI,...,tk) t t \ j I =0 jk =0 i=i J

2

xdtf 1)... df

f / T ^ / p i Pk k \

£ mH ... K(ti,...,tk) ..^Cjk...ji n^ji (tiH x

(tI,...,tk) l Vi t V j =0 jk =0 i=i /

2

xdft(i 1) ...dft(kik)

t tk

T ^ / Pi Pk k \ 2

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£ / ... K (ti,...,tk) n^ji (tin dti •••dtk

(ti,...,tk) t t \ j i=0 jk=0 1=1 /

/. / P 1 Pk k \ 2

= J (K(t1,...,tk) ...^C, ...ji n^ji (tin dti ...dtk =

[tT]fc V ji=o jk=o i=i /

^ pi Pk

= J K2(ti,...,tk)dti...dtk•••ECU•

[t,T]k j1=0 jk=0

1.3 Expansion of Iterated Itô Stochastic Integrals Based on Generalized Multiple Fourier Series. The Case of Complete Orthonormal with Weight r(ti)... r(tk) Systems of Functions in the Space L2([t,T]k)

In this section, we consider a modification of Theorem 1.1 for the case of complete orthonormal with weight r(t1)... r(tk) > 0 systems of functions in the space L2([t,T]k), k G N.7

Let {^j (x)}°=0 be a complete orthonormal with weight r(x) > 0 system of functions in the space L2([t,T]). It is well known that the Fourier series of the

function f(x) (^f(x)^r(x) G Lo([t,T})^ with respect to the system

converges to the function f (x) in the mean-square sense with weight r(x), i.e.

T

T P \ 2

lim i if (x) - ^ Cj^ (x)^ r(x)dx = 0, (1.132)

{ v j=0 '

where

T

Cj = I f(x)tfj(x)r(x)dx (1.133)

is the Fourier coefficient.

The relations (1.132), (1.133) can be obtained if we will expand the function

/(x)y/r(x) G Lo{[t,T}) into a usual Fourier series with respect to the complete

7 The results of this section are generalized to the case of an arbitrary complete orthonormal with weight r(x) > 0 system of functions {^j(x)\/r(x)} 0 in the space Z/2([i, T]) and ipi(x)-\/r(x), ..., tpk(x)s/r(x) G L2([t,T]) in Sect. 1.13 (see Theorems 1.20, 1.21).

orthonormal with weight 1 system of functions

in the space L2([t,T]). Then

T

lim / ( f{x)\Jr{x) — y^ Cj^j(x)\/r(x) ) dx =

V \ 2

P

t j =0

T P 2

= lim i (f (x) — (j^j(xM r(x)dx = 0, (1.134)

M j=0 J

where (j is defined by (1.133).

Let us consider an obvious generalization of this approach to the case of k variables. Let us expand the function K(t1,... ,tk) such that

k

1=1

using the complete orthonormal system of functions

k

¿ = 0,1,2,..., / = 1,.. ., A:

1=1

in the space L2([t,T]k) into the generalized multiple Fourier series.

It is well known that the mentioned generalized multiple Fourier series converges in the mean-square sense, i.e.

2

// /•' Pi Pk k \

wi- //)]! V^MII^(/iîx/wm x 1=1 j 1=0 jfc=° 1=1 /

xdti... dtk =

2

// P 1 Pk k \ k

k(ti—tk) - E... E cC'jfc...j i n ** (ti) n )

,T1k V j 1=0 jk=0 1=1 J 1=1

x dt1... dtk = 0, (1.135)

where

Cjfc...j = J K(t/)r(t/))dt1 ...dtk.

[t,T]k 1=1

Let us consider the following iterated Ito stochastic integrals

T t2

J[^k)ht = i utk)^itk)-- ■ i ■ ■ (1.136)

'ti vvtfc t t

where every ^(t) (l = 1,..., k) is a nonrandom function on [t,T], wTi) = f() for i = 1,..., m and w[0) = t, i1,..., = 0,1,..., m. So, we obtain the following version of Theorem 1.1.

Theorem 1.5 [13]-[16], [28], [40]. Suppose that every ^/(t) (l = 1,...,k)

is a continuous nonrandom function on [t,T]. Moreover, let {^fj(a;)y/'r(a;)}°^0 (r(x) > 0) is a complete ort.honormal system of functions in the space Lo{[t,T}), each function ^ j(x)yjr(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then

Pi Pk / k

(-) ji

./[^>]t,( = J^V ..-ECM n <j.

ji=0 jfc=0 \/=1

-l.i.m. £ ^^Jv^Aw^.-.^^Jv^Aw^) , (1.137)

(ii,...,ifc )GGfc

where

Gk = Hk\Lk, Hk = {(/!,...,«: I1,..., Ik = 0, 1,...,N - 1}, Lk = {(/1,...,/k) : l1,..., Ik = 0, 1,...,N -1; lg = ¿r (g = r); g,r = 1,...,k}, l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,... ,m,

T

cf = / ^(s)v^dwW t

are independent standard Gaussian random variables for various i or j (in the

4-:+, - w':> (i = 0,1,..., m), {Tj}N=0

case when i = 0), Aw— = w[j+j — w— (i = 0,1,..., m), {Tj}N=0 is a partition

of [t, T], which satisfies the condition (1.9),

Cj ...ji = j K (t1,...,tk ) n (ti )r(ti^ dt1 ...dtk (1.138)

[t,T ik 1=1

is the Fourier coefficient,

^1^1).. .^k(tk), t1 < ... < tk K(t1,...,tk)H , t1,...,tk G [t,T], k > 2,

0, otherwise

and K(t1) = ^1(t1 ) for t1 G [t,T].

Proof. According to Lemmas 1.1, 1.3 and (1.24), (1.25), (1.36), (1.37), we get the following representation

T t2 k

•>>•*:/•./ = E J ■■■J •••>**) n ... ^ (ii,--,ik ) t t 1=1

pi Pk t t2

1k

E - Er- ' /•••/ E n(^^)v^)

ji=0 jk=0 t t (ti,...,tk) M=1

+JRP1'-'Pk -

T,t

Pi Pk

= E... E (^jk-jix ji=0 jk=0

N-1

xi-i-m. £ ^(r/j ... (r/fc) v^K) AwJ;;)+

N^œ— 1 k

ii,...,ik = 0 iq = 1r; q=r; q,r = 1,...,k

DPi,--,Pk

Pi Pk

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= E • • • E X

ji=0 jk=0

N-1

xii.i.m. £ i^rjv^Awi;;)... ^(rjv^Awi;;)-

l.i.m. £ vVK)Aw(;;)... V^JAw^) +

(ti,...,ifcJGGfc

nPi,...,Pfc

Pi Pk

E • • • E Cjk...ji X

ji=0 jk=0

k

XI n ci;° - l.i.m. £ %(rh) v^K)Aw[;;)... (.Jv^jAwS;)) + l=1 (li,...,lk)GGk

+RT1r.'Pk w. p. i,

where

T t2 / A

i ,...,tk) + + \ l = l

RPi,...,Pk

(ti v-^k) t t \ l =

Pi Pk k

ji=0 jk=0 l=1

" i dwti .. -dwtk

where permutations (tl5..., tk) when summing are performed only in the values dwt(ii)... dwt(ik). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with in the permutation (t1,..., tk), then ir swapped with in the permutation (i1?..., ik).

Let us estimate the remainder Rpt'''Pk of the series. According t° Lemma 1.2 and (1.38), we have

t t2

, -pA i ^ c, v^

, RP2

T h ( k J (ti'...'ifc) { 1 V 1=1

Pi Pk k \

E • • • EII(^f'^) <//....,//,, = (i.i39) ji=0 j =0 1=1

2

/. / pi Pk k \

Ck J k(ti,...,tk)— £...£C, ,n**(ti) x

t ] ^ ji=0 jk=0 i=i /

X I n r(ti)) dti ...dtk — 0 (1.140)

,i=i /

if p1,... ,pk —to, where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral (1.136). Theorem 1.5 is proved.

Let us formulate the version of Theorem 1.4.

Theorem 1.6 [14]-[16], [28], [40]. Suppose that every ^(t) (l = 1,...,k)

is a continuous nonrandom function on [t,T]. Moreover, let {^j(x)^r(x)}°cL0 (r(x) > 0) is a complete ort.honormal system of functions in the space Lo{[t,T}), each function ^ j(x)yjr(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the estimate

m{ J [^(k)]Tt - J[^(k)]Tt'^)2] <

/ r. / k \ P i Pk ^

I K2(ti,..., tk) IIr(t,) dti... rftk — £... £ Ci..j ,

^[t'T]k \i=^ / j i=0 jk=0 y

(1.141)

is valid for the following cases:

1. i1,...,ik = 1,...,m and 0 <T — t< to,

2. i1,..., ik = 0,1,..., m, ¿2 + ... + ik > 0, and 0 < T — t < 1,

< k!

where J[^(k)]T,t is the stochastic integral (11. 136)), J[^(k)]TTi,t ,Pk is the expression on the right-hand side of (1.137) before passing to the limit l.i.m. ; another

Pi,...,Pk ^TO

notations are the same as in Theorem 1.5.

1.4 Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures Based on Generalized Multiple Fourier Series

In this section, we consider the version of Theorem 1.1 connected with the expansion of iterated stochastic integrals with respect to martingale Poisson measures.

1.4.1 Stochastic Integral with Respect to Martingale Poisson Measure

Let us consider the Poisson random measure on the set [0,T] x Y (Rn = Y). We will denote the value of this measure at the set A x A (A C [0, T], A c Y) as v(A, A). Assume that

m {v(A, A)} = |A|n(A),

where |A| is the Lebesgue measure of A, n(A) is a measure on a-algebra B of Borel subsets of Y, and B0 is a subalgebra of B consisting of sets A c B that satisfy the condition n(A) < to.

Let us consider the martingale Poisson measure

R(A, A) = v(A, A) - |A|n(A).

Let (Q, F, p) be a fixed probability space, let {Ft, t G [0,T]} be a non-decreasing family of a-algebras Ft c F.

Assume that the following conditions are fulfilled:

1. The random variables v([0,t),A) are Ft-measurable for all A C B0, t G [0,T].

2. The random variables v([t,t + h), A), A C B0, h > 0 do not depend on events of a-algebra Ft.

Let us define the class H(n, [0, T]) of random functions ^ : [0, T] x Y x Q ^ R1 that are Ft-measurable for all t G [0,T], y G Y and satisfy the following

condition

T

0 Y

m

{|p(t, y)|<} n(dy)dt

< oo.

Consider the partition {Tj}N=0 of the interval [0, T], which satisfies the con-

dition (1.9), and define the stochastic integral with respect to the martingale Poisson measure for ^>(t, y) G H2(n, [0,T]) as the following mean-square limit [95]

T T

i /^(t,y)R(dt,dy) =f l.i.mj /VN)(t,y)R(dt,dy), (1.142) J J N^TO J J

0 Y 0 Y

where ^(N)(t,y) is any sequence of step functions from the class H2(n, [0,T]) such that

T

lim / / m

N^œ J J 0Y

p(t, y) - >(t, y) |>n(dy)dt ^ 0

It is well known [95] that the stochastic integral (1.142) exists, it does not depend on selection of the sequence ^>(N) (t, y) and it satisfies w. p. 1 to the following properties

m | j j p(t, y)R(dt,dy) F01 =0,

T

J y*(a^1(t, y) + M(t, y))R(dt,dy) =

0Y

T T

= a/ / (t, y)R(dt,dy) + W / ^(t, y)R(dt,dy),

0Y

m

T

0Y

<^(t, y)z>(dt, dy)

T

m = // m

0Y

{lv(t, y)|2

Fo

where a, ^ G R1 and (t,y), ^2(t,y), ^(t, y) from the class H2(n, [0,T]).

2

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The stochastic integral

T

J J p(t, y) v(dt,dy)

0 Y

with respect to the Poisson measure will be defined as follows

T T T

J J p(t, y)v (dt,dy) = y J p(t, y)i> (dt,dy) + ^ J p(t, y)n(dy)dt, (1.143)

0 Y 0 Y 0 Y

where we suppose that the right-hand side of the last equality exists. According to the Ito formula for Ito processes with jumps, we get [95]

t

(zt)p = J J{(zT— + y (t, y))p — (zT—)p) v(dT,dy) w. p. 1, (1.144)

0Y

where p £ N and zT — means the left-sided limit value of the process zT at the point t ,

t

zt = j j Y(t, y)v(dT,dy).

0Y

We suppose that the function y (t, y) satisfies the conditions of existence of the right-hand side of (1.144) [95].

Let us consider the useful estimate for moments of stochastic integrals with respect to the Poisson measure

OpCT) < .max m / (((bp(r,y))1/p + l)" - 1) n(dy)dr L (1.145)

jejp, 1}

",0 Y

where

Op(i) = sup m{|p), bp(T,y) = m{|y(t,y)|p).

0<r<t ^ } ^ }

We suppose that the right-hand side of (1.145) exists. According to (see (1.143))

t t t

J J Y(T y)v>(dT,dy)^y J Y(t, y)v(dT,dy) J Y(T y)n(dy)dT

0 Y 0 Y 0 Y

j

and the Minkowski inequality, we obtain

m

{|«|*})1/* < (m{n2»})1/2p + (m{|i(|^1/2p

(1.146)

where

t

¿t = J J Y(t, y)i>(dr,dy)

0 Y

and

t

def

^ = J J Y(T, y)n(dy)dT-

0Y

The value m j |iT |2P| can be estimated using the well known inequality [95]

m{|it|2p} < t2p-1 / m

Y (t, y)n(dy)

2p

dT,

(1.147)

where we suppose that

m

Y (t, y)n(dy)

2p '

dT < oo.

1.4.2 Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures

Let us consider the following iterated stochastic integrals

P

T,t =

T t2

= / I Xk (tk, yk) .J I X1(t1, y1)R(ii)(dt1,dy1) ...R^(dtk,dyk), (1.148)

t X t X

where ¿1,..., ik = 0,1,...,m, Rn =f X, Xi (t, y) = (t (y) (l = 1,...,k), every function ^ (t) : [t,T] ^ R1 (l = 1,...,k) and every function ^ (y) : X ^ R1 (l = 1,..., k) such that

t

Xi (T, y) G H (n, [t, T ]) (l = 1,...,k),

where definition of the class H2(n, [t,T]) see above, v(i)(dt,dy) (i = 1,... ,m) are independent Poisson measures for various i, which are defined on [0, T] x X,

V(i)(dt,dy) = v(i)(dt,dy) - n(dy)dt (i = 1,... ,m)

are independent martingale Poisson measures for various i, £(0)(dt,dy) =

n(dy)dt, v(0)(dt, dy) =f n(dy)dt.

Let us formulate an analogue of Theorem 1.1 for the iterated stochastic integrals (1.148).

Theorem 1.7 [1]-[16], [40]. Suppose that the following conditions are hold:

1. Every (t) (/ = 1,...,k) is a continuous nonrandom function at the interval [t,T].

2. (x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fy(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7).

3. For I = 1,..., k and q = 2k+1 the following condition is satisfied

J (y)|qn(dy) < to. x

Then, for the iterated stochastic integral with respect to martingale Poisson measures P[x(k)]T,t defined by (1.148) the following expansion

pl pfc / k

(g^s)

p i,...,pfc^ ^^ M 11%

ji =0 jfc =0 \g=i

P [x(k)]T,t = l.i.m. £ i n^j^

l.i.m. E H j(Tis) / ^(y)^(is)([Tis,Tis+i),dy) ) (l ......lk)GGk g=i X /

(1.149)

that converges in the mean-square sense is valid, where {Tj}N=0 is a partition of the interval [t,T] satisfying the condition (1.9),

Gk = Hk\Lk, Hk = {(/i,...,/k) : 1i,...,1k = 0, 1,...,N - 1},

Lk = {(1i,...,1k): li,...,1k = 0, 1,...,N-1; lg = lr (g = r); g,r = 1,...,k},

l.i.m. is a limit in the mean-square sense, ii,..., ik = 0,1,..., m, random variables

T

nfg) = / h (T) J ((y)£(ifl)(dT,dy) t X

are independent for various (if = 0) and uncorrected for various j,

k

C —

^ j 1 —

J K (ti,...,tk)n j (ti)dti ...dtk

— I K(t1,...,tk)n^(^«H

[t,T]k 1=1

is the Fourier coefficient,

!^i(ti).. .^k(tk), ti < ... < tk

, ti,...,tk G [t,T], k > 2,

0, otherwise

and K(ti) = ^i(ti) for ti G [t, T].

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Proof. The scheme of the proof of Theorem 1.7 is the same with the scheme of the proof of Theorem 1.1. Some differences will take place in the proof of Lemmas 1.4, 1.5 (see below) and in the final part of the proof of Theorem 1.7.

Lemma 1.4. Suppose that every ^ (t) (/ = 1,..., k) is a continuous function at the interval [t, T] and every function ((y) (/ = 1,..., k) such that

J (y)|2n(dy) < oo.

X

Then, the following equality

N —i j2 i k „

P[X(k)]T,t = l.i.m. £ .^11 / Xi(j,y^fe,Tji+i),dy) (1.150)

_n _A , T J

j=0 ji=0 l=1

is valid w. p. 1, where {rj}N=0 is a partition of the interval [t,T] satisfying the condition (1.9),

X

i>(i)([r, s), dy) z/(i)([r,s),dy) ^ (i — 0,1,...,m).

v(i)([t, s),dy)

In contrast to the integral P[x(k)]T,t defined by (1.148), z/(i1 )(dtl, dy/) is used in the integral P[X(k)]T,t instead of ¿>(i1 )(dtl, dy/) (/ = 1,..., k).

Proof. Using the moment properties of stochastic integrals with respect to the Poisson measure (see above) and the conditions of Lemma 1.4, it is easy to notice that the integral sum of the integral Pcan be represented as a sum of the prelimit expression from the right-hand side of (1.150) and the value, which converges to zero in the mean-square sense if N ^ to. Lemma 1.4 is proved.

Note that in the case when the functions ^(t) (l = 1,... ,k) satisfy the condition (*) (see Sect. 1.1.7) we can suppose that among the points Tj, j = 0,1,..., N there are all points of jumps of the functions ^(t) (l = 1,..., k). Further, we can apply the argumentation as in Sect. 1.1.7.

Let us consider the following multiple and iterated stochastic integrals

n-1

l.i.m. J] •■■•■j )ul (y^'fe > rji+i)i dy) = />[<!>]$,

j 1,-jfc =0 l=1 X

N-^oo

T t2

def

J ..J t* )J ^(y)^ 1 ]№,dy) ...J (yKfc](dt* ,dy) =

t t X X

where ): [t,T]* ^ R1

is a bounded nonrandom function and the sense of notations of the formula (1.150) is remaining.

Note that if the functions ^(y) (l = 1,...,k) satisfy the conditions of Lemma 1.4 and the function $(t1,..., t*) is continuous in the domain of integration, then for the integral P^lT*] the equality similar to (11.150) is valid w. p. 1.

Lemma 1.5. Assume that the following representation takes place:

gi y) = h/ (t M(y) (1 = 1,•••,k),

*

where the functions h(t) : [t,T] ^ R1 (l = 1,..., k) satisfy the condition (*) (see Sect. 1.1.7) and the functions ^/(y) : X ^ R1 (l = 1,..., k) satisfy the condition

j (y)|pn(dy) < to for p = 2*+1. X

Then

T

II I I gi(s, y)^(il)(ds,dy) — P[$]$ w. p. 1,

(1.151)

i=1

t X

where i\ — 0,1,..., m (l — 1,..., k) and

$(*!,..., ik ) —JJ h (ti).

i=1

Proof. Let us introduce the following notations

N-1 ,,

J[gi]N = ^ / g(Tj , y)^(il)([Tj, Tj+1), dy),

j=0 X

T

J fe ]T,t1—1

tX

de^ / gi(s, y)V'(il) (ds, dy),

where {t,}N=0 is a partition of the interval [t,T] satisfying the condition (1.9). It is easy to see that

U [gi ]n -II J [gi ]

T,t —

l=1

k //-i \ / k

= E n J(J[Pl]N - J[gi]T,t) n J&]n /=i \q=i / \q=/+i

Using the Minkowski inequality and the inequality of Cauchy-Bunyakovsky together with the estimates of moments of stochastic integrals with respect to the Poisson measure and the conditions of Lemma 1.5, we obtain

m

IP [gi ]n -n J [gi ]

T,t

i=1

2^ \ 1/2

k

< ( m

i=1

J [^]N - J ]T,i

1/4

(1.152)

where Ck < to. We have

N1

Jfe]n - J]T,t — ^ J[Agi]

Tq+1

q=0

k

4

where

Tq+1

J[AflL+1>T, = (hi(Tq) - h(s)) / 0 (y)z>(il)(ds,dy).

Let us introduce the notation

h(N)(s) = hi (Tq), s G [Tq ,Tq+i), q = 0,- 1.

Then

N-1

f[A

JTg+l >

J [#i]n - J [gi]T,t = Y^ J [Agi

q=0

T

= / (h(N)(s) - hi(s)) |(y)z>(il)(ds,dy).

t X

Applying the estimates (1.145) (for p = 4) and (1.146), (1.147) (for p = 2) to the value

4'

m

T

hi

t X

h((N)(s) - hi(s)) / (y)z/(il)(ds,dy)

taking into account (1.152), the conditions of Lemma 1.5, and the estimate

|hi (Tq) - hi (s)| <£, s G [Tq ,Tq+i], q = 0,1,...,N - 1, (1.153)

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where £ is an arbitrary small positive real number and |Tq+1 - Tq| < £(e), we obtain that the right-hand side of (1.152) converges to zero when N ^ to. Therefore, we come to the affirmation of Lemma 1.5.

It should be noted that (1.153) is valid if the functions hi (s) are continuous at the interval [t,T], i.e. these functions are uniformly continuous at this interval. So, |hi(Tq) - hi(s)| < £ if s G [Tq,Tq+1], where |Tq+1 - Tq| < £(e), q = 0,1,..., N - 1 (£(e) > 0 exists for any £ > 0 and it does not depend on points of the interval [t,T]).

In the case when the functions hi (s) (l = 1,..., k) satisfy the condition (*) (see Sect. 1.1.7) we can suppose that among the points Tq, q = 0,1,..., N there are all points of jumps of the functions hi(s) (l = 1,..., k). Further, we can apply the argumentation as in Sect. 1.1.7.

Obviously, if ii = 0 for some I = 1,..., k, then we also come to the affirmation of Lemma 1.5. Lemma 1.5 is proved.

Proving Theorem 1.7 by the scheme of the proof of Theorem 1.1 using Lemmas 1.4, 1.5 and moment properties of stochastic integrals with respect to the martingale Poisson measures, we obtain

m I (Rir**)2! < C n/v?(y)n(dy)x

v ) /=i v

T } / pi pk k \

x E .. K(ti,...,tk)..j,nhji(t/H x (ti,...,tfc) { { V ji=0 j =0 /=i /

xdti... dtk =

k „ „ / pi pk k \ c^/(2(y)n(dyW K(ti,...,tk)...£c, ..^n (t/)

/=i X [t,T]^ ji=0 jk=0 /=i J

2

X

xdti... dtk <

/. / pi pk k \ 2 < Ck J K(ti,...,tk) hji (t/H dti ...dtk ^ 0

[t T]k ji=0 jk =0 / = i /

if pi,... ,pk ^ o, where constant Ck depends only on k (k is the multiplicity of the iterated stochastic integral with respect to the martingale Poisson measures). Moreover, Rrt ''^ has the following form

T r / pi pk k \

E /.../ k(ti,...,tk)...Ec*hji(t/Hx (tl,''',tk) t t \ ji=0 j=0 /=i /

x / (i(y)i>(ii)(dti,dy)... i (k (y)z>(ik) (dtk, dy), (1.154)

where permutations (ti,... ,tk) when summing in (1.154) are performed only in the values (i(y)i>(ii)(dti, dy)... (k(y)^ (ik)(dt k,dy). At the same time, the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (ti,... ,tk), then ir swapped with in the permutation (ii,... ,ik). Moreover,

(y) swapped with (y) in the permutation (^i(y),..., (y)). Theorem 1.7 is proved.

Let us consider the application of Theorem 1.7. Let i1 = i2 and i1, = 1,..., m. Using Theorem 1.7 and the system of Legendre polynomials, we obtain

t t2

J J J J ^l(yi)i>(il)(dti,dyi)i>(i2)(dt2,dy2) =

t X t X

T — ti Î1 (9 V^ 1

i=1

T

t X

where

T

= j(T) / (My)z>W(dr, dy) (l = 1, 2)

j - / rj

tX

and (t)}j=o is a complete orthonormal system of Legendre polynomials in the space L2([t, T]).

1.5 Expansion of Iterated Stochastic Integrals with Respect to Martingales Based on Generalized Multiple Fourier Series

1.5.1 Stochastic Integral with Respect to Martingale

Let (Œ, F, p) be a fixed probability space, let {Ft, t G [0, T]} be a non-decreasing family of a-algebras Ft c F, and let M2(p, [0,T]) be a class of Ft-measurable for each t G [0,T] martingales Mt satisfying the conditions

s

m{(Ms - Mt)2} = j p(t)dT, (1.155)

t

m{ |Ms - M/} < Cp | s - t|, p = 3,4,...,

where 0 < t < s < T, p(r) is a non-negative and continuously differentiable nonrandom function at the interval [0,T], Cp < to is a constant.

Let us define the class H2(p, [0,T]) of stochastic processes £t, t G [0,T], which are Ft-measurable for all t G [0, T] and satisfy the condition

T

m

{|6|2} P(t)dt

<.

For

any partition {t(nh of the interval [0,T] such that I j J j=0

0 — t0(n) < t1N) < ... < riN) — T, max

0 1 N 0<j<N -1

t(N) _ t(N)

^ 0 if N ^ o (1.156)

we will define the sequence of step functions £(N) (t,w) by the following relation €(N '(t , w) = (w) w. p. 1 for t G

t(n) t(n)

where £(N)(t,w) G H2(p, [0,T]), j = 0,1,...,N - 1, N = 1, 2,...

Let us define the stochastic integral with respect to martingale from the process £(t,w) G H2(p, [0,T]) as the following mean-square limit

N1

T

Um. £ £ <N (rf\w) (m (T<N?,W) - M (rfU) j =f J irdMr, (1.157)

j=0 0

where £(N)(t, w) is any step function from the class H2(p, [0, T]), which converges to the function £(t, w) in the following sense

T

lim

N

m

£(N)(t,^) - £(t,w) p(t)dt — 0.

It is well known [95] that the stochastic integral (1.157) exists, it does not depend on selection of the sequence £(N)(t,w) and it satisfies w. p. 1 to the following properties

m

£t dMt

F0 — 0,

m

T

CtdMt

T

Fr

m = m < J ttp(t)dt

T TT

j(att + ß^i)dMt = a j ttdMt + ß j fadMu 0 0 0 where tt, ^t G H2(p, [0,T]), a,ß G R1.

1.5.2 Expansion of Iterated Stochastic Integrals with Respect to Martingales

Let Q4(p, [0,T]) be the class of martingales Mt, t G [0,T], which satisfy the following conditions:

1. Mt, t G [0,T] belongs to the class M2(p, [0,T]).

2. For some a > 0 the following estimate is correct

4 '

m

g(s)dMs

< K4 |g(s)|ads,

(1.158)

where 0 < t < t < T, g(s) is a bounded nonrandom function at the interval [0,T], K4 < œ is a constant.

Let Gn(p, [0, T]) be the class of martingales Mt, t G [0,T], which satisfy the following conditions:

1. Mt, t G [0,T] belongs to the class M2(p, [0,T]).

2. The following estimate is correct

m

g(s)dMs

< œ,

where 0 < t < t < T, n G N, g(s) is the same function as in the definition of the class Q±(p, [0,T]).

Let us remind that if )n G H2(p, [0, T]) with p(t) = 1, then the following estimate is correct [95

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m

ts ds

2 n

< (t - t)2n-1 ^ |ts|2^ ds, 0 < t<T < T. (1.159)

2

n

T

Let us consider the iterated stochastic integral with respect to martingales

t t2

J[^Mt = / ^k(tk) . . .j ^i(ti)dMt(i1'il).. .d<'ik\ (1.160)

i i

where il5..., ik = 0,1,... ,m, every ^(t) (l = 1,..., k) is a continuous non-random function at the interval [t, T], Ms(r,i) (r = 1,..., k, i = 1,..., m) are independent martingales for various i = 1,..., m, Ms(r'0) = s. Now we can formulate the following theorem.

Theorem 1.8 [1]-[16], [40]. Suppose that the following conditions are hold:

1. Every ^(t) (l = l,...,k) is a continuous nonrandom function at the interval [t,T].

2. {fy(x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fy(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7).

3. Mp1) G Q4(p, [t,T ]), Gn (p, [t,T]) with n = 2k+1, ¿/ = 1,...,m, l = 1,..., k (k G N).

Then, for the iterated stochastic integral Jwith respect to martingales defined by (1.160) the following expansion

Pi Pk / k

J^ = J.^ ECjk^mif

ji=0 jk=0 \/=1

-l.i.m. £ j (Tii)A<;n)... j (Tik)AMTf;i

(ti,...,ikJGGk

ik)

that converges in the mean-square sense is valid, where ii, . . . , ik = 0, 1, . . . , m,

iN j }j=0

A A/f(r'i) _ A/

'"j+i

Gk = Hk\Lk, Hk = {(/i,...,/k) : li ,...,1k = 0, 1,...,N - 1},

{Tj}N=0 is a partition of the interval [t,T] satisfying the condition similar to (0561), AMj'° = Mj+i - Mj'° (i = 0,1,... ,m, r = 1,..., k),

Lk ={(/i,...,/k ): h,..., Ik = 0, 1,...,N -1; lg = 1r (g = r); g,r = 1,...,k}, 1.i.m. is a limit in the mean-square sense,

T

if1 ) = J j (s)dMf'il) t

are independent for various il (if ii — 0) and uncorrelated for various j (if p(r) is a constant, il — 0) random variables,

C —

/k

K (tu...,tk )n 1 (ti )dti ■■■dtk

jk-n — J K (ti,--.,tk) vji (t )dti

[t,T ]k 1=1

is the Fourier coefficient,

^i(ti) ...^k (tk), ti <...<tk K(ti,...,tk)—< , ti,...,tk e [t,T], k > 2,

0, otherwise

and K(ti) = ^i(ti) for ti e [t,T].

Remark 1.4. Note that from Theorem 1.8 for the case p(r) = 1 we obtain the variant of Theorem 1.1.

Proof. The proof of Theorem 1.8 is similar to the proof of Theorem 1.1. Some differences will take place in the proof of Lemmas 1.6, 1.7 (see below) and in the final part of the proof of Theorem 1.8.

Lemma 1.6. Assume that M(r,i) e M2(p, [t,T ]) (i — 1,...,m), M(r,0) — s (r — 1,... ,k), and every (t ) (I — 1,... ,k) is a continuous nonrandom function at the interval [t,T]. Then

N —i j2 i k

J[i,(k% — l.i.m. £ (j )AMil;") w.p. 1, (1.161)

1k =0 11=0 i=i

where {tj}N=0 is a partition of the interval [t,T] satisfying the condition similar to (1.156), il — 0,1,... ,m, I — 1,... ,k; another notations are the same as in Theorem 1.8.

Proof. According to the properties of the stochastic integral with respect to martingales, we have [95

m ^dm^'m m{ &|2} p(s)ds, (1.162)

m

&dsj l< (t — t) i m{ fa |2} ds, (1.163)

where e H2(p, [0, T]), 0 < t < t < T, i/ = l,...,m, l = 1,...,k. Then the integral sum for the integral Junder the conditions of Lemma 1.6 can be represented as a sum of the prelimit expression from the right-hand side of (1.161) and the value, which converges to zero in the mean-square sense if N —y to. More detailed proof of the similar lemma for the case p(t) = 1 can be found in Sect. 1.1.3 (see Lemma 1.1).

In the case when the functions ^(t) (l = 1,..., k) satisfy the condition (*) (see Sect. 1.1.7) we can suppose that among the points Tj, j = 0,1,..., N there are all points of jumps of the functions ^(t) (l = 1,..., k). So, we can apply the argumentation as in Sect. 1.1.7.

Let us define the following multiple stochastic integral

N-1 k

) d=f T №iLk)

Nj

l.i.m. £ $(TJi,...,TJjnAAi<;;'<) = T[*]<î>, (1.164)

jiv-Jfc =0 /=1

where {t.}n=0 is a partition of the interval [t, T] satisfying the condition similar to (1.156) and $(t1, • • • ,tk) : [t,T]k ^ R1 is a bounded nonrandom function; another notations are the same as in Theorem 1.8.

Lemma 1.7. Let Ms(Ml) G Q4(p, [t, T]), Gn(p, [t, T]) with n = 2k+1, k G N (i/ = 1, • • •, m, l = 1, • • •, k) and the functions g1(s), • • •, gk(s) satisfy the condition (*) (see Sect. 1.1.7). Then

k «

n / g/(s)dMs(/'il) = T[$]$ w. p. 1, /=1 t

where i/ = 0,^•••,-m, l =

k

^(t1, • • •, tk) = J!g/(t/)• /=1

Proof. Let us denote

T

N-1

J [g/ In = E g/ (t. )AMfl ), J [g/1 T,t d=f I g/ (s)dMs(/'il ), . =0

where {t.}n=0 is a partition of the interval [t, T] satisfying the condition similar to (1.156).

Note that

IP [gl ]n — Ü J [9i ]

T,t =

1=1

k /1- 1

E IIJ[9q]Tt) (J[gi]N - J[91 ]T,t) II J[gq]N

1=1 \q=1 / \q=1+1

Using the Minkowski inequality and the inequality of Cauchy-Bunyakovsky as well as the conditions of Lemma 1.7, we obtain

m

II J [91] N —

II J [91 ]

T,t

l=1

1/2

<

c*ë m

l=1

J [9i ]n — J [9i ]T,t

1/4

where Ck < œ is a constant. We have

N1

J [91 ]n — J [91 ]T,t = E J [A91]

Tq+1

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q=0

Tq+1

J [A91]

Tq+1,Tq

(9i(Tq) — 91 (s)) dM(1,i

a)

(1.165)

Let us introduce the notation

9/N)(s)= 91 (Tq), s G [Tq,Tq+1), Q = 0, 1,...,N — 1

Then

N1

J[91 ]n — J[91]T,t = ^ J[A91]

Tq+1 ,Tq

q=0

T

9/N)(s) — 91 (s)) dM().

k

2

Applying the estimate (1.158), we obtain

m

T

g\N)(s) - 91 (s)) dM()

<

T

< k4

9iN)(s) - 9i(s)

ds =

N -1 Tq+1

K4^ / |9i(Tq) - 9i(s)|a ds<

__n J

= K4

q=0

N1

< KAeaY, (Tq+i - Tq) = ^(T - t).

q=0

(1.166)

Note that we used the estimate

|gi (Tq) - gi (s)| <e, s G [Tq,Tq+i], q = 0,1,...,N - 1 (1.167)

to derive (1.166), where |Tq+1 — Tq | < 5(e) and £ is an arbitrary small positive real number.

The inequality (1.167) is valid if the functions gi(s) are continuous at the interval [t,T], i.e. these functions are uniformly continuous at this interval. So, |gi (Tq) — gi (s)| < e if s G [Tq ,Tq+i], where |Tq+i — Tq | < 5(e), q = 0,1,...,N — 1 (5(e) > 0 exists for any e > 0 and it does not depend on points of the interval

[t,T ]).

Thus, taking into account (1.166), we obtain that the right-hand side of (1.165) converges to zero when N ^ to. Hence, we come to the affirmation of Lemma 1.7.

In the case when the functions gi(s) (l = 1,... ,k) satisfy the condition (*) (see Sect. 1.1.7) we can suppose that among the points Tq, q = 0,1,... ,N there are all points of jumps of the functions gi (s) (l = 1,... ,k). So, we can apply the argumentation as in Sect. 1.1.7.

Obviously if ii = 0 for some l = 1,... ,k, then we also come to the affirmation of Lemma 1.7. Lemma 1.7 is proved.

Proving Theorem 1.8 similar to the proof of Theorem 1.1 using Lemmas 1.6, 1.7 and moment properties of stochastic integrals with respect to martingales

4

a

(see (1.162), (1.163)), we obtain

m ^ i ^ <

2'

T ll / Pi Pk k \

<Ck £ ... K(t!,...,tk)..,n(t/) x

(ii,...,ifc) t t V ji=0 jk=0 /=1 /

X pi(ti)dti . ..pk (tk )dtk < (1.168)

T "If Pi Pk k

< Ck^ ... K (ti,...,tk)..^Cjk.^n ^ (t/H dti ...dtk

(ti,...,tk) { t V ji=0 jk=o /=i /

/. / Pi Pk k \ 2

= Ck J K(ti,...,tk) ..^Cjk...jin (t/H dti ...dtk — 0

[t T]k V ji=0 jk=0 /=i /

if pi,... ,pk — to, where constant Ck depends only on k (k is the multiplicity of the iterated stochastic integral with respect to martingales) and p/(s) = p(s) or p/(s) = 1 (l = 1,..., k). Moreover, RT t" Pk has the following form

T ti ( Pi Pk k \

RT;t...'Pk = E /..^k(ti.....tk)-E..^Cj,...j in^(t/Hx

(t ,...,tk) t t j =0 jk=0 /=i

x dMt(i'i i) ...dMt(kk'ik), (1.169)

where permutations (ti,..., tk) when summing in (1.169) are performed only in the values dM^ i)... dMt(kk'ik). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (ti,..., tk), then ir swapped with in the permutation (ii,... ,ik). Moreover, r swapped with q in the permutation (1,..., k). Theorem 1.8 is proved.

1.6 One Modification of Theorems 1.5 and 1.8

1.6.1 Expansion of Iterated Stochastic Integrals with Respect to Martingales Based on Generalized Multiple Fourier Series. The Case p(x)/r(x) < to

Let us compare the expressions (1.139) and (1.168). If we suppose that r(x) > 0 and

<- f< ^ -

—— < c < to, r(x)

where p(x) as in (1.155), then

2

I \ -*- / I >v / s j s j JK"'JLWWJL\V/ I ^

\ A__H A__H 7_1 /

[t,T ]

// Pi Pk k \

k (ti—tk ) -E ---y,0»-jill ** (ti )

V ji=0 jk=0 i=1 /

xp(ti)dti. ..p(tk)dtk =

// Pi Pk k

k(ti—tk)-.^(^jk...jill^ji(ti)) x

V ji=0 jk =0 i=1

[t,T

r(ti) r(tk )

Pi Pk k x 2

< C'k I (K(ti,...,tk) .. (7jk...ji n ^ (tiU x

ji=0 jk=0 i=1

[t,T

k

X I n r(ti ) dti ...dtk ^ 0

J=1 /

if pi,... ,pk ^ to (see (1.140)), where Ck is a constant, {^j (x)}°=0 is a complete orthonormal with weight r(x) > 0 system of functions in the space L2([t,T]), and the Fourier coefficient (Âjk...j1 has the form (11.138).

So, we obtain the following modification of Theorems 1.5 and 1.8.

Theorem 1.9 [13]-[16], [40]. Suppose that the following conditions are

fulfilled:

1. Every (t) (I = 1,... ,k) is a continuous nonrandom function at the interval [t,T].

2. {^j(x)}°=0 is a complete orthonormal with weight r(x) > 0 system of functions in the space L2([t,T]), each function ^j(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Moreover,

<r r ^ ^ —— < C < oo.

r(x)

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3. Ms(/,i1) G Q4(p, [t,T ]), Gn (p, [t,T]) with n = 2k+1, i/ = 1,...,m, l = 1,..., k (k G N).

Then, for the iterated stochastic integral Jwith respect to martingales defined by (1.160) the following expansion

Pl Pk / k

J [«=pi,1...^ £ .•• £ M nif)

j 1=0 jfc =0 \/=1

-l.i.m. £ ttji (t/ i)AMTi1;i 1 )... ttjk (r/k)AMTJi

(/ i,...,/k )GGk

ik )

that converges in the mean-square sense is valid, where ii,... ,ik = 1,... ,m,

iN j }j=0

A A/f(r'i) _ A/

ji

{rj}N=0 is a partition of the interval [t,T] satisfying the condition similar to (1.156), AM™ = Mji) - M™ (i = 1,... ,m, r = 1,... ,k),

Gk = Hk\Lk, Hk = {(li,...,lk) : li ,...,lk = 0, 1,...,N - 1}, Lk = {(li,...,lk): li,...,lk = 0, 1,...,N-1; lg = lr (g = r); g,r = 1,...,k},

l.i.m. is a limit in the mean-square sense,

T

j0 = J ^ (s)dMs(/'ii) t

are independent for various i/ = 1,... ,m (l = 1,..., k) and uncorrelated for various j (if p(x) = r(x)) random variables,

Cjk ...ji = i K (ti,...,tk) n (t/)r(t/)) dti ...dtk

/=i

k

is the Fourier coefficient,

^1(^1) (tk ), t1 <...<tk

K (t1,...,tk)

t1,...,tk G [t,T], k > 2,

0,

otherwise

and K(ti) = ^i(ti) for ti G [t,T].

Remark 1.5. Note that if p(x),r(x) = 1 in Theorem 1.9, then we obtain the variant of Theorem 1.1.

1.6.2 Example on Application of Theorem 1.9 and the System of Bessel Functions

Let us consider the following boundary-value problem

where the functions p(x), q(x), r(x) satisfy the well known conditions and a, 3, y, 5, A are real numbers.

It is well known (Steklov V.A.) that the eigenfunctions $0(x), $i(x), ... of this boundary-value problem form a complete orthonormal with weight r(x) system of functions in the space L2([a,b}). It means that the Fourier series of the function ^/r(x)f(x) G L/2{[a,b]) with respect to the system of functions ^r(x)&o(x)i yjr{x)$>i(x)i... converges in the mean-square sense to the function ^/r(x)f(x) at the interval [a, b}. Moreover, the Fourier coefficients are defined by the formula

It is known that when solving the problem on oscillations of a circular membrane (general case), a boundary-value problem arises for the following Euler-Bessel equation

r2R''(r) + rR(r) + (A2r2 - n2) R(r) = 0 (A G R, n G N). (1.171)

The eigenfunctions of this problem, taking into account specific boundary conditions, are the following functions

(p(x)^'(x))' + q(x)$(x) = -Ar(x)$(x), a$(a) + ß $'(a) = 0, 7 $(b) + ö&(b) = 0,

b

a

(1.172)

where t E [0, L] and ^j (j = 0,1, 2,...) are positive roots of the Bessel function Jn(n = 0,1, 2,...) numbered in ascending order.

The problem on radial oscillations of a circular membrane leads to the boundary-value problem for the equation (1.171) for n = 0, the eigenfunctions of which are the functions (1.172) when n = 0.

Let us consider the system of functions

*'(T) = Tj^~yJ" №r) • i =

where

»

J„(X) = J>i r (f)

2) r(m + 1)r(m + n + 1)

m=0 \ / \ /

is the Bessel function of the first kind,

»

r(z>=i

0

is the gamma-function, ^j are positive roots of the function Jn(x) numbered in ascending order, and n is a natural number or zero.

Due to the well known properties of the Bessel functions, the system {^j (t)}CC=0 is a complete orthonormal with weight t system of continuous functions in the space L2([0,T]).

Let us use the system of functions (1.173) in Theorem 1.9.

Consider the following iterated stochastic integral with respect to martingales

T s

J J dM^ dMs(2), 00

where

s

Mf = J VfdfW (¿ = 1,2), 0

fii} (i = 1, 2) are independent standard Wiener processes, Ms(i) (i = 1, 2) are martingales (here p(t) = t), 0 < s < T. In addition, Ms(i) has a Gaussian distribution.

It is obvious that the conditions of Theorem 1.9 are fulfilled for k = 2. Using Theorem 1.9, we obtain

T s

P 1 P2

//> f 1 -F2

j dM^dM® = U.^ £ C*> 1 jj,

0 0 > 1=0 j2=0 where

T

> = J « (T)dM«

0

are independent standard Gaussian random variables for various i or j (i = 1, 2, j = 0,1, 2,...),

T s

=/ ^2 (s)/T «>1 (T )dTdS

00

is the Fourier coefficient.

It is obvious that we can get the same result using the another approach: we can use Theorem 1.1 for the iterated Ito stochastic integral

T s

J V7s J y/ïd^df®,

00

and as a system of functions (s)}°=0 in Theorem 1.1 we can take

y/2s ffij

As a result, we obtain

T s

r r m P2

0 0 j1=0 j2=0

where

T

j = J to(tf

0

are independent standard Gaussian random variables for various i or j (i = 1, 2, j = 0,1, 2,...),

T s

cnn = J V-stoM J ^toAr)drds

00

is the Fourier coefficient. Obviously that Cj2j1 = Cjj. Easy calculation demonstrates that

is a complete orthonormal system of functions in the space L2([t,T]). Then, using Theorem 1.1, we obtain

T s

P1 P2

= l.i.m.

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P1'P2^0 H O t t j1=0 j2 =0

//> f 1 1J2

where

T

j = J h(Tf

t

are independent standard Gaussian random variables for various i or j (i = 1, 2, j = 0,1, 2,...),

T s

c]m = J J V^Ï4n(T)dTds tt

is the Fourier coefficient.

1.7 Convergence with Probability 1 of Expansions of Iterated Ito Stochastic Integrals in Theorem 1.1

1.7.1 Convergence with Probability 1 of Expansions of Iterated Itô Stochastic Integrals of Multiplicities 1 and 2

Let us address now to the convergence with probability 1 (w. p. 1). Consider in detail the iterated Itô stochastic integral (1.96) and its expansion, which is corresponds to (1.97) for the case ii = i2

j-ihh) _ T — t I (i^ (i2) 1 ( Ah) An) A^A^A I M1 7/H

7(00)T,i - Ko Co ^-V )•

First, note the well known fact [99].

Lemma 1.8. If for the sequence of random variables <^p and for some a > 0 the number series

c

EM {& ia}

p=1

converges, then the sequence <^p converges to zero w. p. 1. In our specific case (ii = i2)

T — t » 1 / \

r(n«2) _ T-(n«2)p , ¿r £ _^ V^ 1 /V(nM»2)

J(oo)T,i ~~ 1(oo)T,t "t" sp; sp — 2 Z^ v * * / '

«=p+1

where

7-(H«2)P _ ^_^ / a(H) ^(«2) i Y^ ^ (A^A^) _ A^A^A \ ii 1

J(oo)T,t~ 2 \ /a.-2 _ i V^"1^ ^ ^"V /' v1-1'0;

. , V4i2 - 1

«=1

Furthermore,

.. r,, ,21 (T -1)2 X» 1 (T -t)2 r 1 ,

M {l&l-} = ^ E * ^ J =

i=p+1 p

i^iln

2 4

1- 2

2p +1

< (1.176)

P

where constant C is independent of p.

Therefore, taking a = 2 in Lemma 1.8, we cannot prove the convergence of to zero w. p. 1, since the series

c

£m{ i?p |2}

p=1

will be majorized by the divergent Dirichlet series with the index 1. Let us take a = 4 and estimate the value m {|£p|4}.

From (1.73) for k = 2, n = 2 and (1.176) we obtain

M{l^|4}<§ (1.177)

p

and

cc

EMH&i 1}<*"Ep<00- <L178>

p=1 p=1

where constant K is independent of p.

Since the series on the right-hand side of (1.178) converges, then according to Lemma 1.8, we obtain that £p ^ 0 when p ^ to w. p. 1. Then

twit ^ when P ^ w- p. 1-

Let us analyze the following iterated Ito stochastic integrals

T

T s

I,

(i1^2) (01)T,t

(t - s) df(il)df(si2), I

(i1i2) (10)T,t

(t - T)df(il)df(si2)

tt

whose expansions based on Theorem 1.1 and Legendre polynomials have the following form (also see Chapter 5, Sect. 5.1)

I

(i 1 i2) _ T t T(iii2)p (T t)2 / Z0 )

(01)T,t

2 JmT,t

A (a^ (i-iK/^r

tiV \/ {2i + l)(2i + 5)(2i + 3) (2i — l)(2i + 3)

Z (i1)Z (i2) Zi Zi

+£P01),

I

(ili2) (10)T,t

_^ Ahh)p

2 (00)^

2 (i2) (i1)

(T -t)2 / z0i2)Z1

+

Z(i1)Z(i2) Zi Zi

I VW + WT5)(2i + 3) ~(2i - 1)(2j + 3)

+ iP10),

where

£ (01) p

(T-i)2/ ^ 1 ^(hM^) >(il)>(i2)\

/ v /TTo-— ^ Si—IS» Si Si-ij-r

^i=p+1

V4i2 - 1

+ E

i=p+1

v/(2/- • 1 )(2/ • 5)(2/ + 3) (2i - l)(2i + 3)

(i1) (i2)

Z (i1)Z (i2) Zi Zi

£ (10) p

(T -1)

1

\i=p+1

V4i2 - 1

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E1 ( Ah) Aii) _ Ah) An)

/7^5-rl^-lS« Sj Sj-l |T

s

f

+

i=p+1

\J(2i + l)(2i + 5)(2i + 3) (2i - l)(2i + 3)

z («i)z («2)

Then for the case i1 = i2 we obtain

m

£ (01) p

16

x

x r ( } I 'i + 2)2 + '' + 1)2 I

( 4 i

i=p+1 \

4i2 - 1 (2i + 1)(2i + 5)(2i + 3)2 (2i - 1)2(2i + 3)2

oo

^ v- 1 K

<K } -<—, i2 p i=p+1 1

<

(1.179)

where constant K is independent of p. Analogously, we get

m

£ (10) p

<

K p,

where constant K does not depend on p.

From (1.73) for k = 2, n = 2 and (1.179), (1.180) we have

m

£(01) p

+m

£ (10) p

p2

and

where constant K1 is independent of p.

According to (1.181) and Lemma 1.8, we obtain that £p01), £jp10) ^ 0 when

(1.180)

(1.181)

p ^ » w. p. 1. Then

I(i1«2)p I(i1«2) I(i1«2)p I(i1«2)

I(01)T,t ^ I(01)T,t, I(10)T,t ^ I(10)T,t

where i1 = i2.

Let us consider the case i1 = i2

when p ^ » w. p. 1,

I

(«1«1) (01)T,t

(T - t)2 (T - t)2

c0i1))

2 /(«1V(i1) ^ + Co Ci

^o si

2

1

2

4

z («1)z («1) Z («1)z («1)

Si Si+2 Si Si

, V ___^ ^ , „(01)

+ ti V V(2i + l)(2i + 5)(2i + 3) (2i — l)(2i + 3) I I ^^

rinn) _(T-t)2 (T-tf ((An)\> ,

Vs

rWHj _ - > _ - > / \ 1 -su M I

J(10)T,t ~ A A U> j + /7T +

4- V I U 4- ^ ^ I I + „(10)

+ \/(2i 4 l)(2i 4 5)(2i 4 3) (2i — l)(2i 4 3) j j p

where

(T -1)2 ^ / Ci(i1)c£2 Ci(i1)Ci(i1)

„(01) _ ^ V

4 \ \/(4 l)(2i 4 5)(2i 4 3) (2i-l)(2i + 3)

p

„(10) _

Lp ~ A

E

i=p+1

z (i1)z (i1) Si Si+2

+

cfi}dn) \

\J (2i 4 l)(2i 4 5)(2i 4 3) (2i - l)(2i 4 3)^'

Then

Mn«ioi»V2l = MI frf 1 =

16

00

2

X

c1

(2i 4 l)(2i 4 5)(2i 4 3)2 + (2, - 1)2(2, 4 3)2

+

1

^i=p+1

(2i - 1)(2i + 3)

K <-

p2

and

E (m{

p=1 v ^

00

Mp )

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+m

mp )

<KJ2~< 00, (1.182)

p=1

p

where constant K is independent of p.

According to Lemma 1.8 and (11. 182), we obtain that Mp01), Mp10) ^ 0 when p ^ » w. p. 1. Then

I(i1i1)p V I(i1i1) I(i1i1)p v I(i1i1) when p OO w p 1

I(01)T,t ^ J(01)T,t, J(10)T,t ^ J(10)T,t when p ^ » w. p. 1.

2

Analogously, we have

I(n^p I(^2) I(ili2)p I(i1 »2) I(ili2)p I(ilia) when p V ^ w p 1

1 (02)T,t ^ 1 (02)T,t' J(11)T,t ^ J(11)T,t' 1 (20)T,t ^ J(20)T,t when p ^ 0 W. p. 1,

where

T s T s

= /(* - «)2 / df^'dfi'21, /(2$, = / /(« - t}2df<'l)df<'2),

t t t t

T s

I'iiw = /(t - »)/(* - T)dfT':i)dfs(!2),

tt

i1,i2 = 1,... ,m. This result is based on the expansions of stochastic integrals

/(o2)2r,t' 1(20)2^, 1(1i)T,t (see the formulas (527H529) in Chapter 5). Let us denote

T

¡^t = A« - s)'dfi*'1,

where l = 0,1, 2 ...

The expansions (I5.7l)-(I59), (15.30), (15.38) (see Chapter 5) for stochastic integrals ¡((0;T t, ¡(i)r ¡(2)T t, ¡(3)T t, ¡(or t are correct w. p. 1 (they include 1, 2, 3, 4, and l + 1 members of expansion, correspondently).

1.7.2 Convergence with Probability 1 of Expansions of Iterated Ito Stochastic Integrals of Multiplicity k (k G N)

In this section, we formulate and prove the theorem on convergence with probability 1 (w. p. 1) of expansions of iterated Ito stochastic integrals in Theorem 1.1 for the case of multiplicity k (k G N). This section is written on the base of Sect. 1.7.2 from [14]-[16] as well as on Sect. 6 from [30] and Sect. 9 from [28].

Let us remind the well known fact from the mathematical analysis, which is connected to existence of iterated limits.

Proposition 1.1. Let |xn , m

}rm=1 be

a double sequence and let there exists

the limit

lim xn m = a < 00.

n, m—>00 '

Moreover, let there exist the limits

lim xn,m < 0 for any m, lim xn,m < 0 for any n.

Then there exist the iterated limits

and moreover,

lim lim xn,m, lim lim xn,m

n m m n

lim lim xn,m = lim lim xn,m = a.

n m m n

Theorem 1.10 [14]-[16], [26], [28], [30], [31]. Let ^(t) (l = 1,...,k)

are continuously differentiable nonrandom functions on the interval [t,T] and (x)}0=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then

J[^(k)]Trp ^ J[^(k)]T,t if p ^ o

w. p. 1, where J[^(k)]y't''p is the expression on the right-hand side of (11.10) before passing to the limit l.i.m. for the case p1 = ... = pk = p, i.e. (see

Pl,'",Pk

Theorem 1.1)

p p / k J №<k>]--p = £ ...£CW1 n cT

ji=0 jk=0 \Z=1

l.i.m. ^ j (Tii^w^... j (Tifc)AwTifckM , (ii,''',ifc)eGk 1 k J

/(ii) YjiV liJ^-"^

(1i,'",1fc)GGk

where i1,..., ik = 1,..., m, rest notations are the same as in Theorem 1.1. Proof. Let us consider the Parseval equality

pi Pk

j k^...,ik)dt1...dtk=pi,i^E... E Ck''ji, (1.183)

[t,T ]

where

k ji=0 jk=0

(t1)...^k(tk), t1 <...<tk k k-1 K (t1,...,tk ) = { = J] ^ (ti )]! 1{t<t+i},

0, otherwise 1=1 1=1

(1.184)

where t1,... ,tk E [t,T] for k > 2 and K(t1) = ^1(t1) for t1 E [t,T], 1A denotes the indicator of the set A,

k

C =

J K(t1,..., tk) j (ti)dt1... dtk (1.185)

= I K(t1,---,tk^(t)dt1

[t,T ]k l=1

is the Fourier coefficient. Using (1.184), we obtain

T t2

Cjk...>1 = j > (tk)^k (tk) - - ^ > (t1^1 (t1)dt1 - - -dtk -t t

Further, we denote

P1 Pk œ

72 def

lim V ,, Vc2 ■ =f V C2 .,

Pk -œ v v >k'"> 1 v >fc-->1

>1=0 jk =0 >1,'",>k =0

If p1 = ... = pk = p, then we also write

p p c

lim Y^ ... y^ C2 . = V^ C2 . .

p / v / v 1 / v •••j1

j1 =0 jk =0 j1,-,jk=0

From the other hand, for iterated limits we write

p1 pk » »

p1-CCo . . . pi—Co E . . . E Cjk •j1 E . . . E Cjfc .j1, j1=0 jk =0 j1=0 jk=0

p1 pk » » lim lim V .. .V Cj 31 d=f V V Cj j1

p1 -c p2 ,...,pk -c k 1 k 1

j1 =0 jk =0 j1=0 j2 ,...,jk =0

and so on.

Let us consider the following lemma.

Lemma 1.9. The following equalities are fulfilled

œ œ œ

y2

>1 / v ' ' ' / v ""'jk'..>1

>1,''',jk =0 >1=0 jk=0

œ œ œ œ

V^ V^ /^2

E-Ejj = £ ■ ■.£CU (1.186)

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jk =0 j1=0 J91 =0 jqk =0

for any permutation (q1,..., qk) such that {q1,..., qk} = {1,..., k}.

Proof. Let us consider the value

p p

£•..£ Cj' ,ji (1.187)

J'„ =0 jqk =0

for any permutation (q,..., qk), where I = 1, 2,..., k, {q1,..., qk} = {1,..., k}.

Obviously, the expression (1.187) defines the non-decreasing sequence with respect to p. Moreover,

p p p p p

r C2

ji —

.. . Cjk'''ji — ^^ .. . Cjk'''j —

jq =0 j9k =0 jqi =0 j92 =0 j?k =0

OO

— E Ck • ' ji < o

ji, ' ' ' ,jk=0

Then the following limit

jk ji jk ji

jH =0 j9k =0 jn , ' ' ' 'j9k =0

exists.

Let p/,... ,pk simultaneously tend to infinity. Then g,r ^ o, where g = min{p/,... ,pk} and r = max{p/,... ,pk}. Moreover,

g g pi pk r r

.. .53 Cjk'' ji — 53.. .53 Cjk'' ji — 53.. .53 Cjk ' ' ji.

jqi =0 j'?k =0 jqi =0 j9k =0 j'?i =0 j?k =

This means that the existence of the limit

pp

C 2

Jqi = 0 jqk =0

implies the existence of the limit

pi pk

2

pi^oo 53.. 53Cjk' ''ji (1.188)

pi, ' ^ E- E ^ ■ ' 'ji i1-189)

jqi =0 jqk =0

and equality of the limits (1.188) and (1.189).

o

Taking into account the above reasoning, we have

q p p p p

lim y^ y^ ... y^ c2 . = lim y^ .. Vc2 . =

p,q—» Z—/ Z—/ ' ' ' Z—/ jk...J'1 p—» Z—/ ' Z—/ jk•••j1

j9i =0 jqi+1 =0 j9k =0 j9i =0 j9k =0

pi pk

p;.Jjs—TO 53 ... 53 Cjk•••j1. (1.190)

jqi =0 j?k =0

Since the limit

oo

V^ C2

/ v jk."j1

=0

exists (see the Parseval equality (1.183)), then from Proposition 1.1 we have

œ œ q p p

V V C2 ■ = lim lim V y,, Vc2 =

>91 =0 >92 >"> =0 >91 =0 >92 =0 >9k =0

q p p œ

= E-ECU = E CLj1 - (1-191)

j91 =0 >92 =0 >9k =0 >1 vJk =0

Using (1.190) and Proposition 1.1, we get

œ œ q p p

E E <

jk...j1

>92 =0 >93 '."'>9k =0 >92 =0 >93 =0 >9k =0

q p p œ

2

"j1

= Jpm 1= 1= - - -1= CU = E Ck > - f1'192)

>92 =0 >93 =0 >9k =0 >92 '."'>9k =0

Combining (1.192) and (1.191), we obtain

c c c c

C2 = C2 .

/ v / v / v jk•••j1 / v jk•••jr

jq1 =0 jq2 =0 jq3 vj =0 j1 v,jk =0

Repeating the above steps, we complete the proof of Lemma 1.9. Further, let us show that for s = 1, . . . , k

œ œ œ œ œ

2

.. j 1

>1=0 js-1=0 js=p+1 js+1=0 jk=0

53- - - 53 53 53- - - 53 Cjk .>1

œ œ œ œ œ

y2

= E E-EE-E^.. *■ (1.193)

js=P+1 js-i=0 ji=0 js+i=0 j=0

Using the arguments which we used in the proof of Lemma 1.9, we have

n n p n n

nlim 53 ' ' ' 53 53 53 ' ' '53 Cj •• ji

C 2

ji =0 js-i =0 js=0 js+i=0 j=0

p o p o o

= E E Cjk ' , j, = -EC? , ' 'ji (1.194)

js=0 ji , ' ' ' js-i js + i, ' ' j =0 js =0 jqi =0 jqk-i =0

for any permutation (q1,..., qk-1) such that {q1,..., qk-1} = {1,..., s - 1, s + 1,..., k}, where p is a fixed natural number.

Obviously, we obtain

p o o o p o

53 53.. .53 Cjk '' ji = 53.. .53.. .53 Cjk '' ji =... =

js=0 jqi =0 jqk-i =0 jqi =0 js=0 jqk-i =0

o o p

= E ^ E E^• • j■ (1.195)

jqi =0 jqk-i =0 js=0

Using (1.194), (1.195) and Lemma 1.9, we get

53... 53 53 53.. .53 Cjk'' ji = 53... 53 53 53.. .53 Cjk '' ji

œ œ p œ œ

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■2

•ji

ji=0 js-i=0 js =0 js+i =0 j =0

53--- 53 53 53 •••53Ci • • j

œ œ œ œ œ p œ œ œ œ

22

■ ji

js=0 js-i=0 ji=0 js+i=0 jfc=0 js=0 js-i=0 ji=0 js+i=0 j=0

53 53 ■ ■ ■ 53 53 ■ ■ ■ 53 Cj • • ji 53 53 ■ ■ ■ 53 53 ■ ■ ■ 53 Cj • • j

œ œ œ œ

2

■ ji'

js=P+1 js-i=0 ji=0 js+i=0 j=0

53 53 ■ ■. 53 ■ ■ ■ 53 • • ji

So, the equality (1.193) is proved.

Using the Parseval equality and Lemma 1.9, we obtain

/p p

k 2(t!,..., tk )dti... dtk - ^... ^ c? ...j

_A _n

[i,TP ji=0 jk=0

TO p p

2

ji / V • • • / V "jk...ji

ji,...,jk=0 ji=0 jk=0

...ji 53.. .53 ...J

toto p p

v Y^ c 2 - V^ Y^ C 2

/ v • • • / v jk...ji / v • • • / v jk...ji

ji =0 jk =0 ji=0 jk =0

p TOTO tototo pp

53 53.. .53 Cjk...ji + 53 53.. .53 Cjk...ji— 53.. .53 Cjk...ji

ji=0 j2=0 jk=0 ji=p+1 j2=0 jk=0 ji =0 jk =0

pp TO TO p TOTO TO

2

ji

ji=0 j2=0 j3=0 jk =0 ji =0 j2=p+1 j3=0 jk=0

EEE-ECU+ E E E-E +

TO TO TO p p

^ _ y^ y^ c 2

ji — jk ...ji

ji=p+1 j2 =0 jk =0 ji=0 jk =0

+ ^ v 53.. .53 Cjk...ji 53.. .53 Cjk...j

TO TO TO p TO TO TO

22

ji

ji=p+1 j2=0 jk=0 ji=0 j2 =p+1 j2 =0 jk =0

53 53..^53Cjk...ji+ E E 53...53Cjk...ji+

p p TO TO TO p p TO

1 C2

ji —

+EE E E-ECU +...+E... E E CLj —

ji=0 j2=0 j3 =p+1 j4 =0 jk =0 ji=0 jk-i =0 jk =p+1

TO TO TO TO TO TO TO

— 53 E-E^i...ji+ E 53 E...ECi...ji+

ji =p+1 j2 =0 jk =0 ji=0 j2=p+1 j2=0 jk=0

TO TO TO TO TO TO TO TO

2

.j i

ji =0 j2=0 j3=p+1 j4=0 jk=0 ji =0 jk-i=0 jk =p+1

+EE E E-ECU + .■■ + £..■ E E CL

k /to tototo to

2

E E- E E E -ECU). (1.196)

s=1 \ji =0 js-i =0 js=p+1 js+i=0 jk =0

Note that we use the following

p p TO TO TO

53 ■ ■ ■ 53 53 53 ■ ■ ■ 53 —

ji=0 js-i=0 js=p+1 js+i=0 jfc=0

mi ms_i to to to

2

•ji —

— E-EE E-ECLj —

ji=0 js-i=0 js=p+1 js+i=0 j'fc=0

mi ms-i to to to

'C 2

ji

— mii^TO 53 ■ ■ ■ 53 53 53 ■ ■ ■ 53Cjk . i

ji=0 js-i=0 js=p+1 js+i=0 jk =0

mi ms-2 to to to to

— E ■ ■' ^3 53 53 ■ ■ ■ 53 Cjfc • ji—

ji=0 js-2=0 js-i=0 js=p+1 js+i=0 j'fc=0

— ... —

TO TO TO TO TO

— 53 ■ ■ ■ 53 53 53 ■ ■ ■ 53 Cjk . j i

ji =0 js-i=0 js=p+1 js+i =0 j'fc=0

to derive (I!. 196), where m1,..., ms-1 > p. Denote

T ¿2

Cjs...ji j (ts)$S(ts) . J j (t1 ^1(t1)dt1

t t

where s — 1,..., k — 1.

Let us remind the Dini Theorem, which we will use further.

Theorem (Dini). Let the functional sequence un(x) be non-decreasing at each point of the interval [a,b]. In addition, all the functions un(x) of this sequence and the limit function u(x) are continuous on the interval [a, b]. Then the convergence un(x) to u(x) is uniform on the interval [a, b].

For s < k due to the Parseval equality and Dini Theorem as well as (1.193) we obtain

TO TO TO TO TO

2

■.ji

ji=0 js-i=0 js=p+1 js+i=0 j'fc =0

53 ■ ■ ■ 53 53 53 ■ ■ ■ 53 Cjfc . . ji

TO TO TO TO TO

2

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.ji

js=p+1 js-i=0 ji=0 js+i=0 j'fc =0

E E •••E E •••ECi .j

T

oo oo oo oo oo

E E-EE-E M(t*)(Cjk-i .ji(tk))2dtt.

js =p+1 js-i=0 ji=0 js+i=0 jk-i=0

T

TO TO TO TO TO „ TO

E E E ". E Mt*) E j . ji(t*))2dt*

__i 1 _A _A _A ' _A J A _A

-i.

js=p+1 js-i=0 ji=0 js + i=0 jk-2 =0 *t jk-i=0

Tt

CO CO CO CO CO k

E E ."E E ••■ E h*(tk) ^2—x(t*—1)(Cjk_2 . ..ji0)2x

____i 1 ; _n ; _n ; _n __n

js=p+1 js-i=0 ji=0 js+i =0 jk-2=0 't t

xdt*_ 1dt* —

T

TO TO TO TO TO

— C E E -E E " E / (Cjk-2..j(T))2dT —

__i 1 _A _A _A ~ _A J

-2-..J! (T 1 ^

js =p+1 js-i=0 ji=0 js + i=0 jk-2=0 t

T

TO TO TO TO TO „ TO

^ E-E E-^ E (Cjk-2..ji(T))2dT

-2-Ji ( T ' '2

js =p+1 js-i=0 ji=0 js + i=0 jk-3=0 t jk-2 =0

T T

TO TO TO TO TO c c

C E E E I h>2-2(9) (Cjk-3...j1 (902d9dT —

js=p+1 js-i=0 ji=0 js+i=0 jk-3 =0 t t

T

TO TO TO TO TO „

— * E E E " E J (Cjk-,.ji(T))2dT —

js=p+1 js-i =0 ji=0 js+i =0 jk-3=0 t

—... —

T

TO TO TO

— C* E E^E/ (Cjs . . ji(T))2dT

js=p+1 js-i=0 ji=0 t

T

TO TO TO „ TO

TO TO TO TO

C* E E(CjW.(T))2dT, (1.197)

js=p+1 js-i=0 j2=0 t ji=0

where constants C, * depend on T — t and constant C* depends on k and T — t.

Let us explane more precisely how we obtain (1.197). For any function g(s) E L2([t,T]) we have the following Parseval equality

t \ - / T

00 / „ \ OO '

œœ

50 ( J ^j (s)g(s)ds ) = 50 ( J 1{s<t}^j (s)g(s)ds 1 =

T t

2 2 2

(1{s<t}) g2(s)ds = g2(s)ds- (1.198)

is

tt

The equality (1.198) has been applied repeatedly when we obtaining (1.197). Using the replacement of integration order in Riemann integrals, we have

T t2

Cjs."j1 (t) = j (ts)^s(ts) --- j (t1 ^1(t1)dt1 - - - dts =

= y j (t1)^1(t1 )j j (t2)^2(t2 ) ...J j (ts)^s(ts )dts ...dt2 dt1 =

t ¿1 t s — 1

= <5j,...j1 (t ).

For l = 1,..., s we will use the following notation

Cjs...ji M) =

T T T

= J j (ti M(ti) ^ j (t/+1)^/+1(t/+1).. .J j (ts)^s(ts)dts.. .dt/+1dt/.

0 tl ts—1

Using the Parseval equality and Dini Theorem, from (1.197) we obtain

» » » » »

... 53 53 53.. .53 Cjk . —

j1=0 js—1=0 js=p+1 js+1=0 jk=0

T

oo oo oo „ oo

œ œ œ œ

^Ck E E --E/E(c>...>(T))2dT

js =p+1 >s-1=0 >2=0 t >1=0

T

œ œ œ „ œ 2

ck E E ---E/Ej(t0 dT

js=p+1 js-1 =0 >2=0 t >1=0

2

TT

to to to „ T 2

js=p+1 js-i=0 j2=0

C* E Y ^Y / ^2(t1) (<j...jM1)) dt1dr — (1.199)

tt

T t

to to to „ „ to 2

— C* E E •Yj /^(t^E {pjj(T,t1)) dt1 dT — (1.200)

js=p+1 js-i =0 j3=0 t t j2=0

T t t

to to to „ „ „ 2

— C* E Y •••Y J J ^2(t1^ (t2) (^.j (T,t2^ dt2dt1 dT —

js=p+1 js-1=0 j3=0 t t ti

T T T

to to to „ „ „ 2

— C* E Y -Y / ^2(t1 ) / ^2(t2) ((7j-s..jM2)) dt2dt1dT —

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js=p+1 js-i=0 j3=0 { { {

T t

to to to „ „ 2

— C* E Y -Y /^2(t2) (Csj(T,t2^ dt2dT —

js =p+1 js-i=0 j3 =0 { {

—... —

TO T T TO2

— C* E J y^2—1(ts—1) (<j (T,ts —dts—1 dT — js =p+1 t t

to T T / T \2

— C* E / / / j(Ws(9)d9 dudT, (1.201)

js=p+1 t t \ U /

where constants C*, C*', C* depend on k and T — t.

Let us explane more precisely how we obtain (1.201). For any function g(s) E L2([t,T]) we have the following Parseval equality

2 T 2

00 1 T \ 00

E / ^j'(s)g(s)ds — E / (s)g(s)ds | —

j=0 V e J j=0 \ 1

T T

2 2 2

;i{,<s<T}) g2(s)ds — J g2(s)ds. (1.202)

e

The equality (1.202) has been applied repeatedly when we obtaining (1.201).

Let us explane more precisely the passing from (1.199) to (1.200) (the same steps have been used when we derive (1.201)).

We have

T t

/(* »» 2 / ^2 (t1)E (pjaj (r,t1^ dt^T -

4 (61) lCjs t t >2=0 T T

E II ^2(t1 ) (Csj(T,t1))2 dt1 dT =

j2=0 t t T T

E (Cj-.j (T,t1^ dt1dT

tt

>2=n+1

N-1 Tj œ 2

N^E / ^2(t1^ E >>(Tjdt1Ar>, (1.203)

_A __I 1

>=0 t >2=n+1

where {t>}N=0 is a partition of the interval [t,T] satisfying the condition (1.9).

> J>

Since the non-decreasing functional sequence un(Tj, t1) and its limit function u(Tj,t1) are continuous on the interval [t,Tj] C [t,T] with respect to t1, where

n 2

un(Tj ,t1)^E ((^js•••j2 (Tj ,t1^ , j2=0

Tj

» 2 r 2

U(Tj ^0 = ^ ((^js•••j2 (Tj = (t2) (Cjsj (Tj ,t2^ d^

j2=0 t1

then by Dini Theorem we have the uniform convergence of un (Tj, t1) to u(Tj, t1) at the interval [t,Tj] C [t,T] with respect to t1. As a result, we obtain

» 2

E (Csj(Tj,t1)) <e, t1 E [t,Tj] (1.204)

j2 =n+1

for n > N(e) E N (N(e) exists for any £ > 0 and it does not depend on t1). From (1.203) and (1.204) we obtain

N-1 ^ » 2

N—CoE I ^1) E (Cjsj(Tj^1)) dt1ATj —

j=0 t j2=n+1

N — i Tj T t

.2/, A___ I I „/,21

- £ Niin E J ^2(ti)diiATj = sj J ^2(ti)dtidr. (1.205)

j=0 t t t From (1.205) we get

T t

(6D / .

lim / / ^2(ti) E (C-jMi)) dtidr = 0.

nu J

t i j2=n+i

This fact completes the proof of passing from (1.199) to (1.200). Let us estimate the integral

t

J j (0)^s(0)d0 (1.206)

u

from (1.201) for the cases when (s)}1=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]).

Note that the estimates for the integral

j t

(0)^(0)d0, j > p + 1, (1.207)

where ^(0) is a continuously differentiable function on the interval [t, T], have been obtained in [6]-[16], [21], [32] (also see Sect. 2.2.5).

Let us estimate the integral (1.206) using the approach from [21], [32 . First, consider the case of Legendre polynomials. Then (s) is defined as

follows

where Pj (x) (j = 0,1, 2 ...) is the Legendre polynomial. Further, we have

I 0,(0).0(0)d0=XO^I±I I Pj{y)iKu{y))dy

v z (v)

JT — f

■ ' (pJ+i(z(x)) - Pj.Mxmw - (pj+Mi o) - p3-i{z{vm{v)-

2VWTT v

z(x)

T f [ ((Pj+i(y) - Pj-iiyWuiyVdy I, (1.208)

2

z (v)

where G (t,T), j > p + 1, u(y) and z(x) are defined by the following relations

, , T -1 T +1 . . ( T +1 \ 2

'«(y) = —-—y H---—, z{x) = [ x —

2 ' w V 2 / T -1'

is a derivative of the function ^(0) with respect to the variable u(y).

Note that in (1.208) we used the following well known property of the Legendre polynomials

- d~^r(x) = (2j + 1)Pj{x)' = 2' • ■ ■

From (1.208) and the well known estimate for the Legendre polynomials 106] (also see [109])

K

]PM<VJ+Ki-v*)'/" »e(-1-1»'

where constant K does not depend on y and j, it follows that

x

(h mme

v

< j [ (1 " (z(x)Y)^ + (1 - (z(v)T-)^ + Cl 1 • (1'209)

where j G N, z(x),z(v) G (—1,1), G (t,T) and constants C, C1 do not depend on j.

From (1.209) we obtain

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2

where j G N, constants C2, C3 do not depend on j.

x

Let us apply (1.210) for the estimate of the right-hand side of (1.201). We

have

<

K

T t

t t u

i

j(0)^s(0)d0 dudr -

i x

i

dy

+

dy

j \Ji (1 — y2)i/2 —i _—i

K3 j2

(1 _ y2)i/2

dx + KJ -

(1.211)

where js G N, constants Ki, K2, K3 are independent of js.

Now consider the trigonometric case. The complete orthonormal system of trigonometric functions in the space L2([t,T]) has the following form

(0) =

1,

j=0

y/T^t.

\/2sm (2tit(0 - t)/{T -t)), j = 2r - 1, (1.212) \/2cos (27tt(0 — t)/(T — t)), j = 2r

where r = 1, 2,...

Using the system of functions (1.212), we have

02r-i(W(0)d0 =

2

T-t

sin T _ ifj(Q)dQ

'T _ t 11,, , 2nr(x _ t) ,, , 2nr(v _ t) ■ip(x) cos——--'ip(V)COS-

2 nr

Tt

- f cos2-^iIlmde

(1.213)

02r (0)^(0)d0 =

I 2 i 2vr(0-t) t(û,ÂÛ T~t. / cos T _ f '0(A)d0 =

2

1

x

IT -t if,,,. 2?rr(x -t) ,, , . 2ttr(v - t) 1 ■ip(x) sin——--■ip(v) sin-

2 nr \ r v y T -1 r w T -1

- isin2wrT{0_tt)^(e)d0\, (1.214)

where (0) is a derivative of the function ^(0) with respect to the variable 0. Combining (1.213) and (1.214), we obtain for the trigonometric case

x \ 2

(1-215)

where j G N, constant C4 is independent of j. From (1.215) we finally have

T

<f>ja(6)^s(0)d9 I dudr<^, (1.216)

js

t t "

where js G N, constant K4 is independent of js.

Combining (1.201), (1.211), and (1.216), we obtain

TO TO TO TO TO

E... E E E ... E Cjk -ji -

ji=0 js-i=0 js=p+1 js+i=0 jfc=0

CO TO

^ E = y (1-217)

js=p+1 js p X P

where constant Lk depends on k and T — t.

Obviously, the case s = k can be considered absolutely analogously to the

case s < k. Then from (1.196) and (1.217) we obtain

p p ^

/ ^ ^ G

...,^...'//, E---Ecl*^y. t1-218)

[t,T]k ji=0 jk=0

where constant Gk depends on k and T — t.

2

T

For the further consideration we will use the estimate (1.73). Using (1.218) and the estimate (1.73) for the case pi = ... = pk = p and n = 2, we obtain

m 4 J

T,t

- J

T,t I > ^

< C2,k

p p

J K2(t1,...,tk)dt1 ...dt*-E-ECU

<

\i,T

,1=0 , =0

/

H

2,k

< , p2

where

= (k!)2n(n(2n - 1))n(k-1)(2n - 1)!!

(1.219)

and = GkC2,k.

Let a and in Lemma 1.8 be chosen as follows

a = 4, 4 =

J

T,t

J

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p,...,p T,t

From (1.219) we obtain

E m{ (j

p=i i v

T,t

J

\41 o 1

oo. (1.220)

Using Lemma 1.8 and the estimate (1.220), we have

TJ

T,t if p ^ oo

w. p. 1, where (see Theorem 1.1)

pp

J [Ä-*

E-E^ .,1 n z

(ii)

,1=0 ,fc=0

J=1

l.i.m. E , (Til )AwT;;) (Tik )AwT;kM (1.221)

(i;.....ik № 1 k )

4

2

p

k

or (see Theorem 1.2)

p p / k [k/2] j [#>]-p = £ ...Ecj,.^ cf ^(-i)

ji=0 j=0 \Z=1 r=1

r k—2r

X

E II 1iiS2s-1 = »32« =0}1 j32s-1 = jS2s } il ^

({{31,32}>--->{32r-1>32r }}>{91>--->9fc-2r}) S = 1 1=1

{31,32>'">32r-1>32r>91>'">9fc-2r } = {1>2>'">fc}

(1.222)

where ..., ik = 1,..., m in (1.221) and (1.222). Theorem 1.10 is proved. Remark 1.6. From Theorem 1.4 and Lemma 1.9 we obtain

lim Tim ... Tim M ( (J[i/j{k)}Tt ~ J^ZT^)2 \ <

pg1 ^TO pq2 ^TO pqfc ^TO ' ' / J

/ pfc ^

< k! • lim ... lim

p91 ^0 pgfc ^TO

/n rk

k 2(t1,...,tk )dt1 ...dtk — ^ •••eck....

\[i,t]k j1=0 jk=0 )

= k!

^ „ TO TO ^

J K 2(ii.....tk № ...dik — I=CLj1

\[t,T ]K jq1 =0 =0 )

=0

for the following cases:

1. i1,...,ik = 1,...,m and 0 <T — t< to,

2. i1,..., ik = 0,1,..., m, i2 + ... + ik > 0, and 0 < T — t < 1.

At that, (q1,...,qk) is any permutation such that {q1,...,qk} = {1,...,k}, J[^(k>]r,t is the stochastic integral (11.5), J[^(k>]TT1t'"'pfc is the expression on the right-hand side of (11.101) before passing to the limit l.i.m. , lim means

p1,'.',pfc ^to

lim sup; another notations are the .same as in Theorem 1.1.

Remark 1.7. Taking into account Theorem 1.4 and the estimate (1.218), we obtain the following inequality

^ k\Pk{T t)

M | (/forfc - J[$[hX7P) j < ' , (1-223)

where i1,..., ik = 1,..., m and constant Pk depends only on k.

The estimate (1.223) can be written in a slightly different form. Let us consider this question in more detail.

By analogy with (1.126) we have

lim V^ Cj, Cj n = 0,

p^TO

(1.224)

j'ivjfc =o

where (mi,... ,mk) is any permutation of the set {1,..., k} such that (mk,..., m1) = (k,..., 1); braces mean an unordered set, and parentheses mean an ordered set.

Further, using (1.224) and the estimate (1.218), we obtain

TO Cjfc-jiC

ji vJfc =0

jm^ '"jmi

TO Cjfc-jiC j1,-'',jfc =0

i jmk '''j

m^'Jmi

P

TO Cjfc'"jiC

- 'Jmi

j'ivjfc =0

<

<' E — E

ji vJfc =0 ji,'-',jfc=0/

TO p

Cj ... j Cj ... j

-'Ji '''jmi

<

1

2

-

j'ivJfc =0 ji,'-',jfc =0^

TO p

C2 . + C2

E — E J cjk

jir''Jfc =0 jivjfc =0

ji

/p G

[t,T ]k

where constant Gk depends on k and T — t. Combining (1.75), (1.79), (1.218), and

j'ivJfc =0

, we get

m ^ J

T,t

- J

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^ M- P

k

(1.225)

where i1,..., ik = 1,..., m and constant Pk depends only on k.

Remark 1.8. The estimates (1.73) and (1.218) imply the following inequal-

ity

M^ J

T,t

- J

2n"

< (k\)2n(n(2n - 1 ))n{k~1){2n - 1)!! ^ , (1.226)

where ii,..., = 1,..., m, n G N, and constant Pk depends only on k.

1.7.3 Rate of Convergence with Probability 1 of Expansions of Iterated Itô Stochastic Integrals of Multiplicity k (k G N)

Consider the question on the rate of convergence w. p. 1 in Theorem 1.10. Using the inequality (1.226), we obtain

mJ

T,t

J

T,t

2n '

1/2n

<

Qn,k

(1.227)

where n G N and

Qn>Ä = fc! (n(2n - l))(*-D/2 ((2n - 1)!!)1/2" (^ ~~ t)k'2.

According to the Lyapunov inequality, we have

mJ

T,t

J

T,t

1/n

<

Q

(1.228)

for all n G N. Following [100] (Lemma 2.1), we get

J- J[^fc*

p

1/2-e

P

J

<

p1/2-e pGN

sup p

1/2-e

J

T,t

J

ne

p

1/2-e

(1.229)

n

1

w. p. 1, where

ne = sup ( p

peN

1/2-£

J

-J

T,t

and £ > 0 is fixed.

For q > 1/e, q e N we obtain (see (1.228)) [100]

m {|n£|q} = m i (sup ( p1/2-£

^peN

J

T,t

J

p,...,p

M{ sup I p(1/2-£)q peN

J

p,...,p T,t

<

oo

<

mp

(1/2-£)q

. p=1

J- J[^(k)fc*

oo

^p(1/2-£)^m p=1

J- Jty^fc*

<

p=1

p

oo.

p=1

p

(1.230)

From (1.229) we obtain that for all £ > 0 there exists a random variable % such that the inequality (1.229) is fulfilled w. p. 1 for all p £ N. Moreover, from the Lyapunov inequality and (1.230), we obtain m {||q} < ^ for all q > 1.

1.8 Modification of Theorem 1.1 for the Case of Integration Interval [t, s] (s e (t,T]) of Iterated Ito Stochastic Integrals

1.8.1 Formulation and Proof of Theorem 1.1 Modification

Suppose that every ^ (t) (l = 1,..., k) is a continuous nonrandom function on [t,T]. Define the following function on the hypercube [t,T]k

q

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KT(tl, . . . ,s) = 1{tfc<s}K(ti, . . . ),

(1.231)

where the function K(ti,... , tk) is defined by (1.6), s £ (t, T] (s is fixed), and 1A is the indicator of the set A. So, we have

A(ti,...,tk, s) = 1{t1<...<tfc <s} ^i(ti) ...^k (tk) =

^l(ti) ...^k (tk ), ti <...<tk <s

(1.232)

0,

otherwise

where k > 1, ti,... ,tk £ [t,T], and s £ (t,T].

Suppose that (x)}°=0 is a complete orthonormal system of functions in the space L2([t, T]).

The function i?(ti,... ,tk, s) defined by (1.232) is piecewise continuous in the hypercube [t, T]k. At this situation it is well known that the generalized multiple Fourier series of K?(ti,..., tk, s) £ L2([t, T]k) is converging to this function in the hypercube [t,T]k in the mean-square sense, i.e.

lim

Pi Pk

A(ti,... ,tk, s)

-£ ...£c-k ...ji wn ^ (ti )

ji=0 jk =0

1=1

L2([t,T ]k )

where

Cjk...ji (s) =

/k

KT(ti,...,tk (ti )dti ...dtk

i=i

is the Fourier coefficient, and

m

L2([t,T ]k)

( \i/2 J f 2(ti,...,tk )dti ...dtk

\[t,T ]k /

= 0, (1.233)

[t,T ]k

S t2

= J ^k (tk ) j (tk ) .J ^i (ti ) j (ti )dti ...dtk (1.234)

t t

Note that

k

t2

J [#fe j]s,t = j ^k (tk ) - J ^i(ti)dw(;i) ...dw^) = (1.235) t t T t2

= J 1{tfc<s|^k (tk) ..J W^dw^0... dw£k) w. p. 1, tt

where s G (t, T] (s is fixed), ii,..., ik = 0,1,..., m.

Consider the partition {Tj}N=0 of [t,T] such that

t = t0 < ... < = T, AN = max ATj ^ 0 if N ^ œ, ATj = Tj+i — Tj.

0<j<N—i

(1.236)

Theorem 1.11 [15], [16], [28]. Suppose that every ^(t) (l = 1,...,k) is a continuous nonrandom function on [t,T] and {0j(x)}°=0 is a complete orthonormal system of continuous functions in the space L2([t,T]). Then

Pi Pfc / k

S( i' )

ji=0 jfc=0 \/=i

jk>]s,t = u-m- £ •.. £ Cjk...ji(s) ncj,

l.i.m. £ j(Tii)Aw[:;) ...j(Tik)Aw(:fckM , (1.237)

^ (1i,...,1k)GGk i k J

where J[^(k)]s,t is defined by (11.235), s £ (t,T] (s is fixed),

Gk = Hk\Lk, Hk = {(/i,...,/k) : 1i,...,1k = 0, 1,...,N - 1}, Lk = {(1i,...,1k): li,...,1k = 0, 1,...,N-1; lg = lr (g = r); g,r = 1,...,k},

l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,... ,m,

T

j

j = h (T )dwT:)

are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjk...ji (s) is the Fourier coefficient (11.234), AwTj =

s

w— wj (i = 0, 1,..., m), {rj}N=0 is a partition of [t, T], which satisfies the condition (1.236).

Proof. Let us consider the multiple stochastic integrals (1.16), (1.23). We will write J[$]$ and /'[^ (s G (t, T], s is fixed) if the function $(tb ..., tk) in (11.161) and (11.231) is replaced by 1{i1)...)ifc<s}$(ti,... ,tk).

By analogy with (1.24), we have

T t2

/[*]<? = J..Ji{tfc<s^ ($(ti,...,tk)dwi;i)w.p.1, t t (t1,...,tfc)

(1.238)

where

£

(ti,...,tk)

means the sum with respect to all possible permutations (t1,..., tk). At the same time permutations (t1,... ,tk) when summing are performed in (1.238) only in the expression, which is enclosed in parentheses. Moreover, the nonrandom function $(t1,... ,tk) is assumed to be continuous in the corresponding closed domains of integration. The case when the nonrandom function $(t1,..., tk) is continuous in the open domains of integration and bounded at their boundaries is also possible.

Let us write (1.238) as

T t2

/ws?=/.../ £ (i{tfc<S}*(t1,...,tk)dwt:i)...dwt:fc)) w.p.1,

t t (ilv,ifc)

(1.239)

where permutations (ti,... ) when summing are performed in (1.239) only in the expression $(t1,..., tk)dwt(|l)... dw^.

It is not difficult to notice that (1.238), (1.239) can be rewritten in the form (see (1.25))

T t2

J = E J: J *(ti, •.., tk )1{tk<«}dwi;i)... dw«:k) w. p. 1, (1.240)

(t1,---,t&) t t

where permutations (t1,... ,tk) when summing are performed only in the values 1{tk<s}dwt(il)... dwt(:k). At the same time the indices near upper limits of

integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (ti,..., tk), then ir swapped with in the permutation (ii,..., ik).

According to Lemma 1.1, we have

N —i 12-i

J[^(k)]s,t = l.i.m. ^... £ iK<S}^i(rii)... ^(Tik)Aw(;;)... AwJ;;) =

¡*=0 ¡1=0

N-1 N- i

= E. ■ . E 1(T.i<«>K(T'i,.. ■ -T'k)AwT;i) ■ ■.Aw«;;;) = ¡*=0 ¡1=0 1 *

N—i

= lNi.m. E 1{T«k <«>K (Tii...., Tik )AwT;;)... Aw';;;) =

N^to * 1 *

;i,...,ifc=o

iq =ir; q,r=1,...,fc

T t2

j.J E (l{tk <s>K (ti,...,tk )dwt;i) ...dw^) w.p. 1, (1.241)

t t (i1,...,ifc)

where K(ti,..., tk) is defined by (1.6) and permutations (ti,..., tk) when summing are performed only in the expression K(ti,... ,tk)dwt( ;i)... dwt(;*).

According to Lemmas 1.1, 1.3 and (1.24), (1.25), (1.239)-(1.241), we get the following representation

Pi Pfc n n

E.. . E (su . -. / E (j(ti)... j(tk)dw<;i)... dw<;k)) +

A _A _A J J (-L -L \

J'.|s,t =

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T t2

j1=0 j'fc =0 t t (ti,...,tfc)

+ RT,t,s

Pi Pfc

E... E Cj*-ji(s)x ji=0 j*=0

N-1

x l.i.m. £ j (Tii)... j (Ti*)AwT;;) ... Awi;*) + Rit/*

N^to 1 *

ii,...,ifc=0

iq = ir; q=r; q,r = 1,...,fc

Pl Pk I N—1

l.i.m.

N

£ • • • £ Cjk---jl

(s) l.i.m. E j (Til) • • • j (Tik)AwT:;)... Aw£

ji=0 jk =o

l.i.m. £ j (Til)AwT:i)... j (Tik)Aw<:;M +

(il,---,ik )GGk

T?Pl,---,Pk

Pl Pk

E^E^---jl (s)x

jl=o jk=0

k

X 'II j) — l.i.m. £ j(Til^w^ • • • j (Tik)AwT:fck^ +

i=1 (il,---,ik )GGk

I DPli->Pk „ 1

where

RT,t,s -

T t2 / Pl Pk

E /•••/ U{tk<s}K (t1,...,tk) — £ ••^Cjk---jl (s)JI (ti) I X (tl,---,tk) t t V jl=0 jk=0 1=1

Xdwt(il) ...dwt(:k) -

T t2

r(:l) J,T,(:k)

£ J---JK (t1, • • • ,tk )1{(k <«}dwf(l" • •. dw«> — (1.242)

(t1,---,tk) t t

T t2 Pl Pk

//» ^l rk

/ E^E^-jl (s)II ^ (ti )dw(:l) ...dwí:') (1.243)

«/ _A • _A 7 1

(tl,---,tk) t t j'l=0 jk=0 1=1

k

w. p. 1, where permutations ) when summing in (1.242) are per-

formed only in the values 1{tk<s}dwt(| l} ...dwt(ifc). At the same time permutations (tl5... ,tk) when summing in (1.243) are performed only in the values dwt(|l)... dwt(ik). Moreover, the indices near upper limits of integration in the iterated stochastic integrals in (1.242), (1.243) are changed correspondently and if tr swapped with tq in the permutation (t1,..., tk), then ir swapped with in the permutation (i1,..., ik).

Let us estimate the remainder Rrt"s'Pk of the series. According to Lemma 1.2, we have

m (rtT/^ ^ <

T T ( P1 Pfc k \

< Ck £ ... K(ti,...,tk)1{ifc<s}^ ..^Cj-kj (s)H (ti) x (ti,...,tfc) t t \ j 1 =0 j=0 1=1 /

x dt1... dtk, (1.244)

where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral J[^(k)]s,t and permutations (t1,... , tk) when summing in (1.244!) are performed only in the values 1{tk<s} and dt1... dtk. At the same time the indices near upper limits of integration in the iterated integrals in (1.244) are changed correspondently.

Since K(t1,..., tk) = 0 if the condition t1 < ... < tk is not fulfilled, then

m <! (k ] j. <

2

T ^ / P1 Pk k

<Ck £ /.../ K(i1,...,ik)i{tk<-}-£..^Cjk...j 1 (s)n(tin x

(ti,...,tk) t t \ j 1=0 jk=0 1=1 /

x dt1... dtk, (1.245)

where permutations (t1,... ,tk) when summing in (1.245) are performed only in the values dt1... dtk. At the same time the indices near upper limits of integration in the iterated integrals in (1.245) are changed correspondently.

Then from (1.38), (1.233), and (1.245) we obtain

m< (RTT/ M % <

T r / P1 Pk k

<Ck £ ... K(t1,...,tk)i{ifc<S}^...£Ck...ji(s^(t,)i X (ii,...,ifc) t t \ ji=0 j=0 1=1

xdt1... dtk =

2

/. / Pi Pk k \

= Ck J Ufa ,...,tk ,s) ..^Cjk ...ji (s^ (ti ) dti ...dtk — 0

[tT]k V ji=0 jk=0 1=1 /

if p1,... ,pk —to, where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral J[^(k)]s,t. Theorem 1.11 is proved.

Remark 1.9. Obviously from Theorem 1.11 for the case s = T we obtain Theorem 1.1.

Remark 1.10. It is not difficult to see that Theorem 1.11 is valid for the case when {fy (x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fy(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7 for details).

From Theorem 1.11 for the case of pairwise different numbers i1,...,ik = 1,..., m we obtain

Pi Pk

Jw(k)]" = » ^. ■ ^ Cj'-ji(s)Zi:i).. ■ ijk'' (L246>

ji=0 jk=0

Note that the expression on the right-hand side of (1.246) coincides for the case k = 1, ^1(t1) = 1 with the right-hand side of the formula (6.2) (approximation of the increment of the Wiener process based on its series expansion).

Remark 1.11. Note that by analogy with the proof of estimate (1.218) we obtain the following inequality

P P ( \

Gk (s)

J ..., tk, s)dh ...<ll, Cl^(s) < (1.247)

[t,T]k ji=0 jk=0

where , s) and Cjk ...j 1 (s) are defined by the equalities (I! .2,31) and

(1.234), respectively; constant Gk(s) depends on k and s — t (s £ (t,T], s is fixed).

The following obvious modification of Theorem 1.4 takes place.

Theorem 1.12. Suppose that every ^(t) (l = 1,...,k) is a continuous nonrandom function on [t,T] and {fyj(x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fy (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the estimate

m ((j— J[^(k)]pr,pfcv! <

< k!

( p p i Pk ^

I K2(ti,..., tk, s)dti... dtk — ^ ... £ Cj ...ji (s) V[i,T]k j1=0 jk=0 J

(1.248)

is valid for the following cases:

1. i1,...,ik = 1,...,m and 0 <T — t< to,

2. i1,...,ik = 0,1,..., m, i2 + ... + ik > 0, and 0 <T — t< 1,

where J[^(k)]s,t is the stochastic integral (1.235), J[^(k)]p1t''"'Pfc is the expression on the right-hand side of (1.237) before passing to the limit

l.i.m. ,

PlvvPfc ^TO

^(t1,... ,tk, s) and Cj^"^ (s) are defined by the equalities (1.231) and (1.234), respectively; s £ (t,T] (s is fixed); another notations are the same as in Theorem 1.11.

Remark 1.12. Combining the estimates (1.247) and (1.248), we obtain

p

M - < V ' , (1-249)

where i1,..., ik = 1,..., m, constant Pk depends only on k; another notations are the same as in (1.247) and (1.248).

Remark 1.13. An analogue of the estimate (1.73) for the iterated Ito stochastic integral (1.235) has the following form,

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X

m{ J[^(k)W - J^(k)iïrpfc) \ <

< (k!)2n(n(2n - 1))n(k-1)(2n - 1)!! x

^ p Pi Pk

J kk2(ti,..., tk, s)dti... dtk - £... £ c2k...ji(s)

V[t,T]k j1=0 j=0 /

, (1.250)

where J[^(k)]P?it''"'Pk is the expression on the right-hand .side of (1.237) before passing to the limit

l.i.m. ,

Ki(t1,... ,tk, s) and Cjk'''j■1 (s) are defined by the equalities (11.2311) and (11.234), respectively; s £ (t, T] (s is fixed); i1,..., ik = 1,..., m.

Remark 1.14. The estimates (1.247) and (1.250) imply the following inequality

m | J[^(k)- J[^(k)]P;r;P) 2n[ <

< (k\)2n(n(2n - 1 )y<k-1\2n - 1)!! (Pfc)?? ^

p'

where il5..., ik = 1,..., m, n G N, and constant Pk depends only on k.

1.8.2 Expansions of Iterated Ito Stochastic Integrals with Multiplicities 1 to 5 and Miltiplicity k Based on Theorem 1.11

Consider particular cases of Theorem 1.11 for k = 1,..., 5

pi

J i^s;, = l.i.m. E Cji (sj, (1.251)

ji=0

Pi P2 /

Ji/2)]-',' = £ jM ( cfcf - 1{!i=,2=0}1{3i=j2} ). (1.252)

ji=0 j2=0

'

s,t

Pi P2 P:

= EEE cji wi jVW-

jl j2 j

ji=0 j2=0 j:=0

1{ii=i2=0}l{ji=j2}Cj:) - 1{i2=i:=0}1{j2=j:}C]1il) - 1{ii=i:=0}l{ji=j:}Cj22) I , (1-253)

Pi

P4

J (4)i*=Pi^E -E^ mi Z

(11)

ji=0 j4=0

J=1

1 1 /■ (1:V (14) 1 1 t (12^ (14)

1{ii=i2=0}1{ji=j2}Zj: j - 1{1i=1:=0} 1{ji=i:} zj2 Zj4

1 1 Ah) A1:) n 1 z^iMm)

-1{1i=14=0}1{ji=j4}Sj2 Zj: - 1{12=1:=0}1{j2=j:}Sji j -

1 1 Ah) A1:) ^ 1 A1i)A12)_1

-1{12=14=0}1{j2=j4}Zji j - 1{1:=14=0}1{j:=j4}Zji j + + 1{1i=12=0}1{ji=j2}1{1:=14=0}1{j:=j4} + 1{1i=1:=0}1{ji=j:}1{12=14=0}1{j2=j4} +

+ 1{1i=14=0}1{ji=j4}1{12=1:=0}1{j2=j:} (1'254)

J [^(5)]

Pi

P5

s,t = l.i.m.

E-.-E^i c

(11) j'i

ji=0 j5=0

+ 1{1i=12=0} + 1{1i=12=0} + 1{1i=1:=0} + 1{1i=14=0} + 1{1i=14=0}

{1i=12=0} {1i=14=0} {12=1:=0} {12=15=0}

{1:=15=0} {ji=j2} {ji=j2} {ji=j:} {ji=j4} {ji=j4}

Z (1:)Z (14) Z (15

{ji=j2}Zj: Zj4 Zj5

Z (12) z (1:)Z (15

{ ji=j4} Z j2 Zj: Zj5

Z (1i)Z (14) Z (15

{j2=i:}Zji Z j4 Zj5

Z (1i)z (1:)Z (14

{j2=j5} Zji Zj: Zj4

Z(1i)Z(12)Z(14 {j:=j5}Zji Zj2 Zj4

1 Z(15

{1:=14=0}1{j:=j4}v

{14 = 15=0}1 {j4=j5} Zj:

1 Z(14

{12 = 15=0} 1 {j2=j5} Z j'4

1 Z(15

{12=1:=0}1{j2=j:}Zj5

1 Z(12 {1:=15=0}1{j:=j5}Zj2

j5 (1:

+ + + + +

1

{1i=1:=0}

1{1i=15=0}

1{12=14=0}

1

{1:=14=0}

{14=15=0}

1

{1i=12=0}

1

{1i=1:=0}

1

{1i=1:=0}

1

{1i=14=0}

1

{1i=15=0}

\/=1

Z(12 Z(14) Z(15)_ Zj4 Zj5

Z(12 =j5}Zj2 Z (1:)z (14)_ Zj: Zj4

{j2 = ■ iZ(1i =j4}Zji Z(1:)Z(15) Zj: Zj5

{j:= ■ iZ(1i =j4}Zji Z(12)Z(15) Zj2 Zj5

Z(1i =j5} Zji Z(12)z(1:) 1 Zj2 Zj: +

{ji=j2}1{1:=15=0} {ji=j:}1 {12=14=0} {ji=j:}1{14=15=0} {ji=j4}1{12 = 15=0} {ji=j5}1{12=1:=0}

{i:=j5}Zj4 +

Z (15) 1

{ j2 = j4 } Z j'5 +

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{i4=j5}Cj;1 ) +

Z(1:) I {j2=j5}Zj: +

{j2=i:}Cj.1 +

4

5

+1 {i 1 = i5 =0} 1 {ji =j5 } 1 { i2 =i4=0} 1 {j2 =j4 } Cjs'3 ) + 1{i1=i5=0}l{ji=j5}1{i3=i4=0}l{j3=j4}C]2i2) + + 1{i2=i3=0}l{j2=j3}1{i4 = i5=0}l{j4=j5}C] ^ + 1{i2 = i4=0} 1{j2=j4} 1{i3 = i5=0} 1{j3=j5}Zj ^^

+ 1{i2=i5=0}1{j2=j5}1{i3 = i4=0}1{j3=j4}Cj ^

where 1A is the indicator of the set A, C^.j (s) (k = 1,..., 5) has the form (O34), s e (t,T] (s is fixed).

Consider a generalization of the above formulas

Pi Pk / k [k/2]

J[V>(% = l.i.m ^ ■■^Cjk..J-i(s) I]Cji'> + £(-1)"x

r k—2r

X E n 1{ig2s-i = ig2s =°}1{jg2s-i = jg2s } n C

2s-1 a2s ' '-"^s —1 "^28-

({{31>32}v>{32r-1>32r }},{91v>9fc-2r}) S = 1 1=1

{31,32v>32r-1>32r>91v>9fc-2r } = {1>2v>fc}

where k e N, C^.j (s) has the form (11.2341); another notations are the same as in Theorem 1.2.

1.9 Expansion of Multiple Wiener Stochastic Integral Based on Generalized Multiple Fourier Series

Let us consider the multiple stochastic integral (1.23)

N-1 k

l.i.m. £ * (Tj1,... ,rjfc) J] Aw j =f J'[*]$, (1.255)

N^^ • " n "" ft *

J1>->Jfc = 0 1=1

jq =jr ; q=r; q,r=1,...,fc

where for simplicity we assume that *(ti,...,tk ) : [t,T]k ^ R1 is a continuous

} }j=0

nonrandom function on [t,T]k. Moreover, {t}}N _0 is a partition of [t, T], which

satisfies the condition (1.9).

The stochastic integral with respect to the scalar standard Wiener process (i1 = ... = ik = 0) and similar to (1.255) was considered in [101] and is called the multiple Wiener stochastic integral [101]. Note that *(t1,... ,tk) G

L2([t,T]k) in [101] (this case will be considered in Sect. 1.11, 1.12).

Consider the following theorem on expansion of the multiple Wiener stochastic integral (1.255) based on generalized multiple Fourier series.

Theorem 1.13.8 Suppose that $(t1? ..., tk) : [t, T]k ^ R1 is a continuous nonrandom function on [t, T]k and {fy (x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fy (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the following expansions

Pi Pk / k

J'[< = E-Ec.incf

ji=0 jfc=0 M=1

l.i.m. £ j (Til )Aw(;;) ...j (Tik )Aw(;fckM, (1.256) (ii,...,ik)GGk 1 k )

Pi Pk / k [k/2]

j= E.-En1 ) + E(-1)

Pi,...,Pk^œ

ji=0 jk =0 M=1 r=1

r k—2r

X

E n 1{ig2s-i = ig2s =0}1{jg2s-i = j»2. ^ )

({{Si,S2}>'">{S2r-1>S2r }}>{91>--->9fc-2r}) S=1 1 = 1

{31,32v>32r-1>32r>91>--->9fc-2r }={1>2>--->fc}

(1.257)

converging in the mean-square sense are valid,, where

Gk = Hk\Lk, Hk = {(li,...,lk): li,...,lk = 0, 1,...,N - 1},

Lk = {(li,..., Ik): li,..., Ik = 0, 1,...,N -1; lg = /r (g = r); g,r = 1,...,k},

l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,... ,m,

T

j = | & (s)dw<!)

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0),

k

C —

J $(ti,..., tk) fyji(ti)dti... dtk (1.258)

[t,T]k 1=1

8 Theorem 1.13 will be generalized to the case of an arbitrary complete orthonormal system of functions

(x)}~0 in the sPace L2([t,T]) and $(ti,..., tk) G L2([t,T]k) (see Sect. 1.11, Theorem 1.17).

is the Fourier coefficient, Awj = wj+1 — wj (i = 0, 1,... ,m), {t,}N=0 is a partition of [t,T], which satisfies the condition (1.9); [x] is an integer part of a real number x; another notations are the same as in Theorem 1.2.

Proof. Using Lemma 1.3 and (1.24), (1.25), we get the following representation

J =

P1 Pk T t2

l k f f £... £ Cjk...}1 /... / £ (0,1 (t1)... } (tk)dw<;1)... dwii'> I +

_A _A J J {-L -L \

j1=0 ,k=0 t t (t1,...,tk )

«T,t

P1 Pk N—1

£ ... £ Cjk...}1 l.i.m. £ 0,1 (T11 )... }(Tik)Aw(;;)... AwTik) +

—' —' N^œ— 1 k

,1=0 jk =0 i1,...,ik=n

lq =lr ; q=r; q,r=1,...,k

+RT1t""Pk

P1 Pk i N—1

£ ... £ Cjk...}1 ( l.i.m. £ 0,1 (Ti1 )... 0,k(Tik)Aw[;;)... AwTk) — ,1=0 jk=0

l.i.m. £ 0,1 (Ti1 )Aw(;;)... 0,k (Tik )Aw(;fck M +

( Î1,...,tk )GGk

+ Ri?1,...,Pk

LT,t

P1 Pk

= £... £ Cjk...}1x

,1=0 jk=0

k

x "II j ) — l.i.m. £ 0,1 (Ti1 )Aw[;;)... 0,k(Tik)Aw[;k) ) +

1=1 (l1,...,1k )GGk

where

t t2

+RT1t""Pfc w. p. 1,

T 7 / P1 k \

RT^ = £ /•••/ i^Cti.....tk)-£ ^ (ti) x

, i V ji=o jfc=o 1=1 /

(ti,...,tfc) t t \ ji=o jk=o

xdwt(;i} ...dwt(ifc),

where permutations (t1,..., tk) when summing are performed only in the values dwt(il)... dwt(ik). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,. • •, tk), then ir swapped with in the permutation (i1,..., ik).

Let us estimate the remainder R^t^ of the series using Lemma 1.2 and (1.38). We have '

m^ (RTT1t % <

T ^ / Pi Pk k \

< Ck £ ••• U(i1,...,ik) -£ ..^Cjk ...j^n (tin x (ti,...,tk) t t \ ji=o jk=o 1=1 /

xdt1... dtk =

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// Pi Pk k \

U(t1,...,tk) -£• ••£cj-k...j^n(ti)

[t,T]k V ji=o jk=o 1=1 /

x

xdt1 • • • dtk ^ 0

2

if • • • ,pk ^ oo, where constant Ck depends only on the multiplicity k of the multiple Wiener stochastic integral J'^]!^. The expansion (11.256) is proved. Using (1.256) and Remark 1.2, we get the expansion (1.257) (see Theorem 1.2). Theorem 1.13 is proved.

Note that particular cases of the expansion (1.257) are determined by the equalities (11.45I)-(ITT5T1). in which the Fourier coefficient Cjk ...ji (k = 1,..., 7) has the form (OLE).

1.10 Reformulation of Theorems 1.1, 1.2, and 1.13 Using Hermite Polynomials

In [102] it was noted that Theorem 3.1 ([101], p. 162) can be applied to the case of multiple Wiener stochastic integral with respect to components of the multidimensional Wiener process. As a result, Theorems 1.1, 1.2, and 1.13 can be reformulated using Hermite polynomials. Consider this approach [102 using our notations. Note that we derive the formula (1.262) (see below) in two different ways. One of them is not based on Theorem 3.1 [101 .

We will say that the condition (**) is fulfilled for the multi-index (¿i... ¿k) if mi,..., mk are multiplicities of the elements ii,..., correspondingly. At that, mi + ... + mk = k, mi,... , mk = 0,1,..., k, and all elements with nonzero multiplicities are pairwise different.

For example, for (ii,¿2,¿i,¿i,¿6) we have mi = 3, m2 = 2, m3 = m4 = m5 = 0, m6 = 1 (¿i = ¿2, ii = ¿6, ¿2 = ¿6).

In this section, we consider the case ¿i,...,¿k = 0,1,..., m. Note that in 102] the case ¿i,..., ¿k = 1,... ,m was considered.

Consider the multiple Wiener stochastic integral J' [j ... j ]T

(n-.-ifc)

($(ti,...,tk ) = j (ti)... j (tk )) defined by (1.23) (also see

where

(x)}°=0 is a complete orthonormal system of functions in the space L2([t, T]), each function (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7).

It is not difficult to see that

J

, ](ii...ifc) ji ... j]T,t

=J

j . . . ^jSm1

. . . J

mi ( ...ii )

J

m2 ( ¿2...«2 ) T,t

mfc ( ifc...ifc )

(1.259)

w. p. 1, where we suppose that the condition (**) is fulfilled for the multi-index

(¿i... ¿k) and

, . . . , jgm1+m2 + ...+mfc J* = {jg1 , . . . , jgfc } = {ji , . . . , jk},

J '

...

ml ( ¿/...¿/ )

= 1 for mi = 0; (1.260)

T,t

braces mean an unordered set, and parentheses mean an ordered set. Let us consider the following multiple Wiener stochastic integral

J '

^jgmi+m2 + ---+m;_i + 1 ^jgmi+m2+---+m;

m; ( i; ...i; )

(mi > 0),

where we suppose that

J,'â,m1+m2+...+m|_1 + l ' ' ' J,S,m1+m2 + ... + m| J j J^'1'' ' ^ ' ' ^ ' • • • jhd^l • • • jhrf^l

n1,l n2,l

(1.261)

where ni,i + n2,i +... + ndl,i = mi, ni,i,n2,i,... ,ndi)i = 1,... ,mi, di = 1,... ,mi, l = 1,..., k. Note that the numbers m1,..., mk, g1,..., depend on (i1,..., ik) and the numbers n1;i,..., ndl;i, h1;i,..., hdl;i, di depend on j..., ). Moreover, {jgi ,...,jgk} = {j1,...,jk}.

Using Theorem 3.1 [101], we get w. p. 1

J '

^jgmi+m2 + ---+mi_i+1 ^jgmi+m2 + ---+mi

ml ( il...il )

ku j)..-«n-l,(C), if ii=

0

=

(mi > 0), (1.262)

z(on ni 'l (0) A j , J . . . ^W, J

n-l ,l

if ii = 0

where Hn (x) is the Hermite polynomial of degree n

Hn(x) = (—1)nex

2/2_il fe-^/2 dxn

or

[n/2]

Hn(x) = n! ^

(—1)mx

m~,n—2m

m=0

m!(n — 2m)!27

(n G N),

(1.263)

and Z(i) (i = 0,1,..., m, j = 0,1,...) is defined by (11.111).

For example,

Ho(x) = 1, Hi(x) = x, H2(x) = x2 - 1, H3(x) = x3 — 3x, H4(x) = x4 — 6x2 + 3, H5(x) = x5 — 10x3 + 15x.

From (1.260) and (1.262) we obtain w. p. 1

J'

m;

( i;...i; ) T,t

(O(c£L). if i =

= l{m;=0} + l{m;>0} <

z(on ni'; ^(0)

l , ;

0

if il = 0

(1.264)

where 1A denotes the indicator of the set A. Using (1.259) and (1.264), we get w. p. 1

J' [ j • • • j ITi

(il...ik )

n

i=i

(

j) ...^nd;,; ) , if il = 0

1{m;=0} + 1{m; >0} <

V

VJhiJ VJhdlJ

nd;,;

if ii = 0

/

(1.265)

where notations are the same as in (1.261) and (1.262).

The equality (1.265) allows us to reformulate Theorems 1.1, 1.2, and 1.13 using the Hermite polynomials.9

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9Theorems 1.14, 1.15 (see below) will be generalized to the case of an arbitrary complete orthonormal system of functions {^ (x)}~0 in the space L2([t,T]) and ^1(r), ...,^fc(r) G L2([t,T ]), $(t1,...,tfc) G L2([t, T]k) in Sect. 1.11 (see Theorems 1.16, 1.17).

Theorem 1.14 [28] (reformulation of Theorems 1.1 and 1.2). Suppose that the condition (**) is fulfilled for the multi-index (¿i... ) and the condition (1.261) is also fulfilled. Furthermore, let every ^ (t) (l = 1,..., k) is a continuous nonrandom function on [t,T] and {fy(x)}°=0 is a complete orthonormal system of functions in the space L2([t, T ]), each function fy (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the following expansion

J

(¿1-ifc)

Pi

Pk

l.i.m.

...

Cjk ...jix

ji=

jfc=0

X

n

/=i

1

H

{m;=0} + 1{ml>0} <

V

z(0) \ni-1

.. H

ndl>l I j

,z(0) . . 1 ^.l.

Ci), if ¿1 = 0

if ¿/ = 0

/

(1.266)

converging in the mean-square sense is valid, where we denote the stochastic integral (O) as J[^^T^^); ni,/ + n2,i + ... + n^ = mi, ni,/, n2,i,..., n^,/ = 1,..., m/, d/ = 1,..., m/, I = 1,..., k; the numbers mi5..., mk, gi,..., gk depend on (ii,..., ) and the numbers ni;/,..., ndl;/, hi;/,..., hdl;/, d/ depend on (ji,...,jk); moreover, {jgi,...,jgk} = {ji,...,jk}; another notations are the same as in Theorem 1.1.

Theorem 1.15 [28] (reformulation of Theorem 1.13). Suppose that the condition (**) is fulfilled for the multi-index (ii.. .¿k) and the condition (1.261) is also fulfilled. Furthermore, let <&(ti,..., ) : [t, T]k ^ R1 is a continuous non-random function on [t,T]k and {fy(x)j 0 is a complete orthonormal system of functions in the space L2([t,T]), each function fy (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the following expansion

j w

(ii...ife)

Pi

Pk

l.i.m.

Pi,...,Pk ^^

...

ji=0 jk=0

Cjk...ji X

X

n

/=i

(

1{ml=0} + 1{ml>0} <

V

[Hni,i (j^ ...Hndi,i (j)J , if ¿/ = 0

, if i/ = 0

z(0) \ ni-1 /z(0) . . . VZjhdl,l

/

0

converging in the mean-square sense is valid, where we denote the multiple Wiener stochastic integral (11.255) as J'[^jTi"^; n1;/ + n2j/ + ... + nd;?/ = m/, n1;/, n2;/,..., nd; / = 1,...,m/, d/ = 1,...,m/, l = 1,...,k; the numbers m1,...,mk, g1,...,gk depend on (i1,...,ik) and the numbers n1;/, ...,nd;;/, h1,/,..., hd;,/, d/ depend on (j,...,jk); moreover, j,... j } = {j,..., j}; another notations are the same as in Theorem 1.13.

From (1.264) we have w. p. 1

H (cf), if i1 = 0

(k > 0). (1.267)

1 (j)k , if i1 = 0

k

J'[ j... j iTii""1) = <

Let us show how the relation (1.267) can be obtained from Theorem 1.2. To prove (1.267) using Theorem 1.2 we choose i1 = ... = ik and j = ... = jk (i1 = 0,1,... ,m) in the following formula (see (1.39) and (1.54))

k

j [j... j ft"'=n <j;;)+

/=1

[k/2] r k—2r

+ S( —11)r S n 1{is2s-1 = Vs =0}1{jS2s-1 = ¿32. }H j

r=1 ({{31,32}>--->{32r-1>32r }}>{91>-.->9k-2r}) S = 1 /=1

{si,S2>'">S2r-1>S2r>91>'">9k-2r } = {1.2.-".k}

(1.268)

w. p. 1, where notations are the same as in Theorem 1.2.

The case i1 = 0 of (1.267) is obvious. Simple combinatorial reasoning shows that

r k—2r

Z n 1{ig2s-i = Ss =0}1{jg2s-i = j»2. }H j =

({{31,32}>--->{32r-1>32r}}>{91v>9k-2r}) s = 1 /=1

{31,32v>32r-1>32r>91>--->9k-2r } = {1>2>--->k}

Ck • Cf—2 • ... • (r—1)2 / J;1)^ k—2r

r!

Zf) " r, (1.269)

where i1 = ... = ik, j = ... = jk (i1 = 1,... ,m), and

/!(k — /)!

is the binomial coefficient. We have

Ck • Ck-2

. C 2

• Ck-(r-1)2

k !

r !

r!(k - 2r)!2r

Combining (1.268), (1.269), and (1.270), we get w. p. 1

/ [j • • • jfe'^1 )

(1.270)

[k/2]

(cjii0 + k'E

r=1

-1)r

r!(k - 2r)!2'

4;i))

k-2r

[k/2]

k!

(-1)r

r=0

r!(k - 2r)!2r

(zj;1')k-2r = Hk j

The relation (1.267) is proved using (1

From (1.265) and (1.268) we obtain the following equalities for multiple Wiener stochastic integral

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J'

ji •

j Jrt

nzf +E<-D

[k/2]

1)r

1=1

r=1

X

E

ni

k 2r

({{si >S2}>--->{S2r-1>S2r}}>{91>--->9k-2r }) S=1 {S1 ,S2>'">S2r-i>S2r>9i>'">9fc-2r }={1,2>'">k}

{;S2s-i = ;S2s =0}1{jS2s_i = jS2s } il

1 = 1

(;q)

n

1=1

/

^ j)(z^), if il = 0

\

1{m1=0} + 1{m1 >0} <

V

/

:cr-.w. if=0

(1.271)

w. p. 1, where notations are the same as in Theorem 1.2 and (1.261), (1.262).

k

Let us make a remark about how it is possible to obtain the formula (1.262 without using Theorem 3.1 [101].

Consider the set of polynomials Hn(x, y), n = 0,1,... defined by [103

/

H"{x'y) =

ax—a2y/2

(Ho(x,y) =f 1).

a=0

It is well known that polynomials Hn(x,y) are connected with the Hermite polynomials (1.263) by the formula [103]

[n/2]

(—1)ixn—2iyi

Vv,

¿=0

i!(n — 2i)!2i

(1.272)

For example,

Hi(x, y) = x, H2(x,y) = x2 - y, H3(x, y) = x3 — 3xy, H4(x, y) = x4 — 6x2y + 3y2, H5(x,y) = x5 — 10x3y + 15xy2.

From (1.263) and (1.272) we get

Hn(x, 1) = Hn(x). Obviously, without loss of generality, we can write

Ui ■ ■ ■ jk) = {ji- ■ -Ji h- ■ -h ■■■ Jr ■ ■ ■],)■

(1.273)

(1.274)

mi

m2

where m1 + ... + mr = k, m1,..., mr = 1,..., k, r = 1,..., k, k > 0, and jb ..., jr are pairwise different.

Analyzing the proof of Theorem 1.1 and applying the orthonormality of (x)}°=0, we can notice that (we suppose that the condition (11.274) is fulfilled)

J' [ j ... j Pt

1 (¿i...ii)

N-1

l.i.m.

N->oo

£ j (rii) ... j (nfc )Aw,

(ii)

.. AwTii)

1 tu

ii,...,ifc=o

N—1

= LLm. Y, j (T'l) ..■ j (T'mi )Aw" ... Aw'« x

l1,...,lm1=0

;q—;3; 9=3; 9?3=1j---jm1

N —1

X E j (TJmi + 1 ) . . . j (r,mi+m2 . . • AW(;i;i ^ X

m

-I +1 I''', 1 - 0

ii + 1?---?;mi+m2 ;q-;3; 9=3; q,3-mi + 1,...,mi+m2

N1

x E j (T/k-mr+1)... (T,k )Aw«;;-mr+1... Aw<;k>

;k-mr + 1>'">;k-0 —;3; 9—3; 9,3-k-m, + 1,...,k

N—1 N—1

l.Lin. (E j(T/1 )Aw<;i>... E j(T/mi)Aw<;m» —

/l=0 /mi =0

E j (T/i )AwT;;» ... j (TJmi )Aw<imi I x

(/i,...,/mi )gG

N—1 N—1

XI E j (T'mi + 1)Aw';,l)i + i ... (T'mi+m2 ^W^

/mi+i=0 /mi+m2 =0

E ' j (T'mi + 1)Aw';:» +i . . . j (T,mi + m2 ^W^

(/mi+i,-,/mi+m2 )GGmi + i,mi+m2

X

N—1 N—1

x

lk-mr+i=0 lk=0

E j(Tik-m,+i)Aw«;;-m,+ij(T,k)Aw<;;>

Y j (T/k-mr + i )Awi*-),„ + i . . . j (T/k )AW<;:»

(/k -m, + i 5...? /k)GG'k

- m, + i,k

where the set Gmn is defined according to the same rule as the set Gk in (1.10). However, the elements of the set G'm n are the numbers 1m,..., 1n (n > m), while the elements of the set Gk are the numbers 1i,..., . Moreover, the set G'm n contains subsets of coinciding elements with the property that the number of elements in these subsets does not exceed two.

We have (see the proof of Theorem 1.1) w. p. 1 (ii = 0)

N —1 N —1

J>iK)Aw<;;>... £ fyji(t„i)Aw<;:) —

/i=0 /mi =0

y, fyji (t/i )aw';i> ... j (t_i )aw«;;) =

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(/i,...,/mi )GG'i

'N — 1 \ mi [mi/2]

J>i („i )Aw';^] +£(—1)r X

,/1=0 / r=i

2

N-1 x r

x E E4 k )(AwT:i^ 1 x

({{si >32}>--->{32r-i>32r}}>{9i>--->9mi-2r }) \/i=0 {Si,S2 >--->32r-i>32r>9i>'">9mi-2r }={i>2>--->mi}

'N-i \ mi—2r

x | ^ fyji (t/i)AwT:;)

/i=0

N—i \mi [mi/2] , /N —i 2\ r

+ £ (E4K> 1 x

/,¥— 1 \ mi—2r x £ j (T/i )Aw<;i»

/i=0 i

[mi/2] / . /N —i 2\ r /N —i \ mi—2r

/N—1 N—i 2\

= Hmi £ fyji(T/i)Aw^;)^ j(T/i) (AwT;;)X /i=0 i /i=0 i

where notations are the same as in Theorems 1.1, 1.2. Similarly we get w. p. 1

N 1 N 1

£ j (T,mi + 1)Aw<;m»i + i ... E j (T<mi+m, ^W^

/mi+i=0 /mi+m2 =0

E ' j (T,mi + 1)Aw'«+i ... j (T,mi+m2 ^wi^

(/mi + i,...5/mi+m2 )GGmi + i,mi+m2

/N—1 N—1 2\

ffm^ j(t/i^w^, E j(T/i)(Awii;»)

\/i=0 /i=0 J

N—1 N—1

E j (Tk-m,. + i )AW<;;-m, + i ...E^ (T'k )AW';I»-

/k-m, + i=0 /k =0

E j (T'k-m,+i )aw«i-)„,+i ... j (t/i )Aw<;;»

(/k -m, + i ?...? /k)GGI

-m, + i,k

/N—1 N—1 A

Hm, E j(t/i)AW<ii>, E4(T/i)(AW«ii»)

/i=0 i /i=0 i

Then

^ [ j . . . j ]

(ii...ii)

/N—1 N—1 2\

Um. Hmi E j(T/i^i», E j(T/i) (Aw<;;>) X

/i=0 /i=0

/N—1 N—1 2

xhm^ j(t/i^w^, £ j(t/i) (aw(;;^ i x

x

/i=0 /i=0

'N— 1 N—1

2

ffm^ j (T/i )Aw<;;i^ j (T/i) (awT;1^ (1.275)

/i=0 /i=0

w. p. 1 for i1 = 0 and

J'[ j • • • j]t,Î"0) =

'N-1 \ mi /N- 1

Nim (E j (t/i )At/ j ... j (t/, )At/,

,/i=0 J \/,=0

T mi T

f j(s)dsl ...(/j(s)ds

= (Zj(0))mi... (j)"" (1.276)

for i1 = 0, where we suppose that the condition (1.274) is fulfilled; also we use in (1.275) and (1.276) the same notations as in the proof of Theorem 1.1.

Applying (1.272), (1.273), Lemma 1.3, and Remark 1.2 to the right-hand side of (1.275), we finally obtain w. p. 1

J' [ j • • • j ]

ii...ii)

T,t

T T

= HmJ / j (s)dwi;i)^ j (s)ds| x

,t t T T

xHmJ i j (s)dwS;i) J j (s)ds| X .t t T T

•• X Hm I / j (s)<iw<;i>, / j (s)ds| =

= Hmi (ci:'1, ^ Hm2 j, ^ ...Hm, (zC'5, 1

= Hmi (cf) Hm, (cj2i) ) ...Hm, (#')

for i1 = 0, where we suppose that the condition (1.274) is fulfilled. Thus, an equality similar to (1.262) is proved without using Theorem 3.1 [101 .

m

r

Consider particular cases of the equality

for k = 1,..., 4 and

ii,..., i4 = 1,..., m (see (OflMOfl)). We have w. p. 1

J [ j = zf = Hi (cj

.(ii) ji

=

J[jj]¥!'^ = C C - 1{;i=;2}1{ji=j2}

H2

(iiA W (A;2)

Ho icj22^ , if ii = ¿2, j = j

ji

(1.277)

H1 j Hil j

otherwise

j ' [ j j j ^r0 = cj;i)cj:i)cj:i) - i{ji=j2} zf - i^zf - i{ji=js}Cj:

(ii) j2

H3 z(ii) , ji H0 z(ii) ^ z j2 H0 ;zj;iO, if ji = j2 = j3

H2 z(ii) , ji H0 z(ii) , j2 Hi ( ;z(i>), if ji = j2 = j3

< Hi z(ii) , ji H2 z(ii) zj2 H0 ;zj:i)), if j2 = j3 = ji

H0 z(ii) , ji Hi z(ii) , j2 H2 ;z(i>), if ji = j3 = j2

Hi z(ii) , ji Hi z(ii) , ji Hi ( ;zj:i)), if ji = j2, j2 = j3, ji = j3

; (1.278)

J'[ jjjIr" = cji;i,cjr2)cj:2) = Hi (zf) Hi (zf) Hi j

where ii, ¿2, are pairwise different;

j' [ j j j ]Tr:)=cj;l)zj:;l,cj;:) - i{ji=j2}Cj;:) = = (zjii)zj2i) -1^}) zj;:) = j' [0ji j if/ [ j ig1 =

http://doi.org/i0.2i638/ii70i/spbu35.2023.ii0 Electronic Journal. http://diffjournal.spbu.ru/ A.190

H2

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(ii)

ji

=

Hi

(ii)

ji

where i1 = i2 = i3;

j j ]?r)=zii^j—1fe=j3}C

Z(;i) fZ(i2 ) Z(i2)

Zj'l \vZj2 ZjS

H1

=

H1

(ii) ji

where i1 = i2 = i3;

J'[ j j j]

Z(i2) Z(ii)Z(ii) Zj2 yZj'i ZjS

H2

(ii)

ji

=

H1

(ii)

ji

H0

H1

(ii)

j2

(ii) j2

H1

(is)

jS

(is)

jS

1 { j2 = jS } ^ = J '[ j

H2

H1

(i2)

j2

(i2)

j2

H0

H1

(i2) j3

(iii2ii) (ii) (i2) (ii)

jj}) = J '[^j2

H1

(i2) j2

H0

H1

(ii)

j3

(ii) j3

if j1 = j2 if j1 = j2

(ii) ji

T^ J [ j jfe2) if j2 = j3

if j1 = j2

= z^u Z^'1 — 1 Z (i2) =

Zj'l Z j2 Zjs 1{j'l=j's}Zj2

(i2) 7"'^ A l(ilil)

J'[0ji j]

if j1 = j3 if j1 = j3

where i1 = i3 = i2;

J [ j r j2 j r j4]T,t = Zji Zj2 Zjs Zj4 —

1 Z (ii)Z ^ - 1 Z (ii) Z ^ - 1 Z (ii)Z (;i )

1{j'l=j2} ZjS Zj4 1{j'l=jS}Zj2 Zj4 1{j'l=j4}Zj2 ZjS

1 Z(ii)Z(ii) 1 Z(ii)Z(ii) 1 Z(ii)Z(ii)+ 1{j2=jS}Zji Zj4 1{j2=j4} Zji Z jS 1{jS=j4}Zji Zj2 +

+ 1 {j' 1 = j*2 } 1 {jS = j4 } + 1 {j' 1 = jS } 1 {j'2 = j4 } + 1 {j' 1 = j4 } 1 {j'2 = jS }

H4 ' z (;i) , ji H0 z(ii) H0 z(ii) H0 z(ii) , , j4 / ' if (I)

Hi ' z (;i) > Hi z(ii) Hi z(ii) Hi z (;i)) j y ' if (II)

H2 ' z (;i) > H0 z(ii) Hi z(ii) Hi z (;i)) j y ' if (III)

H0 ' z (;i) > Hi z(ii) H2 z(ii) Hi z (;i)) , j4 / ' if (IV)

H0 V (;i) , ji Hi z(ii) Hi z(ii) H2 z (;i)) , j4 / ' if (V)

Hi V (;i)" > H0 z(ii) , j2 H2 z(ii) Hi z (;i)) , j4 / ' if (VI)

Hi V(;i)" > H0 z(ii) , j2 Hi z(ii) H2 z (;i)) if (VII)

Hi ' z (;i)" > Hi z(ii) , j2 H0 z(ii) H2 z (;i)) , j4 / ' if (VIII)

H3 ' z (;i)" > H0 z(ii) , j2 H0 z(ii) Hi z (;i)) if (IX)

Hi ' z (;i)" > H3 z(ii) , j2 H0 z(ii) H0 z (;i)) , j4 / ' if (X)

H0 z(ii) > H0 z(ii) , j2 Hi z(ii) H3 z (;i)) if (XI)

H0 z(ii) > Hi z(ii) , j2 H0 z(ii) H3 z (;i)) zj4 y ' if (XII)

H2 z(ii) > H0 z(ii) , j2 H0 z(ii) H2 z (;i)) , j4 / ' if (XIII)

H2 z(ii) > H2 z(ii) , j2 H0 z(ii) H0 z (;i)) zj4 ) ' if (XIV)

H2 V z(ii) > H0 z(ii) , j2 H2 z(ii) H0 z (;i)) , ■j4 / ' if (XV)

where Hn(x) is the Hermite polynomial (I! .26,3) of degree n and (I)-(XV) are the following conditions

(I). j1 = j2 = j3 = j4,

(II). , j2, j3, j4 are pairwise different,

(III). jl = J2 = J3,J4; J3 = J4,

(IV)- j1 = j3 = J2,j4; j2 = j4,

(V)- j1 = j4 = J2,j3; J2 = j3,

(VI). j = J3 = jlj jl = J4,

(VII). j2 = J4 = j1,j3; j1 = J3,

(VIII). J3 = J4 = j1,j2; j1 = J2,

(IX). J1 = J2 = J3 = J4,

(X). J2 = J3 = J4 = j1,

(XI). j1 = J2 = J4 = J3,

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(XII). J1 = J3 = J4 = J2,

(XIII). j = j2 = J3 = j4,

(XIV). j = j = j2 = j4,

(XV). j1 = j4 = J2 = J3. Moreover, from (1.259) we have w. p. 1

j j .fe]Tr:'4) = h (j) h (j) h (j) h (j) •

where i1, i2, i3, i4 are pairwise different;

J'[ j<Aj: jj^f^ = ^j:]Ti,1/'l)H1 (j) H J) , (1.279)

where ¿1 = ¿2 = ¿3, ¿4; ¿3 = ¿4;

j'[jjjjiTfi!4) = J'i^ji^fH (cf) H1 J), (1.280)

where ¿1 = ¿3 = ¿2, ¿4; ¿2 = ¿4;

j'ijjjjiTf:!i) = J[j^TfH (cf) H1 (cf), (1.281)

http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal.http://diffjournal.spbu.ru/ A.193

where ¿i = ¿4 = ¿2, ¿3; ¿2 = ¿3;

J'[ jj jj]T;,f;2;4) = J'[]0,:]T;r)Hi ]) Hi ]) , (1.282)

where ¿2 = ¿3 = ¿i, ¿4; ¿i = ¿4;

J' [0ji 0j 0,: 0j If^ = J' [0,2 0,4 ]Ti2t2)Hi (]>) Hi (]>) , (1.283)

where ¿2 = «4 = ¿i,¿3; ¿i = «3;

J' [0, 0,2 0,: 0, If:;:) = J' [0,: 0,4 If Hi (<j>) Hi (<f) , (1.284) where ¿3 = ¿4 = ¿i, ¿2; ¿i = ¿2;

J'[j0,20,:0,4f/i;i!l) = J'[0,i0,20,:f/i;i)Hi (]>) , (1.285) where ¿i = ¿2 = ¿3 = ¿4;

J'[0,i0,20,:0,4]Tr!2) = J'[0,20,:0,4If^Hi ] ) , (1.286)

where ¿2 = ¿3 = ¿4 = ¿i;

J'[0,i0,20,:0,4]T;r;:!i) = J'[0,i0,20,4If0Hi ]) , (1.287)

where ¿i = ¿2 = ¿4 = ¿3;

J'[0,i0,20,:0,4Ifi!i) = J'[0,i0,:0,4If0Hi ]) , (1.288)

where ¿i = ¿3 = ¿4 = ¿2;

J' [0,i 0,2 0,: 0,4 |f ^ = J' [0,i 0,2 If J' [0,: 0,4 If. (1.289)

where ¿i = ¿2 = ¿3 = ¿4;

J ij j J J ]^Tili:) = J [ j j ^ J [ j j If- (1.290)

where ¿1 = ¿3 = ¿2 = ¿4;

J' i^ji j j j ^^ = J [ j j tffj i^J: j If' (1.291)

where ¿1 = ¿4 = ¿2 = ¿3.

Note that the right-hand sides of (1 .'2791) (1 .'291) contain multiple Wiener stochastic integrals of multiplicities 2 and 3. These integrals are considered in detail in (1.277), (1.278).

It should be noted that the formulas (1.54) (Theorem 1.2) and (1.1

(Theorem 1.14) are interesting from various points of view. The formulas (1.45)-(1.50) (these formulas are particular cases of (1.54) for k = 1,...,6) are convenient for numerical modeling of iterated Ito stochastic integrals of multiplicities 1 to 6 (see Chapter 5). For example, in [52] and [53], approxima-

tions of iterated Ito stochastic integrals of multiplicities 1 to 6 in the Python programming language were successfully implemented using (1.45)-(1.50) and Legendre polynomials.

On the other hand, the equality (1.266) is interesting by a number of reasons. Firstly, this equality connects Ito's results on multiple Wiener stochastic integral ([101], Theorem 3.1) with the theory of mean-square approximation of

iterated Ito stochastic integrals presented in this book. Secondly, the equality (1.266) is based on the Hermite polynomials, which have the orthogonality property on R with a Gaussian weight. This feature opens up new possibilities in the study of iterated Ito stochastic integrals. Note that the indicated orthogonality property is indirectly reflected by the formula (1.84) (see the proof of Theorem 1.3).

1.11 Generalization of Theorems 1.1, 1.2, 1.14, and 1.15 to the Case of an Arbitrary Complete Orthonormal System of Functions in the Space L2([t, T]) and ),

... (T) e ¿2([t,T]), $(ii,..., tk) e L2([t,T]k)

In this section, we will use the definition of the multiple Wiener stochastic integral from [101], [104] to generalize Theorems 1.1, 1.2, 1.14, and 1.15 to

the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^i(r), ... (t) G L([t,T]), $(ti,... ,t*) G ¿2([t,T]k).

Consider the following step function on the hypercube [t,T]k

N-i

(ti)... 1K ,T!fc+i)(tk), (1.292)

Zi,...,Zfc=0

where a1i...1k G R and such that a1i...1k = 0 if = for some p = q,

f 1 if t G A

1a (t ) = | ,

0 otherwise

N

G N, {t,}.=0 is a partition of [t,T], which satisfies the condition (11.9):

t = t0 < ... < tn = T, An = max At,- ^ 0 if N ^ to, At,- = t,-+1 -t,.

0<j<N -i

(1.293)

Let us define the multiple Wiener stochastic integral for (ti,..., tk) [101], 104]

N-i

( ii...;fc) def

/i,...,/fc=0

j'd=f £ aii..JtAwT;;»...Aw<;k», (1.294)

where AwTj = wTj++i — wTj», i = 0,1,...,m, wT0) = t.

It is known (see [104], Lemma 9.6.4) that for any ... ,tk) G L2([t,Tjk) there exists a sequence of step functions (ti,..., ) of the form (1.292) such that

lim i ($(ti,...,tk) — (ti,...,tk ))2 dti ...dtk = 0. (1.295)

N^to j [t,T ]'

We have

N —1

(t1, . . . , tk )= £ ali...lfc l[n1 ,ri1+i)(t1) ... l[nfc ,rifc+i)(tk ) =

llv-jlfc =0

N—i

= £ £ ßliX[Tll ,Tli + l) (t1) ... Ihk +l)(tk), (1.296)

(li ,...,/fc) ii,...,ifc=0

ii<i2<...<ifc

where permutations (11,..., 4) when summing are performed only in the expression 11 < 12 < ... < (recall that a1i...1k = 0 if = for some p = q).

Using (1.296), we get

T t:

£ J.. J(i1,...,ik)dwt(;i)...dwt(;k) = (1.297)

(iivvifc) t t

N-1

= E E a'-'kAw<;;)...iw^J =

(/i,...,/fc) li,...,lfc=0

ii<i:<...<ifc

N- 1

£ aii...ik iw(;;)... iw(;;) =

ii,..,ifc=° =lr; q=r; q,r=i,...,fc

l(;i"-;fc)

= J'[$N]T;V";k) w. p. 1, (1.298)

where permutations (t1,..., tk) when summing are performed only in the values dwt(;i)... dwt(;k) and permutations (11,..., 1k) when summing are performed only in the expression 11 < 12 < . . . < 1k. At the same time the indices near upper limits of integration in the iterated stochastic integrals in (1.297) are changed correspondently and if tr swapped with tq in the permutation (t1,..., tk), then ¿r swapped with ¿q in the permutation (¿1,..., ¿k) (see (1.297)). In addition, the multiple Wiener stochastic integral J'[$N]Ti";k) is defined by (H.294J) and

T t:

/.../ ^n (t1—tk )dwt;i) ...dwt;k) tt

is the iterated Ito stochastic integral.

Using (1.295), (1.298), Lemma 1.2, and (1.38) for Lebesgue integrals, we have

I (;i —ife) T'r.r. ](;i---;fc)

m J[^n]T;,it';k) - J'[*«]&■*0 -

T t:

- Ck Z J "J ($N(*1,...,*k) - ^M(t1,...,tk))2 dt1 ...dtk = (ti,...,t&) t t

= Ck J ($n (t1,...,tk) - (t1,...,tk ))2 dt1 ...dtk =

[t,T ]k

= Ck - |lL2([i,T) < < 2Ck - ]k) + ^ - ||L2([t>T]k^ — 0

if N, M —to, where constant Ck depends only on the multiplicity k of the multiple Wiener stochastic integral.

Thus, there exists the limit

l.i.m. J').

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N—to

We will define the multiple Wiener stochastic integral for ... , tk) £ L([t,T]k) by the formula [101], [104

N-1

J') =f l.i.m. J[$N]T,r!i' = l.i.m. ]T AwT;;)... Aw«,

AT \ —..-v AT \ —..-v 1 fc

l.i.m. / ^¿i... ik1 n—to N—to . , n

il> . . . >'fc =0

(1.299)

where $n(ti,..., tk) is defined by (092), Awj = wij)+1 - wj i = 0,1,..., m,

(0)

wT = t.

It is easy to see that the above definition coincides with (1.23) if the function $(t1,..., tk) : [t, T]k — R is continuous in the hypercube [t, T]k.

Let us prove the following equality

T t2

J' ) = E j.J ^(ti-----tk )dwt(;i) ...dw^) w.p. 1, (1.300)

(t1>--->ifc) t t

where permutations (t1,..., tk) when summing are performed only in the values dwt(il)... dwt(ik). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,..., tk), then ir swapped with in the permutation (i1,... ,ik). In addition, the multiple Wiener stochastic integral J'[$]T>t"ifc) is defined by (1.299) and

T t2

J..J ^(t1—tk )dwt(;i) ...dwi:k) tt

is the iterated Ito stochastic integral.

The equality (1.300) has already been proved for the case $(t1,... ,tk) = (t1,...,tk) (see (1.298)).

From (1.298) we have

T t:

J'[*N]Tr;) = E J. J ^(t1,..., tk)dw«;i)... dw<;k) =

(ti,...,tfc) t t T t:

= E J-J ®(t1,..-.tk )dw';i> ...dwi;;>+

(ti,...,tfc) t t T t:

+ £ j . j (^N (t1,...,tk) - $(t!,...,tk)) dwt;i) ...dwt(;;) w.p.1.

(ti,...,tfc) t t

(1.301)

Passing to the limit l.i.m. in the equality (1.301), we obtain

N ^TO

T t:

J'MTr;) = E J. J—,tk)dwi;i)...dw«;;)+

(ti,...,tfc) t t

T t:

+l.i.m. v i... i($n(t1,...,tk)-$(t1,...,tk))dwt;i)...dw(;;) w.p. 1. N ^TO z—'J J

(ti,...,tfc) t t

(1.302)

Using Lemma 1.2, (1.38) for Lebesgue integrals, and (1.295), we get

T t: 2

m{( £ J . J ($n(t1—tk)-^(t1—tk))dwt;i)...dwt(;;^ ^(ti,...,tfe) t t

T t:

< Ck £ /..^ (^N (t1,...,tk) - $(t!,...,tk))

2

^ dt1... dtk

(ti,...,t&) t t

= Ck y ($N (t1,...,tk) - $(t1,...,tk ))2 dt1 ...dtk ^ 0 (1.303)

[t,T

if N ^ to, where constant Ck depends only on the multiplicity k of the multiple Wiener stochastic integral.

The relations (1.302) and (1.303) prove the equality (1.3

From (1.300) we have

t t2

J

(¿1-ifc ) T,t

= I ^k(tk)... I ^i(ii)dwt(;i)...dw£k) = J'w. p. 1,

|(«1-ifc )

where K = K(tl5...,tk) is defined by (1.6), i.e.

^i(ti) ...^k (tk ), ti <...<tk

K (ti,...,tk ) =

0,

otherwise

where ti,... ,tk G [t,T] (k > 2) and K(ti) = ^(ti) for ti G [t,T]. Applying (1.304), we obtain

J №(k)]Ti

(H---ifc)

= j ' [k ]

(ii-.-ife ) T,t

(1.304)

(1.305)

pi pk

£... £ Ck...ji J'[ j... j]TTk) + J'[Rpi-.-Pkft--") w. p. 1, (1.306)

ji=0 jk=0

where

pi Pk

Rpi-pk(ti,... ,tk) = K(ti,... ,tk) - £ ... £ Cjk...ji n^(ti) (1.307)

ji=0 jk=0

i=i

and

C =

Cjk ...ji =

/k

K (ti,...,tk ^ (ti )dti ...dtk

[t,T ]k 1=i

(1.308)

is the Fourier coefficient corresponding to K(ti,... ,tk).

Using Theorem 9.6.9 [104] (also see [101], Theorem 3.1) and (1.271), we get

J' [ j... j ]T

(ii...ik)

n

i=i

(

iHni,i j) ...^ (j)J , if ii = 0

l{m; =0} + l{m;>0} <

V

.(0) \ni-' />(0) \ndi-'

Z(0) ... Z(

Sjh . I \ Sj

if ii = 0

/

k

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k [k/2] =n ci;-5 +E(-i)r*

'=1 r=1

r k-2r

* E n !{;,:,-i = •„ =0i1{j»„_i = j»„. } II j"> (1.309)

({{si,s:>,...,{s:r-i,s:r }}>{qi>...>9:-:r}) S=1 1=1

{si,s:,...,s:r-i.s:r>9i>...>9k-:r }={i,:,...,;}

w. p. 1, where notations are the same as in Theorems 1.2 and 1.14; the multiple Wiener stochastic integral J'[ j ... j]Tlt";:) is defined by (1.299).

Again applying (1.300), we have

T t: ,

J'[Rpi..p;) = E /"./I K(t1, . . . • tk)-

(ti) t t ^ Pi P: k

i (t') )dw<;i) •..dwi:;), (1.310)

ji=0 j: =0 '=1

where permutations (t1,..., tk) when summing are performed only in the values dwt(;i)... dwt(;:). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,... ,tk), then ¿r swapped with ¿q in the permutation (¿1,..., ¿k). In addition, the multiple Wiener stochastic integral J'[Rpi...p:]T;it...;:) is defined by (1.2

According to Lemma 1.2, (1.7), and (1.38) for Lebesgue integrals, we have

m{ J' [Rpi.p

] (;i ...;:) 2

-P:]T,t 1 ' <

T ^ / Pi P: k \2

< Ck £ ... K (t1-----tk) -£ ..^Cj: „^JI ^ (t'H X

(ti,...,t:) { { \ ji=0 j:=0 '=1 /

xdt1... dtk =

2

// Pi P: k \

K(t1,..., tk) - £ ... £ Cj:...ii n (t') dt1... dtk ^ 0

u^i: V Ji=0 j =0 '=1 /

(1.311)

if p1,... ,pk ^ to, where constant Ck depends only on the multiplicity k of the

iterated Ito stochastic integral J[^(k)]y1rik).

Thus, the following theorem is proved.

Theorem 1.16 [28] (generalization of Theorems 1.1, 1.2, and 1.14). Suppose that the condition (**) is fulfilled for the multi-index (¿i... ¿k) (see Sect. 1.10) and the condition (1.261) is also fulfilled. Furthermore, let ^(t) G L2([t,T]) (l = 1,..., k) and (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then the following expansions

pi pk

J [^<k']TiiVik) = l.i-m £ ... £ Cjkj x

.71=0 ,7fc=0

x

n

/=i

1{ml=0} + 1{ml>0} <

V

(z^) ...^ (.),), if ¿/ =

k 7)"...(zilr, if / = °

J

(il...ik)

l.i.m.

p1 1

j1=0

Pk

k [A/2]

"Ec^in c7r) + £(—1)

jk=0

-i)rx

,/=i

r=i

x

£

1

{i32s-1 = »32« =0} 1{i'g2s-1 = ^ } H

k—2r

n z

(iq;)

({{31.32 }>'">{32r-1>S2r }}>{91>--->9k-2r }) S=1 {31,32.'".32r-1.32r.91.'".9k-2r } = {1.2.-".k}

/=i

(1.312)

converging in the mean-square sense are valid, where [x] is an integer part of a real number x, ni,/ + n2,/ + ... + ndj,/ = m/, ni,/,n2,/,... ,ndj)/ = 1,... ,m/, d/ = 1,..., m/, I = 1,..., k; the numbers m1,..., mk, g1,..., gk depend on (¿i,..., ¿a) and the numbers ni,/,... ,ndj)/, hi,/,..., hdj,/, d/ depend on (ji,...,jk); moreover, {jg1, ...,jgk} = {ji,...,jk}; another notations are the same as in Theorems 1.1, 1.2, and 1.14.

Replacing the function K(ti,..., tk) by $(ti,..., tk) we get the following theorem.

Theorem 1.17 [28] (generalization of Theorems 1.13, 1.15). Suppose that the condition (**) is fulfilled for the multi-index (¿i... ¿k) (see Sect. 1.10) and the condition (1.261) is also fulfilled. Furthermore, let $(ti,... ,tk) G L2([t,T]k)

and {0- (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then the following expansions

Pi P:

J' [*]Tr*> = l.i.m. E —...

ji

X

ii=0 j:=0

X

n

'=1

J) -Hv (C) • if *=0

1{ml=0} + 1{m, >0} <

V

z(0n ni'- (c(0) \

if i' = 0

/

J' №

(;i...;:)

T,t

l.i.m.

Pi,...,P: ^TO

Pi i

ii=0

P:

[k/2]

1)r

j:=0 \'=1

r=1

X

E

1

k 2r

i I I z(;q,)

{;s:s-i = ;s:s =0}1{js:s-i = } II j

({{s:.s:}.....{s:r-:.s:r }}>{9i>...>9:-:r}) s=1 {si,s: ,...,3:r-i>3:r>9i>...>9:-:r }={i>:>...>:}

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zj

'=1

(1.313)

converging in the mean-square sense are valid, where [x] is an integer part of a real number x, m,' + n2,' + ... + ndj,' = m', m,',n2,',... ,ndi)' = 1,... ,m', d' = 1,..., m', I = 1,..., k; the numbers m1,..., mk, g1,..., gk depend on (i1,...,ik) and the numbers m,', ...,nd,d, h^',..., hd,,', d' depend on (j, ...,jk); moreover, {jgi,..., jg:} = {j,..., jk}; the multiple Wiener stochastic integral J'[^Tt ^) is defined by (1.299); another notations are the same as in Theorems 1.13, 1.15.

It should be noted that an analogue of Theorem 1.17 was considered in 102]. Also note that the proof in [102] is slightly different from the proof given in this section.

Note that Theorem 1.16 can also be proved without using the multiple Wiener stochastic integral. To do this, we define the following multiple stochastic integral as the sum of iterated Ito stochastic integrals

T

t:

J'wTr) d=f £

$(t1

.,tk)dwt(;i)...dwt(;:) w.p. 1, (1.314)

(ti,...,t:) t

k

where $(t1,... ,tk) £ L2([t,T]k); another notations are the same as in (1.300). Further, we have

J [^Tr^) = /'[k]^) =

Pi P:

= £ . . . £ Cj: j J''[ j . . . j^ + J''[flPi.„P:^) w. p. 1, (1.315)

j=0 j =0

where K(t1,...,tk) and rp:...p:(t1,...,tk) are defined by (1.3051) and (1.3071) correspondingly. Moreover, J''[ j ... jand J''[RPi...P:) are defined by (Pm ' '

Passing to the limit l.i.m. in (1.315) and using (1.311), (1.314), we

Pi,...,P: ^to

obtain

Pi P:

j[z^fr"> = l.i.m. e... e Cj: .jij''[ j... j]T;r> =

ji =0 j: =0

Pi P: T t}

= l.i.m. E ..^Cj: .j E /i (tk) ••./j (t1)dw(;i) ...dw^,

ii=0 i:=0 (ti,...,t:) t t

(1.316)

where permutations (t1,..., tk) when summing are performed only in the values

dwt(;i)... dwt(;:). At the

same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,... ,tk), then ir swapped with in the permutation (i1,..., ik).

It is easy to see that the equality (1.316) can be written as

Pi P:

1 (h-.;:)

J [Z^Tr^ = l.i.m. E

P ,...,P

j1X

ii=0 j: =0

T t:

X £ J j (tk) ..J j (t1)dwt(;i)... dw(;:), (1.317)

(j:,...J:) t t

where

E

(ji,...J:)

means the sum with respect to all possible permutations ji...,jk). At the same time if jr swapped with in the permutation ji ...,jk), then ¿r swapped with in the permutation.

Further, using the Ito formula, we can prove the following equality

T ^ k [k/2] £ /&)... i(ti)dwt(;1)...dwt(:k)= nzf + £(—1)rx

(.71,7) { { /=i r=i

r k—2r

E n 1{i32s-1 = i32s =0}1{i32s-1 = i32s } H C' (018)

X

({{31.32 }.-■■. {32r-1.32r }}.{q1 .-■-.qk-2r }) S=1 / = i

{31,32,"',32r-1,32r,ql,"',qk-2r }={1,2,■■■,k}

w. p. 1, where notations are the same as in Theorem 1.2 and (1.317).

The main difficulty in proving (1.318) using the Ito formula is related to the need to take into account various combinations of indices ¿i,..., ¿k = 0,1,..., m.

1.12 Generalization of Theorems 1.3, 1.4 to the Case of an Arbitrary Complete Orthonormal System of Functions in the Space L2([t,T]) and ^(t), ..., (t)

e L2([t,T])

In this section, we will use the multiple Wiener stochastic integral with respect to the components of a multidimensional Wiener process to generalize Theorems 1.3, 1.4 to the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^1(t), ... (t) G L2([t,T]).

Theorem 1.18. Suppose that ^1(t),... (t) G L2([t,T]) and {0.(x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then

m

-J

p

T,t

K 2(ti,...,tk )dti ...dtk -

[t,T ]k

ji=0 j =o

T

t2

j£ / j(tk)... j(ii)dft(;i}... f

?(ifc )

(jl ,---,jfe) t

(1.319)

2

p

where

T t:

j[^(k)]T,t=J ^(tk)... I ^1(t1)dft(:i)... dft(;:), tt PP

J = £ . . . £ Cj: „j J' [ j . . . j fr::), (1.320)

j=0 j:=0

J'[ j ... j]T;it...::) is the multiple Wiener stochastic integral defined by (1.299), the Fourier coefficient Ci:...j has the form (11.3081), K(t1,... ,tk) is defined by (1.6),

T

zj;) = / (s)df<;>

t

are independent standard Gaussian random variables for various i or j (i = 1, . . . , m),

E

(ji,...J:)

means the sum with respect to all possible permutations j ..., jk). At the same time if jr swapped with jq in the permutation j ..., jk), then ir swapped with iq in the permutation (i1,..., ik) (see (1.319)).

Proof. First, note that the formula (1.320) appears due to the equality (1.306). Using the equality (1.300), we get

T t:

j'[j...j]T;r::) = £ /i(tk)...|j(t1)dfi;i)...dft(:::) w.p. 1,

(t1,...,t:) t t

(1.321)

where permutations (t1,..., tk) when summing are performed only in the values dft(;i)... dft(:::). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,... ,tk), then ir swapped with iq in the permutation (i1,..., ik).

It is easy to see that the equality (1.321) can be written in the form

T t:

j' [ j... j fr::) = £ J j (tk) ..J J (t1)dft(:i)... df::) w. p. 1,

(j:,...J:) t t

(1.322)

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where

E

(i1,-..,ik)

means the sum with respect to all possible permutations ji...,jk). At the same time if jr swapped with in the permutation ji ..., jk), then ¿r swapped with in the permutation.

Thus, an analogue of the equality (1.85) is proved under the conditions of Theorem 1.18 (compare (1.76), (1.85) and (1.320), (1.322)). Further proof of Theorem 1.18 is similar to the proof of Theorem 1.3. Theorem 1.18 is proved.

Consider the following obvious generalization of Theorem 1.4.

Theorem 1.19. Suppose that ^1(t),... (t) G L2([i,T]) and {0.(x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([i,T]). Then the estimate

m

T,t

J

P1,...,Pk T,t

<

/

< k!

/P1 Pk

K2(ti,...,ik№...¿ik — £.-ECLi

; _n ;__n

\ [t,T]k

is valid for the following cases:

1. ^,...,4 = 1,...,m and 0 <T — i< to,

j1=0 jk=0

/

(1.323)

2. ¿1,..., ¿k = 0,1,..., m, ¿2 + ... + ¿A > 0, and 0 < T — i < 1,

where J[^(k)]T,t is the iterated Ito stochastic integral (11.5), J[^^j1^"^ is the expression on the right-hand side of (1.312) before passing to the limit l.i.m. ;

another notations are the same as in Theorems 1.1, 1.2, 1.16.

In addition, under the conditions of Theorem 1.19 we have the estimate (also see (11.73))

mI (j[^(k)ht — Jty^Tr^) 2"j <

< (k!)2n(n(2n — 1))n(k—1)(2n — 1)!! x

/

x

p1 pk

J k2(ii,...,tk)dti...¿tk— E...ECU

\[t,T ]k

j1=0 jk=0

/

2

n

1.13 Generalization of Theorems 1.5, 1.6 to the Case of an Arbitrary Complete Orthonormal with Weight r(x) > 0 System of Functions in the Space L2([t,T]) and ipi(x)y/r(x), ipk(x)y/r(x) £ L2([t,T])

In this section, we will use the multiple Wiener stochastic integral with respect to the components of a multidimensional Wiener process to generalize Theorems 1.5, 1.6 to the case of an arbitrary complete orthonormal with weight r(x) > 0 system of functions in the space L2([i,T]) and ijji(x)^/r(x), ..., i/j}~{x)-\/r(x) £ L2([t,T]). From the results of Sect. 1.3, 1.11 we obtain the following two theorems.

Theorem 1.20. Suppose that ipi(x)y/r(x),..., ipk{x)\/r{x) £ L2([i,T]), where r(x) > 0. Moreover, let

j=0

is an arbitrary complete orthonormal with weight r(x) system of functions in the space L2([t,T]). Then, for the iterated Ito stochastic integral

t t2

ifc

J[^ik)ht= / Mtk)yffîiï... / (1.324)

the following expansion

Pi Pk / k [k/2]

= £... £ M n Zf+£(-1)

^ . . J"' + > '(-1)rx

Pi,...,Pk^^ z—' z—' " ,

ji =0 jk=0 \Z=1 r=1

r k-2r

X £ n 1{:g:s-i = :g:s =0}1 J:.-i = } II j'

({{s:.s: },...,{s:r-i.s:r }}>{9i>...>9:-:r}) s=1 '=1

{s:,s: ,...,3:r-i>3:r>9i>...>9:-:r }={i>:>...>:}

(1.325)

that converges in the mean-square sense is valid, where i1, . . . , ik = 0, 1, . . . , m,

T

cf = I ^-Wv^dwW

t

are independent standard Gaussian random variables for various % or j (in the case when i = 0),

Ci i = I K(ti,...,tk) I li (i/)r

J K(ti,... ) n^j(ti^dti.. .dtk

j-ji = I K (6i

[t,T ]k 1=1

is the Fourier coefficient, K(ti5... ,tk) is defined by (1.6); another notations are the same as in Theorems 1.1, 1.2, 1.5.

Theorem 1.21. Under the conditions of Theorem 1.20 the following estimate

m <! i jw*^ - J[#fe) !> <

/ / k \ pi Pfc ^

< k!

// k \ pi Pfc

K2(ti,..., tk) m r(tiH dti... dtk - £ ... £ (7?k...j

V[t,T]k W 7 ji=o jk=o y

is valid for the following cases:

1. ¿^...^ = 1,...,m and 0 <T — i< to,

2. A,...,^ = 0,1,..., m, ¿2 + ... + ¿A > 0, and 0 <T — i< 1,

where J[^(k)]Tri is the stochastic integral (11.324), J^^]^"'^ is the expression on the right-hand side of (1.325) before passing to the limit l.i.m. ; another

P1,...,Pk ^to

notations are the same as in Theorems 1.6, 1.20.

Chapter 2

Expansions of Iterated Stratonovich Stochastic Integrals Based on Generalized Multiple and Iterated Fourier Series

This chapter is devoted to the expansions of iterated Stratonovich stochastic integrals. We adapt the results of Chapter 1 (Theorem 1.1) for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of continuously differentiable weight functions and complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([i,T])) and multiplicity k, k G N (the case of convergence of trace series). The mean-square convergence of the mentioned expansions is proved. The considered expansions contain only one operation of the limit transition in contrast to its existing analogues. This property is very important for the mean-square approximation of iterated stochastic integrals. Also, we consider an approach to the expansion of iterated Stratonovich stochastic integrals of multiplicity k (k G N) based on generalized iterated Fourier series converging pointwise.

2.1 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicity 2 Based on Theorem 1.1. The case p1,p2 ^ to and Smooth Weight Functions

2.1.1 Approach Based on Theorem 1.1 and Integration by Parts

def

Let (Q, F, p) be a complete probability space and let f (i, w) = ft : [0, T] x Q ^ R1 be the standard Wiener process defined on the probability space (Q, F, p).

Let us consider the family of a-algebras {Ft, t G [0, T]} defined on the probability space F, p) and connected with the Wiener process in such a way that

1. Fs c Ft c F for s < t.

2. The Wiener process is Ft-measurable for all t G [0,T].

3. The process /t+A — for all t > 0, A > 0 is independent with the events of a-algebra Ft.

Let M2([0, T]) be the class of random functions £(t, u) = : [0, T] xft ^ R1 defined as in Sect. 1.1.2.

We introduce the class Qm([0, T]) of Ito processes nT, t G [0, T] of the form

nr = no + J «ds + J bsd/s, (2.1)

o o

where (aT)m , (bT)m G M2([0,T]) and

lim m{ |bs — bT|4} =0 for all t G [0,T].

The second integral on the right-hand side of (2.1) is the Ito stochastic integral (see Sect. 1.1.2).

Let C2,1 (R1 x [0,T]) be the space of functions F(x,t) : R1 x [0,T] ^ R1 with the following property: these functions are twice differentiable in x and have one derivative in t. Moreover, all these derivatives are bounded.

Let t(N^, j = 0,1,..., N be a partition of the interval [t, T], t > 0 such that

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j

t = t0N) <t<N) <...<tNN) = T, max

0 1 N 0<j<N-1

T(N)_ T(N) Tj + 1 Tj

0 if N 00.

(2.2)

The mean-square limit

N-1 ,1 x *

^ E ^ (2 ('M"'+ ys) 'r) (%;- - /*-») = / nnrM (2.3) j=0 t

is called [105] the Stratonovich stochastic integral of the process F(nT, t), t g

[t,T], where t(N), j = 0,1,..., N is a partition of the interval [t,T] satisfying

the condition

It is known [105] (also see [79]) that under proper conditions, the following

relation between Stratonovich and Ito stochastic integrals holds

,T T T

/1 f dF

F(i]T,r)dfT + - I —(i]T,r)Mr w. p. 1. (2.4) t t t If the Wiener processes in (2.1) and (2.3) are independent, then

= t T

F(nT,t)df = J F(nT,t)df w. p. 1. (2.5)

t

A possible variant of conditions under which the formulas (2.4) and (2.5) are correct, for example, consists of the conditions: £ Q4([t, T]), F(nT, t) G M2([t,T]), F(x,t) G C2'1(R1 x [0,T]).

Note that if F(x,t) = F1(x)F2(t), then the smoothness condition F(x,t) G C 2'1(R1 x [0,T]) can be weakened. Namely, it suffices to replace the condition with respect to t by continuity with respect to this variable.

Further in Chapter 2, in most cases, by {0j(x)}°=0 we will denote the complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]). Also we will pay attention on the following well known facts about these two systems of functions [106].

Suppose that the function f (x) is bounded at the interval [t,T]. Moreover, its derivative f (x) is continuous function at the interval [t,T] except may be the finite number of points of the finite discontinuity. Then the Fourier series

T

to „

y^ Cj (x), Cj = f (x)0j (x)dx j=0 t converges at any internal point x of the interval [t, T] to the value (f (x + 0) + f (x — 0)) /2 and converges uniformly to f (x) on any closed interval (of continuity of the function f (x)) lying inside [t,T]. At the same time the Fourier-Legendre series converges if x = t and x = T to f (t + 0) and f (T — 0) correspondently, and the trigonometric Fourier series converges if x = t and x = T to (f (t + 0) + f (T — 0)) /2 in the case of periodic continuation of the function f (x).

In Sect. 2.1 we consider the case k = 2 of the following iterated Stratonovich and Ito stochastic integrals

* T * t2

J*[^(k)]T,t = / ^k (tk).../ ^(tOdw^ ...dwt(:k), (2.6)

T t2

J = / ^k (tk) ..J ^i(ti)dwt(;i)... dw£k), (2.7)

t t

where every ^ (t) (l = 1,..., k) is a continuous nonrandom function at the interval [t,T], w[i) = f(i) for i = 1,... ,m and wT0) = t, f(i) (i = 1,... ,m) are independent standard Wiener processes.

Let us formulate and prove the following theorem on expansion of iterated Stratonovich stochastic integrals of multiplicity 2.

Theorem 2.1 [8] (2011), [10]-[21], [32]. Suppose that {j(x)}=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). At the same time ^2(s) is a continuously differentiable nonrandom function on [t,T] and ^1(s) is twice continuously differentiable non-random function on [t, T]. Then, for the iterated Stratonovich stochastic integral

* T * t2

J*[^(2)kt = y ^2)/ ^i(ti)dft(;i)dfi;2) (ii,i2 = 1,...,m) tt

the following expansion

Pi P2

J *¥(2)]T,t = l.i.m. EE W

ji=0 j2=0

that converges in the mean-square sense is valid, where

T S2

Cj2ji = J V^Mj (S2^ ^i(si)0ji (si)dsids2 tt

and

T

Cf = / j (sf

t

are independent standard Gaussian random variables for various i or j.

Proof. In accordance to the standard relations between Stratonovich and Ito stochastic integrals (see (2.4) and (2.5)) we have w. p. 1

T

r[iF]T,t = J[iF\T,t + il{il=i2} j Uti)Uti)dh, (2.8)

t

where here and further 1a is the indicator of the set A. From the other side according to (1.46), we have

Pi P2

Zjl Zj2 — l{:i=:2}l{j1=j2}

j= e E cj:i,cj:2) - i{.i=«i

ji=0 j2=0

pi P2 min{pi,p2}

= l.i.m. E ^^ Cjji ifif - lOi^} lim E Cjiji. (2.9)

P1,P2^TO Z-' Z-' J1 J2 L J Z-/

ji=0 j2=0 ji=0

From (2.8) and (2.9) it follows that Theorem 2.1 will be proved if

T

1 r

- Utimt^dt^^Cj^. (2.10)

t ji=0

Note that in this section and in Sect. 2.1.2 we present two different proofs of the existence of a limit on the right-hand side of (2.10) for the polynomial and trigonometric cases.

Let us prove (2.10). Consider the function

K*(t1,t2) = K(t1,t2)^^l{t1^}Mti)Mtih (2-11)

where tbt2 G [t,T] and K(ti,t2) is defined by (1.6) for k = 2.

Let us expand the function K*(t1,t2) defined by (2.11) using the variable t1, when t2 is fixed, into the generalized Fourier series at the interval (t,T)

to

K*(ti,t2) = £ Cji(t2) j(ti) (ti = t, T), (2.12)

ji=0

where

T t2

Cji (t2) = J K*(ti ,t2)0ji (ti)dti = ^2(t2) y (ti)0ji (ti)dti. (2.13)

tt

The equality (2.12) is satisfied pointwise in each point of the interval (t,T) with respect to the variable t1, when t2 G [t,T] is fixed, due to a piecewise smoothness of the function K*(t1,t2) with respect to the variable t1 G [t,T] (t2 is fixed).

Note also that due to well known properties of the Fourier-Legendre series and trigonometric Fourier series, the series (2.12) converges when ti = t,T.

Obtaining (2.12) we also used the fact that the right-hand side of (2.12) converges when ti = t2 (point of a finite discontinuity of the function K(ti, t2)) to the value

i (K(U - 0, to) + K(U + 0, to)) = ^(uyUto) = K*(UM).

The function Cj (t2) is a continuously differentiable one at the interval [t, T]. Let us expand it into the generalized Fourier series at the interval (t,T)

to

Cj(t2) = E Cj2jij(t2) (t2 = t,T), (2.14)

j2=0

where

T T t2

Cj =J Cj (t2)0j2 (t2)dt2 = J ^2(t2)0j2 (t2^ ^i(ti)0ji (ti)dtidt2,

t t t

and the equality (2.14) is satisfied pointwise at any point of the interval (t,T) (the right-hand side of (2.14) converges when t2 = t,T).

Let us substitute (2.14) into (2.12)

to to

K *(ti ,t2) = EE Cj2ji j (ti) j (t2), (ti,t2) e (t,T)2, (2.15) ji=0 j2=0

where the series on the right-hand side of (2.15) converges at the boundary of the square [t,T]2.

It is easy to see that substituting ti = t2 in (2.15), we obtain

1 to to

-UhyUh) = E (2-16)

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j1=0 j2 =0

From (2.16) we formally have

T T

1 P r TO to

- / ik{hYh{ti)dti = / =

t t j1 =0 j2=0

to to „

ji j(t1)^j2(t1)dt

_A _A J

T

1 ) j ( t1 )dt1 =

ji=0 j2=0 t

Pi P2 n

lim lim £ j (t1)0j2 (t1)dt

i^OO P2—>-00 ^ ^ I

■ _A ,,' _A

T

p1^TO

1 ) j

ji=0 j2=0 t

Pi P2 min{pi,p2} to

= lim lim £ £Cj2ji l{ji=j2} = lim lim £ Cjiji = £ Cjiji . ji=0 j2=0 ji=0 ji=0

(2.17)

Let us explain the second step in (2.17) (the fourth step in (2.17) follows from the orthonormality of functions (s) at the interval [t,T]).

We have

T T

/to pi r

(t^j(t1)dt1 / Cji(t^j(t1)dt1 t ji=0 ji=0 t

T T

<y |^2(t1)Gpi (t1)| dt1 < of |Gpi (t1)| dt1,

tt

<

where O < oo and

(2.18)

TO

£ / (s)ds0j (t) = gp(t).

j=p+1{

Let us consider the case of Legendre polynomials. Then

\GPl(ti)\ = \

to ^

£ (2j1 + 1)y ^1(u(y))P,i(y)dyPji(z(t1))

ji=pi+1 _1

where

T - t T + t T + t 2

= —F^V + —ii—» = ~

2

2 T-f

and Pj(s) is the Legendre polynomial.

From (2.19) and the well known formula

dPJ+\{x) _ 12ti(x) = (2j + 1 )Pj(x),

dx

= 1, 2,

(2.19)

(2.20)

(2.21)

T

J

we obtain

\GP1(h)\ = l

E l(pji+i(z(ti)) - Pji-i(z (ti ))) ^i (ti )

ji =pi+i

Tt

z(ti)

(Pji+i(y) - Pji-i(y)) ^i(u(y))dy Pji(z(ti))

i

<

oo

< C0

T-t +——

E (Pji+i(z(ti))Pji (z(ti)) - Pji-i(z(ti))Pji (z(ti)))

ji=Pi+i

+

00

E ^i(ti)

1

ji=pi+i

2 ji + 3

(Pji+2(z(ti)) - Pji (Z(ti)))-

2ji - 1

(Pji (z(ti)) - Pji-2(z(ti))) -

Tt

z(ti)

1

2 J \ 2ji + 3

i

(Pji+2(y) - Pji (y))-

2ji - 1

(Pji(y) - Pji-2(y)) <(u(y))dy Pji(z(ti))

(2.22)

where C0 is a constant, ^ and ^ are derivatives of the function ^i(s) with respect to the variable u(y).

From (2.22

and the well known estimate for Legendre polynomials

|Pn(y)l <

K

-y2)1/4

, y e (-1,1), n e N,

(2.23)

where constant K does not depend on y and n, we have

< C0

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|Gpi(ti)| <

lim E (Pji+i(z(ti))Pji(z(ti)) - Pji-i(z(ti))Pji(z(ti)))

n—>-TO z-'

ji=Pi+i (

+

z(ti)

+Ci E

ji=Pi+i

1

+

dy

1

V

(1 - (z(ti))2)i/2 (1 - y2)i/4 (1 - (z(ti))2)i/4

!

<

1

< o0

lim (Pn+1 (z(t1 ))Pn(z(t1)) - Ppi(z(t1

n—yoo

£ i

72 V ("1 - (z(t ))2)1/2 ji=pi+H1 \(1 (z(t1)) )

< lim I - + —

+ o2

(1 - (Z(t1))2)1/4

+

<

oo

E ji

n pW (1 - (z(t1))2)1/2

+ o2

ji=

71 V(1 - (z(t1))2)

2 1/2

<c4i i- + E 1

1

(1 - (z(t1))2)

1

^1/4

<

>2 I /, ^ v ,i'2

P1 ji=pr+1 J!/ (1 - (z(t1 ))2)

ji =

^71 (1 - (z(t1))2)

2 1/4

<

K

+

pa (1 - (z(t1))2)1/2 (1 - (z(t1))2)1/4/'

where O0, O1,..., O4, K are constants, t1 G (t,T), and

oo

1 f dx 1

. Jl ~ J X2 Pi' ji=p1+1 1 pi

From (2.18) and (2.24) we get

T T

»to pi „

/ EOji(t1) j(t1)dt1 / Oji(t1 ) j(t1)dt1

{ ji=0 ji=0 {

<

(2.24)

(2.25)

<

K

dy

+

dy

PMi (1 -y2)1/2 I (1 -y2)1/4

0

if p1 ^ to. So, we obtain

T T ^

^ [Mhyhit^dt! = f ^dtimtm t t ji=0

T T

to „ to „ to

E J Oji(t1) j(t1)dt1 = Ey EOj2jijfa) j(t1)dt1 =

ji=0 t ji=0 t j2=0

1

T

TO TO „ TO

= E E J Cj2jij (ti) j (ti)dti = E Cjiji. (2.26)

ji=0 j2=0 t ji=0

In (2.26) we used the fact that the Fourier-Legendre series

TO

E Cj2ji j (ti) j2=0

of the smooth function Cji (ti) converges uniformly to this function at the interval [t + e,T - e] for any £ > 0, converges to this function at the any point ti e (t,T), and converges to Cj (t + 0) and Cj (T - 0) when ti = t, T.

More precisely, we have

T T-e

p TO „TO

J ECj'2jij (ti)^ji (ti)dti = J ECj'2ji j (ti)^ji (ti)dti + Ae + Be = t j2=0 t+e j2=0

to T - e

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= E Cj2ji / j (ti)0ji (ti)dti + Ae + Be =

j2=0 t+e

oo ( T t+e T\

= E Cj2ji J- j - j j (ti)^ji (ti)dti + Ae + Be =

j2=0 \t t T-e/

= E Cj2ji (1{ji =j2} - j (A) j (A) + j (0) j (0))) + Ae + Be = j2=0 V '

/to to \

= Cjiji - e E Cj2jij (A) j(A) + E Cj2jij (0) j (0) + Ae + Be, (2.27)

Vj2=0 j2=0 J

where 0 e [t,t + e], A e [T - e,T], and

/t+e TO T TO

E j j (ti)0ji Be = / ECj'2ji j (ti)^ji (ti)dti.

t j2=0 T-e j2 =0

In obtaining (2.27) we used the theorem on the mean value for the Riemann integral and orthonormality of the functions (x) for j = 0,1, 2 ...

Further, we have |Ae| + |B£| < eO, where O < to is a constant. Performing the passage to the limit lim in the equality (2.27), we get

e^+0

T

oo

/ £ Cj'2 j ) j ) — jl * { j2=0

Then (see (2.26

T

œ „ œ œ

E / ECj'2ji j (t0 j (t1)dt1 — £ Cjij

_A _A _n

jl=0 t j2 =0

'1)^71 — / . ^jiji

ji =0

and the relation (2.10) is proved for the case of Legendre polynomials.

Let us consider the trigonometric case and suppose that (x)}°=0 is a complete orthonormal system of trigonometric functions in L2([t,T]).

Denote

Q def

T T

„ œ pi „

/ (t1 ) j(t1 )dt1 / Cji(t1 ) j(t1 )dt1

t ji=0 ji=0 i

t ti

„00 i

/ £ ^fa) j fa) / j (£)* t ji =Pi+1 t

We have

S2pi —

T œ 2pi T

/ fa) j (t1)dt1 / Cji fa) j fa)dt2

t ji =0 ji=0 t

T œ ti

T-t

/ £ ^2fa) jfa) / (0) j(£)*

t ji=2pi+1 { T / ti

t ji=Pi + ^ t

ti

T~t

, / ! ( N 2nj1(s - tK 2nj1 (t1 - t) .

+ / '^i(s)cos———;—as cos———;- I aii

Tt

2

1

n

T

fUmuti) J2 ~sin

i V ji=pi+iji

1 . 2nji(ti - t)

Tt

T — t TO 1

H—^—E -( '0i(ii)-'0i(i)cos

j'i=pi+i ji

2n

2nji{ti~t) T-i

Tt

T

< Ci

= Ci

/ '02(il) E _Sin

i ji=Pi+i ji

T-t 1 . 2nji (ti - t)

<

oo

T

1

T-t . 2nji(ti -1)

dti

pi

E fj 'h(h)sm T_ ji =pi + ^i "t

dti

+

Pi

where constants Ci, C2 do not depend on pi.

Here we used the fact that the functional series

^ 1 . 2nji (ti - t) > —sm-

ji=^i

Tt

(2.28)

(2.29)

converges uniformly at the interval [t + e, T - e] for any e > 0 due to Dirichlet-Abel Theorem, and converges to zero at the points t and T. Moreover, the series (2.29) (with accuracy to a linear transformation) is the trigonometric Fourier series of the smooth function K(ti) = ti - t, ti e [t,T]. Thus, (2.29) converges to the smooth function at any point ti e (t,T).

From (2.28) we obtain

S2pi < C3

T

E i(uT)-ut)- f ji=pi+iji V {

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Tt

C2 C4 + — < —, Pi Pi

(2.30)

where constants C2, C3, C4 do not depend on pi.

t

Further,

S^-1

T ti

«00 i

J £ «t1) j(toy «0) j(0)*

t ji=2pi t

T ti

S2pi + / ^2(t1)02pi fa) / (^)02pi (0)*

<

< S2pi +

2

Tt

T

i

Tt

Moreover,

T

i

. 2np1(t1 -1)

T

■ipo{tijcos-——-- -0i(ii)sm

2np1

Tt

T

- [m^z^de]^.

. (2.31)

(2.32)

The relations (230)-(232 imply that

S2pi-1 <

where constant O5 is independent of p1. From (2.30) and (2.33) we obtain

Cs P1

spi —

T oo ti

J ^2fa) j fa)y (0) j (0)d0dt!

t ji=pi+1 t

(2.33)

K

<--> 0 2.34

P1

if p1 ^ to, where constant K does not depend on p1 (p1 G N).

Further steps are similar to the proof of (2.10) for the case of Legendre polynomials. Theorem 2.1 is proved.

Note that the estimate (2.34) will be used further.

2.1.2 Approach Based on Theorem 1.1 and Double Fourier—Legen-dre Series Summarized by Pringsheim Method

In Sect. 2.1.1 we considered the proof of Theorem 2.1 based on Theorem 1.1 and double integration by parts (this procedure leads to the requirement of double continuous differentiability of the function ^i(t) at the interval [t, T]). In this section, we formulate and prove an analogue of Theorem 2.1 but under the weakened conditions: the functions ^i(t), ^2(t) only one time continuously differentiable at the interval [t,T]. At that we will use the double Fourier series summarized by Pringsheim method.

Theorem 2.2 [13]-[16], [27], [46]. Suppose that { j(x)}=0 is a complete

orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, ^(s), (s) are continuously differentiable functions on [t,T]. Then, for the iterated Stratonovich stochastic integral

■T ^ ¿2

j*[^(2)]t,î =/ «Î2)/ }dft(;i)dft(;2) (il,i2 = 1,...,m)

the following expansion

P1 P2

./•[./2>]r,( = l.i.m. £ £C,,cji-'cir (2.35)

jl=0 j2=0

that converges in the mean-square sense is valid, where

T S2

Cj2ji = V^M j(S2W ^i(si)0ji(si)dsids2 (2.36)

and

T

j = / j (sf

t

are independent standard Gaussian random variables for various i or j.

Proof. Theorem 2.2 will be proved if we prove the equality (see the proof of Theorem 2.1)

T 1 f

- = (2-37)

t ji=0

where Cjj is defined by the formula (1.8) for k = 2 and j = j2. At that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]).

Firstly, consider the sufficient conditions of convergence of double Fourier-Legendre series summarized by Pringsheim method.

Let Pj(x) (j = 0,1, 2,...) be the Legendre polynomial. Consider the function f (x,y) defined for (x,y) G [—1,1]2. Furthermore, consider the double Fourier-Legendre series summarized by Pringsheim method and corresponding to the function f (x,y)

n m i

lim EEôVi2/ + l)(2i + l^^y) j =0 ¿=0

def

where

œ 1

^ £ -V/(2j + l)(2, + l)CJPi(x)P;-(y), (2.38)

i,j=0

C¿ = iV(2j + l)(2i + l) í fix^PiWPjMdxdy. (2.39)

[-1,1]

2

Consider the generalization for the case of two variables [108] of the theorem on equiconvergence for the Fourier-Legendre series [109 .

Proposition 2.1 [108]. Let f (x,y) G L2([—1,1]2) and the function

f (x, y) (1 - x2)—1/4 (1 - y2)—1/4

is integrable on [—1,1]2. Moreover, let

|f(x,y) — f(u, v)| < G(y)|x — u| + H(x)|y — v|,

where G(y),H(x) are bounded functions on [—1,1]2. Then for all (x,y) G (—1,1)2 the following equality is satisfied

/ n m

lim ^^-vWTTK2ÍTT)C¿Pt(xOP^y)

/ OO \

\j=0 ¿=0

— (1 — x2)—1/4(1 — y2)—1/4Snm(arccosx, arccosy, F) ) = 0. (2.40)

At that, the convergence in (2.40) is uniform on the rectangle

[-1 + e, 1 - e] x [-1 + 6, 1 - 6] for any e, 6 > 0,

Snm(#, F) is a partial sum of the double trigonometric Fourier series of the auxiliary function

F($,(p) = \/\sm6\\/\smLp\f(cos6, cos(p), $,(p G [0,7r],

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and the Fourier coefficient Cj is defined by (2.39). Proposition 2.1 implies that the following equality

/ n m \

lim EE o V^J + l)(2i + l^JP^Q/) - /(xyy) = 0 (2.41)

n,m 2

\j=0 «=0 /

is fulfilled for all (x,y) G (-1,1)2, and convergence in (2.41) is uniform on the rectangle

[-1 + e, 1 - e] x [-1 + 6,1 - 6] for any e,6> 0

if the corresponding conditions of convergence of the double trigonometric Fourier series of the auxiliary function

g(x,y) = f (x,y) (1 - x2)i/4 (1 - y2)i/4 (2.42)

are satisfied.

Note also that Proposition 2.1 does not imply any conclusions on the behavior of the double Fourier-Legendre series on the boundary of the square [-1.1]2.

For each 6 > 0 let us call the exact upper edge of difference |f (t ') - f (t'')| in the set of all points t', t'' which belong to the domain D as the module of continuity of the function f (t) (t = (ti,... ,tk)) in the k-dimentional domain D (k > 1) if the distance between t', t'' satisfies the condition p (t', t'') < 6.

We will say that the function of k (k > 1) variables f (t) (t = (ti,... ,tk)) belongs to the Holder class with the parameter a G (0,1] (f (t) G Ca(D)) in the domain D if the module of continuity of the function f (t) (t = (ti,... ,tk)) in the domain D has orders o(6a) (a G (0,1)) and O(6) (a = 1).

In 1967, Zhizhiashvili L.V. proved that the rectangular sums of multiple trigonometric Fourier series of the function of k variables in the hypercube [t, T]k converge uniformly to this function in the hypercube [t, T]k if the function

belongs to Ca([t,T]k), a > 0 (definition of the Holder class with any parameter a > 0 can be found in the well known mathematical analysis tutorials [110]).

More precisely, the following statement is correct.

Proposition 2.2 [110]. If the function f (xi,..., xn) is periodic with period 2n with respect to each variable and belongs in Rn to the Holder class Ca(Rn) for any a > 0, then the rectangular partial sums of multiple trigonometric Fourier series of the function f (x1,..., xn) converge to this function uniformly in Rn.

Let us back to the proof of Theorem 2.2 and consider the following Lemma.

Lemma 2.1. Let the function f (x,y) satisfies to the following condition

lf (x,y) - f (xi,yi)l < Cilx - xi| + C2|y - yi

where Ci, C2 < œ and (x,y), (xi,yi) G [—1,1]2. Then the following inequality

is fulfilled

|g(x,y) -g(xi,yi)| < ^p1^

(2.43)

where g(x,y) in defined by (2.42),

P = \/{x-xi)2 + (y-yi)2,

(x,y) and (xi,yi) G [-1,1]2, K < oo.

Proof. First, we assume that x = xi, y = yi. In this case we have

|g(x,y) - g(xi,yi)| =

(1 - x2)i/4 (1 - yT4(f(x,y) - f(xi,yi)) +

2\i/4

+f (xi, yi^(1 - x2)i/4 (1 - y^i/4 - (1 - x2)i/4 (1 - y2)i/4)

+ C3

where C3 < œ. Moreover,

< Ci |x - xi| + C2 |y - yi| +

(1 - x2)i/4 (1 - y2)i/4 - (1 - x2)i/4 (1 - y2)i/4

(1 - x2)i/4 (1 - yT4- (1 - x2r (1 - y2)

2) 1/4

<

(2.44)

(1 - x2)i/4((1 - y^i/4 - (1 - yOi/^ +

+ (1 - !/?)

2\ i/4

2)i/4 2)i/4

— - - x- '

X )

'D

<

(1 - y2)1/4 - (1 - y2)

2) i/4

+

(1 - x2)i/4 - (1 - x2)i/4

'i

(2.45)

(1 - x2)i/4 - (1 - x2) - x)i/4 - (1 - xi)i/4

2) i/4

<

< k2

+ (1 - xi)i/4 ((1 + x)i/4 - (1 + xi)i/4) - x)i/4 - (1 - xi)i/4| + 1(1 + x)i/4 - (1 + xi)i/4|) , (2.46)

where K2 < to.

It is not difficult to see that

(1 ± x)i/4 - (1 ± xi)i/4 1(1 ± x) - (1 ± xi)|

((1 ± x)1/2 + (1 ± xi)1/2) ((1 ± x)1/4 + (1 ± xi)1/4)

\XI-X\1,A-

|x2 - x|1/2

|x2 - x|1/4

<

(1 ± x)1/2 + (1 ± xi)1/2 (1 ± x)1/4 + (1 ± xi)1/4

< |xi - x|1/4. (2.47)

The last inequality follows from the obvious inequalities

|x2 - x|1/2

(1 ± x)1/2 + (1 ± xi)1/2

|x2 - x|1/4 {l±x)lA + {l±xlyA

< 1,

1.

From (2.44)-(2.47) we obtain

|g(x,y) -g(xi,yi)| <

< Ci|x - xi| + C2|y - yi| + C4 (|xi - x|1/4 + |yi

<

C5P + C6P1/4 < Kp1/4,

where C5, C6, K < to.

The cases x = xi5 y = yi and x = xi, y = yi can be considered analogously to the case x = xi5 y = y1. At that, the consideration begins from the inequalities

|g(x,y) - g(xi,yi)| < K2 (1 - y2)i/4 f (x,y) - (1 - y?)i/4 f (xi,yi)

(x = xi, y = yi) and

|g(x,y)- g(xi,yi)| < K2 (i- x2)i/4f(x,y) - (i- x?)vV(xi,yi)

(x = xi5 y = yi), where K2 < œ. Lemma 2.1 is proved.

Lemma 2.1 and Proposition 2.2 imply that rectangular sums of double trigonometric Fourier series of the function g(x,y) converge uniformly to the function g(x,y) in the square [—1, 1]2. This means that the equality (2.41) holds.

Consider the auxiliary function

>2(ti)«Î2), ti > t2 K(ti,t2)H , ti,t2 e [t,T] (2.48)

^i(ti)^2(t2), ti < t2

2\ i/4

and prove that

|K'(ti, t2) — K'(ti, t*) | < L (|ti — til + |t2 — t2|),

(2.49)

where L < to and (ti,i2), (ti,t2) G [t,T]2.

By the Lagrange formula for the functions ^(¿i), ^2(ti) at the interval

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[min{ii,ii} , max{ii,ii}]

and for the functions ^(¿1), ^2(£2) at the interval

[min {t2, ¿1} , max {t2, ¿1}]

we obtain

<

|K'(ti,t2) — K'(t*,t*)| < >2(ti)^i(t2), ti > t2 f ^2(ti)^i(t2), t* > t2

^i(ti)^2(t2), ti < t2 Uifa^fa), t*< t2

+

+ Li |ti - ti | + L2 |t2 - t2| , Li,L2 < to.

(2.50)

We have

>2(ti)^i(t2), ti > t2 (^2(ti)^i(t2), ti > t2

>i(ti)^2(t2), ti < t2 lXti)^2(t2), ti< t2

0, t2 > t2, ti > t^ or t2 < t2, ti < t2

= <i ^2(ti)^i(t2) - ^i(ti)^2(t2), ti > t2, ti < t2.

>i(ti)^2(t2) - ^2(ti)^i(t2), ti < t2, t2 > t2 By Lagrange formula for the functions ^2(t2), ^2(t2) at the interval

[min{t2, t2}, max{t2, t2}]

we obtain the estimate

>2(ti)^i(t2), ti > t2 f^2(ti)^l(t2), ti > t2 |

|

(2.51)

^l(tl)^2(t2), ti < t2 Ul(tiW>2(t2), ti< t2

0, t2 > t2, tif > ti or t2 < t2, tif < t2 < L3|t2 - ti^ , (2.52)

1, t2 < t2, > or t2 > t2, < t2 where L3 < to.

Let us show that if t2 < t2, tf > t2 or t2 > t2, tf < ti, then the following inequality is satisfied

|t2 - ti| < |ti- ti| + |ti- t2|. First, consider the case t2 > t2, ti < t2. For this case

(2.53)

Then

and (2.53) is satisfied.

t2 + (ti- t2) < t2 < ti.

(ti- ti) - (ti- t2) < t2 - ti < 0

For the case t2 < t2, t| > t2 we obtain

ti + (ti - tl) < ti < t2.

Then

(tl - ti) - (t2 - ti) < tl - t2 < 0 and also (2.53) is satisfied.

From (2.52) and (2.53) we have

>2(tl)^l(t2), ti > t2 ( ^2(tl)^l(t2), ti > t2

^l(tl)^2(t2), ti < t2 I^l(ti)^2(t2), ti< t2

<

0, t2 > t2, ti > ti or t2 < t2, ti < t2

< L3 (|ti - ti| + |t2 -12|)< <

1, t2 < t2, ti > ti or t2 > t2, t* < t2

1, t2 > t2, ti > ti or t2 < t2, ti < t2

< L3 (|ti - ti| + |t2 -12|) I

1, tl < t2, ti > t2 or tl > t2, ti < t2

= L3 (|ti- ti| + |t2 - t2|). (2.54)

From (2.50), (2.54) we obtain (2.49). Let

T - t T + t T - t T + t

where x,y G [-1,1]. Then

(h(x)) ^l (h(y)) , x > y

,

^i (h(x)) ^2 (h(y)), x < y

where x,y G [-1,1] and

T — t T + t

Ms) = —* + -f- (2-55)

The inequality (2.49) can be rewritten in the form

|K''(x,y) - K''(x*,y*)| < L* (|x - x*| + |y - y*|), (2.56)

where L* < to and (x,y), (x*,y*) G [—1,1]2.

Thus, the function K''(x,y) satisfies the conditions of Lemma 2.1. Hence, for the function

K"(x,y) (1 - x2)1/4 (1 - y2)1/4

the inequality (2.43) is correct.

Due to the continuous differentiability of the functions (h(x)) and (h(x)) at the interval [-1,1] we have K''(x,y) G L2([-1,1]2). In addition

/ 1 x

f K»(x, y)dxdy <(l r 1 f 1

J (1 - x2)1/4(1 - y2)1/4 " \ J (1 - X2)1/4 J (1 - y2)1/4 [-1,1]2 V1 -1

+ I -oTTTT [ --T-77-dydx | < OO, C < OG.

J (1 (1 -y2)1/4 * I

-1 x /

Thus, the conditions of Proposition 2.1 are fulfilled for the function K''(x, y). Note that the mentioned properties of the function K''(x,y), x,y G [-1,1] also correct for the function K'(t1,t2), t1 , t2 G [t,T].

Remark 2.1. On the basis of (2.49) it can be argued that the function K'(t1, t2) belongs to the Holder class with parameter 1 in [t, T]2. Hence by Proposition 2.2 this function can be expanded into the uniformly convergent double trigonometric Fourier series in the square [t,T]2, which summarized by Pring-sheim method. However, the expansions of iterated stochastic integrals obtained by using the system of Legendre polynomials are essentially simpler than their analogues obtained by using the trigonometric system of functions (see Chapter 5 for details).

Let us expand the function K'(t1,t2) into a multiple (double) Fourier-Legendre series or trigonometric Fourier series in the square [t,T]2. This series is summable by the method of rectangular sums (Pringsheim method), i.e.

T T

«1 n-2 „ „

K'(t1,t2)= lim / K' (t1,t2)j (i1)j (t2)dt1dt2 • j (i!)j (t2) =

ji=0 j2=0 t t

/ T t2

ni n2 I « 2

= lim EE / (i2W (tl)dtidt2 +

ni,n2^^ j-0 j-0 W y

ji=0 j2=0 \ t t

T T

+ / fe) j (t2M ^2(tl)0ji (tl)dtl I dt20ji (tl)0j2 (t2) =

t t2

ni n2

^^ 1 ^ W j162

lim E E (Cj2ji + Cij2) j(tl) j(t2), (2.57)

ji=0 j2=0

where (tl, t2) G (t,T)2. At that, the convergence of the series (2.57) is uniform on the rectangle

[t + £, T — e] x [t + 5, T — 5] for any £, 5 > 0 (in particular, we can choose £ = 5).

In addition, the series (2.57) converges to K'(tl,t2) at any inner point of the square [t,T]2.

Note that Proposition 2.1 does not answer the question of convergence of the series (2.57) on the boundary of the square [t,T]2.

In obtaining (2.57) we replaced the order of integration in the second iterated integral.

Let us substitute tl = t2 in (2.57). After that, let us rewrite the limit on the right-hand side of (2.57) as two limits. Let us replace j with j2, j2 with j, nl with n2, and n2 with nl in the second limit. Thus, we get

n1 n2 1 lim EE^fe^^^^^^1^^)' he(t,T). (2.58) ji=0 j2=0

According to the above reasoning, the convergence in (2.58) is uniform on the interval [t + £, T — e] for any £ > 0. Additionally, (2.58) holds at each interior point of the interval [t,T].

Let us fix £ > 0 and integrate the equality (2.58) at the interval [t + £, T — e]. Due to the uniform convergence of the series (2.58) we can swap the series and the integral

ni n2 T—'5 1 T—'5

lim E E/ 'MhyHtiWh. (2.59)

ni,n2^^ Z-' Z-'J 2 J

ji=0 j2=0 t+5 t+5

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Lemma 2.2. Under the conditions of Theorem 2.2 the following limit

limV Cj

ji=0

exists and is finite, where Cj is defined by (2.36) if j = j2, i.e.

T t2

Cjiji = J ^(¿2) j (¿2) ^ (ti)0ji (t1)dtidt2.

t t

Lemma 2.2 has already been proved in Sect. 2.1.1. Further, in this section, another proof of Lemma 2.2 is given. This will allow us to obtain useful estimates that will be used later in Chapter 2.

Applying the equality (2.59) for n1 = n2 = n and Lemma 2.2, we get

T-£ n T-£

- / '0i(ii)'02(ii)rfii = lim V Cnn / =

2 J n—z—' J

t+£ j1 'j2=° t+£

n / T t+£

= J——TO ^ ^ji j j (t1)0j2 (t1)dt1 - j j (t0j (t1)dt1-

ji,j2 =0 \t t

- j j (t1 )0j2 (t1)dt1 = T-£

= nlim S Cj'2ji ( 1{ji=j2} - ( j (0)j (0) + j (A)^j2 (A) j S j = 00 ji ¿2=0 V V ^ ^

nn

= lim£ Cjiji - slim £ Cj2ji( j (0)j (0) + j (A)j (A)) , (2.60) ji=0 ji ,j2=0 V 7

where 0 G [t,t + s], A G [T - s,T]. In obtaining (2.60) we used the theorem on the mean value for the Riemann integral and orthonormality of the functions (x) for j = 0,1, 2 ...

Applying (2.60), we obtain

s lim £ Cj2jYj(0)j(0) + j(A)j(A)) =

n—>oo z—* \ /

ji,j2=0 V 7

77.

T-e

It 10 n

lim E Cjiji — lim E Cj2ji / j (t0j (tl)dtb

?.—^oo z—* n—>oo z—' /

j =0 ji?j2=0 t+e

ji=0

where the limits

TT e

/ i/ /1/

lim E Cjiji , lim E j'i / j (t0j (tl)dtl

7—TO z—' n—TO z—' J

-' n ji ,j2=0 t+e

ji=0

exist and are finite (see Lemma 2.2 and the equality (2.59)). This means that the limit

£ lim E Cj2ji( j (0)j (0) + j (A)j (A))

n—>-to z—' v )

ji,j2=0 V 7

also exists and is finite.

Suppose that the following relations

E Cj2 ji j (T) j (T)

j'ij2=0

< K < TO,

E Cj2ji j (t)^ji (t) j'ij2=0

< K < TO

(2.61)

are satisfied for n G N (the relations (2.61) will be proved further in this section); constant K does not depend on n.

Note that

£ lim V Cj2ji ( j (0) j (0) + j (A) j (A)

77—>-00 Z-' V

j'i,j2=0

= lim £

n—>oo

E Cj2ji j (0)^j2 (0)+ E Cj2ji j (A)j (A)

ji,j2=0 ji,j2 =0

(2.62)

Using (2.58) (nl = n2 = n) and (2.61), we obtain

£ lim

n—> CO

E Cj2ji j (0)^j2 (0)+ E Cj2ji j (A)^j2 (A)

ji ,j2=0 ji,j2=0

<

< £ lim

n—> CO

E Cj2ji j (Wj2 (0) ji?j2=0

+

E Cj2ji j (A)j (A) j'i,j'2=°

< 2£Kl — 0 (2.63)

71

n

71

if s — +0, where 0 G [t, t + s], A G [T - s, T], constant K1 is independent on n.

Performing the passage to the limit lim in the equality (2.60) and taking

£—+0

into account (2.62), (2.63), we get

TT

1 C

- = YtCjdl- {2.64:)

t ji=0

Thus, to complete the proof of Theorem 2.2, it is necessary to prove (2.61). To prove (2.61), as well as for further consideration, we need some well known properties of the Legendre polynomials

The complete orthonormal system of Legendre polynomials in the space L2([t,T]) looks as follows

^Wf^iU'-^)^), ¿ = 0,1,2,..., (2.65)

where Pj (x) is the Legendre polynomial.

It is known that the Legendre polynomial Pj (x) is represented as

At the boundary points of the orthogonality interval the Legendre polynomials satisfy the following relations

Pj (1) = 1, Pj (-1) = (-1)j,

Pj+1(1) - Pj(1) = 0, Pj+1(-1) + Pj(-1) = 0,

where j = 0, 1, 2, . . .

Relation of the Legendre polynomial Pj (x) with derivatives of the Legendre polynomials Pj+1(x) and Pj_1(x) is expressed by the following equality

Pj(x) = (p;+1(x) - P^xj) , j = 1,2,... (2.66)

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The recurrent relation has the form

xP{x) = u±lEmj = 1>2>...

jk j 2j + l J

Orthogonality of the Legendre polynomial Pj (x) to any polynomial Qk(x) of lesser degree k we write in the following form

Tl

J Qk(x)Pj(x)dx = 0, k = 0,1, 2,..., j — 1. -l

From the property

1 r 0 if k = j

/ Pk(x)Pj(x)dx = ^

—1 [2/(2j + 1) if k = j

it follows that the orthonormal on the interval [-1 , 1] Legendre polynomials determined by the relation

p;(x) = \/^±pJ(x), j = 0,1,2,...

Remind that there is the following estimate [106

K

+ ve {-1,1), J = 1,2,..., (2.67)

where constant K does not depend on y and j. Moreover,

|Pj(x)| < 1, x G [—1,1], j = 0, 1,... (2.68)

The Christoffel-Darboux formula has the form

±(2j + 1 )PJ(x)PJ(y) = (n + (2.69)

j=0 y x

Let us prove (2.61) (see [27])). From (2.69) for x = ±1 we obtain

¿(2 j + 1 )P3{y) = (n + (2.70)

j=0 y - 1

£(2 j + 1)(-1 ypj(y) = (n + 1)(-1 Y^j^m^m. (2.71)

,Pn+i(y) + Pn(y)

AZJ -t- in-ir^iy; = -t- 1JI-1J

j=0

From the other hand (see (2.66))

E(2j + 1)Pj (y) = 1 + £(2j + 1)Pj (y) =

j=0 j=1

nn

= 1 + Ej (y) - P;:-1(y)) = 1 + (E(Pj+1(y) - Pj-1(y))) =

j=1 j=1

= 1 + (Pn+1 (x) + Pn(x) - x - 1)' = (Pn(x) + Pn+1 (x))' (2.72)

and

nn

E(2j + 1)(-1)j Pj (y) = 1 ^(-1)j (2j + 1)Pj (y) = j=0 j=1

nn

= 1 + £(-1j(Pj+1 (y) - Pj-1(y)) = 1 + (E(-1)j(Pj+1(y) - Pj-1(y)0 =

j=1 j=1

= 1 + (( 1)n(Pn+1 (x) - Pn(x)) - x + 1)' = ( 1)n(Pn+1 (x) - Pn(x))'. (2.73)

Applying (ETDMETB]), we get

(n + = {Pn{x) + P^))', (2.74) y - 1

(n + i)^±iM±^M = {Pn+l{x) _ pn(a0)/. (2.75) Let us prove the boundedness of the first sum in (2.61). We have

n

E Cj2ji (T)^ji (T ) =

ji,j2=0

1 y

1 n n

= I E E^ + i)^! + J UHy))Pj2(y) J Uh{yi))Pn{yi)dyidy = j2=0 ji=0 _1 _1

n

1 y

1nn

= 4 / + / +

— l j2=0 — ji=0 Tl Ty

= \J MKy)) J Uh{yi))d{Pn+l{yi) + Pn(yi)) I d(P„+i(y) + p„(y)) =

Tl Ty

" "...............

= 4 I MHy)) ( I MHyi))d(Pn+i(yi) + P„(yi))J d(Pn+i(y) + P„(y))+

—1 \—l Tl Ty

".....".....- - >

A(%)) ^ ^i(M2/i)KBn+i(2/i) + Pn(2/i))j d(Pn+i(y) + Pn(y)) =

where

T — t T + t

A(%)) = ^2(^(2/)) " %) = —y + (2.76)

Further,

1 2

h = l([ MKyMPn+i(y) + Pn(y))) =

—l

l

T—t

dy < C1 < to,

(T) - [(Pn+i(y) + Pn(yM(%))^-^

2

—l

where ^ is a derivative of the function with respect to the variable y, constant C1 does not depend on n.

By the Lagrange formula we obtain A(%)) = Q(T - ¿)(y - 1) + t) - ipi Q(T - ¿)(y - 1) + t) =

= MT) - MT) + {y-1) (m) - m) -t) =

= Cl + ay (y — 1), (2.77)

where |ay | < to and C1 = (T) — (T).

Let us substitute (2.77) into the integral 12

I2 = I3 + I4,

where

1 / y

I3 = J ay(y — 1) J ^i(%i))d(Pn+i(yi) + Pn(yi)) | d(Pn+i(y) + Pn(y)), 1 / y

I4 = C\H J ^%iMPn+i(yi) + Pn(yi)H d(Pn+i(y) + Pn(y)).

Integrating by parts and using (2.74), we obtain i

— i

y

-J (P??+i(yi) + P??(yi))'0i(h(yi))^(T — t)dyi^dy. —i

Applying the etimate (2.67) and taking into account the boundedness of ay and ^/i(h(yi)), we have that |13| < to.

Using the integration order replacement in 14, we get

I4 = Cij fa(h(yi)W J d(Pn+i(y) + Pn(y)H d(Pn+i(yi) + Pn(yi)) =

—i \yi )

i i

= Ci J^i(h(yi))d(Pn+i(yi) + Pn(yi)) y d(Pn+i(y) + Pn(y)) — —i —i

— Cij ^i(h(yi)) ^ /1 d(Pn+i(y) + Pn(y))j d(Pn+i(yi) + Pn (yi)) =

= — ^6.

Consider I5

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Tl

I5 = 2C1 y ^l(h(yl))d(Pn+l(yl) + Pn(yl)) = —l

= 2Ci (T) - J(Pn+1(yi) + - t)dy 1

Applying the estimate (2.68) and using the boundedness of ^(h(yl)), we obtain that |I5| < to.

Since (see (2.77))

MHy))=1>i(\(T-t)(y-l)+T\ =

= (T) + (y - -t) = C2 + f3y(y - 1),

where |^y | < to and C2 = (T), then

y

1 / yi \

)

I6 = C3 / | / d(Pn+l(y) + Pn(y)) j d(Pn+l(yl) + Pn(yl))+

1 / yi

+Cl y ^yi(yl — 1) y d(Pn+l(y) + Pn(y)H d(Pn+l(yl) + Pn(yl)) =

Tl 2 = T ijd(pn+i(y) + pn{y)) I +

1 / yi

+C\J^(ft-'H'+y-)-^)) i/d(p„+l(!/) + p„(,)) I ^ Tl

= 2C3 + Cl y ^ (n +1)(Pn+l(yl) — Pn(yl))(Pn+l(yl)+ Pn(yl))dyl. l

Using the estimate (2.67) and taking into account the bounedness of ^yi, we obtain that |16| < to. Thus, the boundedness of the first sum in (2.61) is proved.

Let us prove the boundedness of the second sum in (2.61). We have

n

E Cj2ji j (t)^ji (t) =

ji ,j2=0

T1 Ty

1 n n

I ' l ' n n

4 E E^ +1+ 1)(-1)J'1+J'2 / Uh(y))PJ2(y) / UKyi))Ph{yiY j =0 ji=0 -1 -1

xdy1dy =

T1 n Ty

= \ juHy)) E(2j2 + l)Ph(y)(-Vh jMHyi))x 4-1 j2=0 -1 n

x^ (2j1 + 1)Pji (y1)(-1)ji dy1dy = ji=0

J h(h(y)) ( jMh(yi))d(Pn+i(yi) - pn(yi)) I X

xd(Pn+1(y) - Pn(y)) = = \jMh(y)) ^jMHyiMPr+M - Pn(yi)) j d(Pn+1(y) - Pn(y)) +

+iy* A(h(y)) ^jMHyMPr^M - Pn(yi))^ d(Pn+1(y) - pn(y)) =

4/i+zj2'

where A(h(y)), h(y) are defined by (2.76). Further,

T1 2 Jl = l\ J Mh(y))d(Pn+1(y) - Pn(y)) | =

= I " J(Pn+i(y) ~ Pn(y)Wh(y))^d^j < K\ < oo,

(2.78)

where ^ is a derivative of the function with respect to the variable y, constant Kl is independent of n.

By the Lagrange formula we obtain

A(h(y)) = Q(T - t)(y + 1) +1) - ^ Q(T - t)(y + 1) + t) =

= ut) - Mt) + (y+1) (mh) - rt(py) )\(T -t) =

= K2 + Yy (y + 1), (2.79)

where |Yy | < to and K2 = ^2(t) — ^l(t). Consider J2

Tl Tl

J = J A(h(y))d(Pn+l(y) — Pn(y)^ ^l(h(yl))d(Pn+l(yl) — Pn(yl)) — —l —l

— J A(h(y)) | J ^l(%l))d(Pn+l(yl) — Pn(yl)H d(Pn+l(y) — Pn(y)) =

—l y

= J3 J4 — J5.

The integral J4 was considered earlier (see J1 and (2.78)), i.e. it has already been shown that | J4| < to. Analogously, we have that | J3| < to.

Let us substitute (2.79) into the integral J5

J5 = J + J7,

where

J = J Yy (y + 1) J ^l(h(yl))d(Pn+l(yl) — Pn (yl)) | d(Pn+l(y) — Pn (y)),

—l y

J7 = K2 J (Jfa (h(yl))d(Pn+l(yl) — Pn(yl)) I d(Pn+l(y) — Pn(y)).

—l y

Integrating by parts and using (2.75), we get 1

—l

1

" J(Pn+1 (yi) - Pn(yi) WHyi))\{T- t)dy^j dy. y

Applying the etimate (2.67) and taking into account the boundedness of Yy and (h(yl)), we have that | J61 < to.

Using the integration order replacement in J7, we obtain

1 / yi

J7 = K2J ^l(%l)) J d(Pn+l(y) — Pn(y)) I d(Pn+l(yl) — Pn(yl)) =

Tl Tl

= K2J ^l(h(yl))d(Pn+l(yl) — Pn(yl)^ d(Pn+l(y) — Pn(y)) — K2J8 = —l —l

= K2J42(—1)n — K2J8,

where

J = J ^l(%l)) J d(Pn+l(y) — Pn(y)H d(Pn+l(yl) — Pn(yl)).

— 1 \yi

Since (see (2.79))

Mh(y)) = fa(±(T-t)(y + l)+t\ =

= + (y + 1 WAPy)1-^ -t) = K3 + ey(y + 1), (2.80)

where |sy | < to and K3 = ^1(t), then

J8 = K3 / / d(Pn+1(y) - Pn(y)) I d(Pn+1(y1) - Pn(y1)) + -1 Vyi

+ J sy(y + 1) I J d(Pn+1(y) - Pn(y)H d(Pn+1(y1) - Pn(y1)) =

-1 Vyi

1

+ en{yi + l)(n + 1)(P +lfal) + PnM) {pJyi) _ pM)<];y =

y1 + 1

-1

1

= 2K3 + J syi(n + 1)(Pn+1(y1) + Pn(y1))(Pn(y1) - Pn+1(y1))dy. (2.81) -1

When obtaining the equality (2.81), we used (2.75). Applying the estimate (2.67) and taking into account the bounedness of syi, we obtain that | J8| < to. Thus, the boundedness of the second sum in (2.61) is proved. The relations (2.61) are proved. Theorem 2.2 is proved.

Consider another proof of Lemma 2.2. We will prove that

n

Cjiji

ji=0

is the Cauchy sequence for the cases of Legendre polynomials and trigonometric functions.

Consider the case of Legendre polynomials. Below in this section we write lim instead of lim . Fix n > m (n,m G N). We have

n>m

T s

nn

E j = J ^2(s)0ji (s) / )0ji (t)dTds =

ji=m+1 ji=m+1 t t

Tt

n 1 x

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E (2jl + 1) / ^2(h(x))Pji(x) /* ^l(h(y))Pji(y)dydx

ji=m+l _l

1

l

E / ^l(h(x))^2(h(x)) (Pji+l(x)Pji(x) — Pji(x)Pji—l(x)) dx—

j'i=m+^l

n 1 x

£ J fa(h(x))Pn(x) J (PJ1+M-Pn-M) m(y))dydx

j'i=m+l —l —l

1

/n

fa{h{x)yilj2{h{x)) E (^/i+i^)^^) ~ Pn{x)Pn-i(x))dx

_l ji=m+l

n 1 1 E / [ Pn{x)Uh{x))dxdy

8

Tt

8

j'i=m+^l 1

Tt

^l(h(x))^2(h(x)) (Pn+l(x)Pn(x) — Pm+l(x)Pm(x)) dx+

l

(T — t)

2

E

1

8 2jl + 1

ji=m+l J1 ^

(Pji+l(y) — Pji—l(y)) (h(y))x

x (Pji+l(y) — Pji—l(y)) «%)) +

Tt

(Pji+l(x) — Pji—l(x)) ^2(h(x))dx dy,

(2.82)

where , ^2 are derivatives of the functions , with respect to the variable h(y) (see (2.55)).

Applying the estimate (2.67) and taking into account the boundedness of the functions ^l(r), ^2(t) and their derivatives, we finally obtain

EZ Cjiji

j'i=m+l

<0,(1 + 1

dx

n m) J (1 — x2)l/2

l

n

ji=m+1

11

<C3(i + -+ V 4)^0 (2.83)

vn m j-i=m+x -v

if n,m — to (n > m), where constants C1, C2, C3 do not depend on n and m. Now consider the trigonometric case. Fix n > m (n,m G N). Denote

t t2

n n 2

Sn,m = E Cjiji = E / ^2(t2)0ji (¿2M ^(¿Oj (¿1)dt1dt2.

ji=m+1 ji=m+1 t t

By analogy with (2.82) we obtain

S2n,2m = E j ^2(t2)0ji (¿2^ (¿1)^j"i (¿1)^1^2 =

ji=2m+1 t t

= 7^7 £ (JJ

ji=m+A { {

T t2 \

+jut^f^ J ^oJ^^ldtrdtA = tt

~ E i f^iw jm^f^A -

ji=m+1 j1 V V t J

- / '^(iijcos----- '02(T) -'02(il)cos-

T - t T - t

T

- i cos27rJ^~^2 j dt-i

ti

T

, f ,,(. , . 27Tjiih-t) 2TTji(fi-f) ,

+ / '^(iOsin—-—I -02(ii)sm——-—+

1

T

+1 ^(felsin2^^^^ }dh I , (2.84)

ti

where (t), ^2(t) are derivatives of the functions (t), ^2(t) with respect to the variable t.

From (2.84) we get

n

j =m+l jl

if n,m —y to (n > m), where constant C does not depend on n and m. Further,

S2n—1,2m = S2n,2m

T t2

2 fUt,)cos2^-t] [MJcos2^-^, (2.86)

T- t T2K2' T-t 1 T-t

S2n,2m-1 — S2n,2m +

T t2

+ /^(¿2)cos2™(J~^ ifaih) coS2™{t^~ t] dhdu, (2.87)

t t

S2n—1,2m—1 = S2n,2m—1

T t2

2 ? , , n 2nn(t2 — t) / , , , 2nn(tl — t) 7 7

'02 (¿2) cos——- / '^i(ii)cos——-dt\dto =

T — t T — t T — t

tt

T t2

2 f l . . 2nm(t2 — t) / , . , 2nm(tl — t) , ,

= ¿>2n,2m + ^ / '02(^2)COS-- / '0i(ii)cOS--dtidt2~

tt T t2

- ^/ osB^ll J (2.88)

tt

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Integrating by parts in (r2.86l)-(l2881), we obtain

Cl

I^W—1,2 m. | < | So'ti/Ini | H--, (2.89)

n

C

| $2n,2m—l | < | $2n,2m | H--, (2.90)

l^n-l^m-ll < \S2n,2m\ + C\ (--1--J , (2.91)

\m n J

where constant Ci does not depend on n and m. The relations (285), (ES9HE33) imply that

lim |S2n,2m| = lim |S2n-1,2m| = lim |S2n,2m-1| = lim |S2n-1,2m-1| = 0.

(2.92)

From (2.92) we get

lim |Sn,m| =0. (2.93)

The relation (2.93) completes the proof.

2.1.3 Approach Based on Generalized Double Multiple and Iterated Fourier Series

This section is devoted to the proof of Theorem 2.2 but by a simpler method 13]-[16], [36] than in Sect. 2.1.1 and 2.1.2. We will consider two different

parts of the expansion of iterated Stratonovich stochastic integrals of second multiplicity. The mean-square convergence of the first part will be proved on the base of generalized multiple Fourier series converging in the mean-square sense in the space L2([t,T]2). The mean-square convergence of the second part will be proved on the base of generalized iterated (double) Fourier series converging pointwise. At that, we will prove the iterated limit transition for the second part on the base of the Lebesgue's Dominated Convergence Theorem. Thus, let us prove Theorem 2.2 by a simpler method than in Sect. 2.1.1 and 2.1.2.

Proof. Let us consider Lemma 1.1, definition of the multiple stochastic integral (1.16) together with the formula (1.19) when the function ... ) is continuous in the open domain and bounded at its boundary as well as Lemma 1.3 for the case k = 2 (see Sect. 1.1.3).

In accordance to the standard relation between Stratonovich and Ito stochastic integrals (see (2.8)) we have w. p. 1

T

J*№2)]T,t = J[^2)ht + JMtMiMdh. (2.94)

t

Let us consider the function K*(ti,t2) defined by (2.11)

K*{t1,t2) = K{tut2) + ^l{t1^2}Mti№{t2), (2-95)

where

K(ti,t2) = 1{tl<t2}^i(ti)^2(t2), ti,t2 G [t,T]. (2.96)

Lemma 2.3. Under the conditions of Theorem 2.2 the following relation

J [K *© = J *W'(2)]r,t (2.97)

(2)

is valid w. p. 1, where J[K*]yt is defined by the equality (1.16).

Proof. Substituting (2.95) into (1.16) (the case k = 2) and using Lemma 1.1 together with (1.19) (the case k = 2) it is easy to see that w. p. 1

T

J[I<ifl = J[^]Ttt + il{il=i2} JMtiWtiWh = J*['0(2)]ï> (2.98)

t

Let us consider the following generalized double Fourier sum

Pi P2

j (ti ) j (t2 ),

jl=0 j2=0

where Cj2j1 is the Fourier coefficient defined as follows

Cj2ji = J K*(ti,t2)0ji(ii)j(t2)dtidt2. (2.99)

[t,T ]2

Further, subsitute the relation

Pi P2 Pi P2

K*(tl ,t2)^E Cj2ji] (tl )0j2 (t2) + K*(tl, t2) — E E Cj2ji j (tl ) j (t2) ji=0 j2=0 ji=0 j2=0

(2)

into J[K*]Tt. At that we suppose that pl,p2 < to. Then using Lemma 1.3 (the case k = 2), we obtain

Pi P2

j*['ia(2)]t,, = £ £]] ' j;2)+j№,1«]T?t w. p. 1, (2.100)

ji=0 j2=0

(2)

where the stochastic integral J[RPlP2]Tt is defined in accordance with (1.16) and

Pl P2

RP1P2 (t1,t2) = K*(t1,t2) Cj2j1 j (t1)0j2 (t2), (2.101)

jl=0 j2=0

T

j = I h (sf

(i)

j / j

t

T ¿2 T ti

l(2) _ fin (+ + \ Jf(ilUf(i2)

J [RPlP2

f + / / RPl P2

(t1,t2)dft(2i2)dft(lil) +

tt

T

+ 1{il = i2} / RPlP2 (t1,t1)dt1.

Using standard moment properties of stochastic integrals [95] (see (1.26), (1.27)), we get

m ^ ( j[rplp2]rt

m

T ¿2 T ¿l

?(il^f(i2) , / / n (, , \ Jf^ JX^O

rpiP2(i1,i2)dft(;l)dft(;2) + / / rPiP2(i^c^c ^ +

,t t

t t

T

+ 1{il=i2} / RPlP2 (t1,t1)dt1 I <

t

T t2 T tl

2

< 2 I / I (Rplp2(^1,^2)^ dt1dt2 + J J (Rplp2 (t1,t2))2 dt2dt1 | +

t t t t

T

+ 1{il=i2} / Rplp2 (t1,t1)dt1

2

t

T

2

2 J (rpiP2 (t1,t2))2 dt1dt2 + 1{il=i2}(/ rPiP2 (¿1^1)^1 ) . (2.102)

[t,T]2 \t

2

We have

2

(RpiP2 (ti, ^2)) dtidt2 =

[t,T ]2

p / Pi P2 \ 2

: J i K *(ti,t2) "EE Cj2ji j (ti)0j2 (t2H dtidt2

[t,T p ^ ji=0 j2=0 '

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// Pi P2 \ 2

K(ti,t2) ^ X) Cj2jij (ti)0j2 fe) dtidt2.

[t,Tji=0 j2=0 /

The function K(t^t2) is piecewise continuous in the square [t,T]2. At this situation it is well known that the generalized multiple Fourier series of the function K(t^t2) G L2([t,T]2) is converging to this function in the square [t,T]2 in the mean-square sense, i.e.

Pi P2 2

k (ti,t2 ) -EE Cj2 .nil ^ (ti ) =°,

lim

Pi,P2^œ

ji=0 j2=0 1=1

L2([t,T ]2)

where notations are the same as in (1.7). So, we obtain

(RPiP2 (ti ,t2))2 dtidt2 = 0.

lim

Pi,P2^œ

(2.103)

[t,T ]2

Note that

T

1

Pi P2

RPlP2{tiM)dti = I ( ^i/ji{ti)'h(ti) - EE^fci^fclii) )

ji=0 j2=0

T

1 ^ Pi P2 ç

(ti)dti Cj2ji / j (ti ) j (ti)dti

t ji=0 j2=0 t

T

/^i i"2

^i(ti)^2(ti)dti Cj2ji 1

; _n ;__n

Pi P2

■i)W2\li)Uli- ¿^ ^Cj2ji 1{ji=j2} t ji =0 j2=0

1 T min{Pi,P2}

= 2 / '^1(^1)^2(^1)^1 - EZ { ji=0

(2.104)

From (2.104) and Lemma 2.2 we get

T T

lim lim / RPlP2{ti,ti)dt\ = lim lim / RPlP2{ti, t\)dt\ = (2.105)

Pl—TO P2—^TO / Pl — TO P2—^TO /

tt

T

1 f Pl

= o / fa{ti)fa{ti)dti - lim =

2 J pi—TO ±—'

t jl=0

T 00 T

= ^ [Mtiyhit^dh-Y CJU1 = lim f RP1Ati,ti)dti, (2.106)

2 / Z-' Pl,P2 — TO /

t jl=0 t

where lim means lim sup.

If we prove the following relation

T

lim lim / RPlP2(t1, t1)dt1 = 0, (2.107)

pl—>-TOp2—TO J t

then from (2.106) we obtain

T

„ TO

\ f Uti)Uti)dti = Y,Cn^ (2-108)

2

t jl=0

T

lim / Rplp2(t1, t1)dt1 = 0. (2.109)

Pl,P2 — TO J t

From (2.102), (2.103), and (2.109) we get

Ml (J [RP!P2 iTt) 2

Pl,P2 — TO

lim m{ (J[Rplp2]T202) =0

,P2 ^TO \ ' /

and Theorem 2.2 will be proved.

Let us prove (2.107). From (2.101) and (2.15) we obtain

lim lim RPlP2(t1,t1)=0, (2.110)

pl —TO p2 —TO l 2

where t1 £ (t,T). Note that the iterated limit (2.110) exists and is finite for

t1 = t, t1 = T.

Since the integral

T

J Rpip2 (ti, ti)dti

t

exists as Riemann integral, then it is equal to the corresponding Lebesgue integral. Moreover, the following equality

lim lim RPiP2 (t1,t1) = 0 when t1 £ [t,T]

Pi—TO p2—^TO

holds with accuracy up to sets of measure zero. According to (2.101), we have

Rpip2 (ti,t2)= [K*(ti,t2) - Cji (t2 ) j (tiH +

V ji=0 /

/ Pi / P2 \ \

+ E Cji (t2) - £ Cj2jij (t2) j (tiH , (2.111) \ji=0 V j2=0 J J

where the Fourier coefficient Cj(t2) is defined by (2.13).

Applying two times (we mean here an iterated passage to the limit lim lim ) the Lebesgue's Dominated Convergence Theorem and taking into

Pi — TO P2 — TO

account (2.12), (2.14), (2.15), (2.105), and (2.111), we obtain

T T

lim lim / RPiP2(ti,ti)dti = lim lim / RPiP2(ti,ti)dti = 0.

Pi—TO P2 — TO / Pi—TO P2 — TO /

tt

Theorem 2.2 is proved.

2.1.4 Approach Based on Arbitrary Complete Orthonormal Systems of Functions in the Space L2([t,T])

Let us prove the equality (2.10) under weaker restrictions. First suppose that (x)}TO=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]) such that (x) for j < to is continuous at the interval [t,T] except may be for the finite number of points of the finite discontinuity. Furthermore, let (t) = (t) or

T

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^i(t) = ^2(t) J g(0)d0, (2.112)

t

where t G [t,T] and ),^2(t),#(t) are continuous functions at the interval

[t,T ].

Thus we will prove the equality

00 T T

E / MhHAh) J Mti)<l>j(ti)dtidt2 = \j (2.113)

j=0 t t t

under the above conditions.

Using the integration order replacement and the Parseval equality, we have

t t2

œ „ 2

E / (Î2)0j (Î2W (tl)dtidt2 = j=0 t t

T t2 ti

œ ft rt rt

= E / ^2 (t2 (Î2W ^2 (tl)0j (tl ) g(T )dTdtidt2 =

j=0 t t t

T T T

œ « « /»

= e / g (t ) / ^2(tl)0j (tl ) / ^2(t2)0j (t2)dt2 dtldT =

j=0 t t t

i

2

T / T

= l^J 9(r) U Ut^M^dh] dr= (2.114)

T 00 / t \ 2

= \J 9(t)J2 (/ l{r<il}^2(ii)^(ii)diij <*r = (2.115)

T T T T

^ / ^ / V^i}^ (¿O^Mt =IJ 9(r) J ll)\{tx)dtXdT =

t t t t

T ti

= IJ tl) J g(r)drdti = (2.116)

tt T

= 1 [ Mh)Mti)dth (2.117)

where the transition from (2.114) to (2.115) is based on the Dini Theorem (using the continuity of the functions (t) (see below), the nondecreasing property of the functional sequence

, t

q '

uq(t) = y J (ti)dti

and the continuity of the limit function

T

u(T) = i ^|(ti)dti

according to Dini's Theorem, we have the uniform convergence uq(t) to u(t) at the interval [t,T]).

From the other hand, using the integration order replacement and the generalized Parseval equality as well as (2.116), we get

t t2

oo „ 2

^ I ^l(t2)0j (¿2) / ^2(ti)0j (ti)dtidt2 = j=0 t t

^ « 2 2

Y / ^2(Î2)0j (¿2) / g(T )dW ^2(ti)0j (ti)dtidt2 j=0 t t t

to T T y

Y / ^2(ti)0j (tiW ^2(Î2)0j (¿2) / g(T)dTdt2dti j=0 t ti t

oo

T T t2

^ J ^2(ti)0j (ti)dt^ ^2(Î2)0j (¿2^ g(T)dTdt2 j=0 t t t

TO 2 12

Y / ^2(ti)0j (ti) / ^2(Î2)0j (¿2) / g(T )dTdt2dti = j=0 t t t T ti T ti

= J Mh) • Mh) J g{r)drdt Jfo{ti) J g{T)drdtl = t t t t

2

T ii T

\ i foih) j g{r)drdti = i f Mh}h(ti)dh. (2.118)

In addition, for the case ) = ^2(t), using the Parseval equality, we obtain

^ / (t2)0j (t2W (tl)dtldt2 = j=0 { t

TO / T \ 2 T

= ¿E f UhWAt^dh = l- J f^dh. (2.119)

j=0 Vt / t

The equality (2.113) is proved.

By interpreting the integrals in the above formulas as Lebesgue integrals, using Fubini's theorem and Lebesgue's Dominated Convergence Theorem in the above reasoning, we can generalize the equality (2.113) to the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^i(t),«t),g(T) e L2([t,T]).

Thus the following theorem is proved.

Theorem 2.3 [46]. Suppose that (x)}°=0 is an arbitrary complete or-

thonormal system of functions in the space L2([t,T]). Moreover, ^1(t ) = (t ) or

T

WT)= «T) J g(0)d0,

t

where t e [t,T] and ^1(t),^2(t),g(T) e L2([t,T]). Then for the iterated Stratonovich stochastic integral

* T * t2

J*[^(2)]T,t =/ ^2)/ f^f (ii,i2 = 1,...,m)

the following expansion

Pi P2

J*[^(2)]T.t = l.i.m. £ Ci!1'C' (2.120)

ii=0 i2=0

that converges in the mean-square sence is valid, where the notations are the same as in Theorems 2.1, 2.2.

For example, suppose that —2(t) = (t — t)1, g(r) = k(r — t)k—1, where l = 0,1, 2,..., k = 1, 2,... Note that this case is important for applications (see Sect. 4.7 and 4.11).

From (2.112) we obtain

T T

-01 (t) = ^2(t) J g(6)de = k(T — t)1 J(0 — t)k—1dQ = (t — t)l+k. t t

Taking into account (2.117)-(2.119), we get

t t2

to „ %

Yj(t2 — t)1 fa (t2) I (t1 — t)l+k(t1)dt1dt2 =

j=0 t t

T t2 T

= E i(t2-t)l+%(to) J (U-tU^dhdt^1- J(T-t)2l+kdr, (2.121) j=0 t t t

where k, l = 0, 1, 2, . . .

The equality (2.121) was obtained in [107] using other arguments. In addi-

tion, the formula (2.121) was used in [107] to generalize the equality (2.113) to the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and —1(t),—2(t) E L2([t,T]). Thus Theorem 2.3 can be generalized to the case —1(t),—2(t) E L2([t,T]).

2.2 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Theorem 1.1

This section is devoted to the development of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the mean (Theorem 1.1). We adapt this method for the iterated Stratonovich stochastic integrals of multiplicity 3. The main results of this section have been derived with using triple Fourier-Legendre series as well as triple trigonometric Fourier series for different cases of series summation and different cases of weight functions of iterated Stratonovich stochastic integrals.

2.2.1 The Case p1;p2,p3 ^ ^ and Constant Weight Functions (The Case of Legendre Polynomials)

Theorem 2.4 [6]-[16], [34]. Suppose that [0, (x)j 0 is a complete orthonormal

system of Legendre polynomials in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

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iff f^ff (i1,i2,i3 = 1,...,m)

t1

t t t

the following expansion

sjc t3 !(st2

/C C fif2 i-3

I I <>4;2>4;3) = EEj'W

t t t j1=0 j2=0 j3=0

(2.122)

that converges in the mean-square sense is valid, where

T s si

Cj3j2j1 =y j (5^ j (51) y j (52)^52^51^5 t t t

and

T

Cj!) = f j(5)dfs(!)

t

are independent standard Gaussian random variables for various i or j. Proof. If we prove w. p. 1 the following equalities

Pi P3 1 , 1 V

,'i- E E = - + -U(!3)), (2.123)

ji=0 j3=0

Pi P3

pi p3 -1 / 1 N

E E = 7<T - A2 ( ci"' - -U|!l)) , (2.124)

^^ J1 4 \ V3 /

ji=0 j3=0

Pi P3

l.i.m. £ ^CjCf = 0, (2.125)

ji=0 ,3=0

then in accordance with the formulas (I2.123l)-(l2.125l). Theorem 1.1 (see (I! -47)). standard relations between iterated ItO and Stratonovich stochastic integrals as

well as in accordance with the formulas (they also follow from Theorem 1.1)

T t

\ f f dsdf« = \{t - if* (>> + w. p. i,

t t T t

\J J di^dr = \(T- tr- (<£■> - ^ci"1) w. p. i tt

we will have T ts t2

/>/>/> P! P2 P3

11 /^ftíi-'^fíí^^^rfíti3'=^M EEECj.

t t t j'i=0 j2=0 j3=0

T T T T

—X{il=ia}i J J dsd£™ - l{t2=t3}^ J J df^dr w. p. 1. t t t t It means that the expansion (2.122) will be proved. Let us at first prove that

œ 1

E CoM = -{T-t?!\ (2.126) ii=0 œ

= —=(T-tf'\ (2.127)

ii=0

We have

r (T - t)W ^000 — -^-,

T s si

O,,, = /00 (s)/ j (*)/ j =

t t t T / s \ 2

= \[ Ms) I I o;. (*,)</*, ) ds, ji > 1, (2.128)

tt where (s) looks as follows

= J( (, - ^ ) , j > 0, (2.129)

where Pj (x) is the Legendre polynomial.

Let us substitute (2.129) into (2.128) and calculate Cojj (j > 1)

C0jiji —

2ji + 1 2 (T-i)3/2

T / z(s)

\

T — t PjM—z—dy

ds —

V

i

/

T / z(s)

8

1

\

y/T^t. 8(2ji + l)

\—1 T

2j1 + 1

Pji+1(y) — Pji—1(y) dy

ds —

y

(Pji+1(z(s)) — Pji—1(z(s)))2 ds,

(2.130)

where here and further

z(s) — s —

T + A 2

2 J T-t

In (2.130) we used the following well known properties of the Legendre polynomials

P (y) —

1

2j + 1

Pj+1(y) — Pj—1(y) , P (—1) —(—1)j, j > 1.

Also, we denote

dP

def

dy

From (2.130) using the property of orthogonality of the Legendre polynomials, we get the following relation

= &TT) / + (»)) dv =

1

(ri

S(2Ji + 1) \2ji + 3 2j

where we used the property

P?(y)dy —

1

2j + 1

j > 0.

2

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1

2

Then

fv (T-tf/2

Z^ 1 - -Q-+

ji=0

(T - if'2 1 y. 1

oo oo

(T - tf/2 (T-tf/2i^ 1 1 - 1 6 + 8 \^4j2-l 3 + f-4j2-l

j=i ^ A ° ji=1 ^

_ (T - tf/2 (T-i)3/2 (1 _ 1 1 \ _ (T - ¿)3/2 ~ 6 + 8 \2 ~ 3 + 27 4 '

The relation (2.126) is proved.

Let us check the correctness of (2.127). Let us represent C1j1j1 in the form

T / s \ 2

Cijui = \j Ms) I j M(si)d.si ) ds = tt

= + ljjl(v)nP)Ml] dy, jl >i.

Since the functions

^ ¡j Vi)dyi | , Ji>l are even, then the functions

y x 2

Pi(y) / Pji(yi)dyi dy, ji > 1

\-i

are uneven. It means that C1jiji = 0 (j > 1). From the other side

l

V3(T-T2 r f . (T-i)3/2

100 = — 16 J y(y + = •

-1

Then

^ ^ (T — t)3/2

ji=0 ji=1 v The relation (2.127) is proved.

Let us prove the equality (2.123). Using (2.127), we get

Pi P3 Pi (rp ,\3/2 Pi P3

V^ V^ / Ah) , \j__V_Ah) , V^ V^ ^ f

/ , / A ./:■,/ .Z ^/:., / , ( "./ ./ "I" . Si -t- / „ / / ./:■,/ ./ ^

ji=0 j3=0 ji=0 W3 ji=0 j3=2

Pi (t t)3/2 Pi 2ji+2

j3). (2.131)

J 1=0 4\/3 ji=0 j3=2 j'3—even

Since

(T-t)^(2ji + l)^Tl f .( f .

( ./."■./■./■ =-^-J PjM J Pji(yi)dyi | <hj

and degree of the polynomial

y

j pji (y1)dy1

equals to 2j + 2, then Cj3 jj = 0 for j3 > 2j + 2. It explains that we put 2j + 2 instead of p3 on the right-hand side of the formula (2.131).

Moreover, the function

y2

f pji (y1)d^1

is even. It means that the function

y

Pj3 (y) | / Pji (y1)dy1

is uneven for uneven j3. It means that Cjjj = 0 for uneven j3. That is why we summarize using even j3 on the right-hand side of the formula (2.131).

Then we have

pi 2ji+2 2pi+2 pi

Ev^ C Z (is) = v x^ C z (i3) =

/ , Cj3jiji Zj3 = ^ Cj3jiji Zj3 =

ji=0 j3=2,j3-even j3=2,j3—even ji=(j3-2)/2

2pi+2 pi

= £ ECj3jiji j3 (2.132)

j3=2,j3-even ji =0

We replaced (j3 — 2)/2 by zero on the right-hand side of the formula (2.132), since Cjsjiji = 0 for 0 < ji < (j3 — 2)/2. Let us substitute (2.132) into (2.131)

Pi P3 Pi

V^ V^ / Ai3) {T ~ tf/2 fn)

j1=0 j3=0 j1=0 4 3

2pi+2 pi

+ E ECj3ji ji Zj33). (2.133)

j3=2,j3—even ji=0

It is easy to see that the right-hand side of the formula (2.133) does not depend on p3.

If we prove that

2

^ M {(£E^ci' - -(tf" + )) } = o.

(2.134)

then the relaion (2.123) will be proved.

Using (2.133) and (2.126), we can rewrite the left-hand side of (2.134) in the following form

M(f(£-^^ci!3) + E E

I \ \ji=0 J j3=2,j3—even ji=0

pi (^_t)3/A2 2PI+2 /pi x2

Ec»«.——j E E^«.

ji=0 / j3=2,j3—even \ji=0

2pi+2 / pi \ 2 =pl™ E Sew . (2.135)

j3=2,j3—even \ji=0 /

If we prove that

Pi

lim

Pi—

2Pi+2

E (EC

j3jiji

= 0,

(2.136)

j3=2,j3-even \ji=0

then the relation (2.123) will be proved. We have

2Pi+2 / Pi j3=2,j3-even \ji=0

4

2Pi+2

E

j3=2,j3-even

/T

Pi

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2

V'

j(5^ / (51)d51 ) d5 ji=0 \t

4

2Pi+2

E

j3=2,j3-even

/T

/

j(5)

V1

/

22 d5

V

(5 -1) - 53 / j (51)d51

ji=Pi+Ai w y

4

2Pi+2

E

T

j3=2,j3-even

V

/ j(5) 53 / (51)d51

{ ji=Pi+M "t

2

d5

<

1

2Pi+2

/T

E

j3=2,j3-even

oo

| j (5)| E

V'

ji=Pi+1 \ t

J

22 j (51)d5^ d5

y

(2.137)

Obtaining (2.137), we used the Parseval equality

~ (I V T 2

53 / j (51)d5H = (1{si<s^2 d51 = 5 - t j1=0 t t

(2.138)

and the orthogonality property of the Legendre polynomials

T

j(5)(5 - t)d5 = 0, j3 > 2.

(2.139)

2

s

1

s

1

s

1

s

Then we have for j E N

j (51)d5H =

2 / z(s)

(T - t)(2j + 1)' "

\

4

pji (y)dy

V

1

!

T-t

I z (s)

\

4(2j1 + 1) Tt

Pji+1(y) - Pji-1(yn dy

V

1

/

(Pji + 1 (z(5)) - Pji-1 (Z(5)))2 <

<

4(2 j + 1) T — t

(2.140)

Remind that for the Legendre polynomials the following estimate is correct

IP(y)l <

K

VJ+T(l-y2)1/4

, y E (-1, 1), j E N,

(2.141)

where constant K does not depend on y and j.

The estimate (2.141) can be rewritten for the function j(5) in the following form

' 1

(5)| <

2j + 1 K

<

j + 1 VT^t, (1 - z2{s))1/A K1 1

<

(2.142)

y/T=t(l-z2{s))1/v

where Kx = Ky/2, s E (t,T).

Let us estimate the right-hand side of (2.140) using the estimate (2.141)

, / , 1 T -1 / K2 K2

('^1 )ds\ < . . ( —- +

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<

2(2j1 + 1) Vj + 2 3i) (1 - (z(5))2)1/2

(■T-t)I<2 1

2 jl (1 -(z(s)W/^

<

(2.143)

where 5 E (t,T), j E N.

2

s

2

s

1

Substituting the estimate (2.143) into the relation (2.137) and using in (2.137) the estimate (2.142) for | j(s)|, we obtain

2pi+2 / pi \ 2 E ( E Cj » <

j3=2,j3—even \ji=0

(T — t)K 4K2

<

2pi+2

/ T \

„ 7 oo

16

j3=2,j3—even

ds v 1

E / / 0\ 3/4 E ;2

^ (1 — (z(s))2) ji=pi+i jiy

(T — t)3K4K2(pi + 1)/ /* dy W ^ 1

V—1

Since

64 <2,144)

1

^ dy <00 (2.145)

—1

and

(1 — y2)3/4

00 TO

1 f dx 1

„lr+1 Ti-J^^Vi"

E2 + if Pi -!• 00, (2.147)

p1

then from (2.144) we find

2

9 pi

j3=2,j3—even \ji=0 / 1

where constant C does not depend on p1 and T — t. The relation (2.147) implies (2.136), and the relation (2.136) implies the correctness of the formula (2.123).

Let us prove the equaity (2.124). Let us at first prove that

TO

j3=0 4 TO

= =(T-t)W. (2.149)

j3=0 4 3

We have

TOTO

E j 0 = C000 + E Cj3j3 0,

j3=0 j3=1

2

r (T- *)3/2 ^000 — -

6

QaÄO = ^fi) / (V) + Pl~M dy 31

Then

(T - tf/2 f i i \ . >;L

yV CT-tf!2

js=0

, (T -11 A 1

+ ö Z^ ro.io J- 1 -L

8 Vj=1 (2j3 + 1)(2j3 + 3) ' j=1 j2 — 1

, i__i + V

^ ö Z^ /L i? _1 Q ^

{T-tf/2 | {T-tf/2 /jj^ 1 1 v^ 1

Vjs=143 _ x ° jS=1 ^

6 8 lf-4j| — 1 3 — 1

_ (T - tf/2 {T-tf2 f 1 _ 1 1 \ _ (T - ¿)3/2 ~ 6 + 8 \2 ~ 3 + 2J ~ 4 '

The relation (2.148) is proved. Let us check the equality (2.149). We have

T s si

Cjsjsji — y j (s) ^ j (s1^ j (s2)ds2ds1ds — t t t

T T T

— ^ j (s2)ds^ j (s1)ds^y j (s)ds —

t S2 Si

T / T \ 2

= \J <f>hM I J (f>j3{si)dsi I ds2 = = iT-mu + DWrnj (jPiMh > i.

(2.150)

Since the functions

pj3 (yi)dyi I , j3 > 1

12

y

are even, then the functions

Pi (y) ^ J Pjs (yi)dyij dy, j'3 > 1

are uneven. It means that Cj3j31 = 0 (j3 > 1). Moreover,

Vs(T-tf'2 r n _ (T-tf2

cw =---J y(l - y)-dy =

—1

Then

TO TO (T — t)3/2

j3=0 j3=1

The relation (2.149) is proved. Using the obtained results, we get

pi p3 p3 3/2 p3 pi

V^ V^ / Ah) _ {A__lJ_An) , V^ V^ r t

/ j / A ./-,/ ■,/ — / v ^hhOSO , Sl / v ( ./:■,/:■,/ '■>

ji=0 j3=0 j3=0 j3=0 ji=2

ps (' ) (T — t)3/2 (• ) ps 2j3+2 (. )

j3=0 V j3=0 ji=2,ji—even

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Since

—1 y

and degree of the polynomial

! i

pj3 (yi)dyi

12

equals to 2j3 + 2, then Cj3jj = 0 for j > 2j3 + 2. It explains that we put 2j3 + 2 instead of p1 on the right-hand side of the formula (2.151).

Moreover, the function

12

pj3 (y!)dy:

y

is even. It means that the function

1

12

Pji (y) | J Pj3 (y1)dy1

y

is uneven for uneven j. It means that Cj3j3ji = 0 for uneven j! It explains the summation with respect to even j on the right-hand side of (2.151).

Then we have

P3 2j3+2 2P3+2 P3

Ev^ C z (ii) = v^ v^ C Z (ii) =

/ , Cj3j3 ji Zji = Cj3j3ji Zji =

j3=0 ji=2,ji-even ji=2,ji-even j3=(ji-2)/2

2P3+2 P3

= E E^i Cj,!i). (2.152)

ji =2,ji-even j'3 =0

We replaced (j - 2)/2 by zero on the right-hand side of (2.152), since Cj ji = 0 for 0 < j3 < (j - 2)/2.

Let us substitute (2.152) into (2.151)

Pi P3 P3 (T t\3/2

Y^Y^ An)_X^r Ail) A -1) Ah)

ji=0 j3=0 j3=0 V

2P3+2 P3

+ E E jji if. (2.153)

ji=2,ji-even j3=0

It is easy to see that the right-hand side of the formula (2.153) does not depend on p1.

If we prove that

"st,M i (t E - W - tr- (cA - -t?c!!i)N) VU 0,

pIAOo \ ^ ^ 4v ' V >/3

I \ —n ;__n \ V <-»

(2.154)

then (2.124) will be proved.

Using (2.153) and (2.148), (2.149), we can rewrite the left-hand side of the formula (2.154) in the following form

lim m

p3 ^to

p3

y ^ Cj3j3 0 j3=0

(T —1)3/21 ,(ii)

2p3+2 p3

Z0 + ^^ ^^ Cj3j3ji C

(ii) ji

ji=2,ji— even j3=0

pi (T — t)3/2

( E ^--I + lim

p3

p3 ^to

j3=0

p3 ^to

p3

2ps+2

E (EC

j3j3ji

2ps+2

= lim V ( V C

ji=2,ji— even \j3=0 2

ji=2,ji— even \j3=0

If we prove that

2ps+2

p3

j3j3ji

2

pjisro E ECi3j3jJ =0,

ji =2,ji— even \j3=0

(2.155)

then the relation (2.124) will be proved. From (2.150) we obtain

p3

2ps+2

E E

Cj3j3ji

ji =2,ji—even \j3=0

1

4

2ps+2

E

/T

ji=2,ji— even 2ps+2 / T

p3

T

\

V1

j ME ( I j (s1)ds1 ) ds2

j3=0

^2

/

4

E

ji=2,ji— even

/

j (s2)

V

t

V

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(T — S2) — E I J j(si)dsi)

j3=p3+1 \S2 /

/T

4

2ps+2 / T to / T

E J j(s2) E IJ j(si)dsi

2

ds2

y 7

22

\

ji=2,j'i—even 2ps+2

ds2

<

t

/T

j3=p3+1

/

T

\

<

4

E

ji=2,ji— even

V'

| j (s2)| E I I j (s1)ds1 I ds2

j3=p3+1

(2.156)

VS2

/

2

1

2

1

In order to get (2.156) we used the Parseval equality

T \ 2 T

co '

j=0 ( / 0ji(51)d51j = / (1{s<si^2 d51 = T - 5 (2.157)

and the orthogonality property of the Legendre polynomials

T

J j (5)(T - 5)d5 = 0, j3 > 2. (2.158)

t

Then we have for j3 E N

T2

(T t) 2

^3(si)dsi ) = + ^ (P^+1 (z(s2)) - Pj,-l {z{so))f <

T — t

< 2(2^Tl) + ^ ("(S2))) <

T-t ( K2 /\2\ 1

< 2(2j3 + l) + H) (l-(z(s2)W2 <

(T - t)K2 1 , , ,

In order to get (2.159) we used the estimate (2.141).

Substituting the estimate (2.159) into the relation (2.156) and using in (2.156) the estimate (2.142) for | j(52)|, we obtain

2P3+2 /Ps \ 2

53 (53 Cjsjsji' <

ji=2,ji-even \js=0

2

4 ¡^2 2ps+2 / „ oo

El / 2 v^

W (i-^2(s,))3/4 ^ j2

ji=2,ji-even \^ V V js=Ps+1 J

(T — ¿)3/\4iv2(p3 + 1) () dy \ ( l\2

1

Using (2.145) and (2.146) in (2.160), we get

£2 (EC»«,,) <C'T-^ + 1» _>0 with p^oo, (2.161)

ji=2,ji—even \j3=0 / 3

where constant C does not depend on p3 and T — t.

The relation (2.161) implies (2.155), and the relation (2.155) implies the correctness of the formula (2.124). The relation (2.124) is proved.

Let us prove the equality (2.125). Since ^(t), (t), (t) = 1, then the following relation for the Fourier coefficients is correct

( '. . . . ( '. . . . ( '. . . Lf '-(' .

^jum * ^jrnji * — 2 ii J3i

where Cj = 0 for j > 1 and Co = \/T — I. Then w. p. 1

Pi P3 Pi P3 /1 \

l.i.m. \ \ CjljsjlCj-i = l-'-111- ) ) ^ ']■ ( './:■. ( './ ./ ./:■. _ ( './:■,/ ./ Cj-i ■

z—* z—* z—* z—* \ 2 j j3

ji=0 j3=0 ji=0 j3=0 V 7

(2.162)

Therefore, considering (2.123) and (2.124), we can write w. p. 1

Pi P3 .

pi,p3^-œ z—» z—» 73 2

ji=0 j3=0

Pi P3 Pi P3

— l.i.m^ ECjijij3j) — ^m- £ ECj3jijicf =

Pi,P3^œ z—' z—' Pi,Pz—' z—'

ji =0 j3=0 ji=0 j3=0

= 1 (T _ _ 1 (T _ ,)3/2 (^2) _ ^ _

- -(T - i)3/2 fC(Si2) + 4=cf2)1 = 0. (2.163)

4

The relation (2.125) is proved. Theorem 2.4 is proved.

It is easy to see that the formula (2.122) can be proved for the case i1 = i2 = i3 using the Ito formula

T * is * t2 ' T X 3

"^з - â \ I ULs I "fil ^O I - ^OOOSO SO SO

-| If \ 1/ \3

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dfj«> ) = g (orf'1) = Co„oC,i!l)Ci!')^"),

where the equality is fulfilled w. p. 1.

6 W s / 6

t "it \ t

2.2.2 The Case ^ Binomial Weight Functions, and Ad-

ditional Restrictive Conditions (The Case of Legendre Polynomials)

Let us consider the following generalization of Theorem 2.4.

Theorem 2.5 [6]-[16], [34]. Suppose that { j (x)j 0 is a complete or-

thonormal system of Legendre polynomials in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

Cfi' = / (t -(t - t'2)'2/ (t - ti)''f^fff)

i i i the following expansion

Pi P2 PS

ffi)=EEEj,,cC'jj (2.164)

ji =0 j2=0 js=0

that converges in the mean-square sense is valid for each of the following cases

1. ii = ¿2, ¿2 = ¿3, ii = ¿3 and li, /2,13 = 0,1, 2,...

2. ii = ¿2 = ¿3 and li = l2 = l3 and li, l2, l3 = 0,1, 2,...

3. ¿i = ¿2 = ¿3 and li = l2 = l3 and li, l2, l3 = 0,1, 2,...

4. ¿i, ¿2, ¿3 = 1,..., m; li = l2 = l3 = l and l = 0,1, 2,...,

where ¿i, ¿2, ¿3 = 1,..., m,

T s s i

jji =J(t - s)lsj (s^(t - si)l2j (si^(t - S2)li j (S2^dsi^ t t t

and

T

zj:) = J j (s)df«:)

t

are independent standard Gaussian random variables for various ¿ or j.

Proof. Case 1 directly follows from (1.47). Let us consider Case 2, i.e. ¿i = ¿2 = ¿3, li = l2 = l = l3, and li, l3 = 0,1, 2,... So, we prove the following expansion

Pi P2 PS

Ä) = P E E E Cjcj;i)c«2i)cj:s) (¿^,¿3 = l,... ,m),

ji=0 j2=0 js =0

(2.165)

where /1, /3 = 0,1, 2,... (/1 = /) and

T s si

Cj = J j (s)(t - s)1^ (t - si)1 j (*)/(t - S2 ^ j (s2)ds2dsi ds. (2.166)

t t t

If we prove w. p. 1 the formula

t s

Pi P3 1 ç ç

1-i-H.. E E ^ (I / (i - (2.167)

ji=0 j3 =0 t t

where coefficients Cjjj are defined by (2.166), then using Theorem 1.1 and standard relations between iterated Itô and Stratonovich stochastic integrals, we obtain the expansion (2.165).

Using Theorem 1.1, we obtain

. T s 1 21+13+1

1 1 < ,07 , 1 (7 ^(»3)

2 / ' / ' " ' 2

- I (t- s)l> I (t - srfds.d fi*) = - Y, 4C]:3) w. p. 1,

j3=0

t

where

T s

C3 = j j m - st3J (7- «o«*^.

tt

Then

P3 Pi 21+13+1

v^ v^ / - _ X^ r -

0 ji=0 j3=0

2/+/3+W Pi 1 \ P3 Pi

E Ui:s,+ E E^.e

j3=0 \ji=0 / ^3=21+13+2 ji=0

Therefore,

T

P3 Pi 1 ^

M<! I EE C»U4S) .', 11 I a - ■sifdsM-i

t 1 1 / V Zv J3Ji:/i> j3 o

Pi,P3^œ I \ z—' z—' -73 2

,j3=0 ji=0 t t

21+13+1 / Pi 1 ^ 2

J™» E E^-^31 +

j3=0 \ji=0

2

s

P3 pi

+ lim m

pi,p3—ro

53 53 Cj3jiji c

,j3=21+l3+2 ji=0

(i3) j3

(2.168)

Let us prove that

pi

lim

pi—

53 ^Ajiji o ji=0

1

— I

2

= 0.

(2.169)

We have

pi

53 ^Ajiji 2 ji=0

/

i pi T V

¿53 I <f>js(s)(t-s)h [I faMit-sJdsA ds-

\ ji=0t \t J

1

T

- Un{s){t-s)h {t-s.fds.ds ) =

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1

4

/T

/

j (s)(t - s)

l3

V

pi

V

53 / j(si)(t - si)/dsi

ji=0 V i

i 4

/T

/

j (s)(t - s)13

t

- J (t - dsi | ds | =

oo

V

(t - si)21 dsi - I I j(si)(t - si)dsi) -

ji=pi + 1 y t

-J (t - si)21 dsi | ds | =

1

4

/T

2

J j(s)(t - s)13 ^ J hi(si)(t - si)1 dsj ds

yt ji=pi+^ t / y

. (2.170)

2

s

2

s

2

s

2

In order to get (2.170) we used the Parseval equality

¿1 Jj (si) (t - si)1dsj =J K2(s,si)dsi, (2.171)

I I = K2

ji=0 \ t / t

where

1

K(s, si) = (t - si)l 1{si<s}, s,si G [t,T]. Taking into account the nondecreasing of the functional sequence

. s X 2

n

Un(s) = " 1 ' ^ "+ "

,(s) = 53 I j(si)(t - si)ldsi

ji=0 V t

continuity of its members and continuity of the limit function

s

u(s) = J(t - si)2ldsi t

at the interval [t,T] in accordance with the Dini Theorem we have uniform convergence of the functional sequences un(s) to the limit function u(s) at the interval [t, T].

From (12.170) using the inequality of Cauchy-Bunyakovsky, we obtain

53 ^ ~~ ) —

2

ji=0

T T / _ / s \ 2\

-if f I ^ I f '

1 I ,2 , w. ,2K , I X—K I I , , . .l

< 3 / - s)2,3As

t

53 J j(si)(t - si)ldsi

2

ds

T

< ^2(T - J ojjs)<ls(T -t) = i(T - (2.172)

t

when pi > N(e), where N(e) G N exists for any £ > 0. The relation (2.172) implies (2.169).

Further,

pi p3 pi 2(ji +l+i)+l3

53 53 Cj3jiji j3 = 53 53 Cj3jiji j3.

ji=0 j3=2l+l3+2 ji=0 j3=2l+l3+2

We put 2(ji+1+ 1)+13 instead of p3, since Cj-jj = 0 for j3 > 2(ji+1+ 1)+13. This conclusion follows from the relation

T

' './:,/ ./ £ J OjJs)(l ,s-)/:" ( I (f)j1{s1){t-si)ldsi ) d.s

T

= \J <f>j3(s)Q2(j1+l+l)+l3{s)ds, t

where Q2(ji+/+1)+/3 (s) is a polynomial of degree 2(j1 + I + 1) + 13. It is easy to see that

pi 2(ji + /+1) + /g 2(pi + /+1)+/g pi

E E Cjgjijizj33) = E ECjgjijiZj33). (2.174)

ji=0 j3=2/+/3+2 j3=2/+/3+2 ji=0

pi

Note that we included some zero coefficients Cjjj into the sum ^ . From

ji=0

(2.173) and (2.174) we have

Pi P3

M{|E E Cj3j,j,zj33)

ji=0 j3=2/+/3+2

2(pi+/+1)+/3 pi

E EC--,zj33)

j3=2/+/3+2 ji=0

( T

. _ _ Pi T

E

j3=2/+/3+2

2(pi+/+1)+/^ / 1 Pi

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\

2 z^ / (pjs{s){t-s; ji=01

2(pi+/+1)+/^ / ^ pi

r...........3/T

E

■ t ji=0 \t

4

j3=2/+/3+2

V

2(pi+/+1)+/3 13 p1 2

I Cj3jiji 1

j3=2/+/3+2 \ji=0

\ 2

j(s1)(t - s1)/ds1 ds

J

\ 2

j(s1)(t - S1)/ds1 ds

J

. 2(pi+/+1)+/^ / T

J E / M(s)(t~s)k ( ¡{t-S.fdSy

j3=2/+/3+2 W

2

s

2

1 2(pi+l+i)+l3 y t

-j E J •s)/:" 53 (/ <f>hM(t-Sl)

53 I I j(si)(t - si)ldsi

ji=pi+i V {

/T

\

ds

y y

2

j3=2l+l3+2

- si)ldsi ds

y ji=pi+^ t y J

(2.175)

In order to get (2.175) we used the Parseval equality (2.171) and the following relation

T

J j(s)Q2l+i+l3(s)ds = 0, j3 > 21 + 1 + /3, t

where Q2l+i+l3 (s) is a polynomial of degree 2/ + 1 + /3. Further, we have

j(si)(t - si)1 dsi =

(T - t)2l+i(2ji + 1)

/ z(s)

\

22l+2

Pji (y)(1 + y^ dy

i

V

(T -t)

/

2l+i

/

22l+2(2ji + 1)

z(s)

X

\

X

(1 + z(s)) Rji(s) - / / (Pji+i(y) - Pji-i(y))(1+ y)l-i dy

<

V

i

/

X

2(s - t)\2l„2

{T-t)2l+12 " 22/+2(2ji + 1) X

/ z(s)

2

Tt

Rj (s) +

(Pji+i(y) - Pji-i(y))(1+ y)l-i dy

V

i

/

<

y

\2l+i

<JT-t)_ ~ 22l+1(2ji + 1)

2

s

2

s

2

(

z(s)

z (s)

\

X

V

22l+iZji(s) + I2 J (1+ y)2l-2dy ^ (Pji+i(y) - Pji-i(y))2 dy -i -i 2l+i

<

J

<

(T -1)2

22l+i(2ji + 1)

X

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X

JlK J 21-1 T-i

2l i

z(s)

j2i+i(y) + Pw(y)) dy

<

V

<

i

!

(t - t)2l+i („ _ , /2

2(2 ji + 1) where j G N,

2Zji (s) +

2/ — 1

z(s) \

j2+i(y) + P?-i(y)) dy

, (2.176)

V

i

/

Rji(s) =

Zji(s) = Pji+i(z(s

z(s)) - i(z(s)),

j1-i

Let us

using (2.141) (ji G N)

j(si)(t - si)1 dsi <

<

(T - t)2l+1 ( K2 K

(

2(2ji + 1) Vji + 2 ji

(T - t)2l+iK2

2

+

2 z(s)

/2 dy

V

<

2j?

(1 - (z(s))2)

2 /2n

+

2)i/2 2/ - 1j (1 - y2)

2)1/2

<

-1

!

(1 - (z(s))2)

2)1/2 2/ 1

, s G (t,T). (2.177)

From (2.175) and (2.177) we obtain

pi p3

m^IE E c,j3j1j1 cjr <

ji =0 j3=2l+l3+2

<

1

4

2(pi+l+i)+l3

/T

2

E

j3=2l+l3+2

V

(s)l(t - s)1^ / hji(si)(t - si)1 dsil ds

ji=pi+A t J J

<

2

1 2(pi+/+1)+/3 / T œ / s \ 2 \

<-A(T-tf3 E /1^)1 E [ *.)'</*. ds

\t ji=pi+At / y

4

j3=2 /+/3+2

<

//T

<

(T - t)4/+2/3+1K4K2 2(pi++1)+/3 f 2ds

E

16

j3=2 /+/3+2

V

+

^ (1 - (z(s))2)3/4

T

f ds

+

21 -1 / (1 - (z(s))^1/v jiipr+1

\

El

<

2

(T - t)Al+2MKAK2 2pi + l / j 2dy i2TT j dy

~ 64 Vx (l-y2)3/4 + 2r^l7 (l-y2)V4 1 "

< C(T - tf+2M 0 when Pi 00, (2.178)

p1

where constant C does not depend on p1 and T — t.

The relations (2.168), (2.169), and (2.178) imply (2.167), and the relation (2.167) implies the correctness of the formula (2.165).

Let us consider Case 3, i.e. i2 = i3 = i1, 12 = 13 = I = 11, and 11,13 = 0,1, 2,... So, we prove the following expansion

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pi p2 p3

^' = ,,, ii- EEECj«'j' («1,i2,«3 = l,...,m),

pi,p2,p3^œ

ji=0 j2=0 j3 =0

(2.179)

where 11,13 = 0,1, 2,... (13 = 1) and

T s si

Cj3j2ji = J j(s)(t - s)/ /(t - s1)/j (*)/(t - s2)^j (s2)ds2ds1 ds. (2.180) t t t

If we prove w. p. 1 the formula

T s

pi p3 1 » »

l.i.m. J] J] C^cj;1' = - / (t - sf / (t - sit^ds, (2.181)

ji=0 j3=0 t

t

2

where the coefficients Cjjj are defined by (2.180), then using Theorem 1.1 and standard relations between iterated Itô and Stratonovich stochastic integrals, we obtain the expansion (2.179).

Using Theorem 1.1 and the Ito formula, we have

T s T T

\ f{t - sf i(t- s^df^ds = i f(t- Si)li f(t- sfdsdfM =

si

where

Then

1 21+1i+1

' E ^cj;1' w. p. i, ji=0

T T

ô„=fj (s1)(t - s1)1i f(t - s)21

t s,

Pi P3 . 21+1,+1

ji=0 j3=0 ji=0

21+1i+1 / p3 1 \ Pi P3

E E

ji=0 \j3=0 / ji=21+1i+2 j3=0

Therefore,

T

Pi P3

.(ii) 1

Mi EE^cf-J/ft-^ jit-s^di^ds

1 Vi=0 j3=0 *t t

21+1i+1 / P3 1 ^ 2

^ £ £ ^ - i + ji=0 \j3=0

Pi P3 2

+PiJs-^ cf ■ (2'182>

\ji =21+1i+2 j3=0

Let us prove that

2

p3 2

in- E^-Ô^J (2-183)

P3^œ \ z—' 2 '

j3 =0

2

s

We have

P3

y ^ Cjsjaji A | —

j3 =0

1

— I

2

P3

T

E I j (S2)(t - S2)11 dsW j (si)(t - si)1 dsW j (s)(t - s) ;ds-j3=0 t

S2

S1

T

1

- f I <f>h(si)(t - Si)'1 I (t- sfdsdsi ) =

si

T / T

j3=0 t \S2

j(si)(t - si^dsi ds2-

1

T

~2 / feMC* - si) s) dsds 1 ) =

1

4

/T

/

j(si)(t - si)11

t

si

P3

T

E / j (s) (t - s)1 ds

T

\j3=0 \si

- I (t - s)21 ds I dsi I =

si

1 4

/T

j(si)(t - si)

1i

t

V

(t - s)2 1 ds - £ I I j (s)(t - s)ds) -

j3=P3+i

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T

-J (t - s)21 ds | dsi | =

si

1 4

/ T « ( T Jj(si)(t - si)^ ^ y

^t j3=p3+i \si

\

j (s)(t - s);ds dsi

. (2.184)

/

2

In order to get (2.184) we used the Parseval equality

to / t \ 2 T

E -s)lds = /K2(s-si)ds- (2185)

j3=0 \si / t

where

l

K(s, si) = (t - s)1 1{si<s}, s,si G [t,T]. Taking into account the nondecreasing of the functional sequence

T2 n I T

U 's ' = * ■ ' h ,s„t s)1

(si)^E ( I j (s)(t - s)1 ds

j3=0

vsi

continuity of its members and continuity of the limit function

T

u(si) = J(t - s)21 ds

si

at the interval [t, T] in accordance with the Dini Theorem we have uniform convergence of the functional sequence un(si) to the limit function u(si) at the interval [t, T].

From ((2.184) using the inequality of Cauchy-Bunyakovsky, we obtain

p3 1 _ \ 2

53 C'jzhji ~ ) —

2

j3=0

T \2\

1 n /./TO/,, \

<lj ^(s^it - Slf-dSl J tt

53 / hj3 (s) (t - s)1 ds \vj3=p3+i Vsi / y

dsi <

T

< - i)2/l I (pl(si)dsi(T -t) = \{T - tf-+ls2 (2.186) t

when p3 > N(e), where N(e) G N exists for any e > 0. The relation (2.183) follows from (2.186).

We have

p3 pi p3 2(j3+l+i)+li

53 53 Cj3j3jiCj!1 = 53 53 Cj3j3jiCJ!1 . (2.187) j3=0 ji =2l+li+2 j3=0 ji=2l+li+2

2

We put 2(j3 + I + 1) + 11 instead of p1, since Cj3j3 - = 0 when j > 2(j3 + I + 1) + 11. This conclusion follows from the relation

t / t \ 2

<" !/:,/:,/ = <f>hM{t ~ SoJ1 I J (J j., ( -s' | )(/ - sl)^«! i ds2 =

T

= \J <l>h(S2)Q2(j3+l+l)+h{S2)dS2, t

where Q2(j3+/+1)+/i (s) is a polynomial of degree 2(j3 + I + 1) + 11. It is easy to see that

p3 2(j3+/+1)+/i 2(p3+/+1)+/i p3

E E Cj3j3ji Cj/ = E ECj3j3ji j . (2.188)

j3=0 ji=2/+/i+2 ji=2/+/i+2 j3=0

p3

Note that we included some zero coefficients Cj3j3ji into the sum ^ .

j3=0

From (2.187) and (2.188) we have

p3 p1

(:1)

ji

i-3=0 ji=2/+/i+2

m e e Cj3j3ji j

'2(p3+/+1)+/i p3 \2| 2(p3+/+1)+/i / p3

E Ecj cf = £ ECj3j„i

ji=2/+/i+2 j3=0 ) I ji=2/+/i+2 \j3=0 2(p3+/+1)+/i / 1 p3 T /T

ji=2/+/i + 2 \ j3=0 { Vs2 /

1 2(p3+/+1)+/^ T p3 i T \ 2 \

4 E / fewi* - S2)/1 E / fe^x* - ds2

ji = 2/+/i + 2 j3=0 Vs2 ) )

1 2(p3+/+1)+/^ T ( T

4 E / (t-S.fds,-

4 j1=2/+/1+2 t s2

2

T

2

E /j(si)(t - si)1 dsi

J3=p3+i

\

ds2 ) )

/T

4

2(p3+l+i)+lW T to / T

E J hi (s2)(t - s2)1^ J j (si)(t - si)

\

ji=2l+li+2

- si)ldsi ds2

V1

J3=p3+i

s2

In order to get (2.189) we used the Parseval equality lowing relation

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T

/ hji(s)Q2l+i+li(s)ds = 0, ji > 2/ + 1 + /i,

/

(2.189) and the fol-

where Q2l+i+li (s) is a polynomial of degree 2/ + 1 + /i. Further, we have

T

hj3(si)(t - si)1 dsi =

s2

(T - t)2l+i(2j3 + 1)

/ i

\

221+2

J Pj3 (y)(1 + y)1 dy V (s2) (T - t)2l+i

!

22l+2(2j3 + 1)

x

/

\

X

(1 + z(s2)) Qj3(s2) - /y (Pj3+i(y) - Pj3-i(y))(1+ y)l-i dy

V z(s2)

< {T-t)2l+12 ^

<

/

X

'2(So-t)

T-t

2l

22l+2(2j3 + 1) / i

2

Q23 (s2)+ /2

(Pj3+i(y) - Pj3-i(y))(1 + y)l-i dy

<

V

V (s2)

J

1

JT-tf^ - 22'+i(2j3 + l)

2

1

2

i

( i i x

X 221+iHj3 (s2)+ l2 J (1+ y)21-2dy | (Pj3+i(y) - Pj3-i(y))2 dy

\ z(s2) z (S2) /

<

- 22'+i(2j3 + 1)

X

221+iHj3 (s2) +

<

(T -t)

21+i

22ll2 21 - 1

/

1

2(2j3 + 1) where j3 G N,

2Hj3 (s2) +

\s2 -1 T-t

l2

21- iN

(P2i+i (y) + Pti(y)) dy

<

z(s2)

/

2l 1

(P23+i(y) + PUfo)) dy

, (2.190)

V

*(S2)

y

Qj3 (s2) = pj3-i(z(s2)) - Pj3+i(z(s2)),

Hj3 (s2) = Pj^_i(z(s2))+ P^+i(z(s2)).

Let us estimate the right-hand side of (2.190) using (2.141) (j3 G N)

T

j(si)(t - si)1 dsi <

s2

(T - t)2l+1 ( K2 K2^

< 2(2j3 + l) U + 2+ h.

(

2

+

l2

dy

\

(1 - (Z(s2))2)i/2 21 - 1 J (1 - y2)i/2

<

z(s2)

J

<

2

+

l2n

(T - t)2l+1K2 /_

2Jl \(1 — {z(so))2)1^2 ' 21-1

, s2 G (t,T). (2.191)

From (2.189) and (2.191) we obtain

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P3 Pi

M^(E E Cj3j3j, j> <

VJ3=0 ji=21+1i+2

2(p3+1+i)+1i

T

\

<

4

E

ji=21+1i+2

V'

(s2)|(t - s^1 £ I I j (si)(t - si) dsi | ds2

j3=P3+i

<

s2

/

i

2

i

2

1

1 2(p3+l+i)+li / T to / T \2 V

<j(T-t)°Jl E \<f>hM\ E [ UjsMit-sJdsA dso

\t J3=p3+i } )

4

ji=2l+li+2

(T-tf+V'^K'K* / / J 2d*,

16 *Jk+2 11/ (i-(^2))s)s/4+

l2TT f d.So \ ^ 1

<

2

(T - t)Al+2h+zKAK2 2p3 + 1 / j 2dy I27T f dy

~ 64 (l-l/2)3/4 +27^17 (1-y2)1/4 1 "

< C(T - tf+2h+3 0 when 00, (2.192)

p3

where constant C does not depend on p3 and T - t.

The relations (2.182), (2.183), and (2.192) imply (2.181), and the relation (2.181) implies the correctness of the expansion (2.179).

Let us consider Case 4, i.e. /i = /2 = /3 = / = 0,1, 2,... and ii,i2,i3 = 1,... ,m. So, we will prove the following expansion for iterated Stratonovich stochastic integral of third multiplicity

pi p2 p3

Cr'3' = '.i-m. EE ECj3j,j,Cj(::i)C,(22)Cj33) (ii,i2,i3 = 1,...,m),

l,t p1,p2,p3^TO z-' z-' z—' •/1 J2 J3

ji=0 J2=0 J3 =0

(2.193)

where the series converges in the mean-square sense, / = 0,1, 2,..., and

T s si

Cj3j2ji = J hj3 (s)(t - s)1/(t - si)1 hj2 (*)/(t - s2)lhji (s2)ds2dsids. (2.194) t t t

If we prove w. p. 1 the following formula

pi p3

!imTO E ECjij3j,cf = 0, (2i195)

ji=0 J3=0

where the coefficients Cj3j2ji are defined by (2.194), then using the formulas (2.167), (2.181) when 11 = 13 = 1, Theorem 1.1, and standard relations between iterated Itô and Stratonovich stochastic integrals, we obtain the expansion (2.193).

Since ^1(s), ^2(s), (s) = (t - s)/, then the following equality for the Fourier coefficients takes place

1

—I 2

where the coefficients Cj3j2ji are defined by (2.194) and

T

Cji = J j (s)(t - s)/ds. t

Then w. p. 1

pi p3

=(:2) _

( ' . . . _l H. . . . ( ' ■ ■ ■ -( '-(' ■

¿pmœ £ ECjij3ji j

ji=0 j3=0

pi p3

Pi P3 /1 \

= E E oCnCn - Cnnn - C]Ml 4"'. (2.196)

ji=0 j3=0 V 7

Taking into account (2.167) and (2.181) when l3 = li = l as well as the Ito formula, we have w. p. 1

pi p3 -, / /

l.i.m. y y cjmjlCf2) = - y c? y C,3cf

,i,p3^œ Z^Z^ jij3jiSj3 2^ j3Sj3

ji=0 j3=0 ji=0 j3=0

pi p3 pi p3

-n.m- £ ECjijij3j1 - l.^ E ECj3jijizj32)

ji=0 j3=0 ji=0 j3=0

/ T T s

1 ^ i(i- - - I if - cV

2 ^ ji J v y s 2 ji=0 t t t

T s

"ïï Jit-*)21 Jit-s№?ds = tt / T T

^/(i-^^ + 2f2ÏTn /" S)3W^2)-ji=0 Î ( ) Î

T T

J(t-Sl)1 J(t-sfdsdf M =

t s,

l T T

^ E ^ A' " 3^ + 2Î2/TIÏ /" S)3/+ldf"2)-ji=0 Î 1 t

T T

1 1 (T - t)2l+l [(t- s)ldfM + [(t - sf+ldi^ I -

2(2/+ 1)

tt l T T

j.=° -t ' t

= l\èCl- fit - S)"ds] J(t - = 0.

\ji=° t / t

Here the Parseval equality looks as follows

l

_ t

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M = £ c2, = /(t - s)21

_A _A *J

- s)ds =

(T -1)

2l+1

2/ + 1

ji=0 ji=0 t

and

T

|(t - s)ldfs(i2) = £ Cj3Zj32) w.p. 1.

l

, l 7 #»(io ) X

t j3=0

The expansion (2.193) is proved. Theorem 2.5 is proved.

It is easy to see that using the Ito formula if i1 = i2 = i3 we obtain (see (1.60))

* T * s * s i

J (t - S)^ (t - S1)^ (t - S2)ldfs(2i)dfs(ii)dfs(ii) = t t t

11 \t-swA

66

l

t ' ji=0

E Cj3j2ji Cji )Ciii)Cj3i) w.p.1. (2.197)

ji ,j2 ,j3=0

2.2.3 The Case p1,p2,p3 ^ ^ and Constant Weight Functions (The Case of Trigonometric Functions)

In this section, we will prove the following theorem.

Theorem 2.6 [6]-[16], [34]. Suppose that [fa (x)}°= 0 is a complete orthonor-

mal system of trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

J J J fff (M2,i3 = 1,...,m)

t t t

the following expansion

9fT s 2

r r r Pl P2 PS

J J J fff = pi^ EE EC** j'zW (2-198)

t t t ji=0 j2=0 j3=0

that converges in the mean-square sense is valid, where

T s s 1

Cj3j2ji = J (s) J fa2 (si) J j (S2)ds2dsids t t t

and

T

Cf = f fa (sf

t

are independent standard Gaussian random variables for various i or j. Proof. If we prove w. p. 1 the following formulas

T t

Pi P3 T 1

l.i.m. E E C»y4s) / / dsdil»', (2.199)

P1'PS^ fi=0 j=0 2 i i

T T

Pi Ps

1

Ä E E = 211(2-20°)

ji=0 js=0 t t

Pl PS

.m - ' — z I-'2)

E ECjijsjiCjS2) = 0, (2.201)

s

ji=0 js=0

then from the equalities (I2.199I)-(I2.2()H). Theorem 1.1, and standard relations between iterated ItO and Stratonovich stochastic integrals we will obtain the expansion (2.198).

We have

c - ~ ^3/2I

JPl,p3 — / j / a ./■■■/ ./ ' ^ /;; ~~ g so 1-

j3 =0 jl=0

Pi Pi

Ah) y0,2 ji,2ji Zo

Pi

+ E C0,2ji,2jiCo'' + E C0,2ji-1,2ji-lCoi3 + E C2j3,0,oC2j33 +

ji=1 ji=1 j3=1

P3 Pi

+ E T C2j3,2ji,2jij + E £ C2j3,2ji-1,2ji-1C2j33) + ^ C2j3-1,0,0C2i3J)-1 +

P3

j3=1 ji=1

j3 = 1 ji = 1 j3 = 1

P3 Pi

Cn: 1 °7i-1,2ji-

P3 Pi

+ £ E C2j3-1,2ji,2jiC2j33-1 + E E C2j3-1,2ji-1,2ji-1C2j33-1, (2.202)

j3 = 1 ji = 1

where the summation is stopped, when 2ji, 2ji - 1 > pi or 2j3, 2j3 - 1 > p3 and

(T-tf2 „ 3 (T-tf2 „ V2(T-tf2

^0,2/,2/ — -~ oTo-5 M),2Z-1,2Z-1 — -~ oTo-5 ^2Z,0,0 — "

8n2/2 '

C2r-1,2Z,2Z = 0, C2l-1,0,0 =

8n2/2 V2{T - if'2

C2r,2Z,2Z =

4n1

t)3/2/(16n2/2), r = 21

0, r = 21

t)3/2/(16n212), r = 21

4n212

(2.203)

, CW-1,2z-1>2Z-1 = 0, (2.204)

Coroi-ioi-i = { -y/2(T - î)3/2/(4tt2/2), r = I

0,

r = 1, r = 21

Let us show that

(2.205)

(2.206)

LLm. S2p1,2P3 = H.m. S2P1,2P3-1

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Pi,P3^œ pi,p3^œ

l.i.m. S2pl-1,2p3-1 = LLm. S2pl-1,2p3 •

Pi,P3^œ pi,p3^œ

(2.207)

We have

2pi

S2pi,2p3 = S2pi,2p3-1 + E C2p3ji jiC2P3i') • (2.208)

ji=0

Using the relations (2.203), (2.205), and (2.206), we obtain

2pi 2pi E C2p3,ji,ji = C2p3,0,0 + E C2p3,ji,ji =

ji=0 ji = 1 pi

,0,0 + ? v C2p3,2ji —1,2ji — 1 + C2p3,2ji,2ji

pi ^ ji = 1 V

(2.209)

From (2.208), (2.209) we obtain

l.i.m. S2pi,2p3 = l.i.m. S2pi,2p3—1. (2.210)

Further, we get (see (2.203)

2p3 —1

S2pi,2p3 —1 = S2pi — 1,2p3 — 1 + E Cj3,2pi,2pi(j^' (2.211)

j3=0

2p3 —1 2p3

E- _ ^ ^3) , ^ ^ A«3) n A«3) _

Cj3,2pi,2pij = C0,2pi,2piZ0 + Cj3,2pi,2pij — C2p3,2pi,2piZ2p3 =

j3=0 j3=1

p3 / \

C0,2pi,2piZ0 + ¿^ I C2j3 —1,2pi,2piZ2j3 — 1 + C2j3,2pi,2piZ2j3 J — C2p3,2pi,2piZ2p3 =

j3=1 V J

= ^ 8tt2p? + - 1{p3>2p1})cip1)- (2.212)

From (2.211), (2.212) we obtain

l.i.m. S2pi,2p3—1 = l.i.m. S2pi—1,2p3—1- (2.213)

Further, we have

2p3

S2pi,2p3 = S2pi — 1,2p3 + E Cj3,2pi,2pij^ (2.214)

j3=0

2P 2P

En zt(i3) _ n zt(i3) I n Z(i 3) —

Cj3,2Pi,2Pij = C0,2pi,2piZ0 + ^ Cj3,2Pi,2Pij =

j3=0 j3=1

P3

= C z(i '

= C0,2Pi,2Pi z0

j3 = 1

P3 / . \

LC03 + C2j3-1,2Pi,2PiZ2j33-1 + C2j3,2Pi,2PiZ2j3 J ■ (2.215)

From (I22T51). (ESDSHEM) we obtain

VV ^fe) _ (T - f )i/2 Ais) V2{T-t)V2 (i3)

The relations (2.214), (2.216) mean that

l.i.m. S2pi,2p3 = l.i.m. S2pi-1,2p3. (2.217)

Pi,P3^œ pi,p3^œ

The equalities (2.210), (2.213), and (2.217) imply (2.207). This means that instead of (2.199) it is enough to prove the following equality

2pi 2p3 1

T t

E E = ? / / dsdfA p-L <2-218>

/ J / J ~ j3jiji-3j3 o

Pi,P3 ^^ —' z—' 73 2

ji=0 j3=0 "t "t

We have

2p3 2pi

o v^ v^ / A*) _ (T ~ f)3/2 An) ,

^>2P1,2P3 — / „ / / ./:■,/ ./ S— g So

j3=0 ji=0

Pi Pi Pi

+ y C0,2ji,2ji(0i3) + 53 C0,2ji-1,2ji-1C0i3) + 53 C2j3,0,0C2j33) +

ji=1 ji=1 j3=1

P3 Pi P3 Pi P3

+ y y C2j3,2ji,2jiC2j33) + £ S C2j3,2ji-1,2ji-1C2j33) + 53 C2j3-1,0,0C2j^^-1 +

j3=1 ji=1 j3=1 ji=1 j3=1

P3 Pi P3 Pi

+ 53 53 C2j3-1,2ji,2jiZ2j33-1 + £ 53 C2j3-1,2ji-1,2ji-1C2j33)-1- (2.219)

j3=1 ji=1 j3=1 ji=1

After substituting (2.203)-(2.206) into (2.219), we obtain

2P3 2Pi /1 Pi 1

EE wr = v-tf** !<*»>* c

j3=0 ji=0 V ji=1 j1

p3 ^ ^ min{pi,p3} ^ ^ p3 ^

47r" S - 4^2 E 72C2S 72C2S 1 • (2-22°)

j3=1 J3 j3=1 J3 j3=1 J3

From (2.220) we have w. p. 1

2p3 2pi / œ .

(¡3) _ (rp +\3/2 I 1 /■ (¡3) 1 V^ 1 ^(¿3)

,£ E C^ = (T — ¿)3'21 + ^ E A Co

j3 =0 ji=0 \ ji=1J1

-l.i.m. t I • (2.221)

j3 = 1

Using Theorem 1.1 and the system of trigonometric functions, we get w. p. 1

T s T

drdf™ = lf(s- t)df^ =

t t t

<2-222>

X n z—' J3

j3 = 1

From (2.221) and (2.222) it follows that

2p3 2pi

U-m. £ ECj3j,ji Zj33)

j3=0 ji=0

= (t - i)3/2 ^c,(,*3) + ¿c,(,*3) - ^g !<<*>_,

T s

= ljj drdf™, tt

where the equality is fulfilled w. p. 1.

So, the relations (2.218) and (2.199) are proved for the case of trigonometric system of functions.

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Let us prove the relation (2.200). We have

pi p3 (T — t)3/2

s def C Z(

ji =0 j3=0

(ii)

ji

6

z0ii)+

p3

pi p3

+ E C2j3,2j3,0C0«i) + E C2j3 —1,2j3 —1,0C0«i) + E E C2j3,2j3,2ji — 1C2ji — 1 +

j3 = 1

pi p3

ji=1 j3=1

pi

pi p3

+ E E C2j3 — 1,2j3 — 1,2ji — 1Z2ji — 1 + E C0,0,2ji — 1C2ji — 1 + E E C2j3,2j3,2jiC2j"i +

ji=1 j3=1

ji=1

pi p3

+ EE c

ji = 1 j3=1

pi

ji=1 j3=1 (ii)

2j3 —1,2j3 —1,2ji Z2ji + C0,0,2ji Z2ji ,

(2.223)

ji=1

where the summation is stopped, when 2j3, 2j3 — 1 > p3 or 2j1; 2j1 — 1 > p1 and

(T-tf/2 ^ 3 (T-tf'2 „ V2(T-tf/i

<^2/,2/,0 = -o 0 70-5 ^2/ —1,2/—1,0 — -O_o?o-, ^0,0,2r — ■

8n2l2

C2I—1,2/ — 1,2r— 1 = 0, C0,0,2r—1 =

8n2l2 \/2(T - i)3/2

C2/.

2/,2r =

4nr

t)3/2/(16n2l2), r = 2l

0, r = 2l

t)3/2/(16n2l2), r = 2l

4n2 r2

(2.224)

, C2/,2/,2r—1 = 0, (2.225)

(2.226)

C2/-i,2/-i> = ^ -y/2(T - i)3/2/(47r2/2), r = I

(2.227)

0, V 7

r = l, r = 2l

Let us show that

l.i.m. SL, o„„ = l.i.m. S'

2pi,2p3

2pi,2p3 —1

= l.i.m. S'

2pi —1,2p3 —1

= l.i.m. S'

2pi —1,2p3-

We have

2p3

S2pi,2p3 = S2pi —1,2p3

+ 53 Cj3,j3,2pi C:

(ii) iS2pi •

(2.228)

(2.229)

j3=0

Using the relations (2.224), (2.226), and (2.227), we obtain

2p 2p y Cj3,j3,2pi = C0,0,2pi + 53 CJ3,J3,2pi =

ji =0 J3=i

p3

= C0,0,2pi + / v C2j3-1,2j3-1,2pi + C2j3,2j3,2pi

p3 /

C0,0,2pi + 53 ( C2j: J3=1 ^

v/2 (r-f)^. .

4^2 I1 1{p3>pi}j-

(2.230)

From (2.229), (2.230) we obtain

Li.m. S2pi,2p3 = l.i.m. S2pi-1,2p3. (2.231)

Further, we get (see (2.224)-(2.226))

2pi-1

S2pi-1,2p3 = S2pi-1,2p3-1 + 53 C2p3,2p3,jiCj'1 , (2.232)

ji=0

2pi-1 2pi

E- /-('i)_^ /-('i) i ^ A'i) r z('i) —

C2p3,2p3,jij = C2p3,2p3,0S0 + C2p3,2p3,jij - C2p3,2p3,2piz2p1 =

ji=0 ji=1

pi / \

C2p3,2p3,0S0 + ^ I C2p3,2p3,2ji-1S2j1-1 + C2p3,2p3,2jij - C2p3,2p3,2piZ2p1 = ji=1 V '

(T-t)»/' ,„, , (>1)

~~ 87T2^ 0 167r2p| li{pi=2p3} - A{pi>2p3}A4p3 •

From (2.232), (2.233) we obtain

H.m. S2pi-1,2p3 = Li.m. S2pi-1,2p3-1. (2.234)

Further, we have

2pi

S2p1;2p3 = S2p1,2p3-1 + 53 C2p3,2p3,jiCj'1 , (2.235)

ji=0

2pi 2pi

En z('i) — n z('i)

C2p3,2p3,jij = C2p3,2p3,0S0 + C2p3,2p3,jij

ji=0 ji = 1

pi / . \ = C2p3,2p3,0(0i + E C2p3,2p3,2ji-1C2ji-1 + C2p3,2p3,2ji(2^ J • (2.236)

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i-1S2ji

ji = 1 X

From (2.236), (2.224)-(2.226) we obtain

2pi

^ n An) _y± ~ L__

.2

E^^JiCj^ - ^ \ Col} 16TT2^ ^i^ldp^• (2-237) The relations (2.235), (2.237) mean that

Li.m. S2pi,2p3 = l.i.m. S2pi,2p3-1 • (2.238)

The equalities (2.231), (2.234), and (2.238) imply (2.228). This means that instead of (2.200) it is enough to prove the following equality

2pi 2p3

T t

1-i-H.. = 1 I I dfs'l]dr w. p. 1. (2.239)

We have

2pi 2p3

qf AH> - 1 > AH> i

j1=0 j3=0

p3 p3 p1 p3

+ E C2j3,2j3,0C0ii) + E C2j3-1,2j3-1,0C0ïi) + E E C2j3,2j3,2j-1(2^-1 +

j3=1 j3=1 ji=1 j3=1

p i p3 p i p i p3

+E E C2j3-1,2j3-1,2ji-1 c2j i-1 + E C0,0,2j i-1 c2j i-1 + E E C2j3,2j3,2j ic2ji)+

j i=1 j3=1 j i=1 j i=1 j3=1

p i p3 p i

+ E E C2 j3 -1, 2 j3 -1, 2 j i c2ji) + E C0,0,2j i c2jî)- (2.240)

j =1 j3=1 j =1

After substituting (2.224)-(2.227) into (2.240), we obtain

2p i 2p3 ( -, 1 p3 ..

E E = {t - tr- ^+ b E i (A+

j i =0 j3=0 V j3=1 j3

\/2 Pl 1 y/2 min^1':p3J' i y/2 pl i

+ 77^-1 - 4^2 fi&ji -2Q1 1 • (2-241)

ji=iJ1 ji=1 J1 ji=^1

From (2.241) we have w. p. 1

2pi 2p3 / 1 TO

E E wr - - ^ ^ + ± E 3

ji=0 j3 =0 \ j3=1J3

pi^to 4n ji 2ji 1 / ji=iJ1 /

(2.242)

Using the Ito formula and Theorem 1.1 for the case of trigonometric system of functions, we obtain w. p. 1

T t / T T

y J df^dr = 1-l(T-t)J dtp + I(t - sjrffi") I = t t t t

/?: pi

- f)3/s ( ii"1 + l-i-m. ^ E )' <2'243)

4 y pi >-to n j—1 ji J1 1

From (2.242) and (2.243) it follows that

2pi 2p3

li.m^ ^ ECj j"

ji=0 j3=0

T t

i II df^dr, tt

where the equality is fulfilled w. p. 1.

So, the relations (2.239) and (2.200) are proved for the case of trigonometric system of functions.

Let us prove the equality (2.201). Since (t), (t), (t) = 1, then the following relation for the Fourier coefficients is correct

( '. . . . ( '. . . . ( '. . . Lf '-(' .

^jum * ^jrnji * — 2 ji J3-

Then w. p. 1

pi p3

=(¡2) _

O-i r)o—Vnn f * '

.m.

ji=0 j3=0 Pi P3 / i

E E ( 2~ ~ c*** ) « <2'244'

ji=0 j3 =0

Taking into account (2.199) and (2.200), we can write w. p. 1

p1 p3

l. i-m. E ECji Zj32)

ji=0 j3=0

1 Pi P3

= A'oG ' - l.i.m.

2 pi,p3^œ z—* z—* 73

ji =0 j3=0

Pi P3

- l.i.m. E ECj3jijiZj32)

ji =0 j3=0

<« _ J(r - tr Uk) + U.m. £ £ ±4?.

4 \ pi^œ n z—' ji •yi

4 \ pi^œ n ^^ ji

1 j1=1 1

From Theorem 1.1 and (2.199)-(2.201) we obtain the expansion (2.198). Theorem 2.6 is proved.

2.2.4 The Case p = p2 = p3 ^ œ, Smooth Weight Functions, and Additional Restrictive Conditions (The Cases of Legendre Polynomials and Trigonometric Functions)

Let us consider the following modification of Theorem 2.5.

Theorem 2.7 [10]-[16], [34]. Assume that (x)}°=0 is a complete or-

thonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and ^i(s), fa2(s), fa3(s) are continuously differentiate functions at the interval [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

J*[fa(3)]T,t = / fa(to/ «fc)/ ^of^fdft(;s) (¿1,^2,^3 = 1,...,m)

# # #

the following expansion

p

J*[V'(3)]r,t = £ jj (2.245)

j1,j2,jS=0

that converges in the mean-square sense is valid for each of the following cases

1. ¿1 = ¿2, ¿2 = ¿3, ¿1 = ¿3,

2. ¿1 = ¿2 = ¿3 and fa (s) = fa(s),

3. ¿1 = ¿2 = ¿3 and fa(s) = fa(s),

4. ¿1, ¿2, ¿3 = 1,..., m and fa(s) = fa(s) = fa(s),

where

T s si

Cjsj2ji = / fa(s)fas (s) / fa(s1)0j2 (s0/ fa Mj (s2)ds2ds1ds t t t

and

T

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j = 1(s)f

t

are independent standard Gaussian random variables for various % or j.

Proof. Let us consider at first the polynomial case. Case 1 directly follows from Theorem 1.1. Further, consider Case 2. We will prove the following relation

Ts

p P 1 C C

EE^(t3) = 2 J J ^Mds^ w. p. 1,

ji=0 js=0 t t

where

T s si

Cjsjiji = fa(s)fas(s) / fa(s1)0ji (s0 / fa(s2)0ji (s2)ds2ds1ds-

Using Theorem 1.1, we can write w. p. 1

1

p3

tt

2 ps^^ z

js=0

Us) I il>2MdSldfM = ^ l.i.m.

where

T

Cjs = J j(s)Ms) J fa2(s1)ds1ds.

tt

We have

pp

1

M {IE E - 14

js =0 \ji=0

( T

p 1 p f

(is)

pp

ESwi =

js =0 \ji=0

1

— I 2

j =0 2 ji=0 t t

l

\

1

T

J h-MMs) j i/j2{si)dsids I = tt

-T

4

(T

j =0

j (s)fa3(s)

/ P ( s \2 s X X

V

53 I I j(s1)fa(s1)ds^ - / fa2(s1)ds1 ji=0 t t

ds / /

1 p (T \2 \

j =0

ds

(2.246)

V

ji=p+1 \ t

/

In order to get (2.246) we used the Parseval equality

2

where

)2 T s

= K 2(s,s1)ds1 = fa2(s1)ds1, tt

K(s,s1) = fa(s1)1{si<s}, s,s1 G [t,T].

Ts

1

2

s

2

s

2

We have for j £ N

2

^(s1)0ji (s1)ds1 I =

(T-t)(2j i + l) 4

T-t

4(2j1 + 1)

/ z(s) ^

J Ph(y№ + dy —1

(Pj-i+1(z(s)) — Pji—1(z(s))) ^(s) —

z(s) \ 2 T — t T — t T + t

\^-y + ^-)dy] , (2.247)

M ^^ .n r y 2 ° 2

1

where

( \ ( T + A 2

z{s) = -s -

2 JT — t'

and is a derivative of the function ^(s) with respect to the variable

T — t T + t —y + —

Further consideration is similar to the proof of Case 2 from Theorem 2.5. Finally, from (2.246) and (2.247) we obtain

<r K— ( ( dy + [ dy \ <

f [I (i-y2)3/4 y (1 -y2)1/4J -

<--> 0 if p ->■ oo,

P

where constants K, K1 do not depend on p. Case 2 is proved.

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Let us consider Case 3. In this case we will prove the following relation

Ts

p P 1 C C

EE = 2 / ^ / 'Msi)df^ds w. p. 1,

p ^ ji=0 j3=0 { {

2

where

T s si

Cjsjs ji =

J fa(s) j (s) J fa(s1)0js (so/ faM j (s2)ds2ds1 ds.

t t t

Using the Ito formula, we obtain w. p. 1

T s T T

i J fa(s1)di^ds = ^ J msi) J^(s)dsdi^. (2.248)

t t t si

Moreover, using Theorem 1.1, we have w. p. 1

T T

5 / lf(s)dsdiP = i 11m. £ C* <2'249)

t si i ji=0

where

T T

C* = j(s1)^1(s1W fa2(s)dsds1.

t si

Further,

T s si

jsji = / (s) / (s0/ faMj (s2)ds2ds1ds =

t t t

T T T

= J ^lMj (s2) / ^(si)0j3 (si)/ (s)dsdsids2 =

t S2 si

T / T \ 2

= ^ J 'Mso^niso) J fasi)<f>j3(si)dsi ) (2.250)

t s2

From we obtain

p/p \ \ 2 | p/p ix2

M { (E E ^ - iq («;■» V V - Iq

ji=0 Vs=0 / / J ji =0 Vs =0

-T

4

T

j'i=0

T

j (S1)^1(S1)

V'

V

E I I j(s)^(s)ds1

j3=0

vsi

T

— J ^2(s)ds | ds1 | =

si

2

j(s)^(s)ds I ds1

^E f faMMsi) E

ji=0 ^ t j3=p+1 \si /

/

In order to get (2.251) we used the Parseval equality

T

E ( / j(s)^(s)ds ) = I K2(s,s1)ds = I ^2(s)ds, j3=0

vsi

si

where

K(s,s1) = ^(s)1{s>si}, s,S1 £ [t,T].

(2.251)

Further consideration is similar to the proof of Case 3 from Theorem 2.5. Finally, from (2.251) we get

pp

M { (E (E cnnn - iq ) cj

ji =0 Vj3=0

(ii) ■j'i

<

p2

dy

+

dy

<

/ (1 — y2)3/4 J (1 — y2)1/4 1 —1

K1

<--> 0 if p ->■ oo,

p

where constants K, K1 do not depend on p. Case 3 is proved.

Let us consider Case 4. We will prove w. p. 1 the following relation

pp

l.i.m. EE Cjji Zj32) = 0 W1(s),«s),«s) = ^(s)).

j'i=0 j3=0

2

1

In Case 4 we obtain w. p. 1

p

(i2)

Li.m- CjijsjiZ

p—>-TO ' *

ji,js =0

ji,js =0 V y

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1 p p p ji=0 js =0 ji,js=0

p

(i2)

-LLm. Cjsjijij

p—to z—* ji ,js =0

00 T T s

ji=0 t t t

T s T T

i [ Ijj(s) f fa(si)dsidfs(i2) = i f ijj2(s)ds f tjj(s)dii

T T T si

/ / ~IJ ^ J ^(s)dsdi^ =

t si t t

T T T T

= IJ i)2(s)ds J -l-j fas,) J ^(s)dsdi^ = 0, t t t t

where we used the Parseval equality

/ T \ 2 T

to to 1 „ \ r

EC = E / fa(s)0j(s)ds = fa2(s)ds. ji=0 j=0 t t

Case 4 and Theorem 2.7 are proved for the case of Legendre polynomials.

Let us consider the trigonometric case. The complete orthonormal system of trigonometric functions in the space L2 ( [t, T] ) has the following form

fa (0) =

1,

j = 0

Vr^t

\/2sm (27rr(0 - t)/{T -t)), j = 2r - 1, \/2cos (27rr(0 - t)/(T - t)), j = 2r

where r = 1, 2,...

Integrating by parts, we have

far_1(0)^(0)d0 =

y/2

y/T=t

<T - t 11 2nr(s - t)

COS-"7---+

2 nr

Tt

<kr(O)4>(O)d0 = f* [ faO) cos 2lirS0~^de =

y/T^t

Tt

'T -1 1 / ,, , . 2nr(s - t) ys) sm-

2 nr

Tt

where r = 1, 2, . the variable 0.

Then

and ^'(0) is a derivative of the function ^(0) with respect to

far-1(0)^(0)d0

C 2C 2C < — = — <-

r 2r 2r 1

(2.252)

1

s

far (0)fa0)d0

<C_2C

— r 2r '

(2.253)

where constant C does not depend on r (r = 1, 2,...). From (2.252), (2.253) we get

j (0)fa0)d0

<

(2.254)

where constant K is independent of j (j = 1, 2,...). Analogously, we obtain

T

j (0)fa0)d0

<

(2.255)

where constant K does not depend on j (j = 1, 2,...). Using (2.246), (2.251), (2.254), and (2.255), we get

p / p

"{IE Ec

j3=0 \ji=0

_ \ Ais)

jsjljl 2 /

pp

M {( E ( E ^ - ¿q. ) ci

ji =0 \j3 =0

^ Ki n

<--> 0 if p ->■ oo,

p

Ki

<-->■ 0 if p ->■ oo,

p

where constant K1 is independent of p.

The consideration of Case 4 is similar to the case of Legendre polynomials. Theorem 2.7 is proved.

In the next section, an analogue of Theorem 2.7 will be proved without the restrictions 1-4 (see the formulation of Theorem 2.7).

2.2.5 The Case p1 = p2 = p3 ^ œ, Smooth Weight Functions, and without Additional Restrictive Conditions (The Cases of Legendre Polynomials and Trigonometric Functions)

Theorem 2.8 [10]-[16], [21], [32]. Suppose that {fa (x)j 0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the

s

space L2([t,T]). At the same time ^2(s) is a continuously differentiable non-random function on [t,T] and ^i(s), ^3(s) are twice continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

-.T * t3 * ¿2

ti

J*[^(3)]T,t = J ^3(toy «^y ^1(t1)dft(;i)dft(;2)dft(;3) (n,^ = i,...,m) t t t

the following expansion

P

p^œ

J •[^<3)]t,( = U.m. E Cj cfcfcjr (2.256)

ji,j2,j3=0

that converges in the mean-square sense is valid, where

T s si

Cj3j2ji = J ^3(s)0j3 (s) ^ ^2(S1)0j2 Mj (s2)ds2ds1ds

t t t

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and

T

j = / fa

t

are independent standard Gaussian random variables for various i or j.

Proof. Let us consider the case of Legendre polynomials. From (1.47) for the case p1 = p2 = p3 = p and standard relations between Itô and Stratonovich stochastic integrals we conclude that Theorem 2.8 will be proved if w. p. 1

Ts

p P 1 f f

^ E E = 2 J J 'HsiyMs^ds^K (2.257)

ji=0 j3=0 t t

Ts

p p 1 T s

^ E E Chhnqf = 2 / / (2.258)

p œ ji=0 j3=0 { {

p p

^EE zj32) = °- (2.259)

ji=0 j3=0

Let us prove (2.257). Using Theorem 1.1 for k = 1 (also see (1.45)), we can write w. p. 1

T s p

\ ( Ms) f Usi)Usi)dSldi^ = l-U.m. V r. < "' .

2 2 p^TO

t t j3=0

where

T s

Cj3 = J fa ^2(si)^i(si)dsids.

tt

We have

2

^ m J / V^ V^ ^(^3) 1 v^ n A*)

ji=0 j3=0 j3=0

2

p p 2

j3=0 \j'i=0

p p 2

v Vr =

2

j3=0 \ji=0

p / p T s s/i

fa(s)fa (s) / fa^Ofai (siW fa^fa (s2)ds2ds1ds

j3=0 \ji=0 t t t

i ?......;........V2

2

tt

p ( T s / p = E / ^3(s)0j3(s) / X^^Ofai (s1)x

j3=0 \ i { Vji=°

si \ \ 2

X J ■i/>i{s2)<f>j1{s2)ds2 - ifa(si)fa(si) ) rfsirfs ) • (2.260)

Let us substitute t1 = t2 = s1 into (2.12). Then for all s1 G (t,T)

Si

/ 1

/ '01(52)^(52)^9 = -fa (-Si )fa («1) • (2.261)

ji=0 /

From (2.260) and (2.261) it follows that

T

si

Ep = E / ^3(s)j(s) / Y ^(sOj(s^ / ^1(s2)j(s2)ds2ds1ds

j3=A t t ji=p+1 t

(2.262)

Applying (2.262) and (2.24), we obtain

p / t / z(s)

EP<CiJ2 J\MM\l~ J

j3=0 t

T

dy

+

z(s) \ \

dy

P J (1 — y2)1/2 J (1 — y2)1/4 —1 —1

ds / /

<

'/ V (I \faMds) < ^^ t J 4&W

T

2 rs)ds = ^ ^ 0

j.=0\i / p2 jf=0 J 33 p

if p —to, where constants C1, C2, C3 do not depend on p. The equality (2.257) is proved.

Let us prove (2.258). Using the Ito formula, we have

T s T T

i J c(s)c-As) J Usi)di^ds=l- Jfa(si) J fa{s}Us)dsdi^ w. p. 1.

t t t si

Moreover, using Theorem 1.1 for k = 1 (also see (1.45)), we obtain w. p. 1

1p

I -ilhiff) I = - l.i.m. 1

T T

\ [ Ms) [ Msi)Msi)ds!dfM =

ji=0

t

where

T T

Cj = Ws) j(s) / ^3(s1)^2(s1)ds1ds.

(2.263)

We have

pp

ji=0 j3=0 pp

j1=0

M {(E (E c1»«. - ^) cj

ji=0 \j3=0

(ii) j1

2

s

p

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2

P / p i \ 2

E V'- , (2.264)

ji=0 \j3=0 J

T s si

Cj = J (s) ^ fa(sl)0j3 fa(s2)0ji (s2)ds2dsids =

t t t

T T T

= J fa(s2)fai(s2) J fa(si)fa-3(si) J fa(s) j(s)dsdsids2. (2.265)

t S2 Si

From (2.263) (2.265) we obtain

, T T ,

p In f i P

Ep = E / (s2)^ji M / 53^2(s1)0j3 (s1)x

ji=0 \ t «a Vj3=0

x j c:,{s)orA{s)(ls - l03(Sl)^2(Sl) j dsid.s2j • (2.266)

We will prove the following equality for all s1 G (t,T)

T

„ 1

/ C,(s)0,Js)(ls = -r,(.s-|)r:!(.s-|). (2.267)

2

j3=0 si

Let us denote

= + ^l{^2}fa(ii)fa(ii), (2.268)

where

Ki(ti,t2) = fa (tl)^3(t2 )1{ti<t2}, ti ,t2 G [t,T].

Let us expand the function Ki*(ti,t2) using the variable t2, when ti is fixed, into the Fourier-Legendre series at the interval (t,T)

T

to „

K*(ti,t2)^53 ^2(tiW fa^^j (t2)dt2 • j (t2) (¿2 = t,T). (2.269)

j3=0 ti

The equality (2.269) is fulfilled in each point of the interval (t,T) with respect to the variable t2, when t1 £ [t, T] is fixed, due to piecewise smoothness of the function K{i(t1 ,t2) with respect to the variable t2 £ [t,T] (t1 is fixed).

Obtaining (2.269), we also used the fact that the right-hand side of (2.

converges when t1 = t2 (point of a finite discontinuity of the function K1(t1, t2)) to the value

^(IuihAi -0) + K1{t1,t1 + 0)) = ^Uhyuti) = KUhiti).

Let us substitute t1 = t2 into (2.269). Then we have (2.267). From (2.266) and (2.267) we get

/ T T T

p / !> !> to „

Ep = E / ^1(s2)^ji(s2) / E ^2(s1)0j3(s1M ^(s) j(s)dsds1ds2

ji=0 \ t s2 j'3=p+1 si

(2.270)

Analogously with (2.24) we obtain for the twice continuously differentiable function ^3(s) the following estimate

T

;,; * < si

E j / ^3(s)0j3 (s)ds

j3=p+1

< P 1,(1- (^(5i))2)1/2 + (1 - (^(Si))2)1/4J ' (2'271)

where s1 £ (t,T), z(s1) is defined by (2.20), and constant C does not depend on p.

Further consideration is analogously to the proof of (2.257). The relation (2.258) is proved.

Let us prove (2.259). We have

{/pp \2| p/p \ 2 EECjji j2) = E ECjijji , (2.272)

\ji=0 j3=0 / J j3=0 Vji=0 /

T s si

Cj'ij3j'i = ^3(s)0ji (s) / ^2(s0j (s0 / ^1(s2(s2)ds2ds1ds =

T si T

= J fa(s1 (s 1) J fa(S2) j (s2)ds^ fa(s)j (s)dsds1. (2.273)

t t si

After substituting (2.273) into (2.272), we obtain

p ( T P ? T ^ 2

E'p = E I fa2(s 1 (s 1 I fa 1 Wji fa(s)fa (s)dsds 1

j3=0 V t ji=0 t si

(2.274)

The generalized Parseval equality gives

si T

to i „

J fa(0)fafa(s)fa(s)ds

ji=0 t si

T T

to

/ 1{^<si}fa1(^)0ji l|s>si|fa3(s)0ji (s)ds

ji=0 t t T

1{r<si}fa(r )1{r>si}fa3(r )dr = 0. (2.275)

t

Using (2.274) and (2.275), we get

p / T to s^ t

E? = E / j(*) £ / faWji(0)dW fa(s) j(s)dsds1

j3=0 \ { ji=p+1 { si

(2.276)

Let us write the following relation

J ^i{s)(!>h{s)ds = ——+ 1 J Ph{y)fa{u{y))dy =

t -1

y/T^t.

2fa2j7TT

z(x)

(pi+1(z(x)) - pji-1(z(x)))fa1(x)-

T f I ((Pjl+i(y) - Pn-i(y))U(U(y))dy ), (2.277)

2

1

where x £ (t,T), j > p + 1, z(x) and u(y) are defined by (220), is a derivative of the function ^1(s) with respect to the variable u(y).

Note that in (2.277) we used the following well known property of the

Legendre polynomials [109

Pj+1 (—1) = —Pj (—1), j = 0,1, 2,...

From (2.141) and (2.277) we obtain

^1(s)0ji (s)ds

<

C

jA (1 — (z(x))2)1/4

+ C1

(2.278)

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where j £ N, x £ (t, T), constants C, C1 do not depend on j. Similarly to (2.278) and due to

Pj(1) = 1, j = 0,1, 2,...

we obtain an analogue of (2.278) for the integral, which is similar to the integral on the left-hand side of (2.278), but with integration limits x and T.

From the formula (2.278) and its analogue for the integral with integration limits x and T we obtain

T

(s) j(s)ds / ^(s) j(s)ds

<

1

K

7A{i-{z{xm^

+ K1

where j £ N, x £ (t, T), and constants K, K1 do not depend on Let us estimate the right-hand side of (2.276) using (2.279)

(2.279)

e;<

p / T to ^ t

< ¿E J I j (s1)| E J W^j (^)d^ «s) j (s)ds

j3=0 \ t ji=p+1 t si

ds1 <

< L± ( [ ¿if-

j3=0 \ ji=p+1 j1 V

(1 — (z(s1))2)1/2

+ K1 ds1 <

x

1

x

2

1

t "

T

ds

T

1

r — \ J (1 - (z(s1))2)3/4

+ K1

dsi

(1 - (z(S1))2)1/4

L2(T - t-l £

1

dy

4p2 (1 - y2)3/4

+ K1

dy

1

(1 - y2)1/4

<

L3p L3 p2 p

(2.280)

if p —y to, where constants L,L1,L2,L3 do not depend on p and we used (2.25), (2.142) in (2.280). The relation (2.259) is proved. Theorem 2.8 is proved for the case of Legendre polynomials.

Let us consider the trigonometric case. Analogously to (2.34) we obtain

T _ T

„ co „

s 2

53 ^2(s1)0j3(s1) / fa(s)fa3(s)dsds1 j3=P+1

si

<

Ki p

(2.281)

where s2 £ (t,T) and constant K1 does not depend on p. Using (2.34) for T = s and (2.262), we obtain

p / T s s/

Ep < / / 53 fa(s1 )fa(s1M faM j(s2)ds2ds1

j3=0 V t t ji=P+1 t

t ji=P+1

ds

j3=0 V ^ 7 ^ j3=0 ^

(2.282)

if p — to, where constants K, K1, K2, L do not depend on p.

Analogously, using (2.281) and (2.270), we obtain that Ep — 0 if p — to. It is not difficult to see that in our case we have (see (2.254), (2.255))

x T

J fa(s)fai(s)ds^ fa(s)fai(s)ds <

Jx

where j £ N, constant C1 does not depend on j. Using (2.276) and (2.283), we obtain

Ci J?

(2.283)

E" <

p<

2

1

2

rp n rp \ 2

P / « TO

¿3=0 \ t ji=P+1

si T

/s)0j(s)ds

< LE J i^j'3(si)l E J ^ «s)fa,i(s)d.

t si

dsH <

p / to \ 2 L p

¿3=0 V ji=p+^v ^ ¿3=0

l2

< — ^ 0 (2.284)

P

if p — to, where constants L, L1, L2 do not depend on p.

Theorem 2.8 is proved for the trigonometric case. Theorem 2.8 is proved.

2.3 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4 Based on Theorem 1.1. The Case pi = ... = p4 —y to (Cases of Legendre Polynomials and Trigonometric Functions)

In this section, we will develop the approach to expansion of iterated Stratonovich stochatic integrals based on Theorem 1.1 for the stochastic integrals of multiplicity 4.

Theorem 2.9 [8]-[16], [21], [32]. Suppose that {fa(x)}TO=0 is a complete

orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t, T]). Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

J*[^(4)]T,t = II J I dwt(1l)dwt(22)dwi33)dwi44) (ii, i2, ¿3,24 = 0, 1,..., m) t t t t the following expansion

p

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J * [V'(4)]r,f = lpi4m. E Cjji if c^W1 (2.285)

¿ij2j3j4=0

that converges in the mean-square sense is valid, where

T S4 S3 S2

Cj4j3j2ji = fa4 (s4 ) / fa'3 (s3 ) / j (s2) fai (s1 )ds1ds2ds3ds4

and

T

J

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT;) = f() for i = 1,..., m and wT0) = t.

Proof. The relation (1.48) (in the case when p1 = ... = p4 = p — to) implies that

p

- (hMMX^MM)

/ C7473727i z

p—TO

lp—to. E c'C w1 = j[fa(4)]T,j+

ji ,j2 J3J4=°

I 1 /l(;3;4) , n /l(;2;4) , n /l(;2;3) , n /l(;i;4) ,

+ 1{;i=;2=0}A1 + 1{;i=;3=0}A2 + 1{;i=;4=0}A3 + 1{;2=;3=0}A4 +

,1 /l(;i;3) I n /1 (;i;2) 1 1 R

+ 1{;2=;4=0}A5 + 1{;3=;4=0}a6 - 1{;i=;2=0} 1{;3=;4=0}B1-

- 1{;i=;3=0}1{;2=;4=0}B2 - l{;l=;4=o}l{;2=;з=o}Bз, (2.286)

where J [fa(4)]TjJ has the form (2.7) for fa1(s),..., fa4(s) = 1 and i1,...,i4 = 0, 1, . . . , m,

A (;3;4) = l ; m V^ C- ■ ■ ■ Z(;3)Z(;4) A1 =lp—1TO. C74j3jiji Zj3 Zj4 ,

j4j3ji=0

p

^ = lp—ito . E Cjc^41,

j4j3j2=0

p

A(;2;3) =1 : m V^ C . . c(;2)Z(;3)

A3 =lp—TO . C74j3j2j4 cj2 7 ,

j4j3j2=0

p

A(;i;4) = l ; m V^ C. . . . c(;i)Z(;4) A4 = lp—mTO . Z^ C74j3j3ji j j ,

j4j3ji=0

A(;i;3) = l : m C- ■ ■ ■ c(;i)Z(;3)

A5 =lp—1TO. C j4j3 j4 ji j j ,

j4j3ji=0

p

A(;i;2) = l : m C- ■ ■ ■ c(;i)c(;21

A6 = lp—mTO. Z^ j3333i23i j j ,

j3j2ji=0

pp B1 = lim E Cj4j4jiji , B2 = l_im N Cj3j4j3j4 ,

p^œ z—' p^œ z—'

ji ,j4=0 j4,j3=0

B3 = lim V C

p^œ ^^

j4j3j3j4

j4,j3=0

Using the integration order replacement in Riemann integrals, Theorem 1.1 for k = 2 (see (1.46)) and (2.10), Parseval's equality and the integration order replacement technique for Ito stochastic integrals (see Chapter 3) [1]-[16], [72], 111], [112] or Ito's formula, we obtain

A

(«3«4)

p

1

T

si

2

E 2j I ( I Oj.(s.,)<ls2 ) dSids^Xj.

j4,j3,ji=0 t

(i3)z (i4) j4

t

p

= l.i.m. \

p^œ ^—' 2

j4 ,j3=0 t

1

T s p / si

2

(«3^(«4) j4

j M / j O^E / j dS1 Q

{ ji=A t

P 1 ^ s ( œ / si \2\

l.i.m. >

p

T

= l.i.m.

p^œ ^—' 2

j4 ,j3=0 t

j (s) J j (s1) t

V

(s1 - t) - E / (s2)ds2

ji=p+A t J J

2

dsi dsx

/-(«3^(«4) _ XZj3 z j4 =

p

1

T

= l.i.m.

p^œ ^—' 2

j4 J3=° t

E ^ I mm I fa3(-si)(-si - - =

i

2

t

T s

tt tt

T

(S1 - t)dw(;3)dw(i4) +

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+ 91{»3=^o} lim e / / fa3(si)(-siA^4)

p

j3=0 i

T s si T

l [[ [ ds2d™ff>d^+\l{n=H^} [(Sl-t)dSl-Ap4) w. p. 1, (2.287)

î Î Î

1

s

where

A

(;3;4)

P A;3 M;4)

= l i m V . c(;3)c(;4) lp—TO . Z^ aj4 73 7 7 ,

j3j4=0

T

Si

aj4j3 =

^ / / E I /

t t ji="+^ j

ds1ds. (2.288)

Let us consider A

T

(;2;4)

/l(;2;4) A2

s2

lp—1TO. E I 7 (s) / 07 M / 073 (s3)ds3 / 073 (s 1)ds 1 ds2dscj2;21 c

(;2^(;41 74

j4j3j2=0 t

s2

= l.i.m.

P—TO ^—'

j4 73 ,j2 =0

/ T

5/

\ J

2

1

T

074 (s) ( J 073 (s3)ds3 I J 0j2 (s2)ds2ds-t ) t

S2 x 2

-o / / 0^2(s2) / (pn(s3)ds3 ds2ds-

2

tt T s

\

2

074 (sW 072 M / 073 (s1)dsH ds2ds

s2

/

c (;21 c (;4) = 7 7

p

T

1

LLm- E 9 / / (f>n(s2)ds2ds-

= i.i.m.

p—TO ^—' I 2

.?4j2=0 \ t

T

1 2

074 (s) / 072 (s2)(s2 - t)ds2ds-

tt T s

074 (sW 072 (s2)(s - t + t - s^ds | ci;2)c]44 ^

- a2;2;4) + a1;2;4) + A3;2;4) = -A2;2;4) + a1;2;4) + A3;2;4) w. p. 1, (2.289)

p

1

2

s

2

s

p

s

2

s

1

s

1

2

where

p

a2'2!4) = ip——to- E cf cir1,

74,72=0

p

A3,2!4)=¡s- E №.

74,72=0

T ™ / s \ 2 s

1 /> TO ' x

= E [J^jsMdsij l^n(s1)dsids, (2.290)

2 .

*J 73=p+1 \ "J

T s / s

1 /> /> TO '

(?jd2 = - / fa(s) / fa2(s3) E / feM^i ) (2.291)

{ { 73=p+1 Vs3

Let us consider A5

(;i;3)

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A5

p

(;i;3) = 5=

T T T T

(;iM;31

= p——1TO- E J 07i (s3^ 7 (s2^ 073 (s1^ 074 (s)dsds1ds2ds3c];;i)c J47'37'^0 t s3 s2 si

p T T T si

= lp—TO- E I 07 (s3) / 073(s1 ) / 074 (s)ds / 074 (s2)ds2ds^c^^31

74,73,7i=0 t

= l.i.m.

p—TO ^-'

7473 Ji=0

s3 si s3

p / T / T \ 2 T

p1

2 J 07i(s3) I j (pj4(s)ds I j (pj3(si)dsids3-

t s3 s3

T T /si \ 2

\ J 07i(s3) J (pn(si) j J (pj4(s2)ds2 ) dsids3-

t s3 s3

T T / T \ 2 \

I feM I <f>j3(si) I I (f)j4(s)ds ) dsids:i

c (;i)c (;3) =

7 c73

s3 \si

T T

p /1 T T

/

= l.i.m. E 0 / 07i(s3)(T1-S3) / fa3(si)dsids3-

p—TO ^—^ I 2

73,7i=0 \ "J s3

2

T T

J MM) J MM)M - sz)dsidsz-

t s3

T T

-\ JJ4>K(Sl)(T - Sl)dSldS31 c;:"C-

t s3

— A^«3' + A^3' + A^3' = —A^«3' + A^3' + A^3' w. p. 1, (2.292)

where

p

A4!i'3) = E j-i j'j31,

j3 ,ji=0 p

(i i«3) _ 1 • ^ V^ „p A«iM«3)

A(i i«3) = i i m V ep C(i i'C(«3) A5 = lp—.m. ej3 j i Cj i Zj3 ,

j3,j i =0

p

Aiii"3) = e ji,z!33).

j3,j i =0

T / T \ 2 T

1 ^ to

4m = IJ MM) E / / MMdsds,, (2.293)

t j4=p+1 \s3 / s3

T T ^ / s \ 2

1 /» /» ^^

ej3ii = 2 1 0111' 1 1S

i sj j4=p+1 \S3

2

t t oo / s \

5 / feM I MM E / (2.294)

T T CO / T

fjdi = \f mm) J MM) E / ds2ds3 =

t s3 j4=p+1 Vs2 /

T to / T \ 2 s2

= 1 [ MM) E [ «feM^i [ MM)ds,ds2. (2.295) 2 i j4=p+1 \?2 / i

Moreover,

^3 + a5

A(i2«3) + A5i2i3)

p

= E (Cj4j3 j2j4 + Cj4j3 j4j2) j Cj^. =

j4,j3,j2 =0

p

T s si si

(;2)c (;31 73

p—TO

= p—m- E J 074(s) J 073 (s1^ 072 (s2^ J 074 (s3)ds3ds2ds1dsQ2;2)c

7'4,73,7'2=0 J J J J

p T si si T

E J 073 (s1) J 072 (s2) J 074 M^^J 074 M^s^2'^31

74,73J2=0 J J J si

p / T si T T

= lp—TO- E I J 073(s^y 072(s2)J 074(s3)ds^y 074(s)dsds2ds1-

74J3,72 =0 \ t t t si

2

T si / T \ 2 \

-J 073 My 072 (s2 ) ( I 074 (s)ds I ds2 ds1

t t \si

c(;21c(;3) =

c72 c73 =

/

p T ? f p Y^

= lp;i—1TO- E J 073 (s0 / 072 (s2) (T - s1) - E J 074 (s3)ds3

^ 7'3,72 =0 t t \ 74=0 \si y y

ds2ds1 X

x<f'cf = 2A6'2!3) w.p. 1-

Then

A

(;2;3) = 2A6;2;3) - a5;2;3) = A4;2;3) - A5;2;3) + A6;2;3) w. p. 1.

(2.296)

Let us consider A4

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(;i;4)

A(;i;4)

T

A4

p

p—]TO- E I 074 (s) / 07i M / 073 (s2) ^ 073 (s1)ds1ds2ds3dsc7(;i)c7(44)

74,73,7i=0 J J s3 s2

2

(;i^(;4)

T s s 2

p 1 T s p s

= 1pi;™- E \J fe(s) J feME y fa3(s2)ds2 rfsarfscj^cjl

74,7i=0 t t 73=0 \s3

T

p 1 T

l.i;m. e ^ I fem / tnMis - s3)ds3dsC^ -

p TO 74,7i =0 { {

s

T s

1p

tt T s

E I M(s) I (f>u(s2,)(s - s3)ds3ds -

P j4=0 t t

T s2 si

= IJ J J dwisl)(isidw^ + ttt ( T s

+^{¿1=^0} E J(s - t)(f)j4(s) J (f)j4(sz)dszds-

\j4=° t t

s

(«i«4)

Ts

T, f j w hs* — t)^j4Mdsads) — A3^ =

J4=0 { { J

T s2 si

= \ f f f dwisl)dsidw^ - A{ilk) w. p. 1. (2.297)

t t t

Let us consider A6

(«i«2)

A(

A6

p

6=

T T T T

(iiM«2 '

^p.^ E J j (s3^ j (s2^ j (s0y j(s)dsds1 ds2ds3Cj(ii'C

j3'j2'j'i=0 t s3 s2 si

T T / T \ 2

= e s /^(s3)/fewe [jM(s)ds\ ds2ds4^

ji ,j2=0 t s3 j3=0 \s2 /

T T

= 1i£- E J<f>hM(T-s2)ds2ds3C^C=

ji ,j2=0 t s3

p T s2

= ¿5/ MM)(T-s2) J Ms^ds^C^ - af2) = ji,j2=0 t t

T s2

tt

T s2

(;i;2)

2 11.' _;-=017 I 07 I s 2 II / - s 2 I I 07 I s3 Ids3ds2 -

1 TO

+ / feW)(T-S2) I (f)n(sz)dszds2 - A

72=0 t t

T si s2 T

= 1111 dw^dw^dsi + il{n=,2^0} J(T- s2)ds2 - A[;ii2) w. p. 1. t t t t

(2.298)

Let us consider B1, B2, B3

p t s / si \ 2

£1 = lim E 9 / feW / fe(si) / feMds2 dsids =

p—TO ^—^ 2

7i,74=0 "J

Ts

p 1 r r p

74=0 t t 74=0

T 1

- (s1-t)dsi- (2'2")

J 74=0

T

s s2

B2 = plim E J 073 (s) J 073 My 074 (s3 )ds^ 074 (s1)ds1ds2ds

74,73=0 t t t

s2

A T

E \ J ^jsis) i J (pj4(so0)dso\ J (f)j3(s2)ds2ds-

o I feW I I <t>.

74,73=0 ^ t \t

T

s2

073 (sW 073 W) / 074 (s3)dsU ds2ds-

tt T s

\

073 (sW 073 W) / 074 (s1)dsU ds2ds

tt

s2

/

s

p

2

s

2

s

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1

2

s

1

2

Ts

TO 1 „ ' p

Eo/ MMis-t) / (f)j3(s2)ds2ds - lim

_A " " _n

2 J rj3K-,y- J p-^TO ^ j3j3

j3=0 t t j3=0

Ts

TO 1 \ * p

/ / lim

j3=0 j3=0

Ts

TO 1 f f p

/ / + lim E^m

j3=0 j3=0

p p p ap

p_i_m to ^ ^ ,?3j3 «—to ^ ^ -p3j3 p—m to ^ ^ ,?3j3 ( )

Moreover,

p—TO ' * •y3-'3 p—TO ' * •y3-'3 p—TO

j3=0 j3=0 j3=0

p

B2 + B3 = lim \ (Cj3j4j3 j4 + Cj3j4j4 j3) = p

p—

j4 ,j3=0

p T s si si

%-m ^ J j (s)J j (s1) J j (s2) J j(s3)ds3ds2ds1ds =

j4 ,j3=01 t t t

p T si si T

= plim E J j (s1) J j (s2) J j (s3)ds3ds^ j (s)dsds1 =

j4,j3=0 t t t si

p / T si T T

Urn ^ (/^j'4(s!)/ jM / j(s2)ds^ j(s)dsds3ds1

j4,j3=0 \ t t t si

T si / T \ 2 \

— J j(s1^ j(s3M J j(s)dsl ds3ds1

t t \si ) j

TO T si

= E / j(s1)(T — s1 ) / j(s3)ds3ds1

j4=0 t t

~ T si

p

p ' j4j4

j4=0 / / * j4=0

f* f* i "E/ j(s1)(T — s1 )J j(s3)ds3ds1 + 2p-imj]/j

= 2 ,l™.E /7474. (2-3°D

Therefore,

p

74=0

B = 2 lim £ /7373 - lim £ a^ - lim £ c^ + lim E ftp^. (2.302)

p—TO ' * j^3 p—TO ' * j^3 p—TO ' * j^3 p—TO ' * j^3

73=0 73=0 73 =0 73=0

After substituting the relations (2.287)—(2.302) into (22861), we obtain

l;i.m- C7473727i c7i c72 7 7

p—TO

7'i,7'2,73,74=0

T s si

= + 11 Jds2dwMdwM+

ttt

T s2 si T si s2

+ JJJ dwMdstfwM + JJJ dw^dw^ds^ t t t t t t

T si

+ jl{i1=i2^0}l{i3=i4^)} J Jdsods1 + R= J*№%t + R w. p. 1, (2.303)

tt

where

R _ I A(;3;4^^1 f A (;2;4) i A (;2;4) i A (;2;4)'\ i

R = -1{;i=;2=0}A1 + 1{;i=;3=0} (v-A2 + A1 + A3 ) +

,1 / A(;2;3) A (;2;3^ A(;2;3^^ 1 A(;i;4) |

+ 1{;i=;4=0} ^A4 - A5 + A6 ) - 1{;2=;3=0}A3 +

I 1 ( A (;i;3^ I A (;i;3^ I /\(;i;3^^ n A (;i;2)

+ 1{;2=;4=0} lv-A4 +A5 +A6 ) - 1{;3=;4=0}A6 -

/ p p p

-1{;i=;3=0}1{;2=;4=0} ( Km E <73 + E Cp373 - ¿m E Trn

\ 73=0 73=0 73=0

pp

-1{;i=;4=0}1{;2=;3=0} ( 2 lifTOE -Trn - E ap373-

\ 73=0 73=0

pp

lim V^ cp 7 + lim V^ bp 7 +

p—TO ^-' 7373 p—TO ^—' 7373 '

73=0 73=0

p

+ 1{2i=22=0}1{23=24=0} pim E ap3j3 . (2.304)

j3=0

From (2.303) and (2.304) it follows that Theorem 2.9 will be proved if

Aj = 0 w. p. 1, (2.305)

p p p p

p

a a A =

limV ap7 = lim V j ■ = lim V ^ 7 = lim V j = 0, (2.306)

p— j3 j3 p— j3 j3 p— j3 j3 p— j3 j3

j3=0 j3=0 j3=0 j3=0

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where k = 1, 2,..., 6, i, j = 0,1,..., m.

Consider the case of Legendre polynomials. Let us prove that A1i3i4) = 0 w. p. 1. We have

\ 2 '

p2

^ ( E a,p4j3j3)cj

j3,j4 =0

p j3—V 2 -A p 2

VV 2ap ■ ap,, + (j + 2ap j. + fap +3V

Z—/ Z—/ 1 j3 j3 j3 j3 V j3j3 J j3j3 j3 j3 V j3 j3 7 / Z—/

j'3=0 ¿3=0 \ / j'3=0

' p \ 2 p j3— 1 2 p 2

Ej. + EE (°p3j3 + j + 2 E (ap3j'3) = i4 = 0),

j=0 / j'3=0 j3 =0 j'3=0

(2.307)

p

2

p

,p C («3)C (M) 5j;

j3,j4 =0 / ) j3 ,j4=0

M E ajj«M = E j)2 (i3 = ^4, i3 = 0, i4 = 0),

(2.308)

m

E

j3,j;=0

ap c(«3'c(«;)

aj;j3 C j3 C j;

(T — t)£ (ap;,0)2 if i3 = 0, i4 = 0

j;=0

=

(T — t)£ KJ2 if i4 = 0, i3 = 0 .

j3=0

[ (T — t)2 (ap0)2 if i3 = i4 = 0

(2.309)

2

Let us consider the case i3 = i4 = 0

» _ (T-t)y(2j4 + l)(2j3 + l)„

■>■->■■■ "32

2

1 y to / yi

X / P74 (y) / P73 fe) £ (2j1 + 1) I Ph (y2)dy^ dy1 dy

-1 -1 7i=p+1 \-1

(T-i)2v/(2j4 + l)(2j3 + l)

32

X

1 TO 1 x / ^s(yi) E ¿"Vr (^1+1(^1) -Ph-M)2f PjMdydy 1 =

-1 7i=p+1 1 yi

_ (r-i)V2jmv

32V2J4 + 1

1 TO 1

X PJS(yi) (Pn-i(yi) - Pj4+i(yi)) E 2TT1 (^1+1(^1) - ^-iCyi))2^!

^ 7i=p+1 j1 +

if j4 = 0 and

_ (T-^y^+T

a7473 32

1 TO 1

X.......

f TO 1

pjsM^-yi) E n^+^-p^y^2^

-1 7i=p+1 j1

if j4 = 0.

From (2.141) and the estimate |P7- (y)| < 1, y G [-1,1] we obtain

\pj(y)\ = \f\P^\-\f\Pjiy)\ < -tt^y~^2)T78' (2-31°)

Using (2.141) and (2.310), we get

HJ^t^Jj^^ 0,^2), (2,n)

KbI + KaI - Co ¿+1 ^ / (T^n - 7 (j3 *0)- (2'312)

1

Kol + I «-So I < Co E J / JT^W- - J {:U - r)' (2'313)

where constants C0, C1 do not depend on p.

Taking into account (23071), (I23m)-(I23T31): we have

p \ 2 I / p \2 p 2

m < i E aj jj' = U + E j. + £ (aj + j) +

j3,j;=0 / J V j3=1 / j'3=1

p j3—1 2 / p 2 \

+ EE j + j +2 E j) + M2 *

j3=1 j3=1 \j'3=1 J

(1 1 p 1 \2 K p j3—1 1 ( 1 1 \2

^ (1 1 p dz \ K1 K3 .A 1

< f- 4 ^2 El (1 f <

~~ 0 I p p3/4 J p P \ J X3/2 J ~

K4 K3 ( 2 \ K5 < —+ — 3--<—

p p V p

if p — to (i3 = i4 = 0).

The same result for the cases (2.308), (2.309) also follows from the estimates (2.311)-(2.313). Therefore,

A1i3i;) = 0 w. p. 1. (2.314)

It is not difficult to see that the formulas

A2i2i;) = 0, A4iii3) = 0, A6iii3) = 0 w. p. 1 (2.315)

can be proved similarly with the proof of (2.314).

Moreover, from the estimates (2.311)-(2.313) we obtain

p

i^E j j3 = 0 (2316)

j3=0

The relations

pp

limE j3 = 0 and lim^ fj = 0 (2.317)

3 3 3 3

p^ œ z—' J3J3 p^œ

j3 =0 j3=0

can also be proved analogously with (2.316). Let us consider a3«2«;)

a3«22;) = a4«22;) + A6i2i;) — a7«22;) = — a7«22;) w. p. 1, (2.318)

where

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p

a7'2!;)=li- E «>,

T s _ / T T

„ „ oo

gp;j2 = M M j (s1) E / j (s2)dsW j (s2)ds^ I ds1ds = { { ji=p+1 Vsi s )

oo T T s T

= E / j (s) / j (s2)ds^ j (s1) f j (s2)ds2ds1ds. (2.319)

ji=p+1 t s t si

The last step in (2.319) follows from the estimate

TO 1 y

^ 1 " ^ 1 / (l-'l/2)1/2 / " y-

Note that

oo ( t T \ 2

= E l\[ MM J MM)dsods\ , (2.320)

ji =p+1 \ t s /

oo T T T T

gp;j2 + gp2j; = E / j (s) / j (s2)ds2ds /* j (s) / j (s2)ds2ds, (2.321)

ji=p+1 t s t s

and

_ (r-i)y(2j4 + l)(2j2 + l)

TO 1 1

x E ottT / (ph-M - Ph+M) *

7i=p+1 j1 + -1 yi

x y P72(y) (Pi-1 (y) - Pi+1 (y)) dydy1, j4, j2 < p. -1

Due to orthogonality of the Legendre polynomials we obtain

(T-i)V(2j4 + l)(2j2 + l) 16

TO 11

x E 27 +1 / P74(2/1) №1-1(2/1)-pi+i(2/i))d2/ix

7i=p+1 j1 -1 1

x J P72 (y) (P71—1 (y) - Pi+1 (y)) dy = 1

1

2

(T - t)2(2p +1) 1 f .

1 if j2 = j4 = P

Pp(y1)dy

1

16 2p + 3 W

-1 0 otherwise

(T - t)2 I 1 if j2 = j4 = P

4(2p + 3)(2p + l) ' ) ' (2'322)

0 otherwise

, = (T — ¿)2(2j4 + 1) Vim IQ

x E lij (ph-1(2/1) - P1+i(l/i)) =

7i=p+1 \-1 /

(T — t)z(2p +1) 1 l)^ „ ^ = '

=-32-W^iJ >iyi) yi

1 0 otherwise

(rp \2 I 1 if j4 = P ( ^ J . (2.323)

8(2p + 3)(2p + 1)

0 otherwise

From (2.307), (2.322), and (2.323) it follows that

m £ jj2 jj

j2 ,j;=0

f p p j3 1 2 p

LjJ + ££ j + j + 2Z (gjkj3

j3=0 / j'3=0 j3=0 \2 \ 2

j'3=0

(T — t)

8(2p + 3)(2p +1)

+ 0+2

(T — t)2

8(2p + 3)(2p + 1)

0

if p — to (i2 = i4 = 0).

Let us consider the case i2 = i4, i2 = 0, i4 = 0 (see (2.308)). It is not difficult to see that

T

s

gp;j2 = j (s) / j (s1 )Fp(s,s1)ds1ds

j;j2 j; j2 1 p 1 1 p t t [t,T]2

Kp(s,s1)0j; (s) j (s1)ds1ds

is a coefficient of the double Fourier-Legendre series of the function

Kp(s,s1) = 1{si<s}Fp (s,s1), (2.324)

where

oo

T

E I j (s2)ds^ j (s2)ds2 = Fp(s,s1).

ji=p+1

si

The Parseval equality in this case looks as follows

pi

T s

lim E (j2)2 = I (Kp(s,s1))2 ds1 ds = i i (Fp(s,s1))2 ds1ds. (2.325)

pi — TO

j;,j2=0

p 1 1 p [t,T]2 t t

From (2.141) we obtain

T

j (0)d0

si

■vS+lvR

pji (y)dy

Ksi)

2

1

2

yfr~t \Pjl-i(z(s1))-Pjl+1(z(s1))\<^--T, 77771' (2-326)

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2\/2jm 1 v v - n (1 - .2(Sl))1/4

where z(si) is defined by (2.20), si G (t,T). From (2.326) we have

C2 1 1

From (2.327) it follows that |Fp(s, s1)| < M£/p in the domain

D£ = {(s,s1): s G [t + e,T — e], s1 G [t + e, s]} for some small e> 0,

where constant M£ does not depend on s,s1. Then we have the uniform convergence

T T T T

p p /» CO /» /»

E / j(0)d0 / j (0)d0 ^ E / j(^)dW j(0)d# (2.328)

j1=0 s si j1=0 s si

at the set D£ if p ^ to.

Because of continuity of the function on the left-hand side of (2.328) we obtain continuity of the limit function on the right-hand side of (2.328) at the set D£.

Using this fact and (2.327), we obtain

T s T—£ s

J J (Fp(s,s1))2 ds1ds = limo J J (Fp(s,s1))2 ds1ds <

t t t+£ t+£

T — £ s

< ^ lim [ f d'Sl d'S

t+£ t+£ TT Ts

C2 f f dsi ds

P2 J { (1 — z2(s1))1/2 (1 — z2(s))1/2

1 y

4 ** W, w, < (2-329)

where constant K1 does not depend on p. From (2.329) and (2.325) we get

p pi TO

,2 V^ i v \2 Ai

pi —TO ^-' ^~j;J2' z-✓ — p2

j2,j;=0 j2,j;=0 j2,j;=0

0 < V G?L)2 < M m V (g>;tjy = V (g>;t]f < ± - 0 (2.330)

if p — to. The case i2 = i4, i2 = 0, i4 = 0 is proved.

The same result for the cases

1) ¿2 = 0, ¿4 = 0,

2) ¿4 = 0, i2 = 0, 3) ¿2 = 0, ¿4 = 0

can also be obtained. Then A7i2i;) = 0 and A3i2i;) = 0 w. p. 1.

Let us consider A5

(ii¿3)

A5ii3' = A4iii3) + A6iii3) — A8iii3) w. p. 1,

where

p

^(iii3) = 1 i m v^ hp C(ii)C(i3' 8 p—

j3,j'i =0

A8iii3) = l . i . m . y j jj',

8 ™ / j j3ji j j '

T T

hp3ji = J j(s3^ j(s)Fp(s3,s)dsds3.

t s3

Analogously, we obtain that A^3' = 0 w. p. 1. Here we consider the function

Kp(s3,s) = 1{s3<s}Fp(s3,s)

and the relation

hjj = J Kp(s3,s)0ji(s3)0j3(s)dsds3 [t,T ]2

for the case i1 = i3, i1 = 0, i3 = 0.

For the case ¿1 = ¿3 = 0 we use (see (2.320), (2.321))

/ t T \ 2

^Li = \ [ J M(s) J (f>j4{si)dsids

j;=p+1 \ t s

oo

T

jji + hjj3 = E J j (s) J j (s2)ds2ds y j (s) j fa; (s2)ds2ds.

j;=p+11 s t s

Let us prove that

lim cjp j = 0.

1-^fY! ' ^ j3j3

We have Moreover,

p

j3=0

cp = f P + dp — gp

j3 j3 f j3j3 + j3j3 gj3j3.

toE fjj = 0, li^E dp3j3 = 0, p— 3 3 p— 3 3

j3 =0 j3=0

(2.331)

(2.332)

(2.333)

where the first equality in (2.333) has been proved earlier. Analogously, we can prove the second equality in (2.333).

From (2.323) we obtain

0 < lim V gj 3 < lim

33

(T — t):

p—>•oo z

j3=0

p—TO 8(2p + 3)(2p + 1)

= 0.

So, (2.331) is proved. The relations (2.305), (2.306) are proved for the polynomial case. Theorem 2.9 is proved for the case of Legendre polynomials.

Let us consider the trigonometric case. According to (2.288), we have

t TO / si \ 2 T

\ J MM) e (/ MM)dso\ J MMdsd.si. (2.334)

v

a=

jij-i 2

t ji=p+1 \ t Moreover (see (2.254), (2.255)),

K

si

fa(s2)ds2

<

j

T

fa (s2)ds2

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si

<

K

(2.335)

where constant K does not depend on j (j = 1, 2,...). Note that

T

T — s1

s)ds =

si

»

p

j

Using (2.334) and (2.335), we obtain

C1 to 1 C1 C1

K,3|<- E -^ — Kl^(2-336)

J4 j2 PJ4 P

where constant C1 does not depend on p.

Taking into account (I23fl71)-(I2^09 and (£236), we obtain that A^ = 0

w. p. 1. Analogously, we get a2'2m) = 0, A4'1i3) = 0, A6'1i3) = 0 w. p. 1 and p p p

lim E aL'3 = 0, li^ j* = 0, lim E /j = 0-

p^TO < ■* j33 P^TO < ■* j33 P^TO < ■* j3j3

j3=0 j3=0 j3=0

Let us consider A3i2i4) for the case i2 = i4 = 0. For the values g24m2 + gjj and gj1 + g2m4—1 (m G N) we have (see (23211))

2m I g2m _

gj4j2 + gj2j4 =

TT TT TT TT

= E f j (s^ j (s2)ds2d^0j2 (s) y j (s2)ds2ds =

j1 =2m+1 t s t s

TT TT TT TT

= E / 0j4 (sW 02r—1(s2)ds2dW j (sW 02r—1(s2)ds2ds+

r=m+^ t s t s

TT TT TT TT

+ J j(s) J hr(s2)ds2d^0j2 (s^ 02r(s2)ds2ds | , (2.337)

t s t s

oo

g 2m— 1 | g2m—1 _

gj4j2 + gj2j4 =

T TT TT TT

E J j (s) J j (s2)ds2dsy j (s^ j (s2)ds2ds =

j1=2m t s t s

= g2m + g2m +

TT TT TT TT

+ J j (s) J 02m(s2)ds2d^0j2 (s^ 02m(s2)ds2ds, (2.338)

t s t s

where

j MM j <far-i{s2)ds2ds = \j^r—t j MM j sin27r^"S^ ^d.sod.s

st

V2VT~t, } ( 27rr(s -1) \

(f)j4{s) cos—;—---1 Ids,

2nr J T — t

t

T T _ T T

2

MM J M-M)dsods= j MM j cos^MiMldsods

t

y/2y/T=t } ( . 2tMs-t)\

- / <P-l s —sin——- as,

2tt r I \ T-t / '

where 2r — 1, 2r > p +1, and j2, j4 = 0,1,... ,p.

Due to orthogonality of the trigonometric functions we have

T T /-, f -1 if H = 0

I fa4(s) J 02r_1(s2)ds2ds = V2(2^ ~ t] ■ i , (2.339)

t s 0 otherwise

T T

i j (s) /* 02r(s2)ds2ds = 0, (2.340)

where 2r — 1, 2r > p + 1, and j4 = 0,1,..., p. From (2.337), (2.339), and (2.340) we obtain

g2m + g2m =

00 (T — t)2 I 1 if j2 = j4 = 0

27T2j12 -

ji=m+1 0 otherwise

n2m = - (n2m + n2m)

■JjAjA 2 1 -JJUa)

j2 =j4

OO

E

ji =m+1

1 if j4 = 0

0 otherwise

Therefore (see (2.25

gjm + gj1 < Ki/(2m) if j2 = j4 = 0

g2m + g2m =0

otherwise

(2.341)

| < Ki/(2m) if j = 0

g2m = 0

otherwise

(2.342)

where constant K1 does not depend on p = 2m.

For p = 2m — 1 from (2.338) and (2.340) we have

g 2m-1 I g2m-1 gj4j2 + gj2j4

E

ji=m+1

(T-tï-2n2j2

1 or 0 if j2 = j4 = 0

0

otherwise

(2.343)

The relation (2.343) implies that

g2m-1

gj4j4

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± (q^-i + o2'7?"1)

j2=j4

OO

E

ji=m+1

(T — t)2 47T2ii

1 or 0 if j4 = 0

0 otherwise

(2.344)

Using (2.343) and (2.344), we obtain

+ gS—11 < K2/(2m — 1) if j2 = j4 = 0

(2.345)

j 1 + j 1 = 0 otherwise

|gj—1| < K2/(2m — 1) if j4 = 0

g2m—1 = 0 otherwise

j4j4

(2.346)

where constant K2 does not depend on p = 2m — 1.

The relations (2.341), (2.342), (2.345), and (2.346) imply the following formulas

|gp4j2 + gp2j4 1 < K3/p if j2 = j4 = 0

, (2.347)

gp4j2 + gp2j4 = 0 otherwise

|g?4j4l < K3/p if j4 = 0

, (2.348)

gp . =0 otherwise

where constant K3 does not depend on p (p G N). Moreover, gjj > 0 (see

(E320)). 44

From (2.307), (2.347), and (2.348) it follows that A^4) = 0 and A^4) = 0 w. p. 1 for i2 = i4 = 0. Analogously to the polynomial case, we

obtain A7i2i4) = 0

and A3i2i4) = 0 w. p. 1 for i2 = i4, i2 = 0, i4 = 0. The similar arguments prove that A5i1i3) = 0 w. p. 1.

Taking into account (2.332), (2.347), (2.348) and the relations

pp

p

limy j = lim y dp7 =0,

p—VfYI ' ^ j3j3 p—VfYI ' ^ j3j3

p—TO z—' •/3-/3 p—^to

j3=0 j3=0

which follow from the estimates

C1 p C1 p C1 p C1

141 0V0), Kol < "A (2.349)

we obtain

p p lim E cp3j3 = - lim E gp3j3'

p—OO ' ' J3J3 p — Q t J j3j3

¿3=0 ¿3=0

p K

0< Mill VVL, < lim —- = 0.

p—oo z—' •/3-'3 p—oo P ¿3=0

Note that the estimates (2.349) can be obtained by analogy with (2.336); constant Ci in (2.349) has the same meaning as constant C1 in (2.336).

Finally, we have

p

lim E cp3j3 =

p^^' ■* J3J3

¿3=0

The relations (2.3°5), (2.3°6) are proved for the trigonometric case. Theorem 2.9 is proved for the trigonometric case. Theorem 2.9 is proved.

Remark 2.2. It should be noted that the proof of Theorem 2.9 can be somewhat simplified. More precisely, instead of (I2.307l)-(l2.3()9l), we can use only one and rather simple estimate.

We have

m He au №

^ \j3j4 =0

= mm Y «L'J Cj(33)C]44) - 1{i3 = i4=0}1{j3=j4} + 1{i3 = i4=0}1{j3=j4} [ \j3 ¿4=0 V

= m H E jC«33)Cj441 - ^^¿j.}) + 1{,'3=*4=0| it ap4,) J =

[ Vj3 ¿4 =0 7 ¿4=0 J )

= m H E0jCi33)Cj441 - 103=!4=0}1{,3=,4})) J +

/ p \2

+ 1{*=u=0} £ ap4j4 • (2.35°)

Vj4=0 J

The expression

E app4j3 ( ¿j - 1{i3=i4=0} 1{j3=j4} ¿3,j4=0 V

can be interpreted as the multiple Wiener stochastic integral (1.255) (also see

(1.23)) of multiplicity 2 with nonrandom integrand function

p

E ap4j3 j (t3)0j4 (t4) .

j3 ,j4 =0

From (1.25) we obtain

T t2

m{Jwit)2}<Ck e /•••/$2(t1,...,tk)dt1 ...dtk=

(t1,--.,tfc) t t

= Ck J $2(t1,...,tk)dt1 ...dtk, (2.351)

[t,T ]k

where J' [$]Tk] is defined by (1.2,3) and Ck is a constant. Then

m (E aujcir' — i{,3=,4=0}i{,3=j,}) i > <

j3j4=0

2

/ p \ 2 p < C^E OUj (t3(t4) dt3dt4 = C^ (apj2 . (2.352)

T12 Vj3j4=0 / j3 ,j4 =0

[t T]2 V3,J4—" / .'/3, j4 =0

From (2.350) and ('2.352) we get

m I i E c]33)cj44^ 1 < C2 E KJ2 + i{.3=i4=0^ a

2

p \ 2 I p / p

p /-(«3^(i4) | ^ ^ ^ V^ ^p A2 i i , . I V^ „p

j4 j4

j3,j4=0 / J j3,j4=0 \j4=0

(2.353)

Obviously, the estimate (2.353) can be used in the proof of Theorem 2.9 instead of (2.307)—(2.309).

The estimate (2.353) can be refined. Using (1.87), we obtain

m{ (, E 0 j zj33)c]44) — i{.3=,1=0}i{J3=j1})

2

p p £ («La)2 + 1ii3=i4=0^ ap3j4 <

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j3,j4=0 j3,j4=0

P i P

< e («'],,f ■ ^ £ ((«/,j~ ' Kjf

j3,j4=0 j3 ¿4=0

p

2

(1 + 1(!3=.,=0^ (j.) . (2.354)

j3j4 =0

Combining (2.350) and (2.354), we have

p 2 p ^ L «PUjjM < (1 +1(.3=.4=0^ («j2+

j3 ¿4=0 / j3,j4=0

2

p ^ 2

+ 1{i3=i4=0} £ j . (2.355)

Vj4=0 J

2.4 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k E N) Based on Generalized Iterated Fourier Series Converging Pointwise

This section is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k (k E N) based on generalized iterated Fourier series. The case of Fourier-Legendre series and the case of trigonometric Fourier series are considered in detail. The obtained expansion provides a possibility to represent the iterated Stratonovich stochastic integral in the form of iterated series of products of standard Gaussian random variables. Convergence in the mean of degree 2n (n E N) of the expansion is proved.

The idea of representing of iterated Stratonovich stochastic integrals in the form of multiple stochastic integrals from specific discontinuous nonran-dom functions of several variables and following expansion of these functions using generalized iterated Fourier series in order to get effective mean-square approximations of the mentioned stochastic integrals was proposed and developed in a lot of author's publications [71] (1997), [72] (1998) (also see

33]). The results of this section convincingly testify that there is a doubtless

relation between the multiplier factor 1/2, which is typical for Stratonovich

stochastic integral and included into the sum connecting Stratonovich and Ito stochastic integrals, and the fact that in the point of finite discontinuity of piecewise smooth function f (x) its generalized Fourier series converges to the value (f (x + 0) + f (x - 0))/2.

2.4.1 Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k E N)

Consider the following iterated Stratonovich and Ito stochastic integrals

* T * ¿2

J V,t = y fak (tk) ..J fai(ti)dw^ ...dw^, (2.356)

i i

T t 2

J [fa(k)]T,t = / fa (tk) ..J ^i(ti)dwt;i}... dw(ik), (2.357)

i i

where every fa (t) (l = 1,..., k) is a continuous nonrandom function on [t, T], w[;) = fi;) for i = 1,..., m and wT0) = t, i1,..., ik = 0,1,..., m.

Let us denote as (faj(x)}°=0 the complete orthonormal systems of Legendre polynomials or trigonometric functions in the space L2([t,T]).

In this section, we will pay attention on the well known facts about Fourier series with respect to these two systems of functions [106] (see Sect. 2.1.1).

Define the following function on the hypercube [t,T]k

|fai(ti) ...fak (tk), ti <...<tk k k-i

= n fa (t) n ^ <ti+1}

0, otherwise l=i l=i

(2.358)

for ti ,...,tk E [t,T ] (k > 2) and K (ti) = fai(ti) for ti E [t,T], where 1a denotes the indicator of the set A.

Let us formulate the following theorem.

Theorem 2.10 [71] (1997), [72] (1998) (also see [5]-[16], [33]). Suppose

that every fa (t) (l = 1,..., k) is a continuously differentiate function at the interval [t,T] and (faj(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, the iterated Stratonovich stochastic integral J * [fa(k)]Tt defined by (2.356) is expanded

into the converging in the mean of degree 2n (n E N) iterated series

oo oo k

J*W'(k)]T,< — £ ... £ Cj,..;, n<f, (2.359)

ji=0 jk=0 7=1

where

T

zj;) — / fa (s)dwS;) t

are independent standard Gaussian random variables for various i or j (in the case when i = 0) and

C — Cjk---ji —

/k

K (t1, (ti)dt1 • • • dtk (2.360)

7 = 1

[t,T]k l=1

is the Fourier coefficient.

Note that (2.359) means the following

{( pi Pk k \2M

1 Z-/ Z-/ I I

\ ji=0 jk=0 l=1 / J

__(2.361)

where lim means lim sup.

Proof. The proof of Theorem 2.10 is based on Lemmas 1.1, 1.3 (see Sect. 1.1.3) and Theorems 2.11-2.13 (see below).

Define the function K*(t1;... , tk) on the hypercube [t,T]k as follows

k-1 / 1

k k-1

K*(tu ....I,-) I] + ^

i=1 i=1 ^

l, •• •, ik) = IIWD HI ijiKii+i} "+" 2Hti=ti+1}

k / k-1 k-1 1 k-1 r k-1 ^

Y[Mti) n^w + E^ E II1^^!} II MtKh+i}

, l=1 r= 1 sr,...,si = 1 l=1 '=1 I

y sr>...>si '=si,...,sr /

l=1

(2.362)

for ti,..., tk E [t, T] (k > 2) and K*(t1) = ^1(t1) for t1 E [t,T], where 1A is the indicator of the set A.

Theorem 2.11 [71] (1997). Let the conditions of Theorem 2.10 be satisfied. Then, the function K*(ti,... , tk) is represented in any internal point of the hypercube [t,T]k by the generalized iterated Fourier series

pi pk k

K *(*!,...,** )= lim ... lim £ ..^Cj-k.^n (ti), (2.363)

ji=0 jk=0 1=i

where (t1,..., tk) E (t, T)k and Cjk...j1 is defined by (12.360). At that, the iterated series (2.363) converges at the boundary of the hypercube [t,T]k (not necessarily to the function K*(t1;..., tk)).

Proof. We will perform the proof using induction. Consider the case k = 2. Let us expand the function K*(t1;t2) using the variable tl5 when t2 is fixed, into the generalized Fourier series with respect to the system {faj (x)}°=0 at the interval (t, T)

to

K*(ti,t2) = E Cji(t2) j(ti) (ti = t, T), (2.364)

j1=0

where

T T

'* I

Cji(t2) = J K*(ti,t2)0ji(ti)dti = J K(ti,t2)j(ti)dti =

t t t2

= fai(ti)faji (ti)dti. t

The equality (2.364) is satisfied pointwise at each point of the interval (t, T) with respect to the variable ti, when t2 E [t,T] is fixed, due to a piecewise smoothness of the function K*(ti,t2) with respect to the variable ti E [t,T] (t2 is fixed).

Note also that due to the well known properties of the Fourier-Legendre series and trigonometric Fourier series, the series (2.364) converges when ti = t,T (not necessarily to the function K*(ti,t2)).

Obtaining (2.364), we also used the fact that the right-hand side of (2.364) converges when ti = t2 (point of a finite discontinuity of the function K(ti, t2)) to the value

i (k(u - 0, to) + k(u + 0, to)) = ^(toyfaito) = K*{t2,t2).

The function Cji (t2) is continuously differentiable at the interval [t,T]. Let us expand it into the generalized Fourier series at the interval (t,T)

Cj (t2) = E Cjj(t2) (t2 = t,T), (2.365)

j2=0

where

T T t2

Cj2ji = J Cji fo) j (t2)dt2 = J ^2^) j (t2^ (tl)dtidt2

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t t t

and the equality (2.365) is satisfied pointwise at any point of the interval (t,T). Moreover, the right-hand side of (2.365) converges when t2 = t,T (not necessarily to Cj (t2)).

Let us substitute (2.365) into (2.364)

oo oo

K *(ti,t2) = EE Cj2ji j (tl) j (t2), (tl ,t2) G (t,T )2. (2.366)

ji=0 j2=0

Note that the series on the right-hand side of (2.366) converges at the boundary of the square [t,T]2 (not necessarily to K*(tl,t2)). Theorem 2.11 is proved for the case k = 2.

Note that proving Theorem 2.11 for the case k = 2 we obtained the following equality (see (2.364))

Mh) (l{il<i2} + ili^}) (2.367)

which is satisfied pointwise at the interval (t, T), besides the series on the right-hand side of (2.367) converges when tl = t,T.

Let us introduce the induction assumption

oo oo oo

EE- E ^-l(tk-l)x

ji =0 j2=0 jk-2=0

tk-i t2 k_2

/n k 2

^k-2(tk-2)fafc-2 (tk-2) .. ^l (tl) j (tl)dtl . . .dtk-2 JJ fa (tl) =

t t 1=l

k-l k-2 , 1 x

= + • (2-368)

Then

to to oo

EE--- E (tk)x

ji=0 j2=0 jk-i=0

tk t2

k-1

x / fak-i(tk-i)0jk-i (tk-i) ••• / fai(ti)j (ti)dti. ..dtk-^JJ (ti)

1=1

TO TO TO ✓ 1 N

= E E • • • E r/'-i///! (+ )

ji=0 j2 =0 jk-2=0 '

tk-i t2 k — 2 x / fak-2(tk-2)0jk-2 (tk-2) ■■■ fai(ti)j (ti)dti . ..dtk-^JJ (ti)

1=1

(1 \ TO TO TO

ife-K^} + EE--- E

' ji =0 j2=0 jk-2 =0

tk-i t2 L 2

k-2

x / fak-2(tk-2)0jk-2 (tk-2) . . /fai (ti)faji (ti)dti ...dtk-^JJ (ti)

1=1

( 1 \ k—1 k—V 1

' '/,(//,) ( l{ifc_1<ifc} + ) n ( 1

V / V

{ij<ij+i} + oMtl^t-l+l}

fe-i<tk} 1 2 {tk-i—tk} I J_J_ r1^1/ J_J_ I 1 2

' 1 = 1 1 = 1 ^

k k—1 / 1 N

= 11^^)11 l1!^} + ^i^i) * (2-369^ 1=1 1=1 ^ '

On the other hand, the left-hand side of (2.369) can be represented in the following form

TO TO k

E... E Cjk •••j'i n^ji (t1)

ji=0 jk =0 1=1

by expanding the function

tk t2

fak(tk) J fak—i(tk—i)0jk-i (tk—1).. fai(ti) j (ti)dti.. .dtk—1

tt

into the generalized Fourier series at the interval (t,T) using the variable tk. Theorem 2.11 is proved.

Let us introduce the following notations

i

-{;sp =^ + 1=0}

J^Ijji^1 = H 1{, =, . .=0} X

P=1

T tSl+3 +2

x^ (tk) ...J ^s;+2 (tsj +2) J ^ s; (ts; + 1 + 1 (ts; + 1) X t t t ^s; + 1 tSi+3 tSi + 2

^s,-1(ts;-1) ...J ^si+2(tsi+2^ ^si (tsi+1)^si+1(tsi+1)x t t t

tsi + i t2

x / ^si-1(tsi-1)... i^ (t1)dwt(;i}...dwt(;s--ii)dtsi+1dwt(;s+^2)...

... dw^X+1dwts;S++2)... dwj;k), (2.370)

where

Ak,i = {(si ,...,S1): si > si-1 + 1,...,s2 > S1 + 1, si,..., sx = 1,...,k - 1},

(2.371)

(si ,...,s1) G Ak)i, l = 1,..., [k/2], is = 0,1,..., m, s = 1,...,k,

[x] is an integer part of a real number x, and 1A is the indicator of the set A.

Let us formulate the statement on connection between iterated Stratonovich and Ito stochastic integrals J*[^(k)]T,t, J[^(k)]T,t of fixed multiplicity k, k G N (see (2.356), (2.357)). ' '

Theorem 2.12 [71] (1997). Suppose that every ^(t) (l = 1,...,k) is a continuous function at the interval [t,T]. Then, the following relation between iterated Stratonovich and Ito stochastic integrals

[k/2] .

r=1 (sr ,...,si)GAfc,r

is correct, where Yh is supposed to be equal to zero.

Proof. Let us prove the equality (2.372) using induction. The case k = 1 is obvious. If k = 2, then from (2.372) we get

= J[^]T,t + \m[2)]h w- P- (2.373)

Let us demonstrate that the equality (2.373) is correct w. p. 1. In order to do it let us consider the function F(x,T) = xfa2(T) and the process F(nT,t, t), where nr,t = J[fa(1)]r,t, t G [t,T]. Then

d f

— (x,t) = fa2(r), drjrt = (2.374)

From (2.374) we obtain that the diffusion coefficient of the process t G [t,T] equals to 1{il=o}fa1(T). Further, using the standard relations between Stratonovich and Ito stochastic integrals (see (2.4), (2.5)), we obtain the relation (2.373). Thus, the statement of Theorem 2.12 is proved for k = 1 and k = 2.

Assume that the statement of Theorem 2.12 is correct for some integer k (k > 2). Let us prove its correctness when the value k is greater per unit. Using the induction assumption, we have w. p. 1

= T

J ■ '|T,i = [k/2]

= I ^i(r) ( J[^k)u + E ......1 dw?^) =

r=l (sr ,...,si)GÂfc,r

* T

= / fak+i(r)J[fa(k)]T,tdwTik+l) +

[k/2]

T

1

+ E j (2.375)

r=1 (sr,...,si)GAfc,r t

Using the standard relations between Stratonovich and Itô stochastic integrals (see (2.4), (2.5)), similarly to (2.373), we get w. p. 1

T

I I, M M I ___Hfc-H I / I ./.I iv-pi. I I I

•fa+1(r) = + ô (2.376)

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* T

J )Ji^]^-'51 dw?^ = t

Jif Sr = k - 1

= < . (2.377)

J[^(k+1)]ft-'si + J[^(k+1)]Tî"'"S1 /2 if sr < k - 1

V ' '

After substituting (2.376) and (2.377) into (2.375) and regrouping of sum-mands, we pass to the following relations, which are valid w. p. 1

[k/2] .

jM{k+1)ht = J[^k+1)ht + e J[^k+i)}sT;rS1 (2.378)

r=1 (sr,... , si)eAfc+ijr

when k is even and

[k'/2]+1 .

= E i E (2.379)

r=1 (sr ,...,si)GAfc'+iir

when k' = k + 1 is uneven.

From (2.378) and (2.379) we have w. p. 1

[(k+1)/2] .

J*№k+^,t = J№k+1)ht+ E E (2.380)

r=1 (sr ,...,si)eAfc+iir

Theorem 2.12 is proved.

For example, from Theorem 2.12 for k = 1, 2, 3, 4 we obtain the following well known equalities [79], which are fulfilled w. p. 1

.T T

* ^(¿Odwf = /^(¿Odwf,

t t *T * t2 T t2

I Mt2)f ^1(t1)dwt;i)dw(;2) = 1 Mt2)f ^1(t1)dwt(;i)dwi;2)+ t t t t

T

+ J >h(h)Mh)dto, (2.381)

t

*T * ¿2 T t2

J fa(h) ..J fai(ti)dwt(il).. .dw(i3) =j ^s(is).. J ^i(ti)dw(il).. .dw(:3) + t t t t

t ts

J Mh) J ih{t2)Mh)dt2dw{tf + tt t ts

+ il{i2=i3^0} J h(h)h(h) J Mt^dw^dh, (2.382) tt

*T * t2 T t2

J uu)..J fai(ii)dwt(;i)...dw(i4) =1 ^4(^4)..J^i(ti)dw(;i)...dw(;4)+ t t t t

T t4 ts

+il{i1=i2^0} [MU) ( hih) [ Mt2)Ht2)dt2dw^dw^b

ttt T t4 ts

+^l{i2=i3^0} [mu) i h(h)Hh) f Mt^dw^dhdw^b

2

t t t

T t4 t2

tt T t4

+ il{il=i2^0}l{z3=i4^0} i i 'h{t2)'fa{t2)dt2du. (2.383)

Let us consider Lemma 1.1, definition of the multiple stochastic integral (1.16) together with the formula (1.19) when the function ... ) is continuous in the open domain and bounded at its boundary as well as Lemma 1.3 (see Sect. 1.1.3). Substituting (2.362) into (1.16) and using Lemma 1.1, (1.19), and Theorem 2.12 it is easy to see that w. p. 1

[k/2] .

J*№{k)}T,t = J№{k)}T,t + E «/['0(A;)]J)i"'Sl = (2.384)

r=1 (sr ,...,si)GAfcjr

where J[K*]T,] is defined by (Œ) and K*(tb... ,i*) has the form (2362 Let us subsitute the relation

pi pk k

K *(t1,...,tk ) = E... E Cjk ...ji n ^ (ti) + K *(t1,..., tk ) ji=0 jfc =0 i=1

Pi Pfc k

- E... E Cjfc...ji n^ (ti ) ji=0 jfc=0 i=1

into the right-hand side of (2.384) (here we suppose that p1,... ,pk < œ). Then using Lemma 1.3 (see Sect. 1.1.3), we obtain

Pi Pfc k

J * №(k)]T,( = £ ... £ Cjk.ji n j ' + J [R>-.P* ]Tkt w. p. 1, (2.385)

ji=0 jfc=0 i=1

where the stochastic integral J[RPi...Pk]Tk] is defined by (1.16) and

Pi Pfc k

(t1,... ,tk) = K*(t1,... ,tk) - E ... E Cjk...ji n^(ti), (2.386)

ji=0 jfc=0 i=1

T

j > = j (s)dw<">.

j.

t

According to Theorem 2.11, we have lim ... lim RPi...Pk(t1,...,tk) = 0 when (tb...,tk) G [t,T]k, (2.387)

Pk ^œ

where the equality (2.387) holds with accuracy up to sets of measure zero. Theorem 2.13. Under the conditions of Theorem 2.10 we have

lim lim ... lim m

J [RPi...Pfc ]T.

(k) 2n"

= 0, n G N.

Proof. At first let us analize in detail the cases k = 2, 3, 4. Using (2.424) (see below) and (1.19), we have w. p. 1

N-1 N-1

J [RPiP2 ]rt = l.i.m. E E Rpip2 (Tii, Ti2)Aw(;;) Aw(;22)

N^to— — i 2

i2=0 ii=0 'N-1 i2-1 N-1 ii-1

/N-1 i2-1 N-1 ii-1\

y.m. E E + E E R-«(Tii.12)Aw«;;»Aw«;2>+

N^ Vi2=0 ii=0 ii=0 12=0/

N1

+li.m. E Rpip2(Tii,Tii^w^Aw^ =

ii=0

T t2 T ti

I /Rip! (t1,t2)dwt(;i)dwt(;2) +J J Rpip2 (t1,t2 )dwt(2^ dwt(i° + t t t t

T

+1 {;i=;2=ow Rpip2(t1, i1)dt1, (2.388)

where we used the same notations as in the formulas (1.16), (1.19) and Lemma 1.1 (see Sect. 1.1.3). Moreover,

Pi P2

Rpip2 (t1,t2)= K*(t1,t2) ^EE Cj2ji^ji (t1)0j-2 (t2), P1,P2 < (2.389)

ji=0 j2=0

Let us consider the following well known estimates for moments of stochastic integrals [95

m

T

2n '

T

< (T - t)n-1 (n(2n - 1))n / m{ |^T|2n} dT, (2.390)

m

T

^T dT

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2n'

T

< (T - t)2n-^ M{ |^T |2n}

di,

(2.391)

where the process such that (£T)n G M2([t,T]) and fT is a scalar standard Wiener process, n = 1, 2,... (definition of the class M2([t, T]) see in Sect. 1.1.2).

Using (2.390) and (2.391), we obtain

m

J [RPi.2 ]T\t

2n"

T t2

< Only J (Rpip2 (ti,t2))2n dtidt2+

tt

T t1 T

\2n

+ / I (Rpip2 (ti,t2))2n dt2dti + 1{;i=;2=0} / (R.i.2 (ti,^))2" dti I , (2.392)

tt

t

where constant On < to depends on n and T — t (n = 1, 2,...). Further, we have

T t2 T ti

J J (Rpip2(ti,t2))2ndtidt2 + ^ J (Rpip2(ti,t2))2n dt2dti = t t t t

T t2 T T

= J J (Rpip2(ti,t2))2ndtidt2 + ^ J (RpiP2(ti,t2))2ndtidt2 = t t t t2

2n

(Rpip2 (ti,t2)) " dti dt2.

(2.393)

[t,T

Combining (2.392) and (2.393), we obtain

m

J [RPi.2 ]T,t

2n'

<

T

< On

>2n

'/(».i^,,,. ^... .......,0,y( ^

\t,T ]2 t

\

/

(2.394)

where constant Cn < to depends on n and T — t (n = 1, 2,...).

Since the integrals on the right-hand side of (2.394) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim (Rpip2(ti?t2))2n = 0 when (ti?t2) G [t,T]

pi —y^O .2—^TO

holds with accuracy up to sets of measure zero. According to (2.389), we have

Rpip2 (t1,t2)= [K*(t1,t2) - jt Cji (t2) j fa)) + V ji=0 J

pi p2 + E Cji(t2) - E Cj2jijfa) jfa) . (2.395)

Vj1=0 V j2=0 J J

Then, applying two times (we mean here an iterated passage to the limit lim lim ) the Lebesgue's Dominated Convergence Theorem and taking into

pi — TO p2—YTO

account (2.364), (2.365), and (2.395), we obtain

lim lim I {Rpip,{tut2))2n dhdt.2 = 0, (2.396)

p1—to p2—TO J

[t,T ]2

T

\2n

lim lim {RPlI»{tut1)yndt1 = 0. (2.397)

p1—to p2—TO J t

From (2.394), (2.396), and (2.397) we get

2n"

lim lim m

J [Rpip2 ]T.

(2) t

= 0, n G N.

Recall that (2.397) for 2n = 1 has also been proved in Sect. 2.1.3.

Let us consider the case k = 3. Using (2.425) (see below) and (1.19), we have w. p. 1

N-1 N-1 N-1

J[Rpip2p3]T3t = l.i.m. EEERpip2p3(Tii,Ti2,Ti3)AwT;ii)AwT;22)AwT;33) =

i3=0 i2=0 ii=0 N-1 i3-1 i2-1 /

= Rpip2p3(Tii,Ti2,Ti3)AwT;ii)AwT;22)AwT;33)+

i3=0 i2=0 ii=A

+Rpip2p3 (Tii, Ti3, Ti2 )AwT;;) AwT;32)AwT;23)+

+Rpip2p3 (Ti2, Tii, Ti3 )AwT;2i) AwT;l2)AwT;зз)+

+Rpip2p3 (Ti2, Ti3, Tii )AwT;2i) AwT;32)AwT;3)+

+Rpip2ps (Ts, Ti2, Tii )AwT;si) AwT;22)AwT;s)+ +»pip2ps en,, Tii, T12 )Aw(;;)Aw(;2) Aw£s)>) +

N —i is —i

+l.i.m. EE Rpip2.s(Ti2,Ti2,Tis)AwT;2i)AwT;22)AwT;ss) +

N—TO 1s=0 l2=A 2 2 s

+Rpip2ps (Ti2, Tis, Ti2 )AwT;2i) AwT;32)AwT;2s)+

+Rpip2ps (Tis, Ti2, Ti2 ^w^Aw^ AwT;;M +

N —i is —i

+l.i.m. EE ( »PiP2.s (Tii ,Tis ,Tis )AwT;ii)AwT;32)AwT;ss) +

N—TO is=0 ii=0 i s s

+»pip2ps (Tis, Tii, Tis )AwT;3i) AwT;i2)Aw(;s)+ +»pip2ps (Tis, Tis, Tii )AwT;3i)Aw[;32) Aw^) +

N—i

(;3) =

/ P2.sV 'is 5 'i.^ 'is^-"'VT7„ ^ TVT/„

Nll

+l.i.m. E Rpip2ps (Tis, Tis, Tis)AwT;i) AwT;32)AwT;:

AT \ o o i

is=0 T ts t2

= III »pip2ps (ti ,t2,,3)dwt(;i)dwt(;2)dwt(;s)+ ttt T ts t2

+ 111 »PiP2Ps(ti,t3,t2)dwt(;i)dwt(;s)dwt(;2) + ttt T ts t2

+ 111 »PiP2Ps (t2,ti 't3)dwi;2)dwt(;i)dwt(;s) + ttt T ts t2

+ 111 »PiP2Ps(t2't3,ti)dwt(;s)dwt(;i)dwt(;2) + ttt T ts t2

+ 111 »PiP2Ps(t3't2,ti)dwt(;s)dwt(;2)dwt(;i) + ttt

T t3 t2

+//1 Rpip2p3(t3,t1 ,t2)dwt(;2)dwt(;3)dwt(;i)+ ttt

T t3

+ 1{;i=,2=0^ yRpip2p3 (t2,t2,t3)dt2dwt(33) + tt T t3

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+ 1{;i=,3=0^ yRpip2p3 (t2,t3,t2)dt2dwt(32) + tt T t3

+ 1{,2=;3=0^ yRpip2p3 (t3,t2,t2)dt2dwt(3i) + tt T t3

+ 1{,2=;3=0^ yRpip2p3(t1 ,t3,t3)dwt(;i)dt3+ tt

T t3

+ 1{;i=,3=0}^ yRpip2p3 (t3 ,t1,t3)dwt(;2)dt3 + tt

T t3

+ W}//(2.398)

tt

where we used the same notations as in the formulas (1.16), (1.19) and Lemma 1.1 (see Sect. 1.1.3). Using (2.390) and (2.391), we obtain from (2.398)

m

J [Rpip2p3 ]T.

(3) 2n

<

T t3 t2 ,

< J J I ( (RpiP2P3 (t1, t2, t3))2n + (Rpip2p3 (t1,t3,t2))2n +

ttt

+ (Rp1p2p3 (t2, t1, t3))2n + (Rp1p2p3 (t2, t3, t1))2n + (Rp1p2p3 (t3, t2, t1))2n +

T ts

+ J J ( 1{;i=;2=0}( (»PiP2Ps(t2,t2,t3)^+ (»pip2ps(t3,t3,t2)^ ) + tt

+ 1{;i=;s=0H (»PiP2Ps(t2513, t2))2n + (»pip2ps(t3, t2, t3))2n +

+ 1{;2=;s=0^ (»pip2ps (t3,t2,t2)^ + (»pip2ps (t2,t3,t3)^J dt2dt3j , Cn < TO.

(2.399)

Due to (2.386) and Theorem 2.11 the function »pip2ps(ti, t2, t3) is continuous in the open domains of integration of iterated integrals on the right-hand side of (2.399) and it is bounded at the boundaries of these domains. Moreover, everywhere in (t, T)3 the following formula takes place

lim lim lim Rpip2ps(ti,t2,t3) = 0. (2.400)

Pi—TO P2 — TO ps—TO

Further, we have

T ts t2 ,

J J J ( (RPiP2Ps (ti,t2,t3))2n + (RpiP2Ps (ti,t3,t2))2n + (RpiP2Ps (t2,ti,t3))2n + ttt

+ (»pip2ps (t2, t3, ti))2n + (»pip2ps (t3, t2, ti))2n + (»pip2ps(t3,ti,t2))^ J dtidt2dt3 =

= J (»PiP2Ps(ti,t2,t3))2ndtidt2dt3, (2.401)

[t,T ]s

T ts

l l ((RPiP2Ps (t2' t2' t3))2n + (»pip2ps (t3,t3,t2))2^ dt2dt3 = tt

T ts T T

= J j (Rpip2ps (t2, t2, t3))2n dt2dt3 + J j (Rpip2ps (t2,t2,t3))2n dt2dt3 = t t t ts

= J (»PiP2Ps(t2,t2,t3))2ndt2dt3, (2.402)

[t,T ]2

2n

T t3

l l ((Rp1p2p3 t3, t2))2n + (Rp1p2p3 (t3,t2,t3 dt2dt3 = tt

T t3 T T

J J (Rp1p2p3 (t2, t3, t2))2n dt2dt3 + J J (Rp1 p2p3 (t2,t3,t2))2n dt2dt3 = t t t t3

( Rp1p2p3 (t2, t3, t2 )) 2n dt2 dt3,

(2.403)

[t,T]2

2n

T t3

I I ((Rp1p2p3 (t3,t2,t2)^ + (Rp1p2p3 (t2,t3,t3 dt2dt3 = tt

T t3 T T

J J (Rp1p2p3 (t3,t2,t2))2n dt2dt3 + J J (Rp1p2p3 (t3,t2,t2))2n dt2dt3 = t t t t3

( Rp1p2p3 (t3, t2, t2 )) 2n dt2 dt3.

(2.404)

[t,T]2

Combining (2399) and (2304), we obtain

/ .

m

J [Rpip2p3 ]T3t

2n

< Cn

(Rp1 p2p3 (t1, t2, t3))2n dt1dt2 dt3+

[t,T]3

+ 1{;1=;2=0} J (Rp1p2p3 (t2, t2, t3))2n dt2dt3 + [t,T ]2

+ 1{;1=;3=0} J (Rp1p2p3 (t2,t3,t2))2n dt2dt3 + [t,T ]2

\

+1{;2=;3=0} J (Rp1p2p3(t3,t2,t2))2n dt2dt3

[t,T]2 y

(2.405)

Since the integrals on the right-hand side of (2.405) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim lim »PiP2Ps(ti,t2,t3)=0 when (ti,t2,t3) G [t,T]3

Pi — TO P2—TO Ps — TO

holds with accuracy up to sets of measure zero.

According to the proof of Theorem 2.11 and (2.386) for k = 3, we have

»pip2ps (ti,t2,t3 ) = K *(ti,t2,t3) — EOi (t2,t3) j (ti) +

V ji=0 J

/ Pi / P2 \ \

+ E Cji (t2,t3) — E Cj2ji (t3)0j2 (t2H j (tiH +

Vji=^ \ j2=0 J J

/ Pi P2 / Ps \ \

+ E E Cj2ji (t3) — E Cj j (t3) j (t2)0ji (tin , (2.406)

\ji=0 j2 =0 V js =0 / /

where

T

Cji (t2,t3)^ / K*(ti,t2,t3)faji (ti)dti,

t

Cj2ji (t3) = J K*(ti,t2,t3)0ji (ti)j (t2)dtidt2. [t,T ]2

Then, applying three times (we mean here an iterated passage to the limit lim lim lim ) the Lebesgue's Dominated Convergence Theorem, we obtain

Pi—TO P2 — TO Ps—TO

lim lim lim / (i?pip9p3(ii, to, h))2n dtidt2dt3 = 0, (2.407)

Pi — TO P2 — TO Ps — TO /

[t,T ]s

lim lim lim / {Rpipr>p3{t0, to, t3))2n dt2dt3 = 0, (2.408)

Pi—TO P2—TO P —TO i 2

[t,T ]2

lim lim lim / {Rpipr>p3{t0, t3, to))2n dt2dt3 = 0, (2.409)

Pi—TO P2—TO P —TO i 2

[t,T ]2

lim lim lim I {Rprpop3{t3, U, U))2n dt2dt3 = 0. (2.410)

pi — TO p2—TO p3 — TO /

[t,T ]2

From (2.405) and (2.407)-(2.410) we get

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J [Rpip2p3 ]T3t

lim lim lim m

pi — TO p2—TO p3 — TO

2n

^=0, n G N.

Let us consider the case k = 4. Using (2.426) (see below) and (1.19), we have w. p. 1

J [Rpip2p3p4 ]T,t =

N-1 N-1 N-1 N-1

N—to

i4=0 i3=0 i

N-1 i4-1 i3-1 i2-1

N

l„i.m: EEEERpip2p3p4(Tii^,T,3,T,4)AwT;;>AwT;22»Aw<!33»Aw<;4»

i4=0 i3=0 i2=0 i1=0

= H.m. E Rpip2p3p4 (Tii ,Ti2 ,Ti3 ,Ti4)x

i4=0 i3=0 i2=0 ii=0 (ii,i2,i3,i4) V

x AwTii) AwTi2)AwTi3)AwTi4M +

Tli Tl2 Tl3 Tl4

N-1 i4-1 i3-1 / \

+ E iR^ (Ti2 ,Ti2 ,Ti3 , Ti4 )AwT;2i) AwT;22) AwTi33) AwT;44M +

N—TO i4=0 i3=0 i2=0 (hMMMr J

N-1 i4-1 i3-1 / \

+1Ni.m.£££ E (Tii,Ti3,Ti3,Ti4)AwT;ii)AwT;3)AwT;33)AwT;44M+

—TO i4=0 i3=0 ii=0 (ii,i3,i3,i4r

N-1 i4-1 i2-1

+l„i.m. E E E E (1.• Ti2, T,4,T,4^w^'aW^'AW^'AW^+

— TO i4=0 i2=0 ii=0 (hMMMV

N-1 i4-1

N

+l.i.m^^E E Rpip2p3p4(Ti3,Ti3,Ti3,Ti4)AwT;3i)AwT;32)AwT;33)AwT;44M +

i4=0 i3=0 (i3,i3,i3,i4)

N-1 i4-1

+lJLm. EE E (Ti2,Ti2,Ti4,Ti4^w^w^^^M +

—TO i4=0 i2=0 (hMUhA

N-1 /4-1

e ee (r

(r/!, T/4, T/4, T/4 )Aw(; ^AwT; 2) Awijf) Aw(J4A +

/4=0 /1=0 (/i,/4,/4,/4)

in '¿4' 'in i4 / T}^ T4 T4 T}4 ,

N-1

+l.i.m. E Rpip2P3P4(t/4, t/4, t/4, t/4)AwT;4!) AwT;42) AwTi43)AwTi44)

N—to 7 ' 14 14 14 14

/4=0

T ¿4 ¿3 ¿2

£

fliW4 (ti, i2, is, i4)dwi(;i)dw(;2)dwi(33)dw(;4) +

t t t t (t1,t2,t3 ,t4)

T t4 t3

+1{;i=i2=o}/// e

t t t (t1,t3,t4)

T t4 t2

+ 1{;i=;3=0}/// E

t t t (t1 ,t2 ,t4)

T t3 t2

+1{;1=;4=o} J J J e

t t t (t1 ,t2 ,t3)

T t4 t2

+1

{;2=;3=0}

E

t t t (t1 ,t2 ,t4)

T t3 t2

+1

{;2=;4=0}

E

t t t (t1 ,t2 ,t3) T t3 t2

+1{;3=;4=0}

t t t (t1 ,t2 ,t3) +1{;1=;2=0}1{;3=;4=0}

Rp1p2p3p4 (t1 5 t1 5 t3, i4)dt1dwt(33) dwt(;4M +

Rp1p2p3p4 (i15 i2, i1, t4)dt1dwt(;2)dwt(44) ) +

Rp1p2p3p4 (i15 i25 i35 t1)dt1dwt(;2)dwt(33M +

Rp1p2p3p4 (i15 ¿25 ¿25 i4)dwt;1 Jdt2 dwt(44M +

Rp1p2p3p4 (i15 i25 i35 i2)dwt;1 Jdt2 dwt(33) +

Rp1p2p3p4 (i15 i25 i35 t3)dwt;1

T t4

Rp1p2p3p4 (i25 ¿25 ¿45 ¿4 ) dt2di4 +

tt

T t4 \

+ J J Rp.p2p3p4(t4, t4, t2, t2)dt2dt4 J +

tt

/ T t4

+1{;i=;3=0}1{;2=;4=0} I J yRpip2p3p4(t2,t4,t2,t4)dt2dt4+

tt

T t4 \

+ J J Rp.P2P3P4(t4, t2, t4, t2)dt2dt4 j + tt

/ T t4

+1{;i=;4=0}1{;2=;3=0} I J yRpip2p3p4(t2,t4,t4,t2)dt2dt4+

tt

T t4 \

+ J J Rp.P2P3P4 (t4,t2,t2,t4)dt2dt4j , (2.411)

tt

where the expression

E

(ai,...,afc)

means the sum with respect to all possible permutations (a1,..., ak). Moreover, we used in (2.411) the same notations as in the proof of Theorem 1.1 (see Sect. 1.1.3). Note that an analogue of (2.411) will be obtained in Sect. 2.6 (also see [10]-[16], [35]) with using the another approach.

By analogy with (2.405) we obtain

m

/ „

J [Rpip2p3p4 ]T,t

(4) 2n'

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<

< cn

J (Rp.P2P3P4 (t1, t2, t3, t4))2n dt1 dt2dt3dt4 + \t,T ]4

+ 1{;1=;2=0} J (Rp.P2P3P4(t2,t2,t3,t4))2n dt2dt3dt4 + [t,T ]3

+ 1{;1=;3=0} J (Rp.P2P3P4(t2,t3,t2,t4dt2dt3dt4 +

[t,T ]3

+ 1{;1=;4=0} J (Rp1p2p3p4(¿2 5 ¿35 ¿45 ¿2))2n ^¿¿3^4 + [t,T ]3

+1{;2=;3=0} J (Rp1p2p3p4(¿35 ¿25 ¿25 ¿4))2n ^¿¿3^4+ [t,T ]3

+ 1{;2=;4=0} J (Rp1p2p3p4(¿35 ¿25 ¿45 ¿2))2n ^¿¿3^4 + [t,T ]3

+ 1{;3=;4=0} J (Rp1p2p3p4(¡¿35 ¿45 ¿25 ¿2))2n ¿¿2^3^ + [t,T ]3

+1{;1=;2=0}1{;3=;4=0} J (Rp1p2p3p4¿¿2^4+

[t,T ]2

+1{;1=;3=0}1{;2=;4=0} J (Rp1p2p3p4¿¿2^4+

[t,T ]2

\

+ 1{;1=;4=0}1{;2=;3=0} J (RP1P2P3P4 ¿¿2^4 5 Cn <

[t,T]2 /

(2.412)

Since the integrals on the right-hand side of (2.412) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim lim lim R«1w2w3w4(¿15 ¿25 ¿35 ¿4) = 0 when (¿15 ¿25 ¿35 ¿4) G [¿5T]

holds with accuracy up to sets of measure zero.

According to the proof of Theorem 2.11 and (2.386) for k = 4, we have

/ P1 \

Rp1p2p3p4 (^¿15 ¿25 ¿35 ¿4) = K^¿15 ¿25 ¿35 ¿4) — (¿2 5 ¿35 ¿4)0j1 (¿1) +

V j1=0 J

/ P1 / P2 \ \

+ E Cj1 (¿25 ¿3 5 ¿4) - E Cj2j1 (¿3 5 ¿4)0^2 (¿2H j (¿1) +

Vj1=0 V j2=0 / /

4

(pi p2 / p3

EE ( Cj2ji (t3,t4) - E Cj3j2ji (t4)0j3 (t3^ j (t2)0ji (t0 ) + ji=0 j2 =0 \ j3=0

pi p2 p3 p4

+ I E E E ( Cj3j2ji (t4) - E Cj4j3j2j1j (t4) J j (t3)0j2 (t2)0ji (t0 ) , Vji =0 j2=0 j3=0 V j4=0

where

T

Cji (t2MM) = J K*(t1,t2,t3,t4)j (t1 )dt1,

t

Cj2ji (t3,t4)= J K*(t1,t2,t3,t4)0ji (t1)j (t2)dt1dt2,

[t,T ]2

Cj3j2ji (t4) = J K*(t1 ,t2,t3,t4)0ji fa) j fa) j (t3)dt1dt2dt3. [t,T ]3

Then, applying four times (we mean here an iterated passage to the limit lim lim lim lim ) the Lebesgue's Dominated Convergence Theorem, we ob-

p.—to p2—TO p3—TO p4—TO

tain

lim lim lim lim / {RPlP2P3Pi{t\, t2, £3, ¿4))2n dtid^dt^dt^ = 0, (2.413)

pi — TO p2—TO p3 — TO p4—TO /

[t,T ]4

lim lim lim lim / (RplP2P3P4(t2, t2, ¿3, ^))2n d^dt^dt^ = 0, (2.414)

pi—TO p2 — TO p3—TO p4 — TO /

[t,T ]3

lim lim lim lim / (RPlP2P3P4(h,h,h,U))2n d^dt^dt^ = 0, (2.415)

pi—TO p2 — TO p3—TO p4 — TO /

[t,T ]3

lim lim lim lim / (RplP2P3P4(t2, h, ¿4, £2))^ dt2dt^dt4 = 0, (2.416)

p.—to p2—TOp3—TOp4—TO J

[t,T ]3

lim lim lim lim / (Rpip2p3p4(h, t2, t2, t/\))Zn dt2dtzdt/± = 0, (2.417)

p.—to p2—TOp3—TOp4—TO J

[t,T ]3

lim lim lim lim / {Rpip2p3p4{h, to, ¿4, h))2n dtodt^dti = 0, (2.418)

[t,T ]

lim lim lim lim / (RplP2P3P4(t3, to, to))2n dtodt^dti = 0, (2.419)

[t,T ]

lim lim lim lim / (RPlP2P3P4(to, to, )2n dtodt^ = 0, (2.420)

[t,T ]

lim lim lim lim / (RPlP2P3P4(to, t.4, to, t.4))2n dtodt.4 = 0, (2.421)

[t,T ]2

lim lim lim lim I (RPlP2P3P4(to, t.4, t.4, to))2n dtodt.4 = 0. (2.422)

[t,T ]2

Combaining (2412 with (2413)-((24m we get

2n>|

lim lim lim lim m

— P2—TO P3 — TO P4—TO

J [RP1P2P3P4 ]T4t

= 05 n G N.

Theorem 2.13 is proved for k = 4.

Let us consider the case of arbitrary k,k G N. Let us analyze the stochastic integral defined by (1.16) and find its representation convenient for the following consideration. In order to do it we introduce several notations. Suppose that

N-1 j2 1

^IvV) = E • • • E E a(j'1,-,jfc),

j'fc=0 j1=0 (j1,...,jfc)

Csr . . . CS1 SNk)(a) =

a r

N-1 jsr +2-1 jsr + 1-1 js1+2-1 jS1 + 1-1 j2 1

E- e E •■• E E E E

j'fc =0 jsr + 1=0 jsr-1=0 jS1 + 1=0 jS1-1=0 j1=0 ^ T .. ..1 = 1

1 1 nlj«1,jS1+1 (j1>->Jfc)

l=1ll

n ijS1,jS1+1 (ji,•••Jfc)'

3

2

where

r

j j + 1 (j1 , . . . , ) = Ijsr jsr + 1 . . . IjSi ,jSi + 1 (j1 , . . . , ) ,

i=1

0

Cso... Csi sNk)(a) = 4%), II j j+1 (j1,... ) = (j1,...,jk),

i=1

j ji+1 (jqi, . . . , jq2 , ji, jq3 , . . . , JV-2 , ji, jqfc-i, . . . , JV ) = = (jqi, . . . , jq2 , ji+1, jq3 , . . . , JV-2 , ji+1, JV-i, . . . , j3fe ) ,

where l = q1,..., q2, q3,..., qk-2, qk-1,..., qk, l G N, aj ;...;j-) is a scalar value, s1,..., sr = 1,..., k - 1, sr > ... > s1, q1,..., qk = 1,..., k, the expression

E

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means the sum with respect to all possible permutations (jqi,..., jqk). Using induction it is possible to prove the following equality

N-1 N-1 k-1 k- 1

£."E°(ji.-j) = E E Csr...C.,sNk)(a), (2.423)

jk =0 ji=0 r=0 Sr,...,si = 1

sr>...>si

where k = 2, 3, . . .

Hereinafter in this section, we will identify the following records

a(ji,...,jk) = a(ji...jk) = aji...jk. In particular, from (2.423) for k = 2,3,4 we get the following formulas

N-1 N-1

E E a(ji,j2) = (a) + (a) = j2=0 ji=0

N-1 j2 — 1 N-1 N-1 j2 — 1

a(jij2) + ^^ a(j2j2) = ^^ 5^(aj'ij2 + aj2j1 j2 =0 j1=0(j1,j2 ) j2=0 j2=0 j1=0

N-1

+ E flj2j2, (2.424)

j2=0

N-1 N-1 N-1

E E E a(j1 j2j3) = sN3)(a) + C1SN3)(a) + C2sfV) + C2C^V) =

j3 =0 j2=0 j1=0

N-1 j3-1 j'2-1 N-1 j3-1

= + EE E a(j2 j2j3 ) +

j3 =0 j2=0 j1=0 (j1,j2,j3) j3 =0 j2=0 (j2,j2,j3)

N-1 j3-1 N-1

+ ^3 53 a(j1 j3j3) + ^3 a(j3j3j3) =

j3=0 j1 =0(j1 ,j3,j3) j3=0

N-1 j3-1 j'2-1

= y ^ ^3 (aj1j2j3 + aj1j3j2 + aj2j1j3 + aj2j3j1 + aj3j2j1 + aj3j1j2 ) +

j3=0 j2=0 j1=0 N-1 j3-1 N-1 j'3-1

+ ^ ^ (aj2 j2 j3 + aj2 j3 j2 + aj3 j2 j2) + ^ ^ 53 (aj1j3j3 + aj3j1j3 + aj3j3j1 ) +

j3=0 j2 =0 j3=0 j1=0

N-1

+ E aj3j3j3, (2.425)

j3=0

N-1 N-1 N-1 N-1

E E E E a(j1 j2J3j4) = ^IvV) + C1SN4)(a) + C2S^)(a) +

j4=0 j3=0 j2=0 j1=0

+C3SN4)(a) + C2C1SN4)(a) + C3C1^N4)(a) + C3C2 ^(a) + C3C2C1S4V) =

N-1 j4 1 j3-1 j'2-1 N-1 j4 1 j3-1

=

a(j1j2j3 j4) + a(j2j2j3 j4)

j4=0 j3=0 j2=0 j1=0 (j1 ,j2 ,j3 ,j4) j4=0 j3=0 j2=0 (j2,j2,j3,j4)

N-1 j4 1 j3-1 N-1 j4 1 j2 1

+

j4=0 j3=0 j1=0 (j1,j3,j3,j4) j4=0 j2=0 j1=0 (j1,j2,j4,j4)

N-1 j4 1 N-1 j4 1

+

a(j2j2j4j4) +

j4=0 j3 =0 (j3 ,j3 ,j3 ,j4 ) j4 =0 j2=0 (j2 ,j2 ,j4 ,j4 )

N-1 j4-1 N-1

+ ^3 ^3 53 a(j1j4j4j4) ^53 aj4j4j4j4 =

j4=0 j1 =0 (j1,j4,j4,j4) j4=0

N-1 j4 1 j3-1 j2-1

= y ^ ^3 53 53 (aj1 j2j3j4 + aj1 j2j4j3 + aj1j3j2j4 + aj1j3j4j2 +

j4=0 j3=0 j2=0 j1 =0

+ aji j4j3j2 + aj1j4j2j3 + aj2j1j3j4 + aj2j1j4j3 + aj2j4j1j3 + aj2j4j3j1 + aj2j3ji j4 + + aj2 j3j4ji + aj3j1j2j4 + aj3j1j4j2 + j j2jij4 + aj3j2j4ji + aj3j4j1j2 + aj3j4j2 ji + + aj4 jij2j3 + aj4j1j3j2 + aj4j2jij3 + j j2j3ji + jj3jij2 + aj4j3j2ji ) +

N-1 j4 1 j3-1

+ ^ ^ (aj2j2j3j4 + aj2j2j4j3 + aj2j3j2j4 + aj2j4j2j3 + aj2j3j4j2 + jj4j3j2 +

j4=0 j3=0 j2=0

+ aj3j2j2j4 + aj4j2j2j3 + aj3j2j4j2 + aj4j2j3j2 + aj4j3j2j2 + aj3j4j2j2) +

N-1 j4 1 j3-1

+ ^ ^ (aj3j3j1j4 + aj3j3j4j1 + aj3j1j3j4 + aj3j4j3j1 + aj3j4j1j3 + jjij4j3 +

j4=0 j3=0 ji=0

+ aj1j3j3j4 + aj4j3j3j'i + aj4j3j1j3 +aj1j3j4j3 + ajij4j3j3 + aj4j1j3j3) + N-1 j4 1 j2 1

+ ^ ^ (aj4j4j'ij2 + aj4j4j2j1 + aj4j1j4j2 + aj4j2j4j1 + aj4j2j1j4 + jjij2j4 +

j4=0 j2=0 ji=0

+ aj1j4j4j2 + aj2j4j4j1 + aj2j4j1j4 + ajij4j2j4 + ajij2j4j4 + aj2j1j4j4) + N-1 j4 1

+ ^ ^ (aj3j3j3j4 + aj3j3j4j3 + aj3j4j3j3 + aj4j3j3j3) +

j4=0 j3=0

N-1 j4 1

+ ^ ^ (aj2j2j4j4 + aj2j4j2j4 + aj2j4j4j2 + aj4j2j2j4 + aj4j2j4j2 + aj4j4j2j2) +

j4=0 j2=0

N-1 j4 1

+ ^ ^ (ajij4j4j4 + aj4jij4j4 + aj4j4jij4 + aj4j4j4j1 ) +

j4=0 ji=0

N-1

+ E aj j. (2.426)

j4=0

Perhaps, the formula (2.423) for any k (k G N) was found by the author for the first time [71] (1997).

Assume that

®(ji,...,jfc) =$(Tji ,...,Tjk ),

i=1

where $ (t1,... , tk) is a nonrandom function of k variables. Then from (1.16 and (2.423) we have

[k/2]

JrnS = E E x

r=0 (sr ,...,S1)GAfc,r

N-1 jsr +2-1 jsr + 1 -1 jS1+2-1 j.S1 + 1-1 j2 1

•■■ E E E E -E E

x l.i.m

N

x

jk =0 jsr + 1=0 jsr-1=0 jS1+1=0 jS1-1=0 j1=0

n ijS1,jS1+1 Cj1,--.,j'fc)

l=1

X

^ ( Tj1 , • • • , TjS1-1 , TjS1 + 1 , TjS1 + 1 , rjS1+2 , • • • , Tjsr-1 , Tjsr + 1 , Tjsr + 1 , Tjsr+2 , • • • , Tjfc ) x

jsr-1' jsr + 1' jsr + 1' 'jsr +2 '

Awj... Aw^^Awj) AwT:s1+1)AwT:s1+2)

'JS1 -1 /Js1 + 1 /Js1 + 1 ' js1+2

j1

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... Aw(isr-1)Aw(isr) Aw(isr+1) Aw(isr+2)... Aw(ik)

Tjsr -1 Tjsr +1 Tjsr +1 Tjsr +2

jfc

[k/2]

E E i№$sWr w. p. 1,

r=0 (sr,...,S1)GAfcjr

(2.427)

where

T tsr +3 tsr +2 tsr tS1+3 tS1+2 tS1 t2

I [$]

(k)s1,...,sr

T,t

E

t t t

x

t t t t nitS1,tS1+1 (t1,...,tfc)

l=1

x

$ I ¿1, • • • , ^-1^ + 1;

, ts1+2, • • • , ¿sr-1, ¿sr+1, ¿sr+1, ¿sr +2, • • • , ¿k x

xdw^... dwf1-1)dwt(isi)idwiisi+1)dwt(isi;2)

t1 ¿s 1 — 1 tS1 + 1 tS1 + 1 tS1+2

• . dwt( v-1)dwt( v) dwt( v+1)dwt( v+2) • • • dwt( ik)

tsr-1 tsr + 1 tsr + 1 tsr +2 tk

(2.428)

where k > 2, the set Ak,r is defined by the relation (2.3711). We suppose that the right-hand side of (2.428) exists as the Ito stochastic integral.

Remark 2.3. The summands on the right-hand side of (2.428) should be understood as follows: for each permutation from the set

r

,tSi+i (tb . . . ,tk) =

i=1

= ^4, . . . , t.i-1, tSi+1, t.1+1, t.1+2, . . . , tSr-1, tSr+1, tSr+1, tSr+2, . . . , tkj

it is necessary to perform replacement on the right-hand side of (2.428) of all pairs (their number is equal to r) of differentials dw^dwj) with similar lower indices by the values 1{;=j=0}dtp.

Note that the term in (2.427) for r = 0 should be understood as follows

T t2

/••./£ (t1,.--,tk)dw<;;'>...dw«;k>),

t t (ti,...,tk)

where notations are the same as in (1.24).

Using (2.390), (2.391), (2.427), and (2.428), we get

m

J [*]Tkt

2n

<

[k/2] f < Cnk E E m\\i[$]Tk

r=0 (sr,...,si)GAfc,r

](k)si,...,sr \t

2n

(2.429)

where

m

i №

(k)si,...,sr

T,t

2n

<

T tsr +3 tsr + 2 tsr ts1+3 ts1+2 ts1 t2

< Cnk

E

t t t

x

t t t t nitsi,tsi+i(ti,...,tk)

l=i

X$2n i t1, . . . , ts1-l, ts1+l, ts1+l, ts1+2, . . . , tsr-b ^+1, tsrtsr+2, . . . , tk ) x

X dt1... dtsi-1dtsi+1dtsi+2... dt.r-1dt.r+1dt.r+2... dtk, (2.430)

where Cnk and Cnk"Sr are constants and permutations when summing are performed in (2.430) only in the values

$2n ( ¿1, • • • , ts1-1, ts1+1, ts1+1, ts1+2, • • • , ¿sr-1, ¿sr+ 1, ¿sr+1, ¿sr+2, • • • , ¿k ) •

Consider ((2329) and (2430) for $(¿1,... ,tk) = RP1...Pk(¿1, • • • ,tk)

m

(k)

2n

<

where

[k/2] . < e m||/ [Rp1.pk ]

r=0 (sr,...,s1)eAfcir

] (k)s1 ,...,sr ]T,t

2n

m

1 [R ](k)s1 '•••'Sr 1 [RP1--vPfc ]T,t

2n

<

(2.431)

T tsr +3 tsr + 2 tsr ts1+3 ts1+2 ts1 t2

s1...sr < Cnk

E

X

t t t t t t t nits1,ts1 + 1 (t1,...,tfc)

1=1 1 1+

xRPn...Pk (^1, . . . , ts1-1, ts1+1, ts1+1, ts1+2, • • • , ¿sr-1, ¿sr+1, ¿sr+b ¿sr+2, • • • , ¿kj x

x di1 • • • ^-1^+1^+2 • • • disr-1disr+1disr+2 • • • dik, (2.432)

where Cnk and CnT^ are constants and permutations when summing are performed in (2.432) only in the values

R

2n

¿1, • • • , ¿s1-1, ¿s1+1, ¿s1+1, ¿s1+2, • • • , ¿sr-1, ¿sr+b ¿sr+1, ¿sr+2, • • • , ¿k ) •

From the other hand, we can consider the generalization of the formulas (2.394), (2.405), (2.412) for the case of arbitrary k (k G N). In order to do this, let us consider the sum with respect to all possible partitions defined by (1.53)

E

a

g1 S,2,...,S,2r-1S,2r,q1...qfc-2r •

({{31>S2}>--->{S2r-1>S2r »>{91 >'">9fc-2r {S1,S2>'">S2r-1>S2r>91 >--->9fc-2r } = {1 >2 >--->k }

Now we can generalize the formulas (2.394), (2.405), (2.412) for the case of arbitrary k (k £ N)

/

2n>|

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m

J [Rpi...pfc ]t.

(k) t

(Rp1...pk(t1, • • •, ^))2n dt1 • • • dtk+

[k/2] +E

E

1{ifli =ig2 —0} ••• l{ifl2r_i =iS2r —0}

X

r—1 ({{si,32 }>•••> {32r-1,32r }},{91v,9fc-2r }) {S1,32v,32r-1,32r,91v,9fc-2r }={1,2,---,fc}

X

R

¿1,• • •, ik

2n

X

[t,T ]

k—r

t31 —132 '...'t32r-1 =t32r.

x ( dt1 • • • dtk

(dt31 dt32 )^dt31..... (dt32r-1 dt32r 2r-1

/

(2.433)

where Cnk is a constant,

^ • • •, ^

t31 —132 '...'t32r-1 =t32r

means the ordered set (t1, • • • ,tk), where we put tg = tg , Moreover,

• , tg2r-1 tg2r '

dt1• • • dtk

(dt31 dt32 )^dt31..... (dt32r-1 dt32r )^dt32r-

means the product dt1 • • • dtk, where we replace all pairs by dtg1, • • •, dtg2 correspondingly.

Note that the estimate like (2.433), where all indicators must be replaced with 1, can be obtained from the estimates (2.431), (2.432).

The comparison of (2.433) with the formula (1.54) (see Theorem 1.2) shows their similar structure.

Let us consider the particular case of (2.433) for k = 4

1

m

J [RP1P2P3P4 ]T,t

2n

< Cn4

(Rp1p2p3p4 (¿1, ¿2, ¿3, ¿4))2n d£1 di2di3 di4+

V[t,T ]4

+ 53 1iig1 =ig2 =0M | RP1P2P3PJ ¿1, ¿2, ¿3, ¿4

({31,32},{91,92}) {31 >32 ,91,92} = {1,2,3,4}

2n

[t,T ]3

t31 =t32 .

X

x di1di2 di3di4

+

(dt31 dt32 )^dt31

+ E 1{i31 =i32 =0}1{i33 =i34 =0}x

({{31>32}>{33>34}})

{31>32,33,34 } = {1,2,3,4}

R

'P1P2P3P4

¿1, ¿2, ¿3, ¿4

2n

X

[t,T ]2

t31 =t32 ,t33 =t34 ,

x ( di1di2 di3di4

(dt31 dt32 ) ^dt31 > (dt33 dt34 ) ^dt33

7

(2.434)

It is not difficult to notice that (2.434) is consistent with (2.412). According to (2.362) and (2.386), we have the following expression

RP1--vPfc , • • • , ¿k) =

k

IIv* (ii ) n 1

/k-1 k-1 1 k-1 r k-1 ^

-{ii<ii+i} + 53 53 II 1{i«i=i«i+i} n 1{+i<ti+1}

l=1 l=1 r= 1 Sr,---,S1 = 1 1 = 1 1=1

\ Sr>--->S1 1 = S1,---,Sr

/

P1 Pk

53 • • .53 Cjk •••j1 (¿i).

j1=0 jk=0 l=1

(2.435)

k

Due to (2.435) the function RPi...Pk) is continuous in the open domains of integration of integrals on the right-hand side of (2.432) and it is bounded at the boundaries of these domains for ... < to.

Let us perform the iterated passage to the limit lim lim ... lim under

pi—TO p2 — TO pk— TO

the integral signs on the right-hand side of the estimate (2.433) (it was similarly performed for the 2-dimensional, 3-dimentional, and 4-dimensional cases (see above)). Then, taking into account (2.387), we obtain the required result. More precisely, since the integrals on the right-hand side of (2.433) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim ... lim (ti,...,tk) = 0 when (ti,...,tk) G [t,T]k

pi—to pk —

holds with accuracy up to sets of measure zero.

According to the proof of Theorem 2.11 and (2.386), we have

Rpi...pfc(ti,... ) = ( K*(ti,... ) - E Cji(t2,... ) j(ti) ) +

V ji=0 /

/ Pi / P2 \ \

+ E Cji (t2, . . . ,tk) - E Cj2ji (t3, . . . ,tk) j fa) j (tiH +

V j2 =0 J J

Pi Pk-i / Pk \

+ (e... e (Cjk-i...jifa) - e Cjk...jijfaM 0jfc-ifa-o... j(ti)

ji=0 jk-i=0 V jk=0 /

where

T

Cji (t2,...,tk ) = ^ K*(ti,...,tk )j (ti)dti, t

Cj2jifa,... ,tk) = J K*fa,... ,tk) j(ti) j(t2)dtidt2,

[t,T ]2

r k-i

Cjk-i...jifa) = K*fa,... ,tk^ fa)dti.. .dtk-i.

^ /=i

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[t,T]k-i /=i

Then, applying k times (we mean here an iterated passage to the limit lim lim ... lim ) the Lebesgue's Dominated Convergence Theorem to the

^—to p2—to pk—to

integrals on the right-hand side of (2.433), we obtain

lim lim ... lim m

Pl—TO p2 — TO Pk —TO

J [RPl---Pfc ]

(k) 2n~ T,t

= 0, n G N.

Theorems 2.13 and 2.10 are proved.

It easy to notice that if we expand the function K*(i1,..., ¿k) into the generalized Fourier series at the interval (¿,T) at first with respect to the variable ¿k, after that with respect to the variable ¿k-1, etc., then we will have the expansion

Pk Pi k

'1, • • • 5 "kj • • • / v • • • / , ^Jk...Ji

Wk—TO w1—TO z—' z—'

jk=0 ji=0 1=1

K*(ti,...,tk )= lim ... lim E •••ECk ...Jill ^ ('1 ) (2-436)

instead of the expansion (2.363).

Let us prove the expansion (2.436). Similarly with (2.367) we have the following equality

T

1 \ TO „

<7,(7/,) ^1.,, . ,,;. • -1.,, // J E j ' V.4///)o/;i7/,)d//,o/; (//,). (2.437)

jk 0 ti„ 1

which is satisfied pointwise at the interval (¿, T), besides the series on the right-hand side of (2.437) converges when = ¿,T.

Let us introduce the induction assumption

T T

oo oo „ „

E ..^^(¿2) / fa3fe) j ('a)... fak ('k ) j ('k )d'k ...dt^ (t )

Jk=0 J3=0 /2 tk 1 ¿=3

+ • (2-438)

7—O 7—O V /

1=2 1=2 Then

TT

to to to „ „ k

E--.EEfai(ti) / fa^fc) j ('2)... fa (tk ) j (tk )dtk ...dt^ (ti )

jk =0 j3=0 j2 =0 ti tk-1 '=2

k

to TO / i \

= E • • • E MMl 1 {tl<t2} + 21{îi=Î2> )

jk=0 j3=0 ^ '

t T k

MMM (ta) .. J ^k (tk )0j (tk)dtk ...dt^ (ti )

¿2 tfc-1

(1 N TO TO

iiiKia} + 2 Y ■ ■ - Y MMX

jk=0 j3=0

T T

k

X y ^3(t3)0j3 (t3) . . .J ^k (tk )0j (tk)dtk . . . dt3 fj (t1) =

¿2 ifc-1 1—3

= '01 (ti) f MhKU} + ^l{i!=i2} ) n'0/(t/) n (^iKi^i} + T^i^+i} ) = ^ ' 1=2 1=2 ^ '

k k—1 ✓ i \ = w.1 + 21™ • (2-439)

1=1 1=1 ^ ^

From the other hand, the left-hand side of (2.439) can be represented in the following form

TO TO k

E-ECk ...jill 0* (ti ) jk=0 ji=0 1=1

by expanding the function

T T

V>l(ti)y ^2(t2)0j2 (t2) - J ^k (tk )0j (tk )dtk . ..dt2

¿1 tk-i

into the generalized Fourier series at the interval (t,T) using the variable t1. Here we applied the following replacement of integration order

T T T

J ^i(ti)0ji (tl)y ^2^2)072 (t2) ...J ^k (tk )0j (tk)dtk ...dt2dti =

t ¿1 tk-1

T t3 t2

= J fa (ik ) j (ik) . . .J fa^j (h) J ) j fa^Mk . . . dik = Cjk...j1. t t t

The expansion (2.436) is proved. So, we can formulate the following theorem.

Theorem 2.14 [10] (2013) (also see [11]-[16], [33]). Suppose that the conditions of Theorem 2.10 are fulfilled. Then

to to k

J*W'(t)]T,f = £ ... £Cj,..;, II<f, (2.440)

jk=0 j1=0 i=1

where notations are the same as in Theorem 2.10. Note that (2.440) means the following

{/ Pk P1 k \2n>|

1l

\ jk=0 j1=0 l=1 / j

where n G N.

2.4.2 Further Remarks

In this section, we consider some approaches on the base of Theorems 2.10 and 1.1 for the case k = 2. Moreover, we explain the potential difficulties associated with the use of generalized multiple Fourier series converging pointwise or converging almost everywhere in the hypercube [¿,T]k in the proof of Theorem 2.10.

First, we show how iterated series can be replaced by multiple one in Theorem 2.10 for k = 2 and n = 1 (the case of mean-square convergence).

Using Theorem 2.10 for k = 2 and n = 1, we obtain

\2

p P \ 2

(hM*2 )

lim (J -[V/2>]T,f j j'C

j1=0 j2 =0

2

p p 2

j1=0 j2=0

(hM«2) i v <

p q

* ^ (2M \(~ E E ) \ +

ji =0 j2=0

\ 2

P q P P 2

= (ii^(i2) V^ n . Aii) Z(i2)

j2

ji=0 j2=0 ji=0 j2=0

2

+2M £ ECj2ji cjrj -EE j c

2

P q 2

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ji =0 j2=P+i

P P q q

p—TO q—TO z

ji=0 ji=0 j2=P+i j2 =P+i

2 lim lim V V C2

P q j

Pq

2

j2j1

P—TO q—TO

ji=0 j2 =P+i

P q P P

2 p1—to q-KTO ( 53 Cj 2j i 53 Cj 2j i

\ji =0 j2 =0 ji =0 j2 =0

(2.441)

P q P P

21 p^EE j- P—TOEE j = <2-442)

ji=0 j2=0 ji=0 j2 =0 /

= J K2(ti, t2)dtidt2 - J K2(ti,t2)dtidt2 = 0, (2.443)

[t,T]2 [t,T]2

where the function K(ti,t2) is defined by (1.6) for k = 2.

Note that the transition from (2.441) to (2.442) is based on the theorem on reducing of a limit to iterated one. Moreover, the transition from (2.442) to (2.443) is based on the Parseval equality.

Thus, we obtain the following Theorem.

Theorem 2.15 [14]-[16], [33]. Assume that (x)}TO 0 is a complete or-

thonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, every ^(t) (/ = 1, 2) is a continuously differentiate

nonrandom function on [t, T]. Then, for the iterated Stratonovich stochastic integral (2.356) of multiplicity 2

J *[^(2)]

* T * ¿2

T,t =/ ^2)/ WOdw^dw^ (ii,i2 = 0,1,..., m)

t

the following expansion

J *

T,t = i-i-m

p—œ

i.i.m. V Cj2j1 C(n)Cf2)

j1,j2=0

that converges in the mean-square sense is valid, where

T t2

(2.444)

is the Fourier coefficient and

T

cf = h (s)dw

j / j

t

(i) s

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT^ = f^ are independent standard Wiener processes (i = 1,..., m) and wT0) = t.

Note that Theorem 2.15 is a modification (for the case p1 = p2 = p of series summation) of Theorem 2.2.

Consider the proof of Theorem 2.2 based on Theorems 1.1 and 2.10. Using Theorem 2.10, we have

0

pi pk

k

'S» Ü5L■ ■ • «5LM{E-EIICÎ1 "r

p1—œ p2—œ pk—œ

< iim iim ... iim

p1—œ p2—œ pk—œ

ji=0 jk=0 p1 pk

1=1

k

<

m^E -Ec^n d" > - j *

¿1=0 jk=0

T,t

< iim iim ... iim m

p1—œ p2—œ pk—œ

1=1

p1 pk

k

J *W'(k)]T,t -E ..^Cik^n Z

¿1=0 jk=0

(il ) II d

i=i

<

p

\ 2^ \i/2

Pi Pk k x 1 x

n(in)

ji=0 jk=0 /=i

* J™ Jte,• • ■ j™, (M{(r'fo-E• • ■ Enc) }] =0.

(2.445)

From the other hand,

/ Pi Pk ( k

lim IE ... IE E ■ • • E II<f - M {J'^tA

ji=0 jk=0 ^ /=i Pi Pk f k

lim lim ... lim £ ... £ (M J] " M { J^fo} •

1—top2—TO pk—TO z—' z—' | 11 | L J

(2.446)

pi—^OO pk—t>00

ji=0 jk=0 ^ /=i

Combining (2.445) and (2.446), we obtain

pi Pfc f A" i

Mi-/ >,} lim Tim ... Tim ^ """ E M II " t2"447)

ji=0 jk=0 I /=i

The relation (2.447) with k = 2 implies the following

T

Pi P2

lim IE EE {cf^'l. <2'448)

Pi — TO P2—TO

ji=0 j2=0

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where 1A is the indicator of the set A. Since

m {j^} = 1{ii=i2=0}1{ji=j2},

then from (2.448) we obtain

Pi P2

l:— l:— \ \ n -i

-{ji=j2}1{ii=«2=0}

Pi P2

M\r[iJjW]TA= lim Tim V V CJ0J11 j ,, = .,■„; 14

I J pi—TO p2—TO z—' z—'

j1=0 j2=0

min{pi,p2} to

l{il=^o} lim lim ^ Cnjl = l{i1=i2Jio}YCM> [2A4Q)

P1—TO P2—TO

j1=0 j1=0

where Cjj is defined by (2.360) for k = 2 and j = j2, i.e.

T t2

Cjiji = j ^2(t2)0ji (^2^ ^l(tl)0ji (tl)dtldt2-t t

From (2.448) and (2.449) we obtain the following relation

T

to 1

E < £ f Us)Us)ds. (2.450)

j1 2 ji=0 t

Combining (1.46) and (2.450), we have

Pi P2 /

JW(2)]T, = ^ E E j № - l{.i = .2=0!l{ji=,2}

ji=0 j2=0 V

Pi P2 TO

=pipmTO ^ s Cj2ji jj - 1{ii=i2=0^ c

{¿1=»2=0} / v Cjiji ji=0 j2=0 ji=0

T

Pi P2

= „¡¿E^ E E cr1 - jif.^^oi I Ms}Ms)d.s. (2.451) ji=0 j2=0 t

Since

T

J*^2^ = + jfa{s}Us)ds w. p. 1, (2.452)

t

then from (2.451) we finally get the following expansion

Pi P2

(2)]t,< = i,i.m. e e^jj.

Pi,P2—TO z-' z—' •yi 72

ji=0 j2=0

Thus, Theorem 2.2 is proved. We have

T

j*[<A(2)]?f = +jus)us)ds=

t

Pi P2 / \ 1 T

EEci;i}cj:2) - i{*i=^o}i{,1=,2} + /

ji=0 j2=0 V / {

min{pi,p2}

Pi P2 /1 T min{pi,p2} \

E E + i{n=^o} 5 / Ms}us)ds ~ y cnn ji=0 j2=0 \ { ji=0 /

(2.453) where

Pi P2

J<2)]Tif2 = £ £ Cj2ji cjr'cjr - 1{!i=,2=0}1{ji=j2}

ji=0 j2=0 \

is the approximation of iterated Ito stochastic integral (2.357) (k = 2) based on Theorem 1.1 (see (1.46)).

Moreover, from (1.73), (1.74), and (2.8) we obtain

m { (2)]T,t - J*[^ (2)]^)= m { (J[^ (2)]T,t - J(2)]Tf)^ — 0

(2.454)

if pi,p2 — to (n G N). Further,

\ 2n

^ (j*[V'(2)]T,f jW

j1=0 j2=0

= Mi | j*[^(2)]T,t - J*[^(2)]T!tP2+

1 T min{p i,p2}

+l{ll=l2^} ( - I ■Ip1{s}ijj2{s)ds - Chh

j =0

< K„ i m 4 i J*[^(2)]T,t - J*[^(2)]Txf2

1 T min{p i,p2}

+ l{ii=i2^o} ( 2 / E ^ ' (2.455)

t j i=0

where constant Kn < to depends on n.

Taking into account (2.450), (2.454), (2.455), we get

{/ Pi P2 \2n>|

-EE jj =o. (2.456)

V j1=0 j2=0 / J

Thus, we obtain the following theorem.

Theorem 2.16. Suppose that {0j(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, fa (t), (t) are continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral (2.356) of multiplicity 2

* T * ¿2

J[fa(2)]T,t = J fa(^ fal(t1)dwt(;i)dwt(;2) (i!,i2 = 0, 1, . . . , m) t t the following expansion

to

j*[fa(2)]T,t = E jzj;i)zj

-(H) Z(«2) j2

jl ,j2=0

that converges in the mean of degree 2n, n G N (see (2.456)) is valid, where the Fourier coefficient Cj is defined by (2.444) and

T

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT^ = f^ are independent standard Wiener processes (i = 1,..., m) and wT0) = t.

Let us consider some other approaches close to the approaches outlined in this section.

Now we turn to multiple trigonometric Fourier series converging almost everywhere. Let us formulate the well known result from the theory of multiple trigonometric Fourier series.

Proposition 2.3 [113]. Suppose that

J 1/(X1,. . . )| (log+1/(x1, ... )|)k log+log+|/(X1,. .. )|x

[0,2n]k

x dxi . ..dxk < oo. (2.457)

Then, for the square partial sums

p P k

... n j (xi) ji=0 jfc=0 l=1

of the multiple trigonometric Fourier series we have

P P k

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lim E ...ji II (x/) = f (xi'---,xk)

p

ji=0 jfc=0 l=1

almost everywhere in [0, 2n]k, where {fa(x)}o=0 is a complete orthonormal system of trigonometric functions in the space L2 ([0, 2n]), log+x = logmax{1, x},

/k

f (xi,... ,Xk) j (xi )dxi.. .dxk

7 = 1

[0,2n]k l=1

is the Fourier coefficient of the function f (x1;..., xk).

Note that Proposition 2.3 can be reformulated for [t, T]k instead of [0, 2n]k. If we tried to apply Proposition 2.3 in the proof of Theorem 2.10, then we would encounter the following difficulties. Note that the right-hand side of (2.433) contains multiple integrals over hypercubes of various dimensions, namely over hypercubes [t,T]k, [t,T]k-1, etc. Obviously, the convergence almost everywhere in [t,T]k does not mean the convergence almost everywhere in [t, T]k 1, [t, T]k 2, etc. This means that we could not apply the Lebesgue's Dominated Convergence Theorem in the proof of Theorem 2.13 and thus we could not complete the proof of Theorem 2.10 (even provided the multiple Fourier series is bounded on [t,T]k). Although multiple series are more convenient in terms of approximation than iterated series as in Theorem 2.10.

Suppose that the conditions of Theorem 2.16 are fulfilled. In the proof of Theorem 2.2 (see (2.58)) we deduced that

ni n2 1

lim = -mhyuti) = k^tuh), ti g (i,T),

jl=0 j2=0

(2.458)

where Cjj is defined by (2.444).

This means that we can repeat the proof of Theorem 2.10 for the case k = 2 and apply the Lebesgue's Dominated Convergence Theorem in the formula (2.433), since Proposition 2.3 and (2.458) imply the convergence almost

everywhere in [t,T]2 and [t,T] (ti = t2 G [t, T]) of the multiple trigonometric Fourier series

p p

lim EE Cj j (*i) j (t2), ti ,t2 G [t,T]2 (2.459)

p^to ' ' ji=0 j2 =0

to the function K*(t1;t2) (the proof of the series (2.459) boundedness is presented in Sect. 2.1.2). So, we can obtain the particular case of Theorem 2.16.

Consider another possible way of the proof of Theorem 2.16, which is based on the function (2.48) and Theorem 2.10. The case ii = i2 follows from (2.453) and (2.454). Consider the case ii = i2 = 0. We have

K *(ti,t2) + K *(t2,ti) = K'(ti,t2), (2.460)

where the functions K'(ti,t2) and K*(ti,t2) are defined by (2.48) and (2.95) correspondingly. Note that the function K'(ti, t2) is symmetric, i.e. K'(ti, t2) = K' (t2 ,ti).

By analogy with (2.388) we get w. p. 1

1 N-i N-i

J[K'/2]M = -l.i.m. Y E -

1

/2=0 11=0 'N-i l2-i N-i /i-r

= (ee+ee)A-v.,ri2)Af<;"Af««+

J2=0 /1=0 /1=0 /2=0, N- i

-2 +-l.i.m. ^A^^Af^

2 N^to V 11 ■

1N

= 9LLm- E E m\th,th) + K'{th,th)) Af£>Af£> + 2 N^ /2=0 /1=0 1 2

1 N-i 2 +-l.i.m. Y K'(rh,rh) (Af^Y =

2 N^to f—' V 11 /

/1=0

N-i /2 i 1 N-i 2

N^to /2=0 /1=0 1 2 2 N^to /1=0 v 1

T t2 t

/1=0

N-i /2 — i

i/jo(to) I Mt^df^df^ + ^ I Uti\Uti)dti

* T * ¿2

= / ^i(ii}dft(il)dft(;i) =f J*[^(2)]T,t, (2.461)

t t

where we used the same notations as in (2.388).

Let us expand the function K'(tl5t2)/2 into a multiple (double) Fourier-Legendre series or trigonometric Fourier series in the square [t,T]2 (see (2.57))

l-K\tlM) =

T T

1 Pl P2 n. n.

= 9 lim EE/ / =

2 1 t

1 p1 P2 / T ¿2

= - lim V V \ Mt2)(f>j2(t2) MhWn{ti)dtidU+ 2 Pl,P2-^ f—' f—' I J J

jl=0 j 2=0 y t t

T T \

+ y fa (^j (t2^ ^2(tl)fai (tl)diH dt20ji (tl)0j2 (t2) =

t t2

1 Pi P2

= o lim + (2-462)

2 Pi,P2 — ^ z—' z—'

ji=0 j2=0

where the series (2.462) converges to K'(t1, t2)/2 at any inner point of the square [t,T]2 (see the proof of Theorem 2.2 for details).

In obtaining (2.462) we replaced the order of integration in the second iterated integral.

Using (2.461), (2.462), and the scheme of the proof of Theorem 2.10 (k = 2), we can obtain the following relation (the proof of the series (2.462) boundedness on the boundary of the square [t,T]2 is omitted (the case p1 = p2 = p see in Sect. 2.1.2))

( / . Pi P2 \2n>|

i M J'A2)h, - 2 E E + Cm) cfc!:1» = 0. (2.463) ^ V jl=0 j2=0 J J

Let us rewrite the sum on the left-hand side of (2.463) as two sums. Let us replace j with j2, j2 with j^, p1 with p2, and p2 with p1 in the second sum.

Thus, we get

Pi P2 x

lirn^ ^ (J -££ Cj2ji

j1=0 j2=0

Theorem 2.16 is proved.

Let us consider another approach. The following fact is well known [110

Proposition 2.4. Let {xni,

r...n=i be

a multi-index sequence and let

there exists the limit

lim xninfc < ro.

ni,... , n k ^ro

Moreover, let there exists the limit

lim Xni,...,nk = yni,... , nk-i < ro for any ni,... ,nk-i.

nk ^ro

Then there exists the iterated limit

lim lim x

and moreover,

ni,...,nk ni,...,nk-i^ro nk^ro

lim lim Xni,...,nk = lim Xni,...,nk •

ni,...,nk-i^ro nk^ro ni,...,nk^ro

Denote

Cjs...j1 (ts+i,... ,tk )= i K (ti,...,tk) n (t/)dti ...dts,

/=1

[t,T]s /=i

where s = 1,..., k - 1 and K (ti,... ,tk) is defined by (1.6). For s = k we suppose that Cjfc. ..j is defined by (11.81).

Consider the following Fourier series

p1 p2

lim £ £ Cj2j1 (t3,..., t*) j (ti) j (t2), (2.464)

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P1,P2 ^TO z-' z-'

j1=0 j2=0

Pi P2 P3

lim^ Ê Ê Ê Cji (t4,..., tk ) j (ti)0j2 fa) j (ta), (2.465)

ji=0 j2=0 j3=0

Pi Pfc-i

lim E--- E Cjfc-i...ji (tk )j (ti) --j (tk-i), (2.466)

ji=0 jfc-i=0

Pi Pk

lim E - - - E Cjk...ji j(ti)... j (tk), (2.467)

ji=0 jk=0

where t1, - - -, tk £ [t,T], {fa(x)}°=0 is a complete orthonormal system of Leg-endre polynomials or trigonometric functions in the space L2([t,T]).

The author does not know the answer to the question on the existence of limits (2.464) (2.467) even for the case p1 = - - - = pk and trigonometric Fourier series. Obviously, at least for the case k = 2 and ^1(r), ^2(t) = 1 the answere to the above question is positive for the Fourier-Legendre series as well as for the trigonometric Fourier series.

If we suppose that the limits (2.464)-(2.467) exist, then combining Proposition 2.4 and the proof of Theorem 2.11, we obtain

to

K *(ti,---,tk ) = E Cji (t2,---,tk ) j (ti) =

ji=0

to to

= E E j(ts, - - -, tk) j(ti) j(t2) = (2.468)

ji=0 j2=0

Pi P2

= lim E ECj'2ji (ts,---,tk ) j (ti)0j2 (t2) = ji=0 j2=0

Pi P2 TO

lim E E E Cj3j2ji (t4, - - - , tk) j (ti)0j2 (t2)0j3 (ts) =

Tin—^rsr, * * * * * *

Pi,P2—TO

ji=0 j2=0 j3=0

Pi P2 P3

li m EE ECji (t4,---,tk) j (ti)0j-2 fe) j (ts) = (2.469) ,p

Pi,P2,P3—TO „ . „ . „

ji=0 j2=0 j3=0

TO TO TO

E E E Cj3j2ji(t4, - - -, tk) j (ti) j (t2)0j-3(ts) = (2.470)

ji=0 j2=0 j3=0

Pi P2 P3 TO

li m EEE ECj4...ji (t5,---,tk)0ji (ti) j (t4) = (2.471) ip

Pi,P2,P3—TO ^

ji=0 j2=0 j3=0 j4=0

P1 Pk

= lim £ ...£Cjfc .j j (ti) ...j (tk). (2.472)

j1=0 jk=0

Note that the transition from (2.469) to (2.470) is based on (2.468) and the proof of Theorem 2.11. The transition from (2.470) to (2.471) is based on (2.469) and the proof of Theorem 2.11.

Using (2.472), we could get the version of Theorem 2.10 with multiple series instead of iterated ones (see Hypothesis 2.3, Sect. 2.5).

2.4.3 Refinement of Theorems 2.10 and 2.14 for Iterated Stratonovich Stochastic Integrals of Multiplicities 2 and 3 (¿1,22,23 = 1,... ,m). The Case of Mean-Square Convergence

In this section, it will be shown that the upper limits in Theorems 2.10 and 2.14 (the cases k = 2, k = 3 and n = 1) can be replaced by the usual limits.

Theorem 2.17 [33]. Suppose that every ^(t) (l = 1, 2,3) is a continuously differentiate function at the interval [t,T] and (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, the iterated Stratonovich stochastic integrals J*[^(2) and J*[^(3)]Tvi (i1,22,23 = 1,...,m) defined by (2.356) are expanded into the converging in the mean-square sense iterated series

lim lim m

pl—TO p2—^TO

J *[^(2)]

Pl P2

EEc

jl=0 j2=0

C(i0C(i2) j2 jl Cjl Zj2

= 0, (2.473)

lim lim m

p2—TO pl — TO

J *[^(2)]

P2 Pl

C (il) C (i2)

j2=0 jl=0

j2

= 0, (2.474)

lim lim lim m

pl—TO p2 — TO P3—TO

lim lim lim m

ps—TO p2 — TO pl—TO

pi p2 p3

C(il) C(i2) C(is)

Z^ Z^ / v CjSj2jl Cjl Cj2 CjS

jl=0 j2=0 js=0

ps p2 pl

C (il) C (i2 ) C (is)

Z^Z^ / v CjSj2jl Cjl Cj2 CjS j3=0 j2=0 jl=0

= 0,

(2.475)

= 0,

(2.476)

where

T

j = J & (sf (i = 1,..., m, j = 0,1,...)

t

are independent standard Gaussian random variables for various i or j and j, Cj3j2 j are defined by (2.360) and (2.358).

Proof. We will prove the equalities (2.473) and (2.475) (the equalities (2.474) and (2.476) can be proved similarly using the expansion (2.436) instead of the expansion (2.363)).

From (2.388) we have w. p. 1

Pi P2

J* [V'(2)]T,f - £ £ j jj = J [rpiP2 ]T2t =

ji=0 j2=0

T t2 T ti

^y rPIP2 (ii,i2)dft(;i)df(;2) + y y Rpip2 (ti,t2 f dfi;i)+ t t t t

T

+ 1{h=i2} J Rpip2(ti,ti)dti, (2.477)

t

where we used the same notations as in (2.388). Uning (2.477), we obtain

T t2 T ti

m{ (j [Rpip2 ]T2t)2} = J y RU (ti ,t2)dtidt2 + J y RpiP2 (ti,t2)dt2 dti +

t t t t

/ T t2 / T \

+ 1{ii=i2} ^y y Rpip2 (ti,t2)Rpip2 (t2,ti )dtidt2 + | y Rpip2 (ti ,ti)dti J t t t

T t2 T T

= J y R2iP2 (ti, Î2)dtidt2 + J y RU (ti,t2)dtidt2 +

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t t t t2 T t2

+ 1{ii=i2} | y y Rpip2 (ti,t2)Rpip2 (t2,ti)dti dt2 + tt

TT \ / T

+ J j Rplp2 (t1,t2)Rplp2 (¿2, ¿1)^2^1 j + 1{il=i2^ J Rplp2 (¿1,^1)^1 t tl /

2

: J Rp1p2(ti,t2)dtidt2+

[t,T ]2 T t2

+ 1*=» I /fc.fc)^ (t2,ti)dtidt2 +

t t

T T \ / T x 2

+ J J Rp1p2(ti,t2)Rp1p2(t2, ti)dtidtn + 1{n=«2} ( J Rp1p2(ti,ti)dti t t2 t

= J r21p2(ti,t2)dtidt2+

[t,T ]2

/ / T \

+ 1{i1=i2} J Rp1p2(ti,t2)Rp1p2(t2, ti)dtidt2 + i J Rp1p2(ti,ti)dti

\[t,T]2 \t / 7

(2.478)

Since the integrals on the right-hand side of (2.478) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim Rp1p2(ti,t2)=0 when (ti,t2) G [t,T]2

p1—>-TO p2—)-TO

holds with accuracy up to sets of measure zero (see (2.387)).

Then, applying two times (we mean here an iterated passage to the limit lim lim ) the Lebesgue's Dominated Convergence Theorem and taking into

p1—TO p2 — TO

account (2.364), (2.365), and (2.395), we obtain

2

[t,T ]2

lim lim / R2ip2(¿1,t2)dt1dt2 = 0, (2.479)

pl—TO p2 — TO / iJliJ2

lim lim / Rplp2(t1,t2)Rplp2(t2,t1)dt1dt2 = 0, (2.480)

pl — TO p2—TO J

[t,T ]2

2

T

lim lim / Rpip2(t1,t1)dt1 = 0. (2.481)

Pi — ro P2—ro J t

The relations (12.4781)—(2.4811) imply the following equality

m<! (J[RpiP2lTztl2

Pi—ro P2—^ro

m pi—s» m{ (J 1=°

The relation (2.473) is proved.

Let us prove the relation (2.475). After replacement of the integration order in the iterated Ito stochastic integrals from (2.398) [1]-[16], [72], [111], [112] (see Chapter 3) we get w. p. 1

Pi P2 P3

J 1^(3)]T,t - Y Y Y Cj3j2 ji Cj;i = J [RP^P2P3 ]T3t =

ji=0 j2 =0 j3=0

T t3 t2

= jj l RPiP2P3 (t1,t2,t3)df(ii)df22)df(3;3) + ttt T t3 t2

+ I I I RPiP2P3 (t1 ,t3,t2)dft(i;i)dft(2;3)dft(3;2) + ttt T t3 t2

+ // j RPiP2P3 (t2 ,t1,t3)dft(i;2)dft(2;i)dft(3;3) + ttt T t3 t2

+ // j RPiP2P3 (t2,t3,t1)dft(i;3)dft(2;i)dft(3;2) + ttt T t3 t2

+ J J I RPiP2P3 (t3,t2,t1)dft(i;3)dft(2;2)dft(3;i) + ttt T t3 t2

+ I I I RPiP2P3 (t3 ,t1,t2)dft(i;2)dft(2;3)dft(3;i) + ttt

T / T

(;3)

+ 1{;i=;2^ I J RPiP2P3 (t2, t2, t3)dt^ J dft33 + tt

}(} \ (n)

+ 1{i2=i3W I / RPiP2P3 (ti , t2, t2)dt^ I +

T / T \

+ l{ii=i3^ J RPiP2P3(ts,t2,ts)dtJ dft(2i2)- (2.482)

(s)

Let us calculate the second moment of J[RPiP2P3]T t using (2.482). We have

m j fJ[RPiP2P3]TS,t) | =

T t3 t2

2

= ///( ^ RUa*(ti,t2,ts ) dtidt2dts + (2.483)

t t t \(ti, t2 ,t3) /

/ T t3 t2

+ 2 ( 1{ii=i2} J J J G]p11P2P3 (ti, t2, tS)dtidt2dtS + ttt T t3 t2

+ 1{i1=i3^^y J GP2P2P3 (tb ^ dt2dtS +

ttt T t3 t2

+ 1{i2=i3^^y J GPSP2P3(tb ^ tS)dt1dt2dtS + ttt

T t3 t2 \

+ 1{i1=i2=i3^^y J GP4P2P3 (tb ^ J +

ttt

+ I (1{ii=i2}RPiP2P3(t1,t1,tS)RPiP2P3(t2,t2,tS) + [t,T ]3

+ 1{i2 = i3}RPiP2P3 (ts, ^ (ts, ^ t2) +

+ 1{i1=i3}RP1P2P3 (t^ ^ ^^^^ (t2, tS, t2) +

+ 2 • 1{ii = «2=i3^ RP1P2P3 (t1 ,t1,tS)RPiP2P3 (ts ,t2,t2) + +RP1P2P3 (tb ^ ^^P^Pij (t2, tS, t2) +

+Rplp2ps(¿3, ¿1, ¿1)Rplp2ps(¿2, ¿3, ¿2) J J dt1dt2dt3, (2.484)

where permutation (i1,i2,i3) when summing in (2.483) are performed only in the value RplP2Ps(¿1, ¿2, ¿3) and the functions GplP2Ps(¿1, ¿2, ¿3) (2 = 1,..., 4) are defined by the following relations

Gpip3 , ¿2, ¿3) = rpiP2Ps ^b^^^p^ps fe^^H + Rplp2ps (¿1, ¿3, ¿2)Rplp2ps (¿3, ¿1, ¿2) + +Rplp2ps (¿2, ¿3,¿1) Rplp2ps (^3, ¿2, ¿1) ,

GP2P2pS , ¿2, ¿3) = Rplp2ps fe^^^p^ps fe^^H

+ Rplp2ps (^b ^ ^^p^^ (^2, ¿3, ¿1) + +Rplp2ps (^2, ¿b ^^p^^ (^3, ¿b ¿2^

GJP3P2PS , ¿2, ¿3) = Rplp2ps ^b^^^p^ps , ¿3, ¿2) + + Rplp2ps (¿2, ¿1, ¿3)Rplp2ps (¿2, ¿3, ¿1) + +Rplp2ps (^3, ¿1)Rplp2ps (^3, ¿b ¿2^

GPlps , ¿2, ¿3) = Rplp2ps fe^^^p^ps fe^^H + Rplp2ps (^b ^ ^^p^^ ¿b ^ + + Rplp2ps (¿1, ¿3, ¿2)Rplp2ps (¿2, ¿1, ¿3) + + Rplp2ps (^b ^ ^^p^^ ^ ¿0 + + Rplp2ps ¿b ^^p^ps ^ ¿0 +

+Rplp2ps(^2, ¿1)Rplp2ps(^3, ¿1, ¿2). http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal. http://diffjournal.spbu.ru/ A.395

Further (see (1.38)), T is t2

J J J i e Rpwivs(ti,t2,t3M dtidt2dt3

t t t \(tl,t2,ts)

= J R2P1P2PS(ti,t2,ta)dtidt2dt3. (2.485)

[t,T ]s

We will say that the function $(t1,t2,t3) is symmetric if

$(ti,tp,t3) = $(ti,t3,tp) = $(tp,ti ,t3) = $(ip,*3 ,ti) = = $(t3,ti,tp) = $(t3,tp ,ti).

For the symmetric function $(ti,t2,t3), we have T is t2 / \

yyyi E ^(ti,tp,t3) I dtidtpdt3 =

t t t \(tl,t2,ts) /

T is i2

= 6//y^(ti,tp,t3)dtidtp dt3 =

ttt

= y $(ti,t2,t3)dtidt2dt3. (2.486)

[t,T ]s

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The relation (2.486) implies that

T is t2

y y j §{tiMM)dtidtodU = i y $(ii,i2,i3)dMMi3. (2.487)

t t t [t,T]s

It is easy to check that the functions GP1P2Ps(t1,t2,t3) (i = 1,...,4) are symmetric. Using this property as well as (2.484), (2.485), and (2.487), we obtain

m{ J[Rpip2Ps]T3t)2} = y RU^(ti,t2,t3)dtidt2dt3+

[t,T ]s

+ 2 J ^{i1=i2}G^P2P3{ti,t2,h)dtidt2dts+

[t,T ]3

+ 1{n=i3}Gp2p2p3 (t1, t2, t3)dtidt2dt3 +

+1 {i2=i3 }Gp3p2p3 (t1, t2, t3)dt1dt2dt3 +

+1{i1=i2=i3}G.p1p2p3 (ti, t2, t3)dtidt2dt^ dtidt2 dt3+

+ I (1{i1=i2}Rp1p2p3(t1,t1,t3)Rp1p2p3(t2,t2,t3) + [t,T ]3

+ !{i2=i3}rp1P2P3 (t3, t1, t1)rP1P2P3 (t3, t2, t2) + + 1{i1=i3}Rp1p2p3 (t1, ^ ^l^p^^ (t2, t3, t2) +

+ 2 • 1{i1=i2=i3^Rp1p2p3 (t1, ^ t3)Rp1p2p3 (t3,t2,t2) + +Rp1p2p3 (tb ^ ^^p^pij (t2, t3, t2) +

+Rp1p2p3(t3, ^ ^^p^(t2, ^ t2^ ^ dt1 dt2dt3. (2.488)

Since the integrals on the right-hand side of (2.488) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim lim Rp1p2p3(t1,t2,t3)=0 when (t1,t2,t3) G [t,T]3

p1 — TO p2—TO p3 — TO

holds with accuracy up to sets of measure zero (see (2.387)).

Using (2.406) and applying three times (we mean here an iterated passage to the limit lim lim lim ) the Lebesgue's Dominated Convergence Theorem

p1 — TO p2—TO p3 — TO

in the equality (2.488), we obtain

lim lim lim m { (j[Rp1p2„3iTD > = 0.

p1—TO p2—TO p3—TO IV T,ty |

The relation (2.475) is proved. Theorem 2.17 is proved.

Developing the approach used in the proof of Theorem 2.17, we can in principle prove the following formulas

which are correct under the conditions of Theorem 2.10 for ¿i,..., = 1,..., m.

2.5 The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k E N) Based on Theorem 1.1

In this section, on the base of the presented theorems (see Sect. 1.1.3, 2.1-2.4) we formulate 3 hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity k (k E N) based on generalized multiple Fourier series converging in L2([t,T]k). The considered expansions contain only one operation of the limit transition and substantially simpler than their analogues for iterated Ito stochastic integrals (Theorem 1.1).

Taking into account (11.44) and Theorems 2.1-2.10, 2.14, and 2.17, let us formulate the following hypotheses on expansions of iterated Stratonovich stochastic integrals of multiplicity k (k E N).

Hypothesis 2.1 [8]-[16], [38]. Assume that {fa(x)j 0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of multiplicity k

* T * ¿2

fcir,, = / ■■•/ dw<;°...dw<:k> (2.489)

t t

the following expansion

p k

C:Sr,t £ Cj:..¿II j' (2.490)

ji,..jfc=0 1=1

that converges in the mean-square sense is valid, where

t t2

Cjfc ...j = J j ).. .J j (ti)dti... t t

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, i1;...,ik = 0,1,..., m,

T

j = / (s)dw<:>

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT:) = fT:) for i = 1,..., m and wT0) = t, A/ = 0 if i/ = 0 and A/ = 1 if i/ = 1,..., m (l = 1,..., k).

Hypothesis 2.1 allows to approximate the iterated Stratonovich stochastic integral I*^ 1 "^jT^. by the sum

p k

IÎ 'X'jT,t = £ M" • (2-491)

j i,--.jfe=0 /=i

where

^ m { (it '-It - I(t =0-

The integrals ('2.489) will be used in the Taylor-Stratonovich expansion (see Chapter 4). It means that the approximations (2.491) may be very useful for the construction of high-order strong numerical methods for Ito SDEs (see Chapter 4 for detail).

The expansion (2.490) contains only one operation of the limit transition and by this reason is convenient for approximation of iterated Stratonovich stochastic integrals.

Let us consider the more general hypothesis than Hypothesis 2.1.

Hypothesis 2.2 [14]-[16], [38]. Assume that (x)}°=0 is a complete or-

thonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, every ^(t) (l = 1,..., k) is an enough smooth non-random function on [t, T]. Then, for the iterated Stratonovich stochastic integral

of multiplicity k

* T * ¿2

J*[fa (k)]T,t = / fa(tk).. j fa(ti)dwi ^... )

the following expansion

p k

J ( k)iT,( = l.Mn. J] C^II Cj;') (2.492)

ji,---jfc=0 1=1

that converges in the mean-square sense is valid, where

T ¿2

Cjfc ---ji = J fa (tk (tk) ...y ^i(ti)0ji (ti)dti •••dtk t t

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, ii,..., ik = 0,1,..., m,

T

<j;) = | j (s)dw<;)

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[;) = fT;) for i = 1,..., m and w[0) = t.

Hypothesis 2.2 allows to approximate the iterated Stratonovich stochastic integral J( k)]T,t by the sum

p k

J % = E Cj.^11 zj;;,), (2.493)

ji,---jfc=0 1=i

where

Hm M | (fc)]T,t - 1^=0.

Let us consider the more general hypothesis than Hypotheses 2.1 and 2.2.

Hypothesis 2.3 [14]-[16], [38]. Assume that {fa (x)}°=0 is a complete or-

thonormal system of Legendre polynomials or trigonometric functions in the

space L2([t,T]). Moreover, every fa(t) (l = 1,..., k) is an enough smooth non-random function on [t, T]. Then, for the iterated Stratonovich stochastic integral of multiplicity k

* T * ¿2

J*[fa(k)]T,t = J fa (tk) ..J fa (ti)dwt(il)... dwi:k) t t

the following expansion

Pi Pk k

ji=0 jk=0 l=1 that converges in the mean-square sense is valid, where

T t2

Cjk...ji = J (tk) j(tk).. y fai(ti)0ji(ti)dti.. .dtk tt

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, ii,..., ik = 0,1,..., m,

T

j = y (s)dw(:) t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[:) = f^ for i = 1,..., m and wT0) = t. Let us consider the idea of the proof of Hypotheses 2.1-2.3. According to (1.10), we have

Pi Pk k

l.i.m. £ ..^Cjk^n j' = JW'(k)lT,t +

ji=0 jk =0 g=i

Pi Pk k

.m. v^ tt^. (t ) aw(:g)

+ l-i-m ^ ..^Cjk...ji l.i.m. E IK(Tig)Aw[;g) w. p. 1,

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(2.495)

ji=0 jk=0 (ii,...,ik)eGk g=i

where notations are the same as in (1.10).

From (2.495) and Theorem 2.12 it follows that

P l Pk k

J*^»]« = l.i.m. £ ...Xji nCjg'» (2.496)

ji=0 jk=0 g=l

if

[k/2]

E^ E ^lir'1

r=1 (sr ,...,s l )eAfc,r

P l Pk k

= l.i.m. El.i.m. y, n-.K)AwT;g» w.p. 1,

P "..'Pk u to N(ii.iu^i »

where notations are the same as in Theorems 1.1 and 2.12.

Note that from Theorem 1.1 for pairwise different i1,..., ik (i1,..., ik = 0,1,... ,m) we obtain (2.496) (compare (1.44) and (2.496)).

In the case p1 = ... = pk = p and (s) = 1 (l = 1,..., k) we obtain from (2.496) the statement of Hypothesis 2.1 (see (2.490)).

If p1 = ... = pk = p and every (s) (l = 1,..., k) is an enough smooth nonrandom function on [t, T], then we obtain from (2.496) the statement of Hypothesis 2.2 (see (2.492)).

In the case when every (s) (l = 1,..., k) is an enough smooth nonrandom function on [t,T] we obtain from (2.496) the statement of Hypothesis 2.3 (see (2.494)).

2.6 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 3 and 4. Combained Approach Based on Generalized Multiple and Iterated Fourier series. Another Proof of Theorems 2.8 and 2.9

In this section, we develop the approach from Sect. 2.1.3 for iterated Stratonovich stochastic integrals of multiplicities 3 and 4. We call this approach the combined approach of generalized multiple and iterated Fourier series. We consider two different parts of the expansion of iterated Stratonovich stochastic integrals. The mean-square convergence of the first part is proved on the base of generalized multiple Fourier series converging in the sense of norm in L2([t, T]k),

k = 3,4. The mean-square convergence of the second part is proved on the base of (1.46), (2.10), Parseval's equality, and generalized Fourier series converging pointwise. At that, we do not use iterated Ito stochastic integrals as a tool of the proof and directly consider iterated Stratonovich stochastic integrals.

2.6.1 Another Proof of Theorem 2.8

Let us consider (2.385) for k = 3, p1 = p2 = p3 = p, and i2, i3 = 1,..., m

p p p

J*W'(3)]T,f = ££ £ Cj cfcfcjr' + J [R^plT3! w. p. 1, (2.497)

j1=0 j2=0 j3=0

where

N-1 N-1 N- 1

J[Rppp]T3t = l.i.m. E E E rppp(tii,^12,Ti3)AfTiil)AfTi2)AfTii3),

/3=0 /2=0 /1=0

ppp

^2^3) - E E E Cj3j2jij (t0 j (t2)0j3 (t3)

ji=0 j2=0 j3 =0

3 f 1

/=1 2

1 1

+ 21{il<i2}1{i2=i3} + ^1{il=i2}1{i2=i3} ) *

Using (2.398), we obtain w. p. 1

7 [R ](3) = R(1)PPP + R(2)PPP J [Rppp]T,t = RT,t + RT,t ,

where

T t3 t2

RT,]PPP = ff J Rppp(t1, ¿2, ¿3 ) dft(1i 1) dft(2i2) dft(3i3) + t t t

T t3 t2

¡¡I Rppp(i1,i3,i2)dft(;1)dft(;3)dft(3i2)-ttt

T is ¿2

+// / RPPP(t2,t1,t3)dfi;2)dfi;i)dfi;s)+ t t t T ts t2

+// / RPPP(t2 ,t3,t1)dfi;s)dfi(;i)dft(;2)+ t t t T ts t2

+// / RPPP(t3 't2,tl)dfi;s)dfi(;2) dft(;i)+ t t t T ts t2

+ // I rppp(t3't1't2)dft(;2)dft(2is)dft(sii), t t t

T ts

R(2)ppp = 1{ti=i2=0^ J Rppp(t2, t2, t3)dt2dftsis) +

t t

T ts

+1 {ii=is=0W / Rppp(t2, t3, t2)dt2dftsi2 + tt

T ts

+ 1{i2=is=0} / / Rppp(t3, t2, t2)dt2dftsi +

tt T ts

+ 1{i2=is=0} J J Rppp(t1, t3, t3)dft1ii)dt3 + tt

T ts

+ 1{ii=is=0} J J Rppp(t3, t1, t3)dfti2)dt3 + t t T ts

+ 1{n=i2=0}^ J Rppp(t3, t3, t1)dft1is)dt3. t t

We have

m <!

J[Rppp]T30^ < 2M { (RT1iPPP)2j + 2M { (rT2)p

(2) ppp

(2.498)

Now, using standard estimates for moments of stochastic integrals [95], we obtain the following inequality

mi (RdM 21 <

T,t

T ts t2

< 6 J J J I (RPiP2Ps (t1' t2' t3))2 + (RPiP2Ps (t1,t3,t2))2 + (RpiP2Ps (t2,t1,t3))2 + ttt

222

+ (Rpip2ps (t2,t3,t1)) + (Rpip2ps (t3,t2,t1)) + (Rpip2ps (t3,t1,t2)) J dt1dt2dt3 =

6 (RPPP (t1'

t2,t3)) dt1dt2dt3.

[t,T ]s

We have

J (Rppp(t1 ,t2,t3))2 dt1 dt2dt3 =

[t,T ]s

// p p p \ 2

K*(t1, t2, t3) - E E E Cjsj2ji j (t1)-j2 (t2)—js (t3M dt1dt2 dt3 [t,T ]s V ji=0 j2=0 js=0 /

2

[t,T ]

where

// p p p . K (t1,t2,t3) - EE ECj'sj2ji j (t1 ) j (t2 )-js (t3 ) ) dt1 dt2dt3 5

s V ji=0 j2 =0 js=0 '

[^1(t1)^2(t2)^3(t3), t1 <t2 <t3 K(t1,t2,t3)=< , t1 ,t2,t3 G [t,T].

0, otherwise

So, we get

piin, m{ (rT,!!"p)2} <

/! p p p > K(¿1, ¿2, h) - E E E Cj3j2ji j (t2) j (t3 M "¿1dt2dt

\ ; _n ;__n ;__n /

(¿2 ) j (¿3 M "¿1 «¿2«t3 =

[t,T ]3

ji=0 j2=0 j3=0

= 0,

(2.499)

where K(i1?i2,i3) G L2([i,T]3).

After replacement of the integration order in the iterated Ito stochastic integrals from R2^ M* [72], [111], [112] (see Chapter 3) we obtain w. p. 1

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R

(2)ppp T,t

1

T t3 T t3

{i1=i2=0} | J J Rppp(^2, ¿2, ¿3^2dft33)+y J Rppp(i3, ¿3, il)"f](1i3)di^ + i i i T t3 T t3

+1{i2=i3=0} | y y Rppp(Î3,Î2,Î2)dÎ2"fi3il) +J y Rppp(ii,i3,i3)df](1il)di^ + ,î Î Î Î T ]3 T ]3

/ ï?

+1 =i

{¿1=i3=0} | y J Rppp(^2, ¿3, ^"^"Îf/ + y J Rppp¿1, ¿3)dft 12 "¿3 i i i i T Î1 T T

1{i1=i2=0} ( y y Rppp(^25 ¿2, ¿1)dÎ2"fii3)^y y ^^^«fir) ) + t t t t1 T Î1 T T

+ 1{i2=i3=0} I / / ^pp^b ^ + / / Rppp(^1, ¿2, ^"^"ft^ | +

.i i T tl

t t1 T T

+ 1{i1=i3=0} I / / Rppp(t2, ¿1, ¿2)"¿2 "if*2) + / / Rppp(t2, ¿15 ¿2^^"ft*2''

,t t t t1 T / T

= 1{il=*2=0} y I y Rpppfe^, ) "ft(3i3) + tt T / T

+ 1{i2=i3=0^y I y ^pp^b ^ ¿2^2 ) + tt

2

+ 1{ii=is=0W I / Rppp(t3,t2,t3)dt3 | dft22

tt

tt ppp

E E E Cjsj2jij (t2) j (t2)-js (t3) ) dt2f +

ji=0 j2 =0 js=0

T T

v2 4

ppp

t t x x '

E E E Cjsj2jij (t1)-j2 (t2)-js (t2) dt2dft(ii) + ji=0 j2 =0 js=0 /

T T ,

+ 1{n=^0} I J {^{t2=t3}Mts)Mt2)Mts)-ppp

E E E Cjsj2jij (t3)-j2 (t2)—js (t3M dt3f2) =

ji=0 j2=0 js=0

^is

{ ji=0 js=0

p p

-EE Cjsjsji j (to) dft(;i)+

/i ji=0 js=0

T

p p

4'2)

t ji=0 j2=0 T ts

^=^0} ( ^ I Uh) I Mt2)Mt2)dt2df^ -EEC^mC]:3) ) +

t t ji=0 js=0

T T

p p

+i{*2=^o} [ ^ I mm I ^2(i2)^(i2)^2^r^EE^1c];i)

t I ji=0 js=0

(-1) E E Cjij2ji j (t2)dft(

1 ^ ^ p P

pp jl=0 j3=0

1{il=i3=0} E E Cjlj3jl Cj

From the proof of Theorem 2.8 we obtain

T t3

M ; I p(2~)PPP^2' I .. I / 1

ivi s i ntj f , _ - , , , ^

tt

p p 2 p p 2 ££Cjcj33) + x^^m £ eCjlj3jlcf } +

jl=0 j3 =0 / J [ \jl=0 j3=0 /

T T

+l{i2=^0}M<J (i i faih) [ fofoWsfoW^ -

t tl p p 2

ÉÉCj-l C^M) ^ 0 (2.500)

jl=0 j3=0 /

if p ^ œ. From (C2.497)-('2.500) we obtain the expansion ((2.256). Theorem 2.8 is proved.

2.6.2 Another Proof of Theorem 2.9

Let us consider (2.385) for k = 4, p1 = ... = p4 = p, and fa(s),..., fa(s) = 1

>jc T ^ t4 ^ t3 ^ t2

t t t t pppp

¿wil1' riw;:^ rfw™ =

= £££ECi^lc^zfc1 + Jw.p. 1, (2.501)

jl=0 j2=0 j3=0 j4=0

where

J [Rpppp]j4,t =

N-1 N-1 N-1 N-1

= N.m. £ £ £ £ , , T,3, TÎ,)Aw«:l)Aw(;2»Aw(;3»Aw(;44),

Nl4=0 l3=0 l2=0 l1=0

Rpppp(tb ^ ^ t4) — K*(t1, t2, t3, t4)

pppp

Y] Y] Y] Y] Cj4jsj2ji-ji (t1 )j (t2)—js (t3 ) j (t4) , (2-502)

ji=0 j2=0 js=0 j4=0

K*(tht2, tz,U) = n ^{iKii+i} + 21it'=t'+i}

11

— I{tl<t2<i3<i4} + 2 1{il=t2<i3<i4} + 2 1{il<t2=t3<i4} +

1 1 1

+ 41{il=i2=i3<i4} + 2 1{il<t2<i3=i4} + 41{il=i2<i3=i4} +

1 1

+ ^l{tl<t2=i3=i4} + g1{il=i2=t3=i4}-

We have

7

|(4) _ V^ r>(;)PPPP

J[Rpppp]j4t ^ ^ rt)Pppp w. p. 1, (2.503)

¿=0

where

n-1 (4-1 (s-1 (2-1

RT0tpppp = l i m. Y^ y y E E ( rPPPP(t'i , T'2 , T's , T'4) X (4=0 (s=0 (2=0 (i=0 ((i,(2,(s,(4)

xAw[ii)AwTi2)AwTis)AwTi4)

'11 ''2 ''s ''4

where permutations (l1, l2, l3, l4) when summing are performed only in the expression, which is enclosed in parentheses,

N-1

RT1tPPPP = 1{ii=i2=0}l.i.m. Y rpppp(t(i ,T'i ,T(s ,T(4 )AT(i Awi;;) Aw^,

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N^to— s 4

li=ls,li = l4,ls=l4

N-1

RT2tPPPP = 1{;i=;s=0}l.i.m. Y Rpppp(T(i ,T(2 ,T(i ,T(4 )AT(i AwT;22) Aw^,

N^to— 2 4

l4,'2,'i = 0

li=l2,li=l4,l2=l4

N-1

4?ppp = 1{i1=i4=0}l.i.m. £ rpppp(t/1 ,t/2 ,t/3 ,t/1 )Ar/1 Aw^Aw^

/1 /2 /3 /1 /1 l2 l3

N—to 2 3

¡3,(2,(1 = 0 ¡1 = 2,*1=3,*2=3

N-1

RT4tPPPP = 1{i2=i3=0}l.i.m. E Rpppp(T/1 ,T/2 ,T/2 ,T/4 )AwT;1)Ar/2 Aw^

/1 /2 /2 /4

N ^TO

l4,l2,l1=0 l1=l2,l1=l4,l2=l4

N-1

R?,tpppp = 1{i2=i4=0}l.i.m. E Rpppp(T/1 ,T/2 ,T/3 ,T/2 )Aw(;1)Ar/2 Aw^,

N—to — 1 3

l3,l2,l1=0 I1=l2,l1 = l3,l2=l3

N-1

46tpppp = 1{;3=;4=0}l.i.m. E Rpppp(T/1 ,T/2 ,T/3 ,T/3 )Aw(;1)Aw(;22)Ar/3,

¡3,12,11 = 0 1 2

11=12,11 = 13,12=13

N-1

N

rt7tpppp = 1{;1=;2=0}1{;3 = i4=0} lim e rpppp(t/2 , t/2 , t/4 , t/4 )at/2 at/4 +

(4,(2=0 l2=l4

N1

+ 1{;1=;3=0} 1{;2=;4=0} lim ^ ^ Rpppp (t/2 , T/4 ,T/2 ,T/4 )at/2 at/4 +

/2 /4 /2 /4 /2

14,12 = 0 12=14

N-1

+ 1{i1 = i4=0}1{i2 = i3=0} lim E RPPPP(T/2 ,T/4 ,T/4 ,T/2 )at/2 at/4 •

/2 /4 /4 /2 /2

N ^TO

14,12 = 0 12 = 14

From (2.501) and (2.503) it follows that Theorem 2.9 will be proved if

- m (RT)' ^ i ^7.

We have (see (1.19), (1.2

T ^ ^ ^

RT0tpppp=J J J J E (Rpppp(i1.i2.i3,i4)dw«;1)dw<;2)dw<33)dw«:4)

t t t t (t1,t2,t3,t4)

where permutations (t1,t2,t3,t4) when summing are performed only in the expression, which is enclosed in parentheses.

From the other hand (see (1.24), (1.25))

T t4 ts t2

rT0)pppp = E J J J / Rpppp(t1,t2,i3,t4)dw<;i)dwi;2)dw';s)dw<;4),

(ti,t2,ts,t4) t t t t

where permutations (t1, t2, t3, t4) when summing are performed only in the values dwt(;l)dwt(;2)dwt(Ss)dwi;4). At the same time the indices near upper limits of integration in the iterated stochastic integrals are changed correspondently and if tr swapped with tq in the permutation (t1,t2,t3,t4), then ir swapped with in the permutation (i1, i2, i3, i4).

So, we obtain

T t4 ts t2

m{ (rT0)PPP^| < 24 Y J J J /(Rpppp(t1,t2,t3,t4))2 dt1dt2dt3dt4 =

(ti,t2,ts,t4) t t t t

= 24 j (Rpppp(t1, t2, t3, t4)))2 dt1dt2dt3dt4 ^ 0 [t,T ]4

if p ^ to, K*(t1,t2,t3,t4) G L2([t,T]4) (see (2.502)).

Let us consider RTt

(1 )pppp

N1

RT1tPPPP = 1{ii=i2=0}l.i.m. E Rpppp (T(i ,T(i ,T(s ,T(4 )AT(i AwT;ss) Aw^ =

N^to— s 4

'4 ,'s,'i = 0

li=ls,li=l4,ls=l4

N-1

= 1{;i=;2=0}l-i-m- V rPPPP(t'i , T'i, T's , T(4 )at'i ^^ ^^ = N^to ^ s 4

'4,'s,'i = 0 ls=l4

N-1 f 1

^^ '4,'s,'i=^ V

ls=l4

1 1 1

p

E Cj4jsj2jij (T(i)-2 (T'i) j(T(s) j(T(4) at'i Aw^ AwT;44) =

j4 ,js,j2,ji=0 /

N-1 '1

Y I-l{Th<Tl3<Tk}

ls=l4

£ Cj j (T'i ) j (T(i ) j (T(s ) j fa) AT, i AwT's AwT'4

j4,js,j2,j'i=0 /

N-1 N-1 N-1 /1

= l{n=i2^0}l-i-m. E E E o1^!^^}-

NITO (4=0 (s=0 (i=A 2

p

E Cj jj(T(i) j (T(i) j(T(s) j(T(4) AT'iAw(;ss)Aw(;44)-

j4,js ,j2,ji=0 /

N-1 N-1 /

-1{;i=;2=0}1{;s=;4=0}l-i-m^ E E 0-

NlTO (4=0 (i=0\

E Cj4 js j2 ji -ji (T'i ) j (T'i ) j (t'4 ) j (t,4 ) ) at'i at,4 j4 ,js,j2,j'i=0

T t4 ts p \

iff di1dw;ss)dw((44) - £ Cj C]ss)C]:4M +

t t t j4,js,ji =0 /

1

20

p

+ 1{;i=;2=0}1{;s=;4=0} E jj'i w p 1

j4,ji=0

When proving Theorem 2.9 we have proved that

P 1 T t2

J™ E MjM = 7 / /

piTO ^ 4

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j4,ji=0 t t

T t4 ts

LLm. £ C-,,C]«C]I4) = I// /'iii'iwi:3,rfw«r» +

j4 ,js,ji=0 t t t

T t2

J J dtidt2 w. p. 1.

tt

Then

lim M* iRT1^2 I =0.

p—y^o

T,t

Let us consider RTt

(2) pppp

N1

42ipppp = 1{:1=;3=0}l.i.m. E Gpppp(T/1 ,T/2,7/1, Tk)Arh Aw^Aw^

N—to— 2 4

14,12,11=0 11=12,11=14,12=14

N-1

= 1{:1=:3=0}l.i.m. E gpppp(t/1 ,T/2 ,T/1 ,T/4 )AT/1 AwT;22)AwT;44) =

N—to— 2 4

14,12,11=0 12=14

1 . N-1 (1 1

—to 14,12,11=^ V

12=14

p

- e C,,j (T/1 ) j (t/2 (T/1 (t/4 ) A^ Aw^ Aw^ =

j4 j3 ,j2 j1 =0 /

N-1 N-1 N-1 p

= 1{:1=:3=0}]Njm; EEE(-1) E Cj4j3j2j1 X

/4=0 /2=0 /1=0 ,4 ,3 ,2 ,j1=0

x j (T/1 (t/2 (T/1 (t/4 )At/1 Aw<: 2)Aw^4)-

iW r^V '11—' T12 — ' 'T14

N-1 N-1 p

1{:1=:3=0}1{:2=:4=0}l.i.m^ E E(-1^ E Cj4j3j2j1 X

N—to f—' ;—' . .

/4=0 /1 =0 j4j3j2j1 =0

X j (T/1 ) j K ) j (T/1 ) j (t/4 ) at/1 at/4 = p

1 V^ c Z (;2)Z (;4) +

1{:1=:3=0} / v Cj4j1 j2j1 Zj Zj4 +

j4 ,j2 j1 =0 p

+ 1{:1=:3=0}1{:2=:4=0} E Cj4j1j4j1 w. p. 1.

j4,j1 =0

When proving Theorem 2.9 we have proved that

p

Urn. £ C,^!Zfcf = 0 w. p. 1,

j4 ,j2 j1=0

lim E C,4,1,4,1 = 0 p—TO z-'

j4,j1=0

Then

TO m{ n = 0.

Let us consider RTt

(3) pppp

N-1

43ipppp = 1{:1=:4=0}l.i.m. E Gpppp(T/1 ,T/2 ,T/3 ,Th )Arh Aw^Aw^ =

N—to— 2 3

13,12,11=0 11=12,11=13,12=13

N-1

1{:1=:4=0}l.i.m. V GppppK ,T/2 ,T/3 ,T/1 )at/1 AwT:22)AwT;!) =

N—to— 2 3

l3,l2,l1=0 l2=l3

N-1 (1

—TO 13,12,11=^ V

l2=l3

e C, ,0,1 (T/1 ), (t/2 ), (t/3 ), (T/1) at/1 Aw^ Awjj)

,4 ,3 ,2 ,,1=0 /

N-1 N-1 N-1 p

= 1{:1=:4=0}Nim.. eee(_1) e ^,4,3,2,1 X

/3=0 /2=0 /1=0 ,4 ,,3 ,,2 ,,1=0 X01 (t/1 ) , (t/2 ) , (t/3 ) , (T/1 )at/1 Aw^Aw^

¿3/r.m ' ¿1/"' t1—'*T12 *T13 N-1 N-1 p

1{:1=:4=0}1{:2=:3=0}l.i.m. e e(_1) e ^4,3,2,1 X

N—to f—' ;—' . .

/3=0 /1 =0 ,4,3 ,,2,1 =0

X (t/1 ) (t/3 ) (t/3 ) (t/1 ) at/1 at/3 =

/1 /1

p

1 V^ C Z (;2)Z (;3) +

■1{:1=:4=0} / v ^,4,3,2,4 , , +

,4,,3,,2=0 p

+ 1{:1=:4=0}1{:2=:3=0} E ^4,2,2,4 w. p. 1.

,4 ,,2=0

When proving Theorem 2.9 we have proved that

LLm. E Cjjj = 0 w. p. 1,

pITO

j4 ,js,j2=0

lim Cj4j2 j2j4 = 0

p—VTO ' *

Then

p—TO z—

j4,j2=0

lim M* (I =0.

P—TO

LT,t J

Let us consider R((4tPPPP

N-1

RT4jPPPP = 1{i2=;s=0}l.i.m. E Gpppp(T(i ,T(2 ,T(2 ,T(4 )AwT;i)AT(2 Aw^

N—to , , , 0 i 4

'4 >'2 >'i = 0 'i = '2 > 'i = '4 >'2 = '4

N-1

1{;2=;s=0}l-i-m- e GPPPP(T'i , T,2 , T,2 , T,4 ^^^= 2 s N—TO ' 1 2 2 4 '1 2 '4

'4

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'1='4

N1

.m. pppp (1 (2 (2 (4

' ' ' =0

N-1 1 N2

'4 '2 '1=0

'1='4

1 1 1

+ 41{Tii=Ti2<T'4} + ~l*-{Th<Th=Th} + g

p

e Cj4jsj2jij (T(i) j (t,2) j(t,2) j(t,4) Aw^A^Aw^ =

j4 ,j's,j2,j'i=0 '

, • N-1 f 1

N2

'4 '2 'i=0 'i='4

p

E Cj4jsj2jij (T(i) j (T(2) j (T(2) j(T(4) Aw^A^Aw^ = j4,j's,j2,j'i=0 /

N-1 N-1 N-1 /1

= l{i2=^o}lfm. e e e o1^!^^}

N—TO (4=0 (2=0 (i=0\ 2

p

e c,,0,1 (t/1 )0,2 (t/2)0,3(t/2)0,4(t/4) awt;;)at/2aw^-

,4 ,,3 ^J^0 /

N-1 N-1 p

-1{:2=:3=0}1{:1=:4=0}l.i.m. >J /J(_1) ^,4,3,2,1 x

N—to f—' ;—' . .

/4=0 /2=0 ,4,,3,,2,, 1=0

X 0,1 (T/4 ) 0,2 (T/2 ) 0,3 (T/2 ) 0,4 (T/4 ) AT/2 at/4 =

p

i<«=**>> |!// jdw^diMf - V c^.cfcf ] +

t t t J4,J2,J1 =0 /

2 ' ' ' IIIOILXV^ - > K/A.A.A.A.i • 1

0

p

+ 1{:2=:3=0}1{:1=:4=0} E ^,4,2,2,4 w. p. 1.

,4,,2=0

When proving Theorem 2.9 we have proved that

p

lim E ^,4,2,2,4 = 0,

p—TO '

,4 ,,2 =0

p T t4 t2

Li.m. ^ = I///«¡wil'Wwii'' w. p. 1.

•^J^0 t t t

Then

to m{ (4rp)2} = 0-

Let us consider R^t^

N -1

45tpppp = 1{:2=:4=0}l.i.m. E Gpppp(T/1 ,t/2 ,t/3 , T/2 ^w^A^ AwT;33)

N—TO — 1 3

13,12,11=0 11=12,11=13,12=13

N-1

= 1{:2=:4=0}l.i.m. e gpppp(t/1 ,T/2 ,T/3 ,T/2 )a<1)at/2 AwT:33) =

N—to — 1 3

l3,l2,l1=0 l1=l3

N-1 (1 1

—TO 13,12,11=^ V

l1=l3

p

Y Cj4jsj2jij (T(i ) j (T(2 ) j (T(s ) j (T(2 ) ) AwT;i)AT(2 j4,js,j2,ji=0

N-1 p

N

1{;2=;4=0}l-i-m- E (-1) E Cj4jsj2jix

's '2 '1=0 j4,js,j2 ,j1=0

'1='s

x-ji (T(i ) j (T(2 ) j (T(s ) j (T(2 )AwT;i) AT(2 AwT;os)

p

= -1{;2=;4=0} yl Cj4jsj4j'i Cj; C

p

= (;i)Z (;s ) js

j4,js,j1=0

N-1 N-1 P

1{;2=;4=0}1{;1=;s=0}l.i.m. (-1) C

N—TO

(s=0 ,2 =0 j4,js,j2,ji =0 X -ji (T's ) j (T'2 ) j (T's ) j (T'2 ) AT(2 AT(s =

(2 (2

p

1 V^ C z (;i)z (;s) +

1{;2=;4=0} / v Cj4jsj4ji Zj j +

j4,j's,j'i =0 p

+ 1{;2=;4=0}1{;i=;s=0} E Cj4jij4ji w' p 1

j4 ji=0

When proving Theorem 2.9 we have proved that

p

1,——to. £ jjiCjjj = 0 w. p. 1,

j4 ,js,ji=0

lim E Cj4ji j4ji = 0.

p—TO

j4ji=0

Then

lim Mi (RT5tPPPP) I =0.

p—TO

Let us consider RTt

(6) pppp

N1

RT6tPPPP = 1{;s=;4=0}l.i.m. Y Gpppp(T(i ,T(2, t,s, t,s ^w^ Aw^ AT(s

N—TO 1 '2

's '2 '1=0 ' i =' 2,' i='s >'2='s

p

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N-1

= 1{;s=;4=0}l-i-m. E GPPPP(T'i ,T'2 ,t's ,t's^^ at's =

N^to— i 2

's '2 '1=0 '1='2

/1

N2

's '2 '1=0 'i = '2

1 1 1

+ + ^-{Th<Th=Th} + g 1 i ^ ± =T"/3 }

p

Y Cj4jsj2j ii (T( i) j (T(2) j (T(s) j (T(s) i Aw^) Aw^ AT's

j4 ,j3,j'2,j' i=0 '

f 1

N2

's '2 ' =0 ' ='2

p

E Cj4jsj2j i i (T( i )-j2 (T'2 )-js (t's ) j (t's ) ] A w(-; i) A w(-;2 ) at"(s

j4,j's,j2,j'i =0 /

p

1(^,^0)1 dw<:'»«iw«f- v C3№J2J1C«;;')C]:2)

j4 ,j2,j' i =0

N-1 N-1 p

-1{;s=;4=0}1{; i=;2=0}l-i-m- y E(-1) y Cj

N—TO

(s=0 ( i =0 j4,js,j2,j i =0

X i (T( i ) j (T( i ) j (T's ) j (T's ) AT( i AT(s =

^P

j4,j2,j' i=0

p

Cj4j4j j

+1{i =i2=0}1{is=i4=0} Cj

j4j i=0

^T ts

dw^dw^dt3 + -l{n=i2^0} / / dtidts-

tt

(i ) (i2)

Ec z(; i )z (;2M +

Cj4j4j2j' i Sj i zj2 J +

j4,j2,j' i=0

p

/ p T t3

+1{*i=*2^o}l{i3=M^o} ( E ('j-.ij-.i- J dtldt^ 1 W" P-

V'4j1=0 t t

When proving Theorem 2.9 we have proved that

p 1 T t3

S = 4 / /

,4,,1=0 t t

p T t3 t2

_ 1 / / /

,4 ,,2,, 1 =0 t t t

T t3

l.Lm. V -III

J J dtidt3 w. p. 1.

tt

Then

.—TO 4 (RT6ippp01 = 0-

Finally, let us consider R(Jtpppp

N-1

(7) pppp

RT,tpppp = 1{:1=:2=0}1{:3=:4=Q}Li.:m. ^ Gpppp^ , T/2 , T/4 , T/4)at/2 at/4 +

N

14,12=0

l2=l4

N-1

+ 1{:1=:3=0}1{:2=:4=0}l.i.m. E Gpppp(T/2 ,T/4 , T/2 , T/4 )at/2 at/4 +

N

l4,l2=0 l2=l4

N-1

+ 1{:1=:4=0}1{:2=:3=0}l.i.m. gpppp(t/2 , T/4 , T/4 ,T/2 )at/2 at/4 =

N

l4,l2=0 l2=l4

N-1 N-1

= 1{:1=:2=0}1{:3=:4=0}Li.m. E E^ppppK ,T/2 ,T/4 , T/4 )at/2 at/4 +

N—TO /4=0 /2=0

N-1 N-1

+ 1{:1=:3=0}1{:2=:4=0}l.i.m. E E^ppppK ,T/4 , T/2 ,T/4)at/2 at/4 +

N—TO /4=0 /2=0

N-1N-1

+ 1{:1=:4=0}1{:2=:3=0}l.i.m. e e gpppp(t/2 , T/4 ,T/4 ,T/2 )at/2 au

/2 5 ' /4) ' /4) ' i2/" ' i2" '/4

N—TO /4=0 /2=0

N-1 N-1 1 1

/4=0 /2=0

E ,2,1 0,1 (t/2 ) 0,2 (t/2 ) 0,3 (t/4 ) 0,4 (t/4 ) ) at/2 at/4 +

J4J3J2J1 =0

N-1 N-1 1

TO /4=0 /2=0 V ,4,,3,,2,,1=0

X 0,1 (T/2 ) 0,2 (T/4 ) 0,3 (T/2 ) 0,4 (T/4 ^ AT/2 At/4 +

N-1 N-1 1 p

+ E E 1 o 1{n0=Tli} - E f ./1./. ./ X

/4=0 /2=0 V ,4 ,,3 ,,2 ,,1=0

X 0,1 (T/2 ) 0,2 (T/4 ) 0,3 (T/4 ) 0,4 (T/2 ) AT/2 At/4 =

1

40

T t4 p \

j j dt2 dt4 - E CJ4J4J1J1 )

t t J4,J1=0 /

p

C,4,1,4,1

1{:1=:3=0}1{:2=:4=0} E C,

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,4,,1=0 p

1{:1=:4=0}1{:2=:3=0} ^^ ,2,4 •

,4 ,,2=0

When proving Theorem 2.9 we have proved that

p 1 T t4

Ji™ S = 4 / /

,4,,1=0 t t

p

lim E CJ4J1J4J1 = 0,

p—TO z-'

p

lim y j j2j4 — 0. p—f

Then

p^to z—

j4,j2=0

lim RT7tPPPP — 0.

Theorem 2.9 is proved.

2.7 Modification of Theorems 2.2, 2.8, and 2.9 for the Case of Integration Interval [t, s] (s E (t, T]) of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 4 and Wong—Zakai Type Theorems

2.7.1 Modification of Theorem 2.2 for the Case of Integration Interval [t, s] (s E (t,T]) of Iterated Stratonovich Stochastic Integrals

of Multiplicity 2

Let us prove the following theorem.

Theorem 2.18 [32]. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, ^i(t), ^2(t) are continuously differentiate functions on [t,T]. Then, for the iterated Stratonovich stochastic integral

J*[^(2)kt — J ^2(t2)J (ti)dfi;i)dft(2i2) (i1,i2 — 1,...,m)

i t

the following expansion

p1 p2

J*^2)].,,f — l.i.m. £ Mjj (2.504)

jl =0 j2=0

that converges in the mean-square sense is valid, where s E (t,T] (s is fixed),

S ¿2

Cjj (s) = I Mh (t2 ) / (ti )dtidt2, (2.505)

and

T

j = J (Tf

t

are independent standard Gaussian random variables for various i or j.

Proof. The case s = T is considered in Theorem 2.2. Below we consider the case s G (t,T). In accordance to the standard relations between Stratonovich and Itô stochastic integrals (see (2.4) and (2.5)) we have w. p. 1

s

J*№i2)]s,t = J[f1]U + I MMMMdh, (2.506)

t

where s G (t,T) (s is fixed), 1A is the indicator of the set A. From the other side according to (1.252), we have

Pi „2

/ / C?2?1 I Z

J l^(2)]s.t = Ji^ E E j W CfC'r - 1(.i=,2}1{ji=j2}

jl=0 j2=0

pi „2 min{pi,„2}

= i.i.m. E wcjij - i^} „m E j(«)• (2-507)

ji=0 j2=0 ji=0

From (2.506) and (2.507) it follows that Theorem 2.18 will be proved if

1 " œ

. MMMMdt^Yc^s). (2.508)

t ji=0

Note that the existence of the limit on the right-hand side of (2.508) for s = T is proved in Sect. 2.1.2 (Lemma 2.2). The case s E (t,T) can be considered by analogy with the proof of Lemma 2.2 (see (2.623), (2.624) for details).

To prove (2.508), we multiply the equality (2.12) by the function

1{t2<s} + 21{i2=s}> t-2 € [t;T],

where s E (t,T) (s is fixed). So we have

K*{ti,t2) ^l{i2<s} + =*}) = ¿^1(^2) (^1{i2<4 + MM,

J1 (2.509)

where ti = t, T,

IC(tut2) = faitfafaiU) ( Mt^u} + lMt1=t2} ), tut2 g [t,T\,

t2

Cj(t2)= fa(t2) y fa(ti)j(ti)dti.

t

The function

QiM (l{i3<-} + ¿l{f3=«}) , G [i, T]

has the same structure as the function K*(ti;t2). Then, by analogy with (2.12), we get

Cjdh) fl{i2<s} + il{t2=s}) ^(fa.(s)<;>,(/2). (2.510)

V 2 J j2=0

where t2 = t,T and the Fourier coefficient Cj2j1 (s) is defined by (2.505). Let us substitute (2.510) into

(1 \ œ œ

' j1=0 j2=0

where (t1,t2) G (t,T)2.

Note that the series on the right-hand side of (2.511) converges at the boundary of the square [t, T]2.

It is easy to see that substituting t1 = t2 in (2.511), we obtain 1 / 1 \ TO TO

^ ' ,1=0,2=0

where t1 = t, T. Denote

Rpip2{tl,t2,s) = K*{ti,t2) ^l{i2<s} + -l{i2=s}^ -p1 p2

EE^ (s)0,1 (t1)0,2 (t2), (2.513)

,1=0 ,2=0

where p ,p2 < œ. Then

RPlP2{ti,ti,s) = i 1{îi<s} + ^1{îi=s}

P1 P2 j1=0 j2=0

Note that

T s

t

T

P1 P2 »

j1=0 j2=0 t

s

- / '0l(Îl)'02(Îl)rfÎl ('S'!1:./ ./Vi =

i j1=0 j2=0

= 2 MMUMdh- È (2.514)

t j1=0

Using (2.514), we obtain

T

1 ^ œ

lim lim / = - / -0i(ii)'02(ii)dii- E ^W5)- (2-515)

Pl^TO / 2 / z—'

t t jl=0

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The equality (2.515) means that Theorem 2.18 will be proved if

T

lim lim / RPlP2 (t1,t1,s)dt1 = 0.

t

Since the integral

T

J Rp1p2(ti,ti,s)dti

t

exists as Riemann integral, then it is equal to the corresponding Lebesgue integral. Moreover, the following equality

lim lim Rp1p2 (t1 ,t1,s) = 0 when t1 G [t,T]

p1—TO p2 — TO

holds with accuracy up to sets of measure zero (see (2.512)). We have

RPlP2(ii,i2, s) = (K*(ti,t2) il{i2<s} + 21{^2=4

YlCfaU) (l{t2<s} + il{i2=s}) +

2

,1=0

+E C,1 fe) 1{t 2<s f + 21{^=-s> ) ~ S (°j-a■>) J (2.516)

,1=0 2 ,2=0

Let us substitute t1 = t2 into (2.516)

+E C,1 fa) 1{t

+ 21iii=s> ) -^ChjM^hiti) ] (2.517)

,1=0 2 ,2=0

Applying two times (we mean here an iterated passage to the limit lim lim ) the Lebesgue's Dominated Convergence Theorem and taking into

p1—TO p2 — TO

account (2.509), (2.510), and (2.513) for the case t1 = t2, we obtain

T

lim lim / Rp1p2 (t1,t1,s)dt1 = 0.

p1 — TO p2—TO J t

Theorem 2.18 is proved.

Let us reformulate Theorem 2.18 in terms on the convergence of the solution of system of ordinary differential equations (ODEs) to the solution of system of Stratonovich SDEs (the so-called Wong-Zakai type theorem).

By analogy with (2.977) for k = 2, = 1,...,m, and s £ (t,T] (s is fixed) we obtain

s p: P2

J u^j 0i(ti)dfi;i)pi dft(;2)p2 = EE Cj2ji (s)cj;i)c]22), (2.518) t t ji=0 j2=0

where p1;p2 £ N and dfT;)p is defined by (12.9741); another notations are the same as in Theorem 2.18.

The iterated Riemann-Stiltjes integrals

s t2 s

v;(;i;2)pip2 = /^2(t2)i^i(ti)dft(:;',p'dft22)p2, xS;t)p' = /^i(ti)df«;')pi

are the solution of the following system of ODEs

'dys(;ti;2)pip2 = 02(s)xs(;^)pidfs(;2)p2, yt,;i;2)pip2 = 0

dxs(;ti)pi = ^i(s)dfi;i)pi, xt(;i)pi = o

From the other hand, the iterated Stratonovich stochastic integrals

* S ^ t2 >jc S

Yir2) = / ^i(ii)dft(;,)dft22), xs;ti) = / wtof;■>

are the solution of the following system of Stratonovich SDEs

dY'r' = fcwxjj' * df<!2), =0

dX'j' = Vi(s) * dfi!l), X<;('> = 0

where * dfs(i), i = 1,..., m is the Stratonovich differential.

Then from Theorem 2.18 and (1.251) we obtain the following theorem.

Theorem 2.19 [32]. Suppose that {j (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, ^i(t), ^2(t) are continuously differentiate functions on [t, T]. Then for any fixed s (s £ (t, T])

l.i.m. Ys(;ii2)pip2 = Ys(i1i2), Xs(itl)pi = l.i.m. X^1).

2.7.2 Modification of Theorem 2.8 for the Case of Integration Interval [t, s] (s £ (t,T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 3

Let us prove the following theorem.

Theorem 2.20 [32]. Suppose that {j(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). At the same time ^2(t) is a continuously differentiable nonrandom function on [t,T] and (t), ^3(t) are twice continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* S * is * ¿2

J*[^(3)kt = y ^(ta)/ «t2)J (tif<i2)fis) (¿i,i2,i3 = i,...,m)

t t t

the following expansion

p

j><»>].,,=1^. £ c^, (s)cj:;',cj;2)c<33)

that converges in the mean-square sense is valid, where s £ (£,T] (s is fixed),

S t3 t2

Cj3j2ji (s) = ^3(£3) j (t3M ^2(£2) j (t2M (tl)0j, (ti)dtidt2dt3

and

T

j = / j (t )dfi"

t

are independent standard Gaussian random variables for various i or j.

Proof. The case s = T is considered in Theorem 2.8. Below we consider the case s G (t,T). First, let us consider the case of Legendre polynomials. From (1.253) for the case pi = p2 = p3 = p and standard relations between Itô and Stratonovich stochastic integrals we conclude that Theorem 2.20 will be proved if w. p. 1

p p I

^ = g I T) I 'Hsi)Usi)dSldf^\ (2.519)

ji=0 j3=0

s

p p 1 n

^ S E = 2 j Mr}h(r) j Msi)dî^dr, (2.520)

ji=0 j3=0 t t

p p

l.i,m. Cj1JSJ1 (s)cj32) = 0. (2.521)

ji=0 j3=0

The proof of the formulas (2.519), (2.521) is absolutely similar to the proof of the formulas (2.257), (2.259). It is only necessary to replace the interval of integration [t,T] by [t, s] in the proof of the formulas (2.257), (2.259) and use Theorem 1.11 instead of Theorem 1.1. Also in the case (2.521) it is necessary to use the estimate (1.209).

Let us prove (2.520). Using Theorem 1.11 for k = 2 (see (1.252) for ¿i = 1,..., m, = 0), we obtain w. p. 1 (also see (2.668), (2.661

s T p

- / Ur)Ur) / fa(sl)df^dT = - LLm. E^iWji t t p ^ ji=0

where

Ci (s) = )fa(r) fa(si)j (si)dsidr =

ji

tt s s

We have

= J fa(si)j(si) J ^a(r)02(r)drdsi. (2.522)

t si

p p p 2

I / * r ï *

=M i ( E E ^.wcj;1' -, E ^Mci:''

Vji=0 j3=0 ji=0

s

2

p / p \ \ 2

M^EIE^w-^wJci:0

ji =0 \js=0

p p i 2 ji=0 Vs=0 /

s e t

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Cjsjsji (s)=/ (0) J MT )js (t ^ ^i(si )jji (si)dsidT^ =

t t t

= J ^i(si) j(si^ ^(t) j(t)J j(0)d»dTdsi. (2.524)

t Si t

From (2.522)-(2.524) we obtain

p / s / p s s Ep (s) = £ i ^i(si)0ji (si) £ i Mt ) j (t ) / «0) j (^t -

ji=0 V t \js=0 si T

~\f Ur)Hr)dr | | . (2.525)

si

Note that, by virtue of the additivity property of the integral, we have

s s

J ^2(T) j (t)J ^(0) j (^T =

si T

s e

= / ^3(0) j (V)J «t) j (T)dTd^-

tt s i e

-j «0) j (0)/ mt ) j (t )dTd^-

tt

s si

- / «0) j(0)^/ «t) j(t)dT. (2.526)

si t

s

Using (2.508), (2.526) and the generalized Parseval equality, we get

E / «t) j(t) I ^(0) j(0)dM

js=0

T =

si

e i ^(0) j (0)1 ^2 (t) j (t)dtdtf

js=0 t t

si

E I ^(0) j(0)1 Mt) j(t)dTd0

js =0 t t

si

e i «0) j(0)d0 / ^2(t) j(T)d

js=0

T =

si

s si

= 2 / ■fa{T}fa{T)dT - - / '02(T)'03(T)rfr-

T

t

T

T =

E I 1{si<e<s}^3(0)jjs (0)d0 / 1{T<si}^2 (t) j (T)d js =0 t t

s T

i J i/jo{r)h{r)dr - J l{si<T<syilj3{r)l{T<Slyilj2{r)dr =

si t

= o / 'ihiMhMdr.

2

(2.527)

si

Combining (2.525) and (2.527), we obtain

Ep(s) = El / «si) j (si) E i ^2(t ) j (t ) / «0) j (0)d0dTdsi )

<

ji=0

j s =p+i

si

< K E ( f |j(si)|

ji=0 V ,

oo

E i Mt) j(t) / «0) j(0)d0d

js =p+i

T

si

dsi

(2.528)

s

e

s

e

s

2

s

where constant K does not depend on p. Let us estimate the value

e i ^(t) j (T) / (^

j3=p+1

t

si

We have (see (2.526

e I Mr) j(t) I (^d

j3=p+1

r

si

<

oo

E J (0)J Mr) j(t

j3=p+1 t t

+

s1

E / j № / «r) j (r)drd^

j3=p+1 { {

+

+ E

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j3=p+1

s1

s1 t

Applying the estimate (2.623) (see Sect. 2.9), we can write

oo

E Cj'i(s) ji=p+1

<-11 1

-pi + (1 - {z{s))2)l,A

(2.529)

(2.530)

where s G (t,T), constant C does not depend on p, z(s) has the form (2.20), and Cjj(s) is defined by (2.505) for the case j = j2.

Applying the estimates (1.209), (2.278), (2.530) to the right-hand side of (2.529) gives

e / «r) j(tW faWj(^dr

j3=p+1

s1

< L / 1

P\ +(1-W«i))s)1/4

X

X 1 +

+

(1 - (z(s))2)1/4 (1 - (z(s1))2)

^1/4 '

(2.531)

where s, s1 G (t,T) and constant L is independent of p.

s

e

s

e

s

1

Combining the estimates (2.142), (2.528), and

L(s)p L(s)

l, we finally obtain

EP(s) <

- p2

p

if p —to, where constant L(s) (s is fixed, s G (t,T)) does not depend on p. The relation (2.520) is proved for the polynomial case. Theorem 2.20 is proved for the case of Legendre polynomials.

For the trigonometric case, by analogy with the proof of Lemma 2.2 (Sect. 2.1.2), we obtain the following analog of (2.530)

<

C

p

(2.532)

si

<

C j '

<j OVO), (2.533)

E Cjiji(s) ji=p+i

where s G [t, T], constant C does not depend on p, and Cj (s) is defined by (2.505) for the case ji = j2.

Note the following obvious estimates for the trigonometric case

si

J (t)dT

t

where s,s1 G [t, T], constant C does not depend on p.

Applying (2.528), (2.529), (2.532), and (2.533), we obtain the assertion of Theorem 2.20 for the trigonometric case. Theorem 2.20 is proved.

Let us reformulate Theorem 2.20 in terms on the convergence of the solution of system of ODEs to the solution of system of Stratonovich SDEs (the so-called Wong-Zakai type theorem).

By analogy with (2.977) for the case k = 3, p1 = p2 = p3 = p, i1,i2,i3 = 1,..., m, and s G (t, T] (s is fixed) we obtain

s t3 t2 p

J*fe)/*fe) j*(ti)<>p4;2f)p = £ Cj3j2j,(s)c«:i)cj22 j

t t t ji'j2 ,j3=0

(2.534)

where p G N and dfT:)p is defined by in Theorem 2.20.

The iterated Riemann-Stiltjes integrals

S t3 t2

'(;i:2:3)p

); another notations are the same as

= I *3(t3) / *2(i2W *1 (tl}dft(i:i)pdfi(^2)pdft(3:3)p,

S ¿2

Ylr" = j Mh)J )<)p df<22)p,

t t s

xSit1 )p = J ^i(ii)dft(,")p

t

are the solution of the following system of ODEs

' dzS;ri3)p = fa (s) Ys(,tli2)p f3)p, Zt(,ri3)p = 0

< dYS(,t1i2)p = fa^xS^f2)p, Yt;t1i2)p = 0 .

k = fa(s)f(:i)p, Xt(il)p = 0

From the other hand, the iterated Stratonovich stochastic integrals

* s * t3 * t2

zi:ris) = / ut3)J ut2) j v-i(ti

t t t * s >jc t2

Y(,;i'2)=j Mh)f ^i(ti)dft(:i)dft(:2),

tt

xi:,i) = / ^(tf

t

are the solution of the following system of Stratonovich SDEs

'dZ^2*0 = fa(s)YS(,ti:2) * dfs(:3), Zt(;i:2:3) = 0

< dYs(;ti:2) = fa(s)Xs(;t° * f2), Y^ =0 , ^ dXiji = fa(s) * dfs(:i), X^ = 0

where * dfs(:), i = 1,..., m is the Stratonovich differential.

Then from Theorems 2.19 and 2.20 we obtain the following theorem.

Theorem 2.21 [32]. Suppose that {fa (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). At the same time fa(T) is a continuously differentiable nonrandom function on [t,T] and fa(t), fa(r) are twice continuously differentiable nonrandom functions on [t, T]. Then for any fixed s (s G (t, T])

l.i.m. Z^2^ = Z^2^, l.i.m. Ys(;ii2)p = Ys(;ii2),

p^œ ' ' p^œ ' '

xiil)p = l.i.m. xSit>.

2.7.3 Modification of Theorem 2.9 for the Case of Integration Interval [t, s] (s G (t,T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 4

Let us prove the following theorem.

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Theorem 2.22 [32]. Suppose that {fa-(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

* S ^ ¿4 ^ ¿3 ^ ¿2

J*[^(4)]s,t = /// / dw^dw^dw^dw™ (M2,is,i4 = 0,1,...,m) t t t t the following expansion

p

J*^4']" = ^ E jiwjjZj33)Zj44)

j1 J2,j3 j4=0

that converges in the mean-square sense is valid, where s G (t,T] (s is fixed),

s S4 S3 S2

Cj4j3j2j1 (s)^/ faj4 y fa3 (s3^ fa2 (s2^ faj1 (sl)dslds2ds3ds4 tttt

and

T

Zj:) = / fa (t)dwT:) t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wT:) = fT:) for i = 1,..., m and wT0) = t.

Proof. The case s = T is considered in Theorem 2.9. Below we consider the case s G (t, T). The relation (1.254) (in the case when p1 = ... = p4 = p — to) implies that

p

Urn. £ ji (s)Cj:',C«22,C«33)Cjr) = J [*(4)]s,t+

p—TO

ji,j2,j3,j4=0 I (:3:4) / \ | ^ ^ /l(:2:4)

+ 1{:i=,2=0}A1' 3!" (s) + 1{,i=,3=0!A2 2"< (s) + 1{:,=,4=0}A3'2!3)(s) + 1{,2=:3=0}A4 i'4' (s) +

[(hm).

+ 1{:2=:4=0}A5:i:3)(s) + 1{:3=:4=0}A6 ^ (s) - 1{:i=:2=0} 1{:3=:4=0}B1 (s)

(:i:2)

-

1{:i=:3=0!1{:2=:4=0!B2(s) - 1{:i=:4=0} 1{:2=:3=0}B3(s) ,

(2.535)

where J[*(4)]s,t has the form (11.2,35) for *1(t),... ,*4(t) = 1 and i1,.. 0, 1, . . . , m,

. ,i4 =

A1:3:4)(s

A2:2:4)(s

A3:2:3)(s

A4:i:4)(s

A5:i:3)(s

= y—to. £ Cj4j3jijiwc№

j4,j3ji=0 p

= li—TO. £ c j(s)cj:2,cj44),

j4,j3,j2=0 p

= l;.Lim: E Cj(s)Cj::)Cj,33,,

P—TO z

j4j3j2=0

= liS; E CjMC^C"^

p—TO z

j4,j3ji=0

= li^ E Cj(s)Cj:')Cj(33),

p—TO

j4,j3ji=0 p

AT^LLm. £ Cj (s)Cjl;',Cj(:2),

p—TO ' * J i J2

j3j2,j'i=0

p p

B1(s)= lim ^ Cj4j4jiji (s), B2(s)= lim ^ Cj3j4j3j4

p—TO ' * p—TO ' *

(s),

ji ,j4=0

j4,j3=0

P

B3(s) = lim E j3334 (s). p —> œ z—'

34,33=0

Using the integration order replacement in Riemann integrals, Theorem 1.11 for k = 2 (see (1.252)) and (2.508), Parseval's equality and the integration order replacement technique for Ito stochastic integrals (see Chapter 3) [1]-[16], 72], [111], [112] or Ito's formula, we obtain (see the derivation of the formula

(2.287))

S T Si

= 1 [ [ [ d.Sodw^dw^ +

2 Si

ttt

+ jl{i3=i4*>} [(Si - t)dSl - Ap4)(.s) w. p. 1, (2.536)

where

p

A1:3i4)(s) = l.i.m. V 3 (s)C1(:3)C(:4),

1 V > p—œ / J 3433 v > 3 3 ' 33,34=0

S T / S i

i

lp ' s ^ ./ :./:■■1 ' 2

Î Î 3'i=p+M t

ÏÏ / ^hir) / fa3(-Sl) E / (2-537)

Let us consider A2i2i4)(s) (see the derivation of the formula (12.2891))

where

2

a2:2:4)(s) = -A2:2:4)(s) + A1:2:4)(s) + A3:2:4)(s) w. p. 1, (2.538)

p

a2:2:4)(s) = l.i.m. y 3 (s)c(:2)c!:4),

2 V ' p — œ / J 3432 V 2 3 3 '

34,32=0 p

A3:2:4)(s) = l.i.m. y 3 (s)C1(:2)Z!:4),

3 p œ 3432 32 34

34,32=0

2

S „ / t \ 2 T

= ^JMt) E J (f)j2{si)dsidT,

1

S, = 2.

t 33=p+1 V t / t

ST

1œ

t { 33=P+1 VS;

p

S

2

Let us consider A5:i:3)(s) (see the derivation of the formula (2292))

A5:i:3)(s) = -A4:i:3)(s) + A5:i:3)(s) + A6:i:3)(s) w. p. 1, (2.539)

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where

A4:i'3)(s) = lp.i—m. E ji(s)Cj:l)Cj,33,,

j3,ji=0 p

A5ii,3,(s) = lp——TO■ E ep3ji (s)Cj(:l)cjзз,,

j3,ji=0 p

A6 (s)='p—TO- E fjji WTO1,

j3 ,ji=0

S OO ( S \ 2 S

^ J MM) J2 / ^(^H J Oj.,(t)(IT(IS:>.

t j4=p+1 \s3 / S3

s S OO / T \ 2

= ^ J MM) J MM J2 /

t S3 j4=p+1 \s3 /

s s OO / s \ 2

./£.;» ^ J MM) J MM) E (/ ^(siKsi) d.sods3 =

t S3 j4=p+1 \S2 /

S / S \ 2 S2

= \J MM) E jy J MM)dszd.s2.

t j4=p+1 \s2 / t

Moreover (see the derivation of the formula (2.296)),

A3:2:3) (s) = 2A6:2:3) (s) - A5:2:3) (s) = A4:2:3) (s) - A5:2:3) (s) + A6:2:3) (s) w. p. 1.

(2.540)

Let us consider a4:i:4'(s) (see the derivation of the formula (2.297))

= I I dw^dsidw^ - A^4)(s) w. p. 1. (2.541)

s S2 Si

Let us consider Ag1i2)(s) (see the derivation of the formula (22

S si S2

4ili3)(s) = UII dw^dw^dSl + t t t

+ jl{i1=i2^0} J(S - s2)ds2 - A^2)(.s) w. p. 1. (2.542)

t

Let us consider B1(s), B2(s), B3(s) (see the derivation of the formulas (2.299), (2.300))

s „

1

Bi(s) = - / (Sl-t)dSl - lim (2.543)

4 t p œ 34=0

p p p B2(s) = lim £ apa33(s) + lim £ c^(s) - lim £ 65333(s). (2.544)

p— œ ' ■* J3J3 p—œ ' ■* j;j3 p—œ < * J3J3

33=0 33=0 33=0

Moreover (see the derivation of the formula (2.301)),

p

S2(s) + B3(s) = 2 1irn£ (s).

»—>•OO

34=0

Therefore (see the derivation of the formula (2.302)),

p p p p

B$(s) = 2 lim V 3 (s) - lim y ap . (s) - lim V 3 . (s) + lim y . (s).

p_^ 00 ^^^^^ 3 3 p_y OO ^^^^^ J3 J3 p_^ OO ^^^^^ J3 J3 p_^ 00 ^^^^^ 3 3

33=0 33=0 33=0 33=0

(2.545)

After substituting the relations (12.5361), (I25381)-(I25451) into (2535), we obtain

p

Urn. £ C3433323i ( s ) Ci : '5 Z3(22 ) Z333 ) Z3(4

3'i,32,33,34=0

S T Si

= J№i%,t + \l{i1=W}f J Jds2dwMdwM +

ttt

S S2 Si S Si S2

J J J dwMdstfwM + J J J dw^dw^ds^ t t t t t t

S

1

T si

ds2dsi + R(s) = J *[#4)]s,t +

t t

+ R(s) w. p. 1,

(2.546)

where

R(s) = -1{,i=,2=o}Al'3!4)(s) + 1{,i=,3=0! (-A^'W + Al!2,4)(s) + +

+1{,i=,4=0} (Af3)(s) - A'*3^) + A^'W) - 1{,2=.3=o!A«'i,4)(s)+ +1{!2=„=o! (-A«':i,3)(s) + A5iii3)(s) + A6'i!3)(s)) - 1{,3=,4=o!A<!1':2)(s)-

P

1{ii=i3=0}l{i2=i4=0^ ME app3j3 (s) + p—TOE Cp3j3 (s) - J—J^E bp3j3 (s)

s) -

j3=0

PP

FP (s) - li m ap (s) -

P—TO ^—'

j3=0 j3=0

P

1{ii=i4=0}1{i2 = i3=0^ 2 J—m E fjP3j3 (s) - p—TO £ ^ (s)

J—*TO e ^3 (s) + P^e jj (s) ) +

P—>-to

j3=0

+ 1{ii=i2=0}1{i3=i4=0} pJ——TO E ^U (s)

(2.547)

j3=0

From

and

it follows that Theorem 2.22 will be proved if

Akij)(s) = 0 w. p. 1, (2.548)

p p p p

lim V^ ap (s) = lim ^ bp! (s) = lim cP (s) = lim /p. (s) = 0,

—TO 3 3 P —TO 3 3 P —TO 3 3 P —TO 3 3

j3=o j3=o

_ (s)= _ P—TO ^-' •/3J3 P—TO

j3=0 j3=0

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(2.549)

where k = 1, 2, . . . , 6, i, j = 0, 1, . . . , m.

Consider the case of Legendre polynomials. Let us prove that

A1:3:4'(s) = 0 w. p. 1.

(2.550)

P

We have

, (T-tUMTmn±T)x j4j3 32

z(s) y TO / yi

x / Pj4 (y) / Pj3 W E (2j1 + 1) i Pji Mdife) dy1dy

-1 -1 ji=p+1 V-1

_ (T — /:)2y/(2j4 + l)(2j3 + 1) w

— -X.

32

z (s) TO z (s) x J Pjs(yi) E JJ^M+i(yi)-Pn-i(yi)f J PjMdydy 1 =

-1 jl=p+1 1 yi

32V2j4 + 1

z(s)

x| Pj3(y1)((Pj4+1(z(s)) - Pj4-1(z(s))) - (Pj4+1(y1) - Pj4 1 (y 1))) X

-1

OO

jl=p+1

if j4 = 0 and

El o

2 ^ + 1 (Pji+i(yi) - Pn-iMY dyi

j4j3 32

z(s)

1

ji=p+1

if j4 = 0, where z(s) is defined by (2.20).

We assume that s G (t,T) (z(s) = ±1) since the case s = T has already been considered in Theorem 2.9. Now the further proof of the equality (2.550) is completely analogous to the proof of the equality (2.314).

It is not difficult to see that the formulas

A2:2:4)(s) = 0, A4:i:3'(s) = 0, A6:i:3'(s) = 0 w. p. 1 (2.551)

can be proved similarly with the proof of (2.550).

Moreover, the relations

p p p

limV ap ■ (s) = 0, lim V № . (s) = 0, lim V . (s) = 0 (2.552)

p—>■ to ' ^ j3.y3 p— œ / j J3J3 p— œ / j J3J3

j3=0 .3=0 .3=0

can also be proved analogously with (2.316), (2.317). Let us consider A^i2i4)(s) and prove that

A3i2i4)(s) = 0 w. p. 1. (2.553)

We have

A3i2i4)(s) = Aii2i4)(s) + Ag2i4)(s) - A7i2i4)(s) = -A7i2i4)(s) (2.554) w. p. 1, where

p

AT'w^pyg. E fawcc1.

j2,j4=0

oo ' s

gj (s) = J j (t) J j (s0 E ( I j (s2)ds2 / j (s2)ds2 ) t t j'l=p+1

VSl

Note that (see (2.3S

œ 1 ' s

= E 2 / / ^2)ds2dr . (2.555)

ji=p+i \{ T J

The proof of (2.553) for the case i2 = ¿4 = 0 differs from the proof of the equality

A^4) = 0 w. p. 1

for the case i2 = ¿4 = 0 (see the proof of Theorem 2.9). In our case we will use Parseval's equality instead of the orthogonality property of the Legendre polynomials.

Using the Parseval equality, we obtain

p p to i / s s \ 2

E^'./:iV) E E 2 [j °i:{r) J ('s-k/'s-i/7

• =0 j4 =0 j1=p+1 \t T J

s

2

s

= E E ^ / M(T) [ <f>jiMds2 - I (f>j1{s2)ds2 ) dr ) <

j4=0 ji=P+1 \{ \{ t

< El / j (t)dn E ( / j (s2)dsj +

j4=0 \t / jl=P^A t )

+ E E I / j (t^ j (s2)ds2dTl =

j4=0 ji=p+1 V t t /

= e / 1{T<s}0j4 (t)dT e / j (s2)dsn +

¿4=0 \ t ) jl=P^A t )

+ E E / 1{T<s}0j4 (t ) / j (s2)ds2dT <

jl=p+lj4=0 \ {{ J

<e i 1{t< s} j (t )dT e i j m^j +

]4=0 \ { J jl=P+A t J

/ T T > 2

TOTO

+ E E / 1{t< S(t ) / j (s2)ds2dT

jl=p+1 ¿4=0 \ { {

T 2 TO O \2

= J (1{t<S^ 2 dT E / j (s2)ds2 +

t ji=P+1 \ t /

oo T / T \ 2

+ E / (1{t<S^ M / j (s2)ds2 I dT =

ji^P+1 t \t J

= (s -1) E /j (s2)dsj + E / / j(s2)dsj dT. (2.556)

ji=p+1 \ t / ji=p+11 \t /

We assume that s G (t,T) (z(s) = ±1) since the case s = T has already been considered in Theorem 2.9. Then from (2.556) and (2.143) we obtain

¿4=0 p

where constant C(s) (s is fixed) is independent of p.

Combining (2.23) and (2.310) with (1.208), we obtain for j e N

0(0)d0

si

I< 1 1

< jl/2+m/A I (1 _ z2(s))m/8 + (1 _ z2(Sl))m/8 ) ' i2"558)

where s, si e (t,T), m = 1 or m = 2, z(s) is defined by (2.20), constant K does not depend on j.

Using the Parseval equality, we get

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p1

lim E (gp4j2(s)) = (Kp(r,si,s))2 dsidr = / / (Fp(r, si, s))2 dsidr,

pi —>• to

where

j4 ,j2 =0

[t,T ]2

tt

(2.559)

T

gplj2 (s)= / 1{T<s}0j4 (t ) / j (si)Fp(r,si,s)dsidT

I -{''<s}^J4V / I rj2V~ ^ pV tt

Kp(r, si, s)0j4(t)0j2(si)dsidr

[t,T

is a coefficient of the double Fourier-Legendre series of the function

Kp(T, si, s) = 1{r<s}1{si<r<s}Fp(T, si, s),

where

oo

E I (s2)ds2 / (s2)ds2 = Fp(T,si,s).

ji=p+i

(2.560)

si

From (2.558) for m = 1 and m = 2 we have

|Fp(T,si,s)| < E

Ki

+

ji=p+i

(ji)7/4 \ (1 - z2(s))i/8 (1 - z2(si))i/8

X

11

+

(1 - z2(s))V4 (1 - z2(t))i/4

<

s

s T

s

1

A. !......, + „ * . !......+ 1

(1 - z2(s))l/8 (1 - z2(si))1/^ ^(1 - z2(s))1/4 (1 - z2(T))1/4y'

(2.561)

where s, s1, t G (t, T), constant K2 is independent of p and we used the estimate (2.714) in (2.561).

The relations (2.559) and (2.561) imply the estimate

£ * W- (2-562)

.2 ,j4=0

for the case s G (t,T ) or z (s) G (-1,1) (the case s = T has already been considered in Theorem 2.9), where constant C1(s) (s is fixed) does not depend on p.

Then from analogue of (2.355) for s G (t, T) (s is fixed), (2.557), and (2.562) we have

| / p \ 2 | p m £ j« (*)№' < (1 + 102=,4=0}) £ j M)2 +

.2,.4=0 .2,.4=0

P

j4=0 / 1

if p — to, where constant C2(s) (s is fixed) does not depend on p. The equality (2.553) is proved.

Let us consider A5iii3)(s)

A5iii3) (s) = A4iii3) (s) + A6iii3) (s) - A8iii3) (s) w. p. 1,

where

A81!3)(s) = l,.i.m. £ (s)cj;i)cjr),

p—œ

.3.1=°

s

. (s)= / . M / 0.3(т)Fp(s3,T,s)dтds3,

.3.1

t S3

where Fp(s3,T, s) is defined by (2.560).

s

Analogously to (12.5531). we obtain that Ag1i3)(s) = 0 w. p. 1. In this case we consider the function

Kp(s3,t, s) = 1{S3<s}1{S3<T<S}FP(S3,T,S) and the relations (see (2.555))

hp3ji (s)= J Kp(s3,T,s)0j 1 Mj(T)dTds3, [t,T ]2

s s

œ i

HnjSs)= E 2 J j M(si)dsidT

j4=p+1 \ t t /

Let us prove that

p

j, (s) = °- (2.563)

¿3=0

We have

Moreover,

(s) = j, (s) + j, (s) - j3 (s). (2.564)

p p pim £ fj(s) = p™ £ ^(s)=(2'565>

¿3=0 ¿3=0

where the first equality in (2.565) has been proved earlier. Analogously, we can prove the second equality in (2.565).

From (2.557) we obtain

p

p—mm ' ^ ( ) ¿3=0

So, (2.563) is proved. The relations (2.548), (2.549) are proved for the polynomial case. Theorem 2.22 is proved for the case of Legendre polynomials.

It is easy to see that the trigonometric case is considered by analogy with the case of Legendre polynomials using the estimates (2.533). Theorem 2.22 is proved.

2

Let us reformulate Theorem 2.22 in terms on the convergence of the solution of system of ODEs to the solution of system of Stratonovich SDEs.

By analogy with (2.977) for the case k = 4, p1 = ... = p4 = p, i1,..., i4 = 0,1,..., m, and s G (t, T] (s is fixed) we obtain

s t4 t3 t2 p

HI/dw!;i)pdw<;2)pdw<33»pdw<44»p = £ C.wcfci:2..4),

tttt

.1 7.2 7.3 7.4 =0

where p e N and dwT)p is defined by as in Theorem 2.22.

The iterated Riemann-Stiltjes integrals

I; another notations are the same

s t4 t3 t2

Vs(ti;2;3;4)p

dwt(;i)p dwt(;2)p dwt(;3)p dwt(;4)p,

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(2.566)

tttt

s t3 t2

Z

(;1;2;3)p

s t

dwt(;i)pdwt(;2)pdwt(;3)p,

(2.567)

ttt

s t2

Y

(ii;2)p

s , t

dwt(;i)pdwt(;2)p,

(2.568)

tt

s

Xiit)p = y dwti

t

(;1)p

(2.569)

are the solution of the following system of ODEs

'dvs(;i;2i3;4)p = Z (;i;2;3)pdw(;4)p V (;i;2;3;4)p

s t

= Z

s t

dw(;4)p, Vt

t t

=0

dZ

(;1;2;3)p = Y( t1;2)p dw(;3)p

s,t

dYs(t1i2)p = Xft^dwp,

dxft )p = 1 • dw(;i)p,

Z(;i;2;3)p = 0

Y(f;i'2)p = 0

x£')p = 0

From the other hand, the iterated Stratonovich stochastic integrals

>}c s ^ £4 ^ £3 ^ £2

VT

t t t t

*S * t3 ^2

Z

(«1«2«3)

s,t

dwt(;i) dwt(i2)dwt(;3),

t t t

>}c S %t2

Y(i1i2) _ Ys,t _

dwi;i)dw<;2),

tt

. s

x _ 1 dw

(ii)

*S,t I tl

t

(2.570)

(2.571)

(2.572)

(2.573)

are the solution of the following system of Stratonovich SDEs

S,t

_ Z

s,t

* dw(i4), Vt

dZ

s,t

dYS(t1i2) _ X^ * dw(i2),

dX*0 _1 * dw(n),

t,t

_0

_ Ys(t1i2) * dw(i3), Z

(¿1«2«3)

t,t

_0

Yt(t1i2) _ 0

Xi0 _0

where * dw(i), i _ 0,1,... ,m is the Stratonovich differential, * _ ds.

Then from Theorems 2.19, 2.21, and 2.22 we obtain the following theorem.

Theorem 2.23 [32]. Suppose that {fy(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([i,T]). Then for any fixed s (s e (t,T])

l.i.m. V

(¿1i2«3«4)p

p—TO

s,t

_V

(h«2«3«4)

s,t

l.i.m. Zs(i^i2i3)p _ Zs(^2i3),

p—to ( (

l.i.m. Ys(t1i2)p _ Ys(t1i2),

p—to ( (

X^1)p _ l.i.m. X

p—to

(»1)

S,t .

2.8 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 4 in Theorems 2.2, 2.8, and 2.9

2.8.1 Rate of the Mean-Square Convergence of Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2

This section is devoted to the proof of the following theorem.

Theorem 2.24 [32]. Suppose that {j(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, ), ^2(t) are continuously differentiate functions on [t,T]. Then, for the iterated Stratonovich stochastic integral

* T * ¿2

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= / fa(t2)/ ^of^f (M2 = i,...,m)

v 2

p \ 2

the following estimate

M J f J*[V'(2)]r,( - £ C^ifcf < H (2.574)

is valid, where constant C is independent of p,

Cj = J ^2(s2)0j2 (s2 ) J ^i(si)0j! (si)dsids2, t t

and

T

if = j (T f

j / t 3 V / t

t

are independent standard Gaussian random variables for various i or j. Proof. Applying (2.8), we obtain

2'

r*L/.(2)l \ ^ ^ /-(«i) Z (¿2)

z32

3i 32=0

M j*[^(2)]T,t - £ C,2jCf j

m^ ( ./[V'(2)]r,( + ii{il=il} j Mh)Hh)dh - £ o^ifcf

t j1,j2=0

2

m (J[^<2)lT,t -Y, Cj2j.( C^if - 1{.,=.2}%=j2} ) +

jij2=0

T x 2 ■

p2

t ¿i=0

^(j[^(2)]t,- E ^¿2^1 (cj;.-i{ii=i2}i{ji=.Mii > + ¿1 ,j2=0

T

p

+ ( ^{n^} I 'Mti)'h{ti)dti - 1 {n=i2}Ea

i = ; 2 } ¿ i ¿ i ¿i=0

2

m Jl^(2)]T,t - J[^(2)K +

/ 1 T p \

+ l{(,-fe> o / ^ltiiW^iiMii ~YC>A \ t ji=0 )

From Remark 1.7 (see (1.223)) we have

2

m < (j[^]T,t - ) \ < j

(2.575)

(2.576)

where constant C1 is independent of p. From Theorem 2.2 (see (2.37)) we get

1

2

T

^1(t1^2(t1)dt1 Cjiji = E Cjiji' ji=0 ji=p+1

(2.577)

Let us consider the case of Legendre polynomials. The estimate (2.83 implies that

oo

ji=p+1

<C2[l-+ E -

(2.578)

where constant C2 does not depend on p.

2

Using (2.25) and (2.578), we have

—

ji=p+i

<

p

(2.579)

where constant C3 is independent of p.

Applying the ideas that we used to obtain the relations (12.851). (I2.89I)-(I2T9T1), we can prove the following estimates for the trigonometric case

S2p —

ji=2p+1

<

Kl p

S2p-1 —

ji=2P

< S2p +

K2 p

(2.580)

(2.581)

where constants K1; K2 do not depend on p.

Using (2.580) and (2.581), we get the estimate (2.579) for the trigonometric case. Combining (r2.575l)-(l'2.577l), (12.579), we obtain (12.5741). Theorem 2.24 is proved.

2.8.2 Rate of the Mean-Square Convergence of Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3

In this section, we consider the following theorem.

Theorem 2.25 [32]. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). At the same time ^2(t) is a continuously differentiable nonrandom function on [t,T] and ^i(t), ^3(t) are twice continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

<r

^3

<Î2

J*[^(3)]T,t — / Wis) / «Î2)/ ^i(ti)dfi(;i)dft(2i2)df4i3) (ii,i2,i3 — 1,...,m)

the following estimate

m

£ c

ji J2J3=0

Z (H)z (»2)Z ( j3j2ji Zji j Zj3

<

c p

(2.582)

p

is valid, where constant C is independent of p,

T S3 S2

C333231 = J fa(s3)033 (S3) J ^2(52)^32 (s2^ (si)3 (si)dsids2ds3, t t t

and

T

Cj!) = J to(T

t

are independent standard Gaussian random variables for various i or j. Proof. We have (see (2.382))

i/ p \ 2

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(j*[^3)]T,f - £ C,3323ic3ii)c(:2)c333)

31,32,33=0

T t3

2 -\h=« 1

tt T t3 p

t t 3i,32,33=0

= m j J[^(3)]T,t - J[^(3)]?p,p+

/ T t3 P

+i{il=i2} u /^3) [ hit2)fait2)dt2di^ - £ ) +

\ i t 3i,33=0

/ T t3 p

t t 3i ,33=0

(i2)

2

p p 2

l{ii=«3^ EC3i333i C332 ) J", (2.583)

3i =0 33=0

where (see (1.47))

p

j[^(3)]Tr = £ (c3:i)c3:2)

3i,32,33=0

1 1 Z(i3 ) — 1 1 Z_ 1 1 z(i2)

1{i1 = i2}±{j1 =j2}Zj3 ±{i2 = i3}±{j2=j3}Zji ±{i1=i3}±{j1=j3} Zj2

Using (2.583) and the elementary inequality

(a + b + c + d)2 < 4 (a2 + b2 + c2 + d2) ,

we obtain

\ 2 '

p x 2

m j *w'(3)]t,, - £ cj cfcfc i ><

^M^M^ )

j1 J2,j3 =0

2

< 4^m | (J[^<3)]>,i - J| + 1{!1=!2}EP1> + 1{!2=,3}EP2) +

+l{.1=i3}Ep3A, (2.584)

where

E{p1] = M{ \ \ l uh) 1 Ut2)Mt2)dhd^ - £ CJ3J1J1

, t t j3=0

T t3 p x 2

ep] = M{ o //^(iodfi^dis- e ^iCj;0

2 I Mhmh) I ih{h)dHl

t t j1 j3=0

\ 2'

p p 2

t(i2) j3

j1=0 j3=0

E3) = M Z ECj1j3j1 Zj3

From Remark 1.7 (see (1.223)) we have

2

M ^ ( JlMht ~ JIMW) | < j, (2-585)

where constant C1 is independent of p.

Moreover, from (2.280) and (2.284) we have the following estimate

M3> < ^ (2.586) pp

for the polynomial and trigonometric cases, where constant C2 does not depend on p.

Using Theorem 1.1 for k = 1 (also see (1.45)), we obtain w. p. 1

T s p

\ f Us) f Usi)Usi)dSldf^ =

t t P 00 33=0

where

T s

(733 = J 3 (s)Us) J ^2(51 )^1(s1)ds1ds. tt Applying the Ito formula, we have

T s T T

I Us)Us) f W^fds = J ^1)/ ^3(s)^2(s)dsdfs(ii) w. p. 1.

t t t si

Using Theorem 1.1 for k = 1 (also see (1.45)), we have w. p. 1

T T p

^ /v(S) [Usi)Usi)ds!dfM = hi.m. irq^

2 ^ 1 s 2 ^ 313

t S 31=0

where

T T

CSl = I Ws)3(s) / ^3(S1 )^2(s1)ds1ds. Further, we get

3i

t

Ep1) < 2Gp1) + 2Gp2), (2.587)

Ep2) < 2Hp1) + 2Hp2), (2.588)

where

^ = M \\ (I^ J Ut2)Ut2)dt2d^ - ¿4ci:3)

\ t t 33=0

p p 2

^'»muIew- e

33=0 31,33=0

T is p

= M \\ ( /^(*3№2(i3) J'MMdi^dh - ¿qcjj15

.t t ji=0 p p \ 2

ji=0 ji,js=0

From Remark 1.7 (see (1.223)) we have

< —, Hi1] < —, (2.589)

p p p p

where constant C2 is independent of p. The estimates

Gf < H® < ^ (2.590)

p p p p

are proved in Sect. 2.2.5 (see the proof of Theorem 2.8) for the polynomial and trigonometric cases; constant C3 does not depend on p.

Combining the estimates (I2.584)-(l2.590l). we obtain the inequality (2.582). Theorem 2.25 is proved.

2

2.8.3 Rate of the Mean-Square Convergence of Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4

This section is devoted to the proof of the following theorem.

. Suppose that (x)}°=0 is a complete orthonormal sys-

Theorem 2.26

tem of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

:T .,.¿3 *Î2

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J *[^4)]t,

t —

(n) 7 (i2) 7 .(«s) 7 r(i4)

¿2

¿s

'¿4

(ïi,Ï2,Ï3,Ï4 — 0,1,... ,m)

t t

the following estimate

m

J ( 4)]

EC Z (ii)Z (i2)Z (is)Z (i4)

Cj4jSj2j1 Zji Z j2 ZjS Zj4

j1j2 jsj4=0

c

< - (2.591) p

is valid, where constant C is independent of p,

T S4 S3 S2

Cj4j3j2j1 =y j j j (s2)J j (si )ds1ds2ds3ds4 ,

t t t t

and

T

j = | to(T)dw«

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[i) = f(i) for i = 1,..., m and wT0) = t.

Proof. First, let us prove that Theorem 2.8 is valid for the case i1,i2,i3 = 0,1,..., m. The case i1, i2, i3 = 1,..., m has been proved in Theorem 2.8. From (1.47) and the standard relation (2.382) between Stratonovich and Ito stochastic integrals of third multiplicity we have that Theorem 2.8 is valid for the following

cases

ii = i2 = 0, i3 = 1,..., m,

11 = i3 = 0, i2 = 1,..., m,

12 = i3 = 0, ii = 1,..., m. Thus, it remains to consider the following three cases

11, i2 = 1,...,m, i3 = 0, (2.592)

12, i3 = 1,..., m, ii = 0, (2.593) ii, i3 = 1,...,m, i2 = 0. (2.594)

The relations (1.47) and (2.382) imply that for the case (2.592) we need to prove the following equality

t t3 t2

W f* f* f*

E / ^3(t3M j(¿2)^2) / j(ti)^i(ti)dtidt2dt3 = ji=01 t t

T t3

= ^ /Wa) i MhyHtfadhdh. (2.595)

Using the relation (2.10), we get

t t3 t2

W f* f* f*

j!=° t t t

T T T

W f* f* f*

= E / j (tl)^l(tlW j (t2)^2(t2W ^3(t3)dt3dt2dti = jl=° t t! t2 ™ T T

W „ „

= E / j (tl)^l(tlW j (^(^«¿l = j!=° t t!

T t2

w „ i

= E / j(^2)^2(^2)/ j(tl)^l(tl)dtldt2 = j!=° t t T

= 5 J MtoJh(to)dto, (2.596)

t

where

T

(t-2) = ^2) y ^3)^3. (2.597)

t2

From (2.596) and (2.597) we obtain

t t3 t2

W

E / ^(tj) / j (¿2)^2) / j (tl)^l(tl)dtldt2 dt3 =

j!=° t t t

T T

= ^ jMt2)h{to) Jh{h)dt3du = t t2 T t3

= ^ y^3(i3) J Mt2)Hto)dtodh. (2.598)

tt

The relation (2.595) is proved. http://doi.org/l°.2l638/ll7°l/spbu35.2°23.ll° Electronic Journal. http://diffjournal.spbu.ru/ A.456

From (1.47) and (2.382) it follows that for the case (2.593) we need to prove the following equality

T is

= \ j h{h)MU) J 'Mti)dtidt3. (2.599)

t t

Using the relation (2.10), we have

T

= \ J Mt3)Mt3)dt3, t

where

t2

)= ^(¿2^ ^l(tl)dti. (2.600)

t

The relation (2.599) is proved.

The relations (1.47) and (2.382) imply that for the case (2.594) we need to prove the following equality

T ts ts

00 „ S S

= jteW^y j(^(tOy ^2(t2)dt2dt1dt3 =

j!=0 t t t! T ts / T T \

E / j (t3)^s(t3^y j (¿1^1 (¿1) J ^2 (¿2 ) dt 2 -J ^(¿2^2 dt1dt3

j!=0 t t Vtl ts /

^ « s «

= E / j(¿3)^3) / j(£1)^1) / ^2(t2)dt2dt1dt3-j!=01 t tl

^ ft s ft

E / j(¿3)^3) / j(£1)^1) / ^2(t2)dt2dt1dt3 =

j!=0 t t ts

TO T t3

= E / j (t3)^s(ts) j (t1)^^1(t1)dt1dts-

j!=01 t T ts

TO „ 5

E / j (t3)^^3(t3W j (¿1)^1(t1)dt1dt3

j!=0

tt

T T

= ^ J MhWiWdh - ^ jMh)Mh)dti = tt T T T T

= 5 J hiMMh) J ihMdtodh - i J'MhyMh) Jihitojdudt! = o, t t! t t!

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where

T

) = ^(¿1)^ ^2(t2)dt2, (2.602)

t!

T

W*3) = ^3(t3^ ^2(t2)dt2. (2.603)

ts

The relation (2.601) is proved. Theorem 2.8 is proved for the case i1, i2, i3 = 0,1,..., m.

Using (2.382) and (2.383), we obtain

m Uj*W'(4)]T,f - £ C,j№;iCi>cfc<33)c

2

(i1)z(i2) Z(i3)Z(i4) j4

j1,j2,j3,j4=0

T S Si

= m <! ( + il{n=^0} [ j [dsodw^dw^ +

ttt

T S2 Si T Si S2

J J J dw^dSldw^ + J J J dw^dw^dSl + t t t t t t T Si p

+^1{n=i2^0}l{i3=i4^0} J J d.sod.si - E Q'^jiCji^C^^j^Cj^

t t jij2j3j4=0

* T ^ s ^ Si

M 4 ( J['0(4)]T, + J J ds2dw^dw^~

t t t

T si * T * S2 * si

il{i1=i2^0}l{i3=i4^0}J J d.sod.si + ^l{i2=i3#)}J J J dw{;i)dsidw{s^) +

t t t t t * T * si * S2 T si

+^{»3=^0}J J J dw{sn)dw{s^dsi — il{il=i2^o}l{i3=i4^o} y J ds2ds! + t t t t t

T S! p X 2

+ 41{H=i2^0}l{i3=i4^0} j j d.Sod.Sl - E f 7 :./:■,/■;./ S' ' C^ C^ Cj.

t t ji,j2,j3,j4=0

* T ^ s ^ Si

M ^ ( J['0(4)]T, + j j

t t t

*T * S2 * Si *T * Si * S2 J J J J J J dw^dw^dS!

t t t t t t

1

T S!

tt

ds2ds1 - E C'C'Cc

j ! ,j2 ,jS ,j4=0

m j[^(4)]T,t - j

;T ^ S >}c S!

1

{i!=i2=0}

ds2dw((is)dw((i4) - S((isi4)p | +

!

{iS=i4=0}

,t t t S T * S2 * S!

71

t t t

sT * S! * S 2

7i

t t t

dw(i!)ds1dw(;4) - Sfi4)p I +

dw (i!)dw (;2)ds1 - sfi2)p

l{i1=i2^0}l{z3=i4^0} I - I I dszdsi —

4

T S!

tt

T

I I MMM ~ t)dsids ) -R

j4=0

(2.604)

where S(isi4)p, S2i!i4)p, S3i!i2)p are the approximations of the iterated Stratonovich stochastic integrals

;T ^ S * S!

:T * S2 >jc S!

ds2dw(;s)dwii4),

dwin)ds1dwii4),

t t t t t t * T ^ S! ^ S2

iff dw^dw^ds!,

t t t

respectively (these approximations are obtained by the version of Theorem 2.8 for the case i1,i2,i3 = 0,1,...,m); J[^(4)]yp'p'p is the approximation of the

2

p

2

s

p

iterated Ito stochastic integral J[^(4)]T,t obtained by Theorem 1.1 (see (1.48))

p

T l"i/i(4)]P'P'P'P = V^ C Z (ii)Z (i2)Z (i3)Z (i4)_

J ]T,t = Cj4j3j2ji Zji Zj2 Zj3 Z j4

jij2,j3,j4=0

-t /l(«3«4)P n /I(«2«4)P n /l(«2«3)P n /l(«i«4)P

-i{ii = i2=0}Ai - i{ii=i3=0}A2 - i{ii = i4=0}A3 - i{i2 = i3=0|A4

1 /l(«i«3)p T /l(«i«2)p^1 1 DP,

-i{i2 = «4=0}A5 - 1{«3=i4=0}A6 + i{ii = i2=0}i{i3=i4=0}Bi +

+ 1{ii=i3=0}1{«2 = i4=0}B2 + 1{ii = «4=0}1{i2=i3=0}B3 ,

where

pp A(i3i4)P = c Z (i3)z (i4) A(i2i4)P = C Z (i2) Z (i4)

Ai = / v Cj4j3jiji Zj3 Zj4 , a2 = / v Cj4j3j2j3 Zj2 Zj4 ;

j4 ,j3 ,ji=0 j4,j3,j2=0

pp

A (i2i3)P = v"^ C Z(i2) z(i3) A (iii4)P = v"^ C Z(n) Z(i4)

A3 / v Cj4j3j2j4 Zj2 Z j3 , A4 / v Cj4j3 j3ji Zji j ,

j4 ,j3,j2=0 j4,j3,ji=0

pp A(«i«3)P = V"^ c Z (ii)Z (i3) A(iii2)P = C Z (ii) Z(i

A5 C j4j3j4ji Zji Z j3 , A6 = Cj3j3 j2 ji Zji Zj2

j4 ,j3 ,ji=0 j3,j2,j'i=0

pp bp = v^ c- ■ ■ ■ bP = v^ c- ■ ■ ■

Bi = / v Cj4j4jiji , B2 = / v Cj3j4j3j4,

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ji ,j4=0 j4 ,j3 =0

p

bP = v^ C- ■ . . .

B3 = / v Cj4j3 j3j4. j4 ,j3=0

Rp is the expression on the right-hand side of (2.304) before passing to the limits, i.e.

R _ "I A(«3i4)p^1 / A (i2i4)p | A(i2i4)p , A (i2i4)p\ ,

Rp = -1{n=i2=0}Ai + 1{ii=i3=0} ^-A2 + Ai + A3 J +

,1 /A(«2i3)p A(i2i3)p , A(«2i3)p^ n A(«i«4)p,

+ I{ii=i4=0} ^4 - A5 + A6 J - l{i2=i3=0}A3 +

,1 / A (iii3)p 1 A(iii3)p I A(«i«3)p\ -, A(«i«2)p

+ 1{*2 = M=0} ( -A4 + A5 + A6 j- 1{i3 = i4=0}A6

1{ii = i3=0}l{i2=i4=0} l Y ap3j3 + Y, cp3j3 - Y j

j3j3

j3=0

where

1{ii=i4=0}1{i2=i3=0^ 2 X] fj3j3 - ^ ap3j3 - S cp3j3 + ^ bp3j3 +

j3=0 j3=0 j3=0

p

+ 1{i1=i2=0}1{i3 = i4=0} Y app3j3 '

j3=0

A

(¿3M)p

Eap Z(i3)Z(i4) A (i2i4)P

aj4j3Zj3 Zj4 ' A

j3 ,j4=0

A

(¿2M)p

EbP Z(i2 ) Z(i4)

j4j2 Z j2 Z j4 '

j4,j2=0

pp

E„P Z (i2) Z (i4) A (i1i3)P _ dP Z (i1)Z (i3)

A

(¿1«3)p

j4j2=0 p

EeP Z(i1)Z(i3) A (i1i3)p

ej3j1 Zj1 Zj3 ' 6

j3j1=0

j3j1=0 p

Efp z(i1)Z(i3)

fj3j1 Zj1 Zj3 '

j3j1=°

where

ap . . cp . dp ■ ep ■ fp ■

j4j3' j4j2' j4j2' j3j1' j3j1' ^^3^1

are defined by the relations (E2SSI), (£220), (E22H), (E223HE225). Using (2.604) and the elementary inequality

we get

(a? + ... + a6)2 < 6 (a? + ... + aj) ,

p

2

m< J*[#>]T,t- E Cj zr'zjr'zrC'l ><

j1 ,j2 j3j4 =0

< 6 (q<1> + QP2) + Qp» + Qj,4) + Q<5> + Q<6>) ,

pp

(2.605)

where

QP11 = m J

T,t

- J

p,p,p,p

p

1

3

5

2

Qii] = 71«

4

{i3=i4=0}

m

T * Si * S2 " 2

J dw(ii)dwi;2)dsi - sf t t t

Q5) = 1{ii=i2=0}1{i3=«4=0}X

T

x ( I I I ^4(5) [ (f>h(si){si -t)dsids

ii= T

2

j4=0 t

Qf = M{ (Rp)2} .

From Remark 1.7 (see (1.223)) we have

Q(P1] < A p p

(2.606)

where constant Ci is independent of p.

Let us prove the version of Theorem 2.25 for the case ii, i2, i3 = 0,1,..., m. The case ii, i2, i3 = 1,..., m has been proved in Theorem 2.25. It is easy to see that, in addition to the proof of Theorem 2.25, we need to prove the following inequalities

T

t3

I I hits) I Mti)'h{ti)dtidh-

T

t3

t2

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-£ / ^(¿3) / j(¿2)^2) / j(ti)^i(ii)dMMi3 ji=01 t t

C

< —,

p

(2.607)

2

S

T

ts

i J ihih)h{h) j Mh)dtidt3-tt p T ts t2

Y J j MMM) j j (t2)fo(t2)J ^1(t1)dt1dt2dt3

js=0 t t t

C

< —,

p

(2.608)

T

ts t2

J2 I j (¿3)^3)/ ^2(t2M j (t1)^1(t1)dt1dt2dt3

j!=0 t

tt

C

< (2.609) p

where constant C is independent of p.

The inequalities (2.607) and (2.608) are equivalent to the following inequalities (see the proof of the cases (2.592), (2.593))

T p T t2

- j 'MtoJh{to)dto-J2 j MM'hito) j ^{hyMt^dhdto 2 t j! =0 t t

<

C

T p T ts

\ J J orJI3)c3(l3) J (pj3{uyh{u)dhdh

t js =0 t t

<

p

(2.610)

C p

(2.611)

where ^2(t2) , ^2(t2) are defined by (2.597) and (2.600), respectively. The inequalities (2.610), (2.611) follow from (2.577), (2.579)-(2.581).

Let us prove (2.609). By analogy with the proof of (2.601) we have

p T ts t2

Y i j (¿3)^3)/ Wt2) / j (¿1)^1 (t 1)dt 1 dt2dt3 =

j!=0 '

T

tt

ts

^ J j (t3)^3(t3W j (t1)^/1(t1)dt1dt3 j!=0 t t p T ts

I] / j (¿3)^3) / j (t1)^1(t1)dt1dt3

j!=0 t t

p

T t3

to „ 3

ji=° t t

to T ^

^ / j fe)^) / j (ti)^i(ti)dtidt3-ji=° t t

T t3

to „ 5

X] / jteW^te) / j(ti)t^i(ti)dtidt3+

ji=p+i t t

to T ^

+ ^ / j(¿3)^3) / j(ti)^i(ti)dtidt3 =

ji=p+i { t

T t3

to „ 5

= - ^ / j(¿3)^3(^3) j(ti)^^i(ti)dtidt3+

ji=p+i t t to T ^

+ X) / j(t3^(¿3) / j(ti)^i(ti)dtidt3, (2.612)

ji=p+i t t

where ^(¿0, ^3(t3) are defined by (2.6021), (l'2.60,3). respectively.

Now the estimate (2609) follows from (26121) and (2579)-(2581). Theorem 2.25 is proved for the case ii, i2, i3 = 0,1,..., m.

Using the version of Theorem 2.25 for the case ii, i2, i3 = 0,1,..., m, we obtain the following estimates

q<2) < —, qp3) < —, q'4) < —, (2.613)

1 p P p P p

where constant C2 does not depend on p. From Theorem 2.2 (see (2.37)) we get

1 T p T s i /(si - J W5) J <f>j4{si){si ~ t)d.sids =

t j4=° t t

T s

to t s

= X) / j(s) / j(si)(si - t)dsids. (2.614)

j4=P+i 1 {

Let us consider the case of Legendre polynomials. From (2.579) and (2.614) we have

oo

T

Y / j (s) / j (s1)(s1 - t)ds1ds

j4=P+1 "t 1

P

(2.615)

where constant C3 is independent of p.

By analogy with (2.580) and (2.581) we have the estimate (2.615) for the trigonometric case. Then

>(5) ^ — (2.616)

QP < % 1 p2

where constant C4 does not depend on p.

Analyzing the proof of Theorem 2.9, we conclude that

C5

Q6) <

p

(2.617)

for the polynomial and trigonometric cases; constant C5 is independent of p.

Combining (2.605), (2.606), (2.613), (2.616), (2.617), we get (2.591). Theorem 2.26 is proved.

s

2.9 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 4 in Theorems 2.18, 2.20, and 2.22 (The Case of Integration Interval [t, s] (s G (t, T]))

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Let us prove the following theorem.

Theorem 2.27 [32]. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, ^1(r), ^2(r) are continuously differentiate functions on [t,T]. Then, for the iterated Stratonovich stochastic integral

* S ^ ¿2

J*[^(2)kt =/«î2)/ ^i(ti)dft(;i)dft;2) (ii,i2 = i,...,m)

the following estimate

m {( j'U'(%,< - £ c„,Mc<;:i)cf < ^ (2.618)

j1j2=0

P

is valid, where s £ (t, T] (s is fixed), constant C(s) is independent of p,

S ¿2

Cjj (s) = J ^2(t2)0j2 (t2) y ^l(tl)0ji (tl)dtidt2,

t t

and

T

(«•>=/ 4>} (t )df«

t

are independent standard Gaussian random variables for various i or j.

Proof. The case s = T has already been considered in Theorem 2.24. Below we consider the case s £ (t,T). By analogy with (2.575) we obtain

m^ (j*W'(2)k, - £ Cj(sj'cf *

j1,j2=0

2

m j [v>(2)]s,f - j i^(2)]P:f +

s

1

+ l{ii=i3} ( 9 I utiyutfadh 1 > (2-619)

where (see

2

t ji=0

Ji^^P = £ Cj2ji (s)( Ciil)Cj22) - 1{ii=i2=0}1{ji=j2} j1:j2=0

From Remark 1.12 (see (1.249)) we have

m < [J[f2)U - j['0(2)e;?y | < (2.620)

where constant C1(s) is independent of p.

Using (2.508), we obtain (the existence of the limit on the right-hand side of (2.508) will be proved further in this section)

s

2 I Mti)Mti)dt1-J2cjljl(s)= E (2.621)

t ji=0 ji=p+1

2

p

Consider the case of Legendre polynomials. By analogy with (2.82) we get for n > m (n, m £ N)

E j (s)= E /«0)j (0)1 )j (t)dTd0

ji=m+1 ji=m+1 t t

T-t

,(s)

^l(h(x))^2(h(x)) (Pn+i(x)Pn(x) - Pm+l(x)Pm(x)) dx-

l

(T -1)2

E

1

,(s)

2? + 1 J

ji=m+1 J1

(Pji+i(y) - Pji-i(y)) (%))x

x (Pji+i(z(s)) - Pji-i(z(s))) ^(s) - (Pji+i(y) - Pji-i(y)) «%))-

T-t

,(s)

(Pji+i(x) - Pji-i(x)) ^2(h(x))dx dy,

(2.622)

where

,, , T -1 T +1 / T + A 2

h(y) = —y + —, = s -

2

2 /T-t'

and ^i, ^2 are derivatives of the functions ^1(t), ^2(t) with respect to the variable h(y) (see (2.55)).

Applying the estimate (2.67) and taking into account the boundedness of the functions ^1(t), ^2(t) and their derivatives, we finally obtain

E Cjiji(s)

ji=m+1

Z (s)

[ dx

nm

-1

+

+C2 E

/ z(s)

ji=m+1

?2

dy

V

+

-i

z(s)

i i

(1 - y2)

^1/2

+

1

(1 - x2)1/2

z (s)

dy

(1 - z2(s))i/4 y (1 - y2)

^1/4

z(s)

dx

\

(1 - y2)1/^ (1 - x2)1/4

dy

(2.623)

!

0

s

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n

1

where constants C1; C2 do not depend on n and m.

We assume that s £ (t,T) (z(s) = ±1) since the case s = T has already been considered in Theorem 2.24. Then

E ^'j'1 (s)

ji=m+1

< C,(«) I 1 + 1 + £ 1

» n m j2

ji =m+1

Jl

where constant C3(s) does not depend on n and m. Thus, the limit

(2.624)

E Cjiji(s)

p—t>00 '

ji=0

(2.625)

exists for the polynomial case. For the trigonometric case, the existence of the limit (2.625) can be proved by analogy with the proof of Lemma 2.2 (Sect. 2.1.2).

The relations (2.624) and (2.25) imply that

oo

E Cjiji(s) ji=p+1

oo

ji=p+1

1

7i

Cfas) P

(2.626)

where constant C4(s) is independent of p.

For the trigonometric case, the analog of the inequality (2.626) can be obtained by analogy with (2.580) and (2.581) (see the proof of Lemma 2.2).

and

Combining (2619)-(2621), (12-626). we obtain the estimate (2618). Theorem 2.27 is proved.

The arguments given earlier in Chapters 1 and 2 of this book allow us to formulate the following two theorems.

Theorem 2.28 [32]. Suppose that (x)}°=° is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). At the same time fa2(T) is a continuously differentiable nonrandom function on [t,T] and fa(t), ^3(t) are twice continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

■M

J *

= / «ta)/ «Î2)/ «t^dw^dw^dw^0

n

p

s

where ¿1, i2, i3 = 0,1,... ,m, the following estimate

MI - E C^wcfcjr'cjr1) h^r1

j1 ,j2 ,j3=0

is valid, where s £ (t, T] (s is fixed), constant C(s) is independent of p,

s is t2

Cj3j2ji (s) = J ^3(t3)0js (*)/ ^2(t2)j (t2^ (ti)0ji (ti)dtidt2dt3,

t t t

and

T

Cj!) = / j (tf

t

are independent standard Gaussian random variables for various i or j.

Theorem 2.29 [32]. Suppose that (x)}°=0 is a complete orthonormal sys

tem of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

= /// I dw^dw^dw^dw™ (M2,i3,i4 = 0,1,..., m)

t t t t the following estimate

2

m < i ./*[<A(4)],f - E cfc!:4) < ^

j1j2 j3j4=0 /

is valid, where s £ (t,T] (s is fixed), constant C(s) is independent of p,

S ¿4 ts t2

Cj4j3j2ji (s)^/ 0j4 (t4^ j3 ^ j y j (t1 )dt1dt2dt3dt4, t t t t

and

T

j = J j(t)dw«

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[i) = f(i) for i = 1,..., m and wT0) = t.

2.10 Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity k (k E N). Proof of Hypothesis 2.2 Under the Condition of Convergence of Trace Series

In this section, we prove the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k (k E N) under the condition of convergence of trace series. Let us introduce some notations.

Consider the unordered set {1, 2,..., k} and separate it into two parts: the first part consists of r unordered pairs (sequence order of these pairs is also unimportant) and the second one consists of the remaining k — 2r numbers. So, we have

({j9h 92}, • • •, {92,-1,92, }}, {ffi, • • •, qk-2r}), (2.627)

part 1 part 2

where

{gi,g2,... ,g2r—I,g2r,qi,... ,qk-2r} = {1,2,... ,k},

braces mean an unordered set, and parentheses mean an ordered set. Consider the sum (1.53) with respect to all possible partitions (2.627)

agig2v,g2r-ig2r ,qi---qfc-2r

({{si>S2}>'">{S2r-l>S2r }},{9i>--->9fc-2r }) {si,S2>'">S2r-1>S2r>91 >'">9fc-2r } = {1>2>'">fc}

and the Fourier coefficient

t t2

Cjk ...j = J fa (tk )fafc (tk ) ..J fa (ti)0j! (ti )dti.. .dtk (2.628) t t

corresponding to the function (1.6), where {fa-(x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]). At that we suppose fa0(x) = 1 /y/T=t.

Denote

' def

C......

Cjfc •••j + ij jiji-2---ji

T tl+2 tl + i

= j ^k (tk ) j (tk ) ... j ^1 + 1(t1+1)jji + i + 0 ^ (t/ )^1-1(t1 )X t t t tl t2 X J ^i_2(t/_2)0ji_2 (ti-2)... J (ti)jji (ti)dti... dt/_2dt/t/+i... dtk = (2.629)

tt

T tl+2 tl+i

= = i J Mtk)M(tk) ■ ■ ■ J ^l+l{tl+l)<f>jl+1{ti+1) J t t t tl t2 x J ^¿-2 (ti-2) (tl—2).. .y ^i(ti)0ji (ti)dti... dti-2dti t/+i.. .dtk =

tt

= VP —

i.e. — is again the Fourier coefficient of type Cjk.„jl but

with a new shorter multi-index jk ... j1+10j1-2 ... j and new weight functions ^l(r), ..., 2(t), - tipi-1(T)'lJJi{T)^ i(r)' • • •, (also we suppose

that {l, l - 1} is one of the pairs {g1, g2},..., {g2r-1, g2r}).

Let

C =f

Cjfc •••jl+i jl jl jl-2-ji =

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(jl jl m

T tl+2 tl+i

= y ^k (tk )jjk (tk ) ... J ^+1(t1+1)jj*+i (t1+1^ (t1 ) 1 (t1 ) jjm (t/ )x t t t

tl t2 x / ^¿-2 (ti—2) jjl-2 (t/—2)... ^i(ti)0ji (ti)dti... dti-2dti tl+i... dtk = (2.630)

_ c.....

i.e. Cjk...j;+1jmj;_2•••j1 is again the Fourier coefficient of type Cjk...j but with a new shorter multi-index ... j/+ijmj/—2 ... ji and new weight functions ), ..., 2(t), ^/-i(r(t), ^/+i(r), ..., (t) (also we suppose that {1 — 1,1} is one of the pairs {01,02}, — , {02r—1,02r}).

Denote

C

(p)

jk -jq -j1

def

q=gi,02v,g2r—i,g2r

oo oo

def

E E • • • E E •••j1

jfl2r -1 =P+1 jfl2r -3 =P+1 j'flS =P+1 j's 1 =P+1

. (2.631)

jgi j32 '•••'jS2r-1 j32r

Introduce the following notation

S^ C,

(p)

jk — jq •••j1

def 1

1

q=01,02,-,g2r—1,g2r

jg2i=g2i-1+1}

£ £

jfl2r-1 =P+1 jfl2r-3 =P+1

■ e e

jS2i + 1 =p+1 jS2i-3 =p+1

OO oo

■ E E Cjk •••j1

jfl3 =P+1 jfl1 =P+1

(jS2i jS2i-1 j32 '•••'jS2r-1 j32r

(2.632)

Note that the operation S/ (/ = 1, 2,..., r) acts on the value

a (p)

jk •••jq •••j1

(2.633)

as follows: S/ multiplies (12.6331) by 1{g2l =g2I-i+1}/2, removes the summation

to

,

Jfl2l-i =P+1

and replaces

with

C ■

ajk •••j1

j31 jS2 '•••'jS2r-1 jS2r

a

jk-j!

(js2l j32I-1 =j32 '•••'jS2r-1 =j32r

(2.634)

Note that we write

a

ajk •••j1

= a

( jg 1 j32 ) ^ ( •) 'js 1 = jS2

jk •••j1

( jg 1 jg 1) ^( •) ,jg 1 = jg2

2

Cjk •••ji

= C

m jgi jS2

jk -ji

( jgi jgi W m 'jS1 jS2

C Cjk...ji

= C

jk ...ji

( jg i jS2 ) •) '( jS3 j34 ) ^ ( •) 'jS i =jS2 'jS3 =jS4

( j'g i j'g i ) ^( •)( jg3 jg3 ) ^( •) ,jg i =jg2 jg3 =jg4

etc.

Since (2.634) is again the Fourier coefficient, then the action of superposition SlSm on (2.634) is obvious. For example, for r = 3

c c c ) n(p)

S3S2S1 ^ <Cjk...jq...ji

q3 l{â'2s=â'2s-i+l}Qfc---ii

2 s=1

( jg2 jg i) •) ( jg4 jg3) ^( •) ( jg6 j'g5) •) jg i =j'g2 jg3 =j'g4 jg5 =j'g6

S3 Si < (7,

(p)

jk ...jq ...ji

1 to

22

jg3 =P+1

( jg2 jg i) ^( •) ( jg6 jg5) ^( •) ,jg i =jg2 'j'g3 =j'g4 'j'g5 = jg6

(p)

jk-jq ...ji

TO TO

1

{54=53+1}

E E Cjk...j'i

jgi =P+1 jg5 =P+1

( jg4 jg3) ^( •) ,jg i = jg2 'j'g3 = j'g4 jg5 =j'g6

Theorem 2.30 [32], [37], [38], [63]. Assume that the continuously differ-entiable functions ) (/ = 1,...,k) at the interval [t,T] and the complete ort.honormal system {(f)j(x)}jt0 of continuous functions {(f>o{x) = 1/y/T — t) in the space L2([t,T]) are such that the following conditions are satisfied:

1

1. The equality

1

2

t2

$l(il )$2(tl)dtl = E / ^(^j fe) / ^l(tl)0j (ti)dtidt2 (2.635)

j=0 { t

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holds for all s £ (t, T], where the nonrandom functions $1(t), $2(t) are continuously differentiable on [t,T] and the series on the right-hand side of (2.635) converges absolutely.

2. The estimates

*i(s) ' T

(t )$i(t )dT

<

j l/2+a '

(t )$2(t )dt

<

ttl(s)

jl/2+a '

oo

e / ^2(t)0j (t) / (0)d0dt

j=p+l i t

<

*2(s)

P

ß

hold for all s £ (t,T) and for some a,^ > 0, where (t), $2(t) are continuously differentiable nonrandom functions on [t, T], j,p £ N, and

T T

J ^2(t)dT < oo, J |^2(t)| dT < oo. tt

3. The condition

p

p^o

lim

p

c c c J (p)

Sl2 . . Cjfc ...jq ...j

9=S1>S2>'">S2r-1>S2r

=0

holds for all possible g1, g2,..., g2r-1, g2r (see (12.6271)) and 11,12,..., such that 11,12,..., £ {1, 2,... ,r}, 11 > 12 > ... > d = 0,1, 2,... ,r - 1, where r = 1, 2,..., [k/2] and

CO Q J n(p)

Sl1 Sl2 . . . Cjfc...jq...j1

=f C(p)

jk ...jq...j1

q=g1,g2,...,g2r-1,g2r

for d = 0.

Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k

s

2

* T * Î2

J*[^(k)]Trik) = / (tk) - J ^1(t1)dwt(;i)... dwtik) (2.636) t t

the following expansion

pk

J * - . " » = l.i.m. £ Cjk . . ¿n j ' (2.637)

ji ,.-,j'k =0 1=1

that converges in the mean-square sense is valid, where

T t2

Cjk-ji = J fa (tk ) j (tk ) ...J (t1 )dt1.. .dtk (2.638)

tt

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, i1,..., ik = 0,1,..., m,

T

j = | & (T )dw<°

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[i) = f(i) for i = 1,..., m and w[0) = t.

Proof. The proof of Theorem 2.30 will consist of several steps.

Step 1. Let us find a representation of the quantity

pk

E Cjk..ji n j)

j'iv-Jfc=0 1 = 1

that will be convenient for further consideration.

Firstly, note that (see (1.257) without passing to the limit l.i.m. and for $(ti,...,tk) = Kpi...pfc(ti,...,tk))

. . . . [k/2]

J[Kpi...pk]T,t = J [KPi...Pk]T,t — ^^( —1) X

r=1

r

X E ni{ig2S-i = ig2s =0}X

({{gi>g2}>'">{g2r-i>g2r}},{qi>--->qk-2r}) S=1

{gi,g2>'">g2r-i>g2r>qi>'">qk-2r }={i,2,---,k}

(i i )

x J[Kgi:::fr'qi-qk-2r]Tqtr"qk-2rJ (2.639)

w. p. 1, where J[Kpi .pk]y,t"ifc) is the multiple Stratonovich stochastic integral

(11.16) (also see (2.994)) and J'[Kpi...pfc^) is the multiple Wiener stochastic integral (03) (also see (OL3)), '

Pi Pk k

Kpi...pfc (ti,... ,tk) = £ ... E Cjk.j n jjl (t/), (2.640)

ji=0 jfc=0 /=1

K ffi...ff2r ,qi...qfc-2r (t t ) =

Kpi...pk (tqi,. . . , 6qfc-2r) =

Pi Pfc r k-2r

E . . . E Cjfc...ji n 1{jg2s-i = jg2s}H j(tql). (2.641)

ji=0 jfc=0 S=1 /=1

Passing to the limit l.i.m. (p1 = ... = pk = p) in (2.639), we get w. p. 1

Piv,Pfc — TO

(see Theorems 1.1, 1.2)

P k [k/2]

l.i.m. £ Cjk Cj(lil) = J [^(k) ) - l.Lm. £(-1)r x

p—to z—» .»..i. ' p—>-to z—»

jivjfc =0 / = 1 r=i

r

x E n1{ig2s-i = ig2s =0}x

({{Si>S2}>'">{S2r-i>S2r }}>{9i .■■■>9fc-2r }) S = 1 {si,S2>'">S2r-i>S2r>9i>'">9fc-2r } = {i>2>'">k}

x J,9i-9fc-2r ]Ti9ti •••igfc-2r ) =

P [k/2]

= J) - am. £ Cjk jE(-1)rx

' p—to z—' z—'

Jij^-^jj^=0 r=1

k-2r

(i9l )

x E n 1{ig2s-i = ig2s =0}1{jg2s-i = j»2. } II ^l

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({{Si>S2}>'">{S2r-i>S2r }}>{9i .■■■>9fc-2r}) S = 1 /=1

{Si,S2>'">S2r-i>S2r>9i>'">9fc-2r } = {i,2>'">k}

(2.642)

r

where J[^(k)]Ti'''ifc) is the iterated Ito stochastic integral

t t2

J№(k)]Ti'''ifc) = / (tk). . . i ^i(ti)dwt(;i)... dw£(2.643)

If we prove that w. p. 1

[k/2] . p [k/2] E ■w(k)rfr!' = -i^m. E Ef-1)''*

r=1 (sr ,'",Si)GAfc,r ji ,'",jfe =0 r=1

r k—2r

X

E n 1iig2s-i = ig2s =0} 1{jg2s-i = j»2. } II C'

(2.644)

({{Si,S2}>'",{S2r-i>S2r }},{qi v>9fc-2r}) S = 1 1=1

{Si,S2>'">S2r-i>S2r>9i>'">9fe-2r } = {i>2>--->k}

then (see (2.642), (2.644), and Theorem 2.12)

p k

^m- £ ^-n z

ji,''',jfc =0 1=1

(il)

[k/2] 1

+ £ £ J[^k)]irai = rW^tt'^ (2.645)

2r

r=1 (sr ,'",Si)GAfcjr

w. p. 1, where notations in (2.645) are the same as in Theorem 2.12. Thus Theorem 2.30 will be proved.

From (2.639) we have that the multiple Stratonovich stochastic integral J[KPi'"Pfcfc'"^ of multiplicity k is expressed as a finite linear combination of

the multiple Wiener stochastic integral J'[KPi'"Pfc^¿''^ of multiplicity k and multiple Stratonovich stochastic integrals

T [K gi'''g2r ,qi'-qfc-2r ](iqi '''iqk-2r )

J [KPi'"Pfc ]T,t

of multiplicities k — 2, k — 4, ..., k — 2[k/2]. By iteratively applying the formula (12.639) (also see (l1.46l)—(fTT50)), we obtain the following representation of the

multiple Stratonovich stochastic integral of multiplicity k as the sum of some constant value and multiple Wiener stochastic integrals of multiplicities not exceeding k

J [K ^i-^ ) = J' [ K ] (i i •••ik ) ++

J [KPl•••Pfc]T,t = J [KPi-Pk]T,t +

[k/2]

+ E E n1{ig2s-i = ig2s =0}x

= 1 ({{Si>S2}>'">{S2r-i>S2r }}>{9i>--->9fc-2r }) {Si,S2>'">S2r-i>S2r>9i>'">9fc-2r } = {l,2>'">k}

(i i )

x J'[Kgl;;;g2r'ql•••qk-2r^•••9k-2r) (2.646)

w. p. 1, where Kp^^(ti,...,tk) and K^-J^2r(t9l, ...,tqk-2r) are defined by the equalities (1227640), (2641).

From (2.646) we have

Pi Pk k Pi Pk

TTY (il) = v^ z (ii) Z (ik) =

... ^j'k-j'i 11 zjl = .../-*/ Cjk •••jiz ji ... j = ji=0 jk=0 /=1 ji=0 jk=0

Pi Pk

= E .-^Cjk • • ji J '[j... jjklife- • '+

ji=0 jk=0

Pi

Pk

+E -Ec

jk • • jl

ji=0

jk=

[k/2] E

r=1

E

({{3l>32}>--->{32r-l>32r }},{9l>--->9k-2r }) {3l,32v>32r-l>32r>9l>--->9k-2r } = {l>2>--->k}

1

S = 1

{i

g2s-1

=0}

X

0

x 17 = 7 J'j ...j fr• • iqk-2r) w. p. 1. (2.647)

{j32s-l = j32s } Lrj9i rj9k-2rJ T,t J^ v /

The formulas (2.646), (2.647) can be considered as new representations of the Hu-Meyer formula for the case of a multidimensional Wiener process [127

(also see [124], [126]) and kernel KPl • • ,Pk(t1,... ,tk) (see (2.640)).

Further, we will use the representation (2.647) for p1 = ... = pk = p, i.e.

p k p

£ cv^nj1 = £ Cjk'jij'[j...jfr"'+

j'ivjfc =0 1 = 1 ji r-'Jfc =0

p [k/2] r

+ E ^jk' ' JiE E ll1{ig2s-i = ig2s =0iX

ji, ' ' ',j'fc =0 r=1 ({{Si,S2},---,{S2r-i.S2r }},{9i .■■■,9fc-2r}) S = 1

{Si,S2>'">S2r-i>S2r>9i>'">9fc-2r } = {i,2>'",k}

X 1(j,2,-i = j»2, } J'[^j«i . ■ ■ jk-2r ]T? ' ' *'k-2' ) w. p. 1. (2.648)

For example, for k = 2, 3, 4, 5, 6 from (2.647) we have w. p. 1

pi p2 pi p2

£ £ Cj2jiZjij = J'[Kp,p2If + £ £ j 1{!,=,2=0)1{ji =j2}, (2.649)

ji =0 j2=0 ji=0 j2 =0

Pi P2 P3

V^ V^ V^ C Z (ii)Z (i2) Z(i3) = 7"'[K ] (iii2i3) +

Cj3j2ji j Zj2 Zj3 = J [KPiP2P3]T,t +

ji=0 j2=0 j3=0

pi p2 p3

+ E E E Cj3j2jJ 1{ii=i2=0}1{ji=j2} J'[ j]T,i + 1{i2=i3=0}1{j2=j3} J'[ j]T,i + ji =0 j2=0 j3=0 \

+1{ii=i3=0}1{ji=j3}J'[0j-2^ | , (2.650)

Pi P4

EV^ C Z (ii)Z (i2) Z (i3)Z (i4) = 7'[K ] (iii2i3i4) +

. . . / v Cj4j3j2ji j Sj2 Sj3 Sj4 = J [KPiP2P3P4]T,t +

ji =0 j4=0

pi p4

+ y^ . . . ^^ Cj4j3j2ji i 1{ii = i2=0}1{ji=j2} J [j j ^ +

ji =0 j4=0 V

+ 1{ii = i3=0} 1{ji=j3} J' [ j j^ + 1{ii=i4=0} 1{ji=j4} J' [j j] Tti3) + + 1{i2 = i3=0} 1{j2=j3} J' [ j j]T,i ^ + 1{i2=i4=0} 1 {j2 = j4} J' [^ji j] T,i ^ + http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal. http://diffjournal.spbu.ru/ A.480

+1 {23 = 24=0} 1 {j3=j4 } [ j j ] Ti,1ti2) + + 1{il=i2=0}l{j1=j2}1{i3=i4=0}l{j3=j4} + 1{il=i3=0}1{j1=j3}1{i2=i4=0}1{j2=j4} +

+ 1{i1=i4=0}1{j1=j4}1{i2 = i3=0}1{j2=j3}

(2.651)

pi

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P5

E\ ^ C Z (il) Z (^2) Z (^3) Z (^4) Z (^5) _ 77 [K ] (21^2232425)

• • ■ / J Cj5j4j3j2j1 Zj1 Zj2 Zj3 Zj4 Zj5 _ J [KP1P2P3P4P5 ]T,t

+

¿1=0 j5=0 P1

P5

+ £ • • • E Cj5j4j3j2j1 ( 1{21 = 22=0}1{j1=j2} [ j j j ^ + ¿1=0 j5=0

+ 1{21 = 23=0}1{j1=j3}j/ j j ^ + 1{21 = 24=0}1{j1=j4}j/ [ j j j fe^5^ + 1{21 = 25=0} 1{j1=j5} j j] T2,2t2324) + 1{22 = 23=0} 1{j2=j3} [ j j j ]T,i ^ +

+ 1{22 = 24=0} 1{j2=j4} j j] T21t2325) + 1{22 = 25=0} 1{j2=j5} [ j j j]T,i ^ +

+ 1{23 = 24=0} 1{j3=j4} j] Tr^ + 1{23 = 25=0} 1{j3=j5} [ j j]T21t2224) +

+ 1{24 = 25=0} 1{j4=j5} j] Tr^ +

+ 1{21 = 22=0} {¿1 = =j2} {23 24=0} {j3 = =j4}J j ]T25t)+

+ 1{21 = 22=0} {¿1 = =j2} {23 25=0} {j3 = =j5}J j ]T24t)+

+ 1{21 = 22=0} j =j2} {24 25=0} { j4 = =j5}J

+ 1{21 = 23=0} j =j3} {22 24=0} {j2 = =j4}J

+ 1{21 = 23=0} j =j3} {22 25=0} {j2 = =Mj/ j ]T24<)+

+ 1{21 = 23=0} j =j3} {24 25=0} {¿4 = =j5}J

+ 1{21 = 24=0} j =j4} {22 23=0} {j2 = ]&'+

+ 1{21 = 24=0} j =j4} {22 25=0} {j2 = =j5}J

+ 1{21 = 24=0} j =j4} {23 25=0} {j3 = =j5}J ^¿2 ]T2,2t) +

+ 1{21 = 25=0} j =j5} {22 23=0} {j2 = =*}J j ]« +

+ 1{21 = 25=0} =j5} {22 24=0} {j2 = =j4}J

+1 {i1 = i5=0} 1 {ji =j5} 1 {i3=i4=0} 1{j3=j4} [j]Zt + +1 {i2 = i3=0} 1 {j2=j3} 1{i4=i5=0} 1 {j4=j5} [0j1]T^ +

+ 1{i2=i4=0}1{j2=j4}1{i3 = «5=0}1{j3 =

Ti ](«i) I

ji ]T,t +

+ 1{i2 = i5=0}1{j2=j5}1{i3 = i4=0}1{j3=j4} [0j1 fei I ,

(2.652)

P1

P6

E\ A C Z (i1)z (i2)z (i3)z (i4)z (i5)z (i6) _ V [K ] (¿1«2i3i4i5«6)

• • • / j Cj6j5j4j3j2j1 Zj1 Zj2 Z j3 Zj4 Z j5 Zj6 _ J [KP1P2P3P4P5P6] T,t

+

j1=0

j6=0 P1

P6

+ • • • Cj6j5j4j3j2j1

j1 =0 j6=0

1{i1=i6=0} 1{j1=j6} [0j2 j j j] T,t

(¿2i3«4«5)

+

+1{i2 = = ¿6=0} {j2 = j J' [0j1 j j 0j5] T^ri4i5 )+1 {¿3 = =¿6=0} {j3 = j } J' [ j j j j] T,^^ +

+1{i4 = = ¿6=0} { j4 = =j6}J [0j1 0j2 j 0j5]Ti,fi3i5)+1{i5 = =¿6=0} {j5 = j J' [ j j j j] T^"^

+1{i1 = = ¿2=0} j =j2}J [ jj j j]Ti,3ii4i5i6)+1{i1 = =¿3=0} j j [ j j 0j5 j]

+1{i1 = = ¿4=0} j =j4}J [0j2 j j j]Ti,2ii3i5i6)+1{i1 = =¿5=0} j j } [ j j j j] ^T^

+1{i2 = =¿3=0} {j2 = =j3}J [0j1 j j j]Ti,fi5i6)+1{i2 = =¿4=0} {j2 = j J/ [ j j j j] STT^^

+1{i2 = = ¿5=0} {j2 = =j5}J [0j1 j j j]Ti,fi4i6)+1{i3 = =¿4=0} {j3 = j J' [ j j 0j5 j]T¿!t¿2¿5¿6)+

+1{i3= = ¿5=0} {j3 = =j5}J [0j1 0j2 j j]Ti,fi4i6)+1{i4 = =¿5=0} { j4 = j } [ j j j j]T¿!t¿2¿3¿6)+

+ 1{¿1 = ¿2=0} =j2} {¿3 ¿4=0} {j3 = =j4}J 0j5 0j6]^6) +

+ 1{¿1 = ¿2=0} =j2} {¿3 ¿5=0} {j3 = =j5}J 0j-4 0,6 ]T¿r)+

+ 1{¿1 = ¿2=0} =j2} {¿4 ¿5=0} {j4 = =j5}J 0j6 ]TT,i ) +

+ ^¿1 = ¿3=0} =j3} {¿2 ¿4=0} {j2 = =j4}J

+ 1{¿1 = ¿3=0} =j3} {¿2 ¿5=0} {j2 = =j5}J j j]?f+

+ 1{¿1 = ¿3=0} =j3} {¿4 = ¿5=0} { j4 = =j5}J 0j2 0j6 ]Tr,i ) +

+ 1{¿1 = ¿4=0} =j4} {¿2 ¿3=0} {j2 = =j3}J

+ 1{¿1 = ¿4=0} =j4} {¿2 ¿5=0} {j2 = =j5}J 0j3 0j6]'Z¿3t 6) +

+1{i6= =«3=0} {j6 = =j3} {«1 = = «2=0} {¿1 = =j2}J' [^4 0j5 ]T«r) +

+1{i6= = «4=0} {j6 = =j4} {«3 «5=0} {j3 = =j5}J' ^2 fe«^

+1{i6= = «4=0} {j6 = =j4} {«2 = «5=0} {j2 = =j5}J' [^¿1 ^¿3 ]T«r) +

+1{i6= = «4=0} {j6 = =j4} {«2 = =«3=0} {j2 = =j3}J [^¿1 ^5 ]T«1t«5) +

+1{i6= = «4=0} {j6 = =j4} {«1 «5=0} {¿1 = =j5}J' [^2 ^3 ]T«"3) +

+1{i6= = «4=0} {j6 = =j4} {«1 =«3=0} {¿1 = =j3}J' [^2 ^¿5 ]T«"5) +

+1{i6= = «4=0} {j6 = =j4} {«1 =«2=0} {¿1 = =j2}J' [^3 ^5 ]T«3«5) +

+1{i6= = «5=0} {j6 = =j5} {«3 «4=0} {j3 = =¿4}/^ ^ ]T«,f) +

+1{i6= = «5=0} {j6 = =j5} {«2 «4=0} {j2 = =¿4}/^ ^¿3 ]T«,f) +

+1{i6= = «5=0} {j6 = =j5} {«2 =«3=0} {j2 = =MJ/ [^¿1 ^¿4 ]T«r) +

+1{i6= = «5=0} {j6 = =j5} {«1 «4=0} j =¿4 } J' [^2 ^3 ]T«2«3) +

+1{i6= = «5=0} {j6 = =j5} {«1 =«3=0} j =MJ/ [^2 ^4 ]T«2«4) +

+1{i6= = «5=0} {j6 = =j5} {«1 =«2=0} j =¿2 } J' [^3 ^4 ]T«"4) +

+ 1{«6 = «1=0} 06= =¿1} {«2 = «5=0} 02 = =¿5 } {«3 «4=0} 03 = =¿4} +

+ 1{«6 = «1=0} 0'6= =¿1} {«2 «4=0} 0'2 = =¿4} {«3 «5=0} 03 = =¿5} +

+ 1{«6 = «1=0} 0'6= =¿1} {«2 «3=0} i?2 = =i3} {«4 = «5=0} 0*4 = =¿5} +

+ 1{«6 = «2=0} 06= =i2} {«1 = «5=0} {¿1 = =¿5 } {«3 «4=0} 03 = =¿4} +

+ 1{«6 = «2=0} 0'6= =i2} {«1 «4=0} {¿1 = =¿4} {«3 «5=0} 03 = =¿5} +

+ 1{«6 = «2=0} 0'6= =i2} {«1 «3=0} {¿1 = =J3} {«4 = «5=0} 0*4 = =¿5} +

+ 1{«6 = «3=0} 0'6= =i3} {«1 = «5=0} {¿1 = =¿5 } {«2 «4=0} 0*2 = =¿4} +

+ 1{«6 = «3=0} 0'6= =i3} {«1 «4=0} {¿1 = =¿4} {«2 «5=0} 0*2 = =¿5} +

+ 1{«3 = «6=0} 03 = =i6} {«1 = «2=0} {¿1 = =¿2 } {«4 = «5=0} 0*4 = =¿5} +

+ 1{«6 = = «4=0} 0'6= =¿4} {«1 = «5=0} {¿1 = =¿5 } {«2 =«3=0} 0*2 = =¿3 } +

+ 1{«6 = = «4=0} 0'6= =¿4} {«1 «3=0} {¿1 = =J3} {«2 «5=0} 0*2 = =¿5} +

+ 1{«6 = = «4=0} 0'6= =¿4} {«1 = «2=0} {¿1 = =¿2 } {«3 «5=0} 03 = =¿5} +

+ 1{«6 = «5=0} 0'6= =i5} {«1 «4=0} {¿1 = =¿4} {«2 =«3=0} 0*2 = =¿3 } +

+ ^=¿5=0^6=5} ^¿^¿2=0} ^j^} ^=¿4=0} ^^^

+ 1{¿6 = ¿5=0}1{j6=j5}1{¿1=¿3=0}1{j1=jз}1{¿2=¿4=0}1{j2=j4^ • (2.653)

Note that the relation (2.651) can be written in the following form

P1 P4 P1 P4

y^ • • • ^^ Cj4j3j2j1 Cj^ Zj2 Cjjj Cj4l _ ^^ • • • ^^ Cj4j3j2j1 J [0j10j2 j j] T,t +

j1=0 j4=0 j1=0 j4=0

P3 P4 /min{p1,p2} \

+ 1{¿l=¿2=0} E E I E Cj4j3j1j1 I [0j3 j]tJ4) + j3=0 j4=0 \ j1=0 /

P2 P4 / min{P1,P3} \

+ 1{¿1 = ¿3=0} E E I E Cj4j3j2j3 I [j j^^

j2=0 j4=0 \ j3=0 /

P2 P3 /min{p1,p4} \

+ 1{¿1 = ¿4=0} E E I E Cj4j3j2j4 l [j j]Zf¿,2t¿3 +

j2=0 j3=0 \ j4=0 /

P1 P4 /min{p2,P3} \

+ 1{¿2 = ¿3=0} E E I E Cj4j3j3j1 I [0j1j]Zz¿!t¿4) + j1=0 j4=0 \ j3=0 /

P1 P3 /min{P2,P4} \

+ 1{¿2 = ¿4=0} E E I E Cj4j3j4j1 l [0j1j] Zг¿,1t¿3 +

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j1=0 j3=0 \ j4=0 /

P1 P2 / min{P3,P4} \

+ 1{¿3 = ¿4=0} E E I E Cj4j4j2j1 l [0j1j Iz^,1!;12 +

j1=0 j2=0 \ j4=0 /

min{p2,P3} min{p1,p4}

+ 1{¿2 = ¿3=0}1{¿1 = ¿4=0} E E Cj4j2j2j4 +

j2=0 j4=0

min{p1,p3} min{p2,P4}

+ 1{¿2 = ¿4=0}1{¿1 = ¿3=0} E E Cj4j3j4j3 +

j3=0 j4=0

min{p1,p2} min{p3,p4}

+ 1{¿3=¿4=0}1{¿1=¿2=0} E E Cj4j4j2j2 W. P. 1

j2=0 j4=0

Step 2. Let us prove that

y ^ Cifc•••¿i+l ¿'(¿'1-1 •••¿s+1ii¿^...¿1 — 0 (2.654)

¿i =0

or

oo

^'¿k•••¿1+1^1 ¿'г-l•••¿s+l¿'г¿'«-^••¿'l ^'¿k•••¿1+1 ¿(¿I-1 •••¿'s+lii¿'«-^••¿'l, (2.655)

¿i =0 ¿i=P+1

where l — 1 > s + 1.

Our further proof will not fundamentally depend on the weight functions ^i(t),... (t). Therefore, sometimes in subsequent consideration we assume for simplicity that ^i(t),... (t) = 1.

We have

C........—

^¿k ••¿+¿1 ¿I-l •••¿«+¿1 ¿«-^"¿l

T ti+2 ti+1 t(

— y ^¿k (tk) •••^¿1 y ^¿i (ti ^ ¿1 (ti—1)... t t t t

^s+2 ts+1 ts

... J ^¿'s + 1 / ^¿l ^¿«-1 (ts—1) ...

t t t

t2

.../<¿¿1 ■..■..—

t

T t« + l ts t2

^¿s+1 ^ y ^¿l ^¿s-l (ts —1) ...J M (t1)dt1 ...dis —1dts X

t t t t

/ T T T T

X ( y ^¿'s+2 (ts+2) ... J ^¿i-l (tl—1) J ^¿i (tl ^ ^¿i+l (tl+1) ...

\ts + 1 ti-2 ti-l ti

T

.. .J ^¿k (t* )dtk ... dti—1... dts+2 ) dts+1 —

tk-1

p

T ts + 1 ts t2

I 0js+1 fc+J / j 0js_1 (ts-1) •••J j (t1)dt1 •••dts-1 dtsx

t t t t

Gjs-1'"j1 (ts)

T T T

xy ^ji (ti ^ 0ji+1 (ti+1) •••J j (tk )dtk •••¿¿1+1 x

ts+1 ti tk-1

Hjfc-J'i+1(ti)

x

ti

ts+3

0ji-1 (t1-1) ••• s+2 (ts+2)dts+2 • • • dt1-1

ts+1 v--

V

Qji-1-"js+2 (ti'ts + !)

dts+1 _

/

T ts + 1

_y 0js+1 fc + J / 0ji (tS )Gjs-1...j1 (tS )dts X tt

T

..ji + 1 (t1 )Qji-1...js+2 (t1 , tS+1)dt1 dtS + 1-

(2.656)

Using the additive property of the integral, we obtain

Qj'i-1-j's+2 (t1, tS+1) _

ti !s+3

_ J j (t1-1) • • •J 0js+2 (ts+2)dts+2 • • • dt1-1 _

!S + 1 !S+1

ti !S+4 !S+3

_ J j (t1-1) •••J 0js+3 (ts+3^ 0js+2 (tS+2)dtS+2dtS+3 • • • dt1-1

!S + 1 !S+1 t

ti !S+4 !S+1

- j (t1-1) ••• 0js+3 (ts+3)dts+3 • ••dt1-1 / 0js+2 (ts+2)dts+2

ts+1

t

l

ts+1

t

1

s

— E "(*<"(ts+l), d< to. (2.657)

m=1

Combining (2.656) and (2.657), we have

p

•¿i+l¿¿i-1 •••¿s + 1¿l¿s-1•••¿1

E C'k•••

¿i =0

d / T P t« + 1

E / ^¿'s + 1 ^O^t^ ^E / ^¿i (tS ^•••¿l (tS )dtS X

¿i =0 t

T

(m)

X / (t)Hjk•••¿i+l(ti¿•^(ti)dtidts+1 . (2.658)

¿i — 1 •••¿s+2

ts+1

Using the generalized Parseval equality, we obtain

ts + l T

to „ „

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E J ^(ts^•••¿l(ts^(ti)Hjk•••¿i+l(ti^(ti)dti

¿l=0 t ts+1

T

1{T<ts+l}Gs-l•••¿1 (t) • 1{T>ts+l}Hk•••¿i+l(t(t)dT — 0. (2.659)

From (2.658) and (2.659) we get

p

■¿i+l¿¿i-l •••¿s + 1¿l¿s-1•••¿1

E C'k•••

¿i =0

d / T TO ts+1

E / ^¿'s+l ^O^L^ (tS+1) E J ^¿'i (ts )G¿s-1•••¿1 (ts )dts X

m=1 V t ¿'i =p+1 t

T

(m)

x / ^(ti)Hjk•••¿i+l(ti(ti)dtidts+j . (2.660)

¿i — 1 •••¿s+2

ts+1

Combining Condition 2 of Theorem 2.30 and (I2.656l)-(l2.658l). (12.6601). we have

p

jl+l jiji-l •••Js + ljJs-l-jl

E •••. ji =0

œ d / T ts+l

EE/ ^js+l^s+oj!^(is+iW fa(ts^s-l-j(ts)dtsx

ji =p+1 m=1 \ t t

x j (ti )Hjk •••ji+l (ti (ti )dti dts+i

oo

T

(m)

'ji-l—js+2 '

ts + l

T t;+2 t; + l ti

E / j (tk) • j j / j (t) / ^ji-l (tl-1) ••• j =p+1 t t t t ts+2 ts+l ts

■J 0js + l ^ / fas-l (tS-1) • • •

t t t

t2

(t1)dt1 .■■M^ <ftm ...«ft, =

t

œ

= — ^ ^ Cjfc •••ji + lji ji-1 •••js+l jijs-l •••jl ■ (2.661)

j =p+1

The equality (2.661) implies (2.654), (2.655).

Step 3. Using Conditions 1 and 2 of Theorem 2.30, we obtain

p

^ ^ Cj'fc •••ji + lji j ji-2 •••jl = ji=0

T tj+2

= y fa (tk ) j (tk) ■■■ J fa (tz+1)0ji+l (ti+1)x tt p ti+l ti

xE / (t^(t) / (ti-1)x ji=o t t

ti-1 t2 X J ^i(ti—2)0¿l-2(ti—2)... y ^(tO^-l(t1)dt1... dti—2dti—1dtidti+1... dtk —

tt

T ti+2

— J ^k (tk ^¿k (tk ) ... J ^ (ti+1)0ii+l (ti+1)x tt TO tl+1 ^

XE / ^(t^¿'i(t) / ^i- 1(ti—1)^ii(ti—1)x ¿i =0 t t tl-1 t2

X y ^(ti—2)0ji-2(t/—2).. (t1)dt1... dt/-2dt/-1dt/dtw.. .dt*-

tt

TO

— ^ ^ C¿k•••¿'l+1¿'l¿'l¿'l-2•••¿'l — ¿'l =P+1

-ic- ■ ~~ 2 ^jfc-ji

^¿k •••¿'l+1¿'l¿'l¿'l —2 •••¿1 . (2.662) ¿¿lWO ¿'l=p+1

Step 4. Passing to the limit l.i.m. in (2.648), we have (see Theorem 1.1)

p—>-TO

p

lp——TO. £ CV^Ci««1'...cir> — J>+

¿'lvJk=0

[k/2] r

+E E n^-l = «g2s=0}X

r=1 ({{S1,S2}>'">{S2r-1>S2r }}>{91>--->9k-2r}) S = 1 {S1,S2>'">S2r-1>S2r>91 >'">9k-2r } = {1>2>'">k}

x >p—^ £ •^11 ¿-l = ¿»2,>J'^ ■.. ¿JST«"-*1 (2'663)

¿'l ,-Jk =0 s=1

w. p. 1.

Taking into account (2.655) and (2.662), we obtain for r — 1

p

E, («. «. ) ^ ••¿1 ^l = ¿»2 } J ^¿.1 . . . ^k-2 9k-2 —

¿'l ,-Jk =0

1

{¿ai = ¿g2 =0}J

*32' J P—to

Li.m. E E Cjk-j1

rt—^rsr, f * * *

jg1 =P+1 Ji>■■■>J9>■■■>Jfc = 0

9=31 >32

1{g2>g1 + 1}X

j31= j32

XJ [j •••0jqk-2 ^ ••■¿9fc-2 ) +

+ 1{¿31 = ¿32 =0}

1

P—TO z—' 2

Ji>■■■>J9>■■■>Jfe = 0

9=31 >32

1

(j32 j31 = j32

{g2=g1 + 1}X

XJ [j •••0j9k-2 ]T,t

1 (¿91 •••¿9fc-2 )

OO

1

¿ = ¿ =0 .m *31~ ^^ p—TO

{¿3i = ¿32 =0}J

'n—^rsr, f * * *

j31 =P+1 j1 >""">j9>""">jk = 0

9=31 >32

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1{g2=g1 + 1}X

j31 j32

xj j ••• ¿V^-2'

OO

1

{¿31 = ¿32 =0}

Li.m. E E Cjk...j1

P—to z-' z-'

j31 =P+1 j1 >■■■ >j9 >""">jk = 0

9=31 >32

X

j31 j32

XJ [j •••0j9k-2 ^ ."¿9k-2 ) +

+1

{¿3l = ¿32 =0} p^TO

p—^ < * 2

j1 >...>j9>.. ■ >jk =0

9=31>32

X J' [j • • • 0j9k-2 ]T,i

(j32 j31 = j32

^91 ...¿9^-2 )

1{g2=g1 + 1}X

(2.664)

2

1

{g2 =

T!t + 1{¿31 = ¿32 =0}l.i.m. RTPt1,g1'g2 w. p. 1, (2.665)

"91- >92^1 p——

where J[^(k)]T!t (g1 _ 1, 2, • • •, k - 1) is defined by (123701),

R

(P)1,g1,g2 T,t

EC (p)

Cjfc--.j9 ...j'!

j1 >■ ■■>j9 >--->jfc =0

9=31 >32

q=g1,g2

J' j ••• jJ?r%-2 '•

p

Let us explain the transition from (2.664) to (2.665). We have for g2 — g1 + 1

1

{«g1 = «g2 =0}

p—to z—' 2

j1 v,j9vj'k =0 9=31,32

X

«¿32 ¿31 M-)^ = ¿32

x/ ^¿91 . . . ^¿'qk-2 ]T,t

(«91 •••«9k-2 )

1p

j1 j---jjq j---jjk = 0

9=31,32

X

XC00)/ ^¿9) ...^-2 ]T,t

(¿32 ¿31 )^0,¿31 = ¿32 («91 • • • «9k-2 )

pp

=0 ¿m1 =0

9=31,32

X

¿32¿31 )Aiml ,¿31 = ¿32

xc«:.) [^¿91 ]T«r • -«^2 '

1 p p

j1,---,j9,---,jk =0 ¿:1 =0 9=31,32

X

X J' [^

:1 ^¿91 . . . ^¿9k-2 ]T,t

«¿32 ¿31 '^¿:) ,¿3) = ¿32

(0«9) • • • «9k-2 '

(2.666)

w.p.l,

(2.667)

where

T

— y ^k (tk ^(t*).. t

C

^k • • ¿1

t31 +3

«¿32 ¿31' 1 '¿3-1 = ¿32 ,g2 =g1+1

r31+2

J (tgl+2)0¿з1+2 (tg)+2^ &gi + 1(tgi (t tt

^31 Î2

x / fa (tgi-i)0jsi_i (tgi-i)... fa(ti)j (ti)dti... dtgi-idtgi dtgi+2 ... dtk,

t t

y y iVT^ ifjmi = o

j) _ J j (T)dwT0) _ J j (T)dT _ , (2.668)

t t [ 0 if jm1 _ 0

fa(r) = (2.669)

The transition from (2.666) to (2.667) is based on (1.43) or (2.1007).

By Condition 3 of Theorem 2.30 we have (also see the property (2.351) of multiple Wiener stochastic integral)

2

_ 0,

lim m ( 4pti,gi'g2)2) < K lim y (j . .

piœ [ \ / I piœ ^^ I jk---jq---ji

q=3i,32

where constant K does not depend on p. Thus

1{i3i = i32 ^¿m S ...jij = j32}J'[ j... 'iqfc-2}

*9r""9fc-2'

j'i,---,j'fc=0

= w. p. 1.

Involving into consideration the second pair {g3,g4} (the first pair is {gi,g2}), we obtain from (2.664) for r = 2

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2 p 2

n 1{i32s-i = i32s =°}lî;i;m. S Qk-ji n 1{j32s-i = j32s }X

s=i j'i,---,j'fe =0 s=i

xJ[j . • • ' =

2

X

1

{i32s-i = »32S =0}'

2s-i a2s •

s=i

£ [M"

j1 ,---,j9■ ,jk =0

9=31,32,33,34

1{

{g2s=g2s-1 + 1}

«¿32¿31 W¿33 MO^ = ¿32 '¿33 = ¿34 s=1

1TO ¿31 =p+1

1

«¿34 ¿33 '^«•',¿31 = ¿32 '¿S3 = ¿34

{g4=g3 + 1}

OO

2 53 ^ ••••'

¿33=p+1

«¿32 ¿31 ' ^ (•' '¿31 = ¿32 '¿33 = ¿34

1{g2=g) + 1} +

OO OO

+ £ £ Ck-•

¿33 =p+1 ¿31 =p+1

¿1

J' ^¿9) . . .

1 («9) • • • «9k-4 '

¿91 ¿9k-4 T't

— (2.670)

¿31 = ¿32 '¿33 = ¿34

12

4 s=1

S2'S1 ^11 1{«32s-1 = «32s =0}lpi.m. RT't

T't

(p)2'g1'g2'g3'g4

s=1

p—TO

(2.671)

w. p. 1, where g3 s2, g1 s1, (s2,s1) G Ak,2, J[&(k)]T2tS) is defined by (12.370) and Ak'2 is defined by (2.3711),

R

(p)2'g1'g2'g3'g4

T't

£

(7

(p)

¿k ¿9 ¿1

j1,...,j9,...,jk =0

9=31,32,33,34

q=g1'g2'g3'g4

-SW (7

(p)

¿k ¿9 ¿1

-S^ (7«

(p)

q=g1'g2'g3'g4

¿k ¿9 ¿1

x

q=g1'g2'g3'g4

XJ' ^¿9) . . . ^¿^k-4 ]T't

(«91 • • • «9k-4 '

Let us explain the transition from (2.670) to (2.671). We have for g2 — g1 + 1,

g4 — g3 + 1

1

l.i.m. -C.j,

p—TO ^ 4 ¿k • • ¿1

J'lv,j9vj'k =0 9=31,32 ,33,34

X

«¿32¿31 '^«•'«¿34¿33 '^«•''¿31 = ¿32 '¿33 = ¿34

X

n 1{«32s-1 = «32s =0} J' №¿91 . . . ^¿9k-4 ]T't

(«91 • • • «9k-4 '

s=1

2

1.

p

= ILkm" E (

4 piœ z—'

9=3i >32>33>34

X

(j32 j3i )^0(j34 j33 )^0,j3i = j32 ,j33 = j34

2

1{

=0}C00)z00)J [j ... jk-4

x ' 1 1{i32s-i = -32s =0}S0 S0 * LVj,i • • • T,t

s=i

(i9i ---i9k-4 )

1

i1^' e e fa-n

Ji,---,J9>'">Jfc =0 jmi ,jm3 9=3i,32,33,34

2

X

(j32 j3i )^jmi (j34 j33 )^jm3 ,j3i = j32 j33 = j34 (i9i ---i9k-4 )

H 1{i32s-i = Ss =0}Zj(îni j) [ j ' ' ' ^9fc-4 ]T,t

s=i

1

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i1^- £ £ Q.-*

ji ,■■■ ,j9 =0 jmi ,jm3 =

9=3i>32>33>34

X

(j32 j3i )^jmi (j34 j33 )Ajm3 'j3i = j32 'j33 = j34

X

2

II 1(-32,-i = -32, =0}J' [j j j . . . ] - ' '"-4 ' = (2-672)

s=i

J [^(k)]T2tSi w.p. 1.

(2.673)

The transition from (2.672) to (2.673) is based on (1.43) or (2.1007). Note that

C

jfc---ji

= C

( j32 j3 i ) ^jm i ,j31 j32

jfc---ji

( j3 i j3 i ) ^jm i ,j31 = j32

is the Fourier coefficient, where g2 = gi + 1. Therefore, the value

C

---ji

(j32 j3i )Ajmi (j34 j33 )A,jm3 ,j3i = j32 J33 = .fa

=C

jfc---ji

(j3i j3i )Ajmi (j33 j33 )ajm3 ,j3i = j32 ,j33 = fa

p

0

p

0

is determined recursively using (2.630) in an obvious way for g2 — g1 + 1 and

g4 — g3 + 1.

By Condition 3 of Theorem 2.30 we have (also see the property (2.351) of multiple Wiener stochastic integral)

lim Ml R

p^œ

>(p)2,gi,g2,g3,g4

<

< K lim V

p^œ ^—'

(p)

jk • • j • • jl

q=31>32>33>34

+

+ SW Cj

(p)

jk-• -jq • • -jl

+ S^ (7

(p)

9=01,02,03,04

jk jq j1

q=01,02,03,04

= 0,

where constant K is independent of p. Thus

p

2

II1!^-! = Ss^lim- . S (jk • • j^n 1{jg2s-1 = j*2*}x

s=1

j1il. VS2s-1_ •ys2s-j'l, • • • ,jk =0 s = 1

XJ'

^j'qk-4]

(iq1 • • • *qk-4)

jq1 jqk-4 T,t

7 il 1{02S=02S-l + l}'^

s2,s1 w p 1

s=1

where =f s2, g1 =f s1; (s2, s1) G Ak,2, J[^(k)]T2tS1 is defined by (12.370) and Ak,2 is defined by (2.371).

Involving into consideration the third pair {g6,g5} ({g1,g2} is the first pair and {g4,g3} is the second pair), we obtain from (2.670) for r = 3

n 1{ig2s-1 = Ss ^lim. S (jk • • • 1 n 1{j's2s-1 = j*2s }X

s=1

j'1, • • • ,jk =0

s=1

XJ' [ j •••0.

](iqr • • iqk-6 )

yjqk-6 ]T,t

n

s=1

1

{i

g2s-1 "g2s

iff,. =0}x

2

3

xlis- E ( ¿Ca-J.

j1 >■■■ >j9 >--->jfc = 0

9=31 >32 >33 >34 >35 >36

X

(j32 j31 W0(j34 j33 M0(j36 j35 WOfa = j32 ,j33 = j34 ,j35 = j36

3

X n 1{g2s=g2s-1 + 1}

s=1

OO

22 £ (

j31 =p+1

(j34j33 W0(j36 j35 M0j31 = j32 ,j33 = j34 ,j35 = j36

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1{g4=g3 + 1}1{g6=g5 + 1}

1 TO

22 ^jk-jl j'33 =p+1

(j32 j31 W0(j36 j35 M0j31 = j32 'j33 = j34 ,j35 = j36

1{g2=g1 + 1}1{g6=g5 + 1}

OO

22 ^••••' j35 =P+1

(j32 j31 )^0)(j34 j33 WOfa = j32 'j33 = j34 ,j35 = j36

1{g2=g1 + 1}1{g4 =

1

OO OO

+ 2 £ £ ^ifc-ii j'33 =p+1 j'31 =p+1

(j36 j35 WOfa = j32 'j33 = j34 ,j35 = j36

!{g6=g5 + 1} +

1

OO

+ 2 £ £ ^ifc-ii j'35 =p+1 j'31 =p+1

(j34 j33 WOfa = j32 'j33 = j34 ,j35 = j36

!{g4=g3 + 1} +

1

OO OO

j'35 =p+1 j'33 =p+1

1

(j32 j31 )^(^),j31 = j32 'j33 = j34 ,j35 = j36

{g2=g1 + 1}

OO OO OO

e £ £ Cjfc-j'1

j'35 =P+1 j33 =P+1 j31 =P+1

X

j31 j32 'j33 j34 ,j35 j36

XJ' [ j

1 (¿91 ."¿9fc-6 )

j91 Tj9fc-^ T,t

{g2s=g2s-1 + 1}

S3,S2,S1

T,t

s=1

+ TT 1 ¿ = ¿ =0\l.i.m. R

1 1 ^32s-1 = ¿32s =0}

(P)3,01,02,-,05,06

T,t

s=1

w.

. p. 1, where g^-i. == i _ 1, 2,3, (s3,s2,s1) G Ak,3, J[fak)]T3tS2'S1 is defined

by (2.370) and Ak,3 is defined by (2.371),

3

R

(p)3,01,02, • • • ,05,06 T,t

£

-â

(p)

jk jq j1

J1j---jJqj---jjk = 0

q=S1,S2>'">S5>S6

+

9=01,02, • • v05,06

+S^ (7.

(p)

jk jq j1

+ S^ (7,

(p)

9=0^02, • • •,05,06

jk jq j1

+

q^l^ • • • ,05,06

+S^ (7,

(p)

jk jq j1

q^l^ • • • ,05,06

—S3S1 < (7,

(p)

jk jq j1

— S3 S2 < (7,

(p)

q^l^ • • • ,05,06

jk jq j1

9=01^ • • •,05,06

— S2 S1 (7

(p)

jk jq j1

J' [ j • •

(iql • • • iqk-6)

9=01^ • • • ,05,06

By Condition 3 of Theorem 2.30 we have (also see the property (2.351) of multiple Wiener stochastic integral)

lim M < ( R

p^œ

jM3^^ • • • ,05,06

p

I < K lim V

I p^œ ^—'

(7

(p)

jk jq j1

jl j-- -,jq j---jjk = 0

q=Sl,S2>'">S5>S6

+

9=01 ^2, • • • ,05,06 .

+ Si < (7(

(p)

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jk jq j1

+ S^ (7

(p)

9=01^ • • • ,05,06

+ S3 < (7 j

(p)

jk jq j1

jk jq j1 2

+

9=01^ • • •,05,06

+

q=0l,02r • • ,05,06

+ | S3S^ j •jq • • •jl

q=0l,02,•••,05,06

+ I S3S^ C7j(p)-jq •••jl

+

9=01^ • • • ,05,06

+ S2S1 < (7(

( p )

jk jq jl

= 0,

9=01^ • • • ,05,06

where constant K does not depend on p.

2

Thus

y—m. n 1{¿32s-1 = ¿32s M—m. E Cj"¿1 II 1{j32s-1 = j32s }X

s=1

2s-1 32s ' - p—TO

j1,...,j'fc=0

s=1

3

t/I" / / I (¿91 ) __"1 f -I

X J [(pjqi . . . <Pjqk_6\tj _ 93 11 1{92s=92s-1

S3,S2,S1 w p 1

s=1

where ^¿-1 = «¿; i _ 1, 2,3, (s3,s2, S1) G AM, J[^(k)]T3;tS2'S1 is defined by (2370) and Ak,3 is

Repeating the previous steps, we obtain for an arbitrary r (r _ 1, 2, • • •, [k/2])

П1{¿32s-1 = ¿32s =0}lp.——m^ ^ jj-1 II 1{j32s-1 = j32s }X

s=1

j'1,...,j'fc =0

s=1

X J' [ j •

I ](¿91 •••¿9k —2r )

rj9fc-2r ]T,t

n

s=1

1

Ks-1 = ¿32s =0}X

P

X l.i.m.

1

p—to z—' 2'

j1 >■ ■■>j9 >■■ ■ >jk = 0 9=31 >32 >■■■ >32r-1 >32r

X

(j32 j31 )^(^)...(j32r j32r-1 WOfa = j32 '•••'j32r-1 = j32r

^ 1fe=S2,—1 + 1} J' [j •■■rj,k-2, iTi ".%-2' ' +

s=1

^ 1{¿32s-1 = ¿32s =0}LLm. RT,t

(P)r,g1,g2,...,g2r-1,g2r

s=1

p—TO

(2.674)

3

'

1 ' '

1{g2s=g2s-1 + 1} J [r ]T,t + H 1{¿32s-l = ¿32s =0}^. ^

S=1 S = 1

def

w. p. 1, where _ s; i _ 1, 2, • • •, r; r _ 1, 2, • • •, [k/2], (sr, • • •, s1) G Ak,r,

(2.675)

1, 2, • • •, [k/2], (sr ,•••,51)

J[fak)]Tr;T'S1 is defined by (2370) and Ak,r is

R

(p)r'g1'g2' • • • 'g2r-1'g2r

T't

£ ( ( —1¿9 • • ¿1

j1 ,...,j9,...,jk =0 9=31,32 ,...,32r-1,32r

+

q=g1'g2' • • • 'g2r-1'g2r

+(—1)r—^ S,A (?«?• ¿9 • • ¿1

i1=1

+

q=g1'g2' • • • 'g2r-1'g2r

r

+ ( —1)r—2 £ Si)SJ if ¿9 , , ¿)

i1,i2=1 l1>l2

+

q=g1'g2' • • • 'g2r-1'g2r

+ ( —11)1 E Si1 Si2 . . . Sir-^ ¿¿k'1 • ¿9• • ¿l

l1,l2,...,lr-1 = 1 11 >l2 > ■ ■ ■ > lr_1

X J ^¿9) . . . ^¿9k-2r ]T't

q=g1'g2' • • • 'g2r-1'g2r («91 • • • «9k-2r '

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/

X

(2.676)

Let us explain the transition from (2.674) to (2.675). We have for g2 —

g1 + 1, . . . ,02r — 02r—1 + 1

p

l.i.m.

p—TO 2

j1vj'9v,jk = 0 9=31,32,...,32r-1,32r

1

X

«¿32¿31 • • • «¿32r¿32r-1 '^«•''¿31 = ¿32 ' • • • '¿32r-1 = ¿32r

^ '1(«3,.-) = «32. =0} J ' [^¿9) ■■•0¿9k-2r]^T«9f)- • "9k-2' '

1{

s=1

1,.

—l.i.m.

2 r p —TO

53 ¿¿k • • ¿1

j1 ,...,j9 ,...,jk =0 9=31,32,...,32r-1,32r

X

«¿32¿31 • • • «¿32r¿32r-1 '^¿fl) = ¿32 ' • • • '¿32r-1 = ¿32r

X

1

s=1

{«32s-1 = «32s =0}

J' [0.

¿91

I ] («91 • • • «9k-2r '

. . ' ¿^k-2r ]T,t

p

r

—l.i.m.

2r p^œ

e E m

=0 jmi ,jm3.. . ,jm2r_i =0 s = 1

9=31>32v>32r_1>32r

{iS2s_1 = »32« =0}

X

xC

jk .. .j i

X

(j32 j31 . . . j32 j32r_1 )^jm2r_1 'j31 = j32 '. . . 'j32r_1 = j32r

'Jm^jm3 jm2r_ 1 LTjq1 r j?fc_2r J T ,t

1,.

—l.i.m.

2r p^œ

e e m

=0 jm1 ,jm3 ...,jm2r_1 =0 S = 1

9=31>32v>32r_1>32r

{i32s_1 = »32s =0}

X

xC.

jk ...j1

X

(j32 j31 )^jm1 ... (j32 j32r_1 )^jm2r_1 'j31 = j32 '...'j32r_1 = j32r

X J'

(00...0iqi ...»qk_2r )

(2.677)

1./ 2r

?,t...'s1 w.p. 1.

(2.678)

The transition from Note that

to (2.678) is based on (1.43) or

C

= C

( j32 j31 ) ^jm 1 ,j31 = j32

jk...j1

( j31 j31 ) ^jm 1 1 = j32

is the Fourier coefficient, where g2 — g1 + 1. Therefore, the value

C

jk...j1

(j32 j31 ...(j32dj32d_1 )^jm2d_1 'j31 = j32 '...'j32d_1 = j32d

— C

— Cjk...j1

(j31 j31 )^jm1 ...(j32d_1 j32d_1 )^jm2d_1 'j31 = j32 '...'j32d_1 = j32d

is determined recursively using (2.630) in an obvious way for g2 — g1 + 1,

02d — 02d—1 + 1 and d — 2,..., r.

By Condition 3 of Theorem 2.30 we have (also see the property (2.351) of multiple Wiener stochastic integral)

lim m ^ R

piœ

)(p)r,gi,g2,---,g2r-i,g2r

T,t

<

p

< K lim

piœ ^—'

ji =0 9=3i,32>--->32r-i>32r

C (P)

Cjfc ---j'9 ---ji

+

q=gi,g2,---,g2r-i,g2r .

+ E ( "M <Cj(p)--j9---j

ll=i

+

q=gi,g2,---,g2r-i,g2r

+ ^ I Sii5/J Cj(p)--j9---ji

¿i ,l2 = i li>l2

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+

q=gi,g2,---,g2r-i,g2r

+

s ) c (p)

Slr-0 Cj'fc ---j'9 ---ji

ii,i2,...,ir-i = i 11 > ¿2 >■■■ > _i

q=gi,g2,---,g2r-i,g2r

/

= 0,

where constant K does not depend on p. So we have

ll1{i32s-i = -32, ^pim. . E Cjk-»¿ill 1{j32s-i = ¿32s }X

s=i

ji,---,jfc=0 s=i

X J [^j9i • • • ^j9fc-2r ]T,t

(i91 ---i9k-2r )

2»' n ^-{928=928-1 + 1} J

sr 7---7S i

T,t

w. p. 1 ,

s=i

(2.679)

def

where g2i-i == s^; i = 1, 2,...,r; r = 1, 2,..., [k/2], (sr J[^(k)]Trt---'Si is defined by (23701) and Ak,r is defined by (2.3

si) G Ak,r,

2

r

1

2

r

2

p

r

Note that

E

({{31,32}v>{32r_1>32r }}>{91v>9fc_2r }) {31>32>--->32r_1>32r>91>--->9k_2r } = {1>2v>k}

A

g1,g3,...,g2r_1

g2=g1+1,g3=g2+1,...,g2r=g2r_1+1

— £ AS1,S2,...,sr, (2.680)

(sr ,...,S1)GÄk,r

where Ag1)g3)...)ff2r_1, As^,...^ are scalar values, g2»-1 — s»; i — 1, 2,...,r; r — 1, 2,..., [k/2] , Ak,r is defined by (2:371):

Ak,r — {(sr, ...,si): sr > sr-i + 1,...,s2 > si + 1, sr,..., si — 1,...,k — 1}.

Using (2.663), (2.679), (2.680), and Theorem 2.12, we finally get

p k p li- E ^..,nj1 — ^ E Ck..jcii1'-..cjr1

j1,...,jk=0 1=1 j1,...,jk=0

[k/2] 1

= JiMtt^ + £ J№ik)]trai = JM{kwk) (2-681)

2r

r=1 (sr ,...,s1)eAfc

r

w. p. 1, where (see (2.370))

r

def

J— n if-p=..,+1=0} X p=1

T tsr +3 isr + 2

><y ^k (tk ) ... J ^Sr + 2(tSr + 2^ y ^Sr (tSr + 1)^Sr + 1 (tSr + 1)X t t t

tsr + 1 ts1+3 ts1+2

X y ^sr — 1(tsr — 1) - j ^s1+2(ts1+2) J ^ s1 (ts1+1)^s1+1(ts1+1) X t t t

tS1 + 1 t2

X / ^S1—1 (ts1—1)... i^ (t1)dwt(;1)...dwt(;;_^1)dts1+1dwt(;;+^2)...

... dwi;;:-1^+idwt(;;+22)... dwt(;k(2.682) Theorem 2.30 is proved.

Remark 2.4. Let us make a number of remarks about Theorem 2.30. An expansion similar to (2.637) was obtained in [127] for the multiple Stratonovich

stochastic integral (2.941). The proof from [127] is somewhat simpler than the proof proposed in this .section. However, the results from [127] were obtained under the condition of convergence of trace series. The verification of this condition for the kernel (1.6) is a separate problem. In our proof we essentially use the structure of the Fourier coefficients (2.638) corresponding to the kernel (1.6). This circumstance actually made it possible to prove Theorem 2.30 using not the condition of finiteness of trace series, but using the condition of convergence to zero of explicit expressions for the remainders of the mentioned series. This leaves hope that it is possible to prove an analogue of Theorems 2.24-2.26, 2.37-2.39 for the case of an arbitrary k (k £ N).

Note that under the conditions of Theorem 2.30 the sequential order of the series

TO TO TO TO

E E ■•■ E E

jS2r-1 =P+1 jS2r-3 =P+1 J'fls =P+1 =P+1

in (2.631) is not important. We also note that the first and second conditions of Theorem 2.30 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]) (see the proofs of Theorems 2.1, 2.2, 2.8, 2.9, 2.27). It is easy to see that in the proofs of Theorems 2.1-2.9 the conditions of Theorem 2.30 are verified for various special cases of iterated Stratonovich stochastic integrals of multiplicities 2-4 with respect to components of the multidimensional Wiener process.

Taking into account Theorem 1.11, we can formulate an analogue of Theorem 2.30 for the case of integration interval [t, s] (s £ (t,T); the case s = T is considered in Theorem 2.30) of iterated Stratonovich stochastic integrals of multiplicity k (k £ N).

Denote

jj ...ji(s)

def

q=0i,02,-,g2r-i,g2r-

def

E £■■•££ ^...ji(s)

jS2r-1 =P+1 j'fl2r-3 =P+1 J'fls =P+1 jS1 =P+1

jS1 j32 '•••'j32r-1 j32r

and introduce the following notation

Si{ Cjp..3,...n(s)

def 1

oo

-1

q=gi,92,-,92r-i,g2r

{921=921-1 + 1}

oo oo

E E

j32r-1 =P+1 j32r-3 =P+1

•■■ E E ■•• E E a»..*(s)

ja2l+1 =P+1 jg2l-3 =P+1 jgs =P+1 jgi =P+1

( j32lj 321-1 )• ) >j31 =j32 >j32r-1 =j32r

where l = 1, 2,..., r,

Cjfc...j1 (s)

is defined by analogy with (2.629),

(j32l j32l-1 MO

t2

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j•j (s) = ^k (tk ) j (tk ) ••• 1 (tx)dix • • .dtk • (2.683)

Theorem 2.31 [32], [37],

. Assume that the continuously differ-entiable functions fa(r) (l = ) at the interval [t,T] and the complete

orthonormal system {faix)}^ of continuous functions {(f>o{x) = 1/y/T — t) in the space L2([t,T]) are such that the following conditions are satisfied:

1. The equality

s OO s t-2

i J = è J <S>o(U)fa(U) J ^(tfafaitfadt^U (2.684)

t j=0 t t

holds for all s G (t, T], where the nonrandom functions $i(r), $2(t) are continuously differentiate on [t,T] and the series on the right-hand side of (2.684) converges absolutely.

2. The estimates

fa (r )$x(r )dr

<

j 1/2+a'

fa (e)§2(o)de

<

^2(s,r )

j 1/2+a '

oo

Y, / ^(r)fa(r) / <^)fa(^)d^dr

j=p+1 t t

<

^s(s)

p

3

s

hold for all s,T such that t < t < s < T and for some a,^ > 0, where $1(r), $2(t) are continuously differentiate nonrandom functions on [t,T], G N, and

s s

J |^1(t)^2(s,t)| dT < TO, J |^3(t)| dT < TO tt for all s G (t,T).

3. The condition

lim

p^œ

E

M jj j (s)

9=31,32>'">32r_1>32r

—0

q=g1,g2,...,g2r_1,g2r

holds for all possible g1, g2,..., g2r—1, g2r (see (12.627)) and l1, l2,..., ld such that l1, l2,..., ld G {1, 2,... ,r}, l1 > l2 > ... > ld, d — 0,1, 2,... ,r — 1, where r — 1, 2,..., [k/2] and

ShSh jj...j1(s)

q=g1,g2,...,g2r_1,g2r

. = Cj (s)

for d = 0.

Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k

]S;i-ifc) = / (tk)... i ^i(ti)dwt(;i)... dw^ (2.685)

the following expansion

J

(»1 ...»k )

— ^ E Cjk- wll j

(»i )

j1,...,jk=0

1=1

that converges in the mean-square sense is valid, where (^„•j (s) is the Fourier coefficient (2.683), l.i.m. is a limit in the mean-square sense, =

0,1,...,m, s G (t,T),

T

j — j (T )dw

jj t

(») T

2

k

p

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wt^ = f^ for i = 1,... ,m and wt0 = t.

It is easy to see that the estimates (1.209), (1.215), (2.254),

I, and

the results of Sect. 2.9 imply the fulfillment of Condition 2 of Theorem 2.31 for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]).

Also the equality (2.508) guarantees the fulfillment of Condition 1 of Theorem 2.31 for these two systems of functions.

It should be noted that (see (2.676))

(-1)r JJ...

Jk-Jq -Ji

+

+ (-1r'£ Shl <t>jq...ji

ll=1

+

r (

+(-1)r-2 Y Sii SiA Jj ...Ji

li,l2 = i I ii>i2

+

+ ( —1)1 ^ Sii Si2 • • • Si

(p)

Jk ...Jq ...Ji

ii,i2,...,ir-i = i 11 >l 2 > .-.>lr_i

q=gi,02v,g2r—i,g2r

e- e c

Jk...Ji

Jgiz

J92r_i =0

Jsi JS2 '•••'J32r_i J32r

2r

ni

1=1

{g2i =g2i_i+1|CJk...Ji

(JS2 Jsi )^( )...(J92r JS2r_i WOfa = J32 '•••'JS2r_i = JS2r

, (2.686)

where the meaning of the notations used in

is preserved.

For example, from (2.686) for the case r = 2 we get

oo oo

E £ (Jk ...J'i

JS3 =P+1 Jgi =P+1

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Jgi Jg2 'J33 JS4

r

0

1

-1

2

{04 =03+ 1}

(jk-j'l

jgl =p+1

( jS4 jS3 ) ^ ( •) ,jg 1 = jS2 ,j33 = jfl4

oo

-1

2

{02=0l + 1}

53 (jk-j'l

j's3 =p+1 pp

( jS2 js 1 ) ^ ( •) ,jg 1 = j32 ,jS3 = jS4

53 53 (jk-

jgl =0 j's3 =0

jl

jSl jS2 ,jS3 jS4

41'

{02=01 + 1} 1{04=03+ 1} ( jk •••jl

(j

(jg2 jgl )^0)(j'g4 jg3)^(^),jgl = jg2 ,jg3 = jg4

As a result, Condition 3 of Theorem 2.30 can be replaced by a weaker condition

lim

p^œ

53 ( 53 • • .53 (j'k-j'l

1

2^

1

1=1

jl,-- -,jq,-- - ,jk = 0 \jgl =0 jg2,

q=gl>g2j--->g2r-ljg2r

{021=021-1 + 1} (jk •••j'l

r-l"

jgl jg2v,jg2r-l jg2r

= 0,

(jg2 jgl )^(•)•••(jg2r jg2r-l)^(0,j'gl = jg2 = jg2r .

(2.687)

where r = 1, 2,..., [k/2].

However, Condition 3 of Theorem 2.30 itself contains a way of proving of the condition (2.687), which is partially realized in the proof of Theorems 2.32-2.35 (see below).

In fact, when proving Theorem 2.34 (the case r = 3 is proved in Theorem 2.35 for ^1(r), • • • ,^6(t) = 1), we proved the following equality

pp

lim 53 53 (jk •jl

r>—vno < * < *

1

41

p^œ

{02=01 + 1} 1{04 =03+ 1}(jk•••j'l

jgl =0 jg3 =0

(j

jgl jg2 ,jg3 jg4

(jg2 jgl)^(0(j'g4 jg3)^0),j'gl = jg2 ,j'g3 = jg4

On the other hand, iterative application of (2.662) gives

53 • • • 53 (jk-j'l

jgl =0 jg2r-l =0

jgl jg2,---,jg2r-l jg2r

0

2

2r it ^-{9,2i=g2i-i+l}(^jk---ji

l=1

(j32 jS1 j32r-1 = j32 ''"'js2r-l = js2r

where r = 1, 2,..., [k/2].

r

2.11 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3. The Case p1 = p2 = p3 ^ to and Continuously Differentiable Weight Functions (t), ^2(r), (t) (The Cases of Legendre Polynomials and Trigonometric Functions)

In this section, we present a simple proof of Theorem 2.8 based on Theorem 2.30.

In this case, the conditions of Theorem 2.8 will be weakened.

Suppose that (x)}°=0 is the same as in the conditions of Theorem 2.8.

First, we show that the equalities

t2 t2 T \ [ $1 (r)^2(r)dr = Y f $2(r)^-(r) [ (e)<f>j(e)d9dT, (2.688)

2

ti j=0 ti ti

1 t2 to t2 t2

- / $1(r)$2(r)dr = E / / $2(r)^(r)drd05 (2.689)

ti j =0 t

hold for all t1,t2 such that t < t1 < t2 < T, where the nonrandom functions $1(t), $2(t) are continuously differentiable on [t,T].

From (2.508) we get

1 tl to tl

- / $1(r)$2(r)dr = £ / $2(r)^(r) / (2.690)

2J ^ J — j

t j=0 t t

1 t2 to t2

- / $!(r)$2(r)dr = / $2(r)^-(r) / (2.691)

j=0 i t

Subtracting (2.690) from (2.691), we obtain 1

t2 oo t2

-j = ^ j $2(r)fa(r) j &i(6)(f)j(6)d6dT =

ti j=0 ti t

to ^ li

= Y / (T(T) (6)d6dT+

j =0 /i {

to T

to 2 „

+ £ / ^2(t(t) / (6)d6dT. (2.692)

j=0 ti /i

Generalized Parseval's equality gives

to ^ li

Y / ^2(T(t)dW (6)d6 =

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j=0 ti t

r^ T T

to „ „

= E / !{ti<T<t2|^2(T(t)dW 1{e<t i(6)d6 = j =0 { t T

= J 1{t i<T<t2}^2(T )1{T<t i }$1(t )dT = 0. (2.693)

t

Combining (2.692) and (2.693) we obtain (2.688). The equality

t2 T t2 t2

j $2(T(T)J (6)d6dT = J (6)J $2(T(T

t i t i t i e

completes the proof of (2.689).

Theorem 2.32 [32], [37], [38], [63]. Suppose that {fa(x)}TO=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^1(t),^2(t),^3(t) are continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * ts * t2

J*[^(3)]T,t = J MU)J ^1(t1)dw(; iW^dw^ (2.694)

ttt

the following expansion

p

J *[V'(3)]t,î = lpi;m. L Cj Cii-'C':2»^31

ji j2 ,j's=0

that converges in the mean-square sense is valid, where ¿1, i2, i3 = 0,1,..., m, T is t2

Cj3j2ji = J ^(^j^(^j(ii)dtidt2dt3 t t t

and

T

j = J to (s)dw<*>

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), Wr ^ = fT^ for i = 1,..., m and wT0) = t.

Proof. As noted in Remark 2.4, Conditions 1 and 2 of Theorem 2.30 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]). Let us verify Condition 3 of Theorem 2.30 for the iterated Stratonovich stochastic integral (2.694). Thus, we have to check the following conditions

p / œ

piœ

i™E E jiii) =0, <2-695)

js=0 \ji=p+1

li^E E Cjsjsji) = 0, (2.696)

ji=0 \js=p+1

p

piœ

j

p

piœ

!™E E cji) =0. (2.697)

j2=0 \ji=p+1

We have

EEc

js=0 \ji=p+1

œ

Cjsjiji

T ts t2

s2

E E J Wtj) j (t3^ ^(¿2) j (Î2)y WO j (Î1)dÎ1dÎ2dÎ3

jS=0 \j'i=p+1 t t t

(2.698)

2

p

2

p

P / T to ts ^

= E M^j (fc) E /

js=A t ji=p+11 t

(2.699)

/ t ts t2 ^ 2

to / „ to 5 i

< E / WOfas (t3) E / ^(¿2(t2) / (t1 )dt1dt2dt3

js=A t ji=p+11 t

(2.700)

t / to ts t2 \2

= f (t3) E / ^(^j (t2 )J fa^Ofaji (t1 )dt1dtJ dt3 < (2.701)

t \ji=p+11 t /

K

< -r ^ 0 2.702

p2

if p ^ to, where constant K does not depend on p.

Note that the transition from (2.698) to (2.699) is based on the estimate (2.626) for the polynomial case and its analogue for the trigonometric case, the transition from (2.700) to (2.701) is based on the Parseval equality, and the transition from (2.701) to (2.702) is also based on the estimate (2.626) and its analogue for the trigonometric case.

By analogy with the previous case we have

p / to x 2

£ £

ji=0 \j's=p+1

p / to T ts ^

EE/ WtOfas (t3) / fa(t2)fas (t2) / WO j (t1)dt1dt2dt3 ji=^js=p+1 { { {

2

pi to T T T

E ( E I WOfe (*0 / WOfas (*2) / WOfas (t3)dt3dt2dt1

(2.703)

ji =0 \jS =p+1 t ti t2

T T T

p / „

E / W^j (t1) E J ^(¿2)fajs (t2) J ^3)fas (t3I < ji=0 \ t js=p+1 ti t2

(2.704)

2

T T T

00 / „ oo „ „

<£ / WOfe (tl) £ / ^ (t2W ^3)^3 (t3)dt3dt2dti

¿1=0 \i j3=p+1t1 /2

2

T / _ T T

oo „ „

^2(ti) ( £ J ^2)^3My ^i(t3)0j3(t3)dt3dt^ dti < (2.705)

Vj3=P+1 ti t2

K

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< — o

p2

(2.706)

if p ^ oo, where constant K is independent of p.

The transition from (2.703) to (2.704) is based on an analogue of the estimate (2.626) for the value

oo

T

£ J ^2(t2)0j3(t2)y ^3(t3)0j3(t3)dt3dt2

¿3 =p+1 ti t2

for the polynomial and trigonometric cases, the transition from (2.705) to (2.706) is also based on the mentioned analogue of the estimate (2.626).

Further, we have

p / o

£ £

¿2=0 \j1=p+1

p / o

T t3 t2

SIS/ ^3(t3)0j1 (*)/ ^2(t2)0j2 fe) / WOfe (t1)dt1dt2dt3

j2=Aj1=P+1 tit

p / o

T

t2

T

J ^2(t2)0j2 fe)y WOfe (t1)dt^ ^3(t3)0j1 (t3)dt3dt2

¿2 =0 \j1 =p+1 t t t2

(2.707)

p i T o t2 T

£ y ^2(t2)0j2(t2) £ J ^1(t1 )0j1 (t1 ^3(t3)0j1 (t3)dt3dt^ <

¿2=0 \ t j1=p+1 t t2

(2.708)

OO I T oo t2 T

< S J ^2(t2)0j2 (t2) £ J (t1)0j1 (t1)dt^ ^j(t3)0j1 (t3)dt3dt2

¿2=0 \ ,t j1=p+1 t t2

2

T / t2 T

7 oo 2 „

= / E I (ti)dtW ^sCts)^,-!(ta)dtJ dt2. (2.709)

t \jl=p+1 t t2 /

The transition from (2.707) to (2.708) is based on the estimate (2.279) and its obvious analogue for the trigonometric case. However, the estimate (2.279) cannot be used to estimate the right-hand side of (2.709), since we get the divergent integral. For this reason, we will obtain a new estimate based on the relation

.710)

From (2.141) and the estimate |Pj(y)| < 1, y £ [-1,1] we obtain

\pM\ = IPM? ■ IPM1-' < l^f«/)!1-5 < juw^^w (2-

where y £ (-1,1), j £ N, £ £ (0,1) is an arbitrary small positive real number. Combining (2.277) and (2.710), we have the following estimate

) j (t )dT

<

C

(j1)1"£/n (1 - Z2 (s)) 1/4_e/4

+ 1

(2.711)

where j1 £ N, s £ (t,T), z(s) is defined by (2.20), constant C does not depend on j1.

Similarly to (2.711) we obtain

T

^3(t ) j (t )dT

<

C

1

(j1)1~£/2\ (1 - Z2(s))1/4"e/4

+1

(2.712)

where j1 £ N, s £ (t,T), constant C does not depend on j1. Combining (2.278) and (2.712), we have

s T

<

L

) j (t )dT fa (t ) j (t )dT

+1

<

(j02-e/2 \ (1 - z2(s))1/4-/4

(1 - z2(s))1/4

+ 1 , (2.713)

where j1 £ N, s £ (t,T), z(s) is defined by (2.20), constant L does not depend on j1.

2

s

1

Observe that

00 œ

1 f dx 1

¿L ^i)2_£/2 ~ J x2~s/2 t1 - (2'714)

Applying (2.713) and (2.714) to estimate the right-hand side of (2.709) gives

P / TO \ 2 K

E E fan • ' " (2-715)

¿2=0 \jl=P+1 J 1

if p fa to, where £ is an arbitrary small positive real number, constant K is independent of p.

The estimation of the right-hand side of (2.709) for the trigonometric case is carried out using the estimates (2.254), (2.255). At that we obtain the estimate (2.715) with £ = 0. Theorem 2.32 is proved.

2.12 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4. The Case p = ... = p4 ^ to and Continuously Differentiable Weight Functions fa(t), ..., ^4(t) (The Cases of Legendre Polynomials and Trigonometric Functions)

Theorem 2.33 [32], [37], [38], [63]. Suppose that [fa (x)j0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let fa(t),...,fa(T) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

fa * ¿4 *Î3 *Î2

= / fa(Î4) / fa(ia) / fa(Î2) / fa^Odwi-W^dw^dw^

(2.716)

the following expansion

p

j 1 ,j2 j3 j4=0

that converges in the mean-square sense is valid, where i1; i2, i3, i4 = 0,1,..., m, T t4 t3 t2

+¿4¿3^2¿1 = J W^fc (t4^ ^3(t3)h3 ^ W^fe M y ^1(t1)^i1 (t1)dt1x tttt

xdt2dt3 dt4

and

T

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Cj° = J h (s)dw«

t

are independent standard Gaussian random variables for various i or j (in the

= fW for i = 1 m w0)

case when i = 0), wT = fT for i = 1,..., m and w^ = t.

Proof. As noted in Remark 2.4, Conditions 1 and 2 of Theorem 2.30 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]). Let us verify Condition 3 of Theorem 2.30 for the iterated Stratonovich stochastic integral (2.716). Thus, we have to check the following conditions

p

piœ El £ C) = f2-717)

jS,j4=0 \j'i=p+1

p

E £ Cj4jij2ji = 0- (2.718)

piœ

¿2*4=0 \j'i=p+1

pœ

plim El £ Cj ) =0- î2'719)

j2,jS=0 \j'i=p+1 p

E E Cj4ji = (2.720)

piœ

ji,j4=0 \j2=p+1

p

E E Cjjji = (2-721)

piœ

ji,js=0 \j2=p+1

pœ

^ el e Cjsjsj2ji ) = (2-722)

ji ,j2=0 \j's=p+1

2

oo oo

lim V V C

t^rsr, I

p^œ

j2jij2ji

= 0,

V?2=p+1 jl=p+1

oo oo

Jim E E Cj1j2j2j1 =

p^œ

Vj2=P+1 jl=p+1

OO oo

ëm ^ S jj =

p^œ

j3=P+1 ji=p+1

oo

lim V C

1—^no I ^-'

p^œ

j3j3jiji

j3=p+1

oo

lim C

l^IYl I -'

p^œ

j3j3jiji

ji=p+1

oo

lim C \ -'

p^œ

jij2j2ji

ji=p+1

(j'ij'iW0,

(j3j3W0 .

C^W-),

= 0,

(2.723)

(2.724)

(2.725)

(2.726)

(2.727)

(2.728)

where in

we use the notation (2.6

Applying arguments similar to those we used in the proof of Theorem 2.32, we obtain for (2.717)

p / œ

2 / T

2 p I œ

t4

E ( E Cj4j3jiji] = E E J ^ j (t4^ ^ (t3)^j3 (t3)X

j3 ,j4=0 \j'i=p+1 / j3 ,j4=0 y ji=p+1 t t

t3 t2

^(£2) j (£2) y ^(£1) j (Î1)dÎ1dÎ2dt3 dt4 ) =

tt

p / T t4

^ J M^j M J faC^j (t3)x

j'3,j4=0 I t t

(2.729)

00

t3

t2

X E / ^(£2) j(£2) / ^(£1) j(£1)dÎ1dÎ2dÎ3d£^ < (2.730)

ji=p+1 { t

2

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2

oo ( T t4

< E / ^4(t4)0j4 (t4^ ^3(t3)0j3 (t3)x

¿3J4=0 \ t t

OO

t3

t2

x £ J ^Mfe(t2^ ^1(t1)hi1 (t1)dt1dt2dt3dt^ = (2.731)

¿1=p+11 t

I{t3<t4}^42 (t4 )^2(t3)x

[t,T

oo

t3

t2

x ( £ J ^2(t2)0j■lM / ^1(t1)hi1 (t!)dMt2 J dt3dt4 < (2.732) ¿1 =p+1t

t

K

<— 0 p2

(2.733)

if p ^ o, where constant K is independent of p.

Note that the transition from (2.729) to (2.730) is based on the estimate

(2.626) for the polynomial case and its analogue for the trigonometric case, the transition from (2.731) to (2.732) is based on the Parseval equality, and the transition from (2.732) to (2.733) is also based on the estimate (2.626) and its analogue for the trigonometric case.

Further, we have for

p / o

2 / T

2 p I o

t4

£ E

+¿4 ¿1 ¿2 ¿1 I = £ I £ J ^4 (t4)h4 My ^3 M^! (t3)x ¿2 ,¿4=0 \j1=P+1 / ¿2 ,¿4=0 V ¿1=p+1 t t

t3

t2

x y ^2(t2)0j2 (t2^ ^1(t1)hi1 (t1)dt1dt2dt3dt4 I =

tt

p / o T ^

= E E J ^4)^4 (t4) y ^2(t2)0j-2 (t2)x

¿2,j'4=0 y'1=p+1 t t

t2 t4 X 2

x J ^1(^)^1 (t1)dt^y ^3^3)^1 (t3)dt3dt2dt4 I =

t t2

(2.734)

(2.735)

2

p / T u

= E / ^4(t4)0j4 faC^fa^ (t2)x

¿2 ¿4=0 \ t t

x E / (ti)^ji (¿i)dti / fa(t3)fa (is< ¿i=p+i { t2

oo / T t4

< E / ^4 (t4)0j4 (¿4^ fa (¿2)^2 (t2)x

¿2 ,¿4 =0 \ t t

TO ^

x E / ^i(ti)^ji (ti )dti / fa(t3)fai (¿3)dt3dt2dt4 ¿i=p+i { t2

= J 1{t2<t4>^4(t4)fa2(t2) x

[t,T ]2

/ to * ^ \2

x E / fai(ti)0ji(ti)dtW fa(i3)fai(¿3)dtJ dt2dt4 < \ji=P+i t t2 /

K

< ^ 0 (2.736)

if p fa to, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.

The relation (2.736) was obtained by the same method as (2.733). Note that in obtaining (2.736) we used the estimates (1.209) and (2.711) for the polynomial case and (1.215) and (2.254) for the trigonometric case. We also used the integration order replacement in the iterated Riemann integrals (see (2.734), (2.735)).

Repeating the previous steps for (2.719) and (2.720), we get

P / TO \ 2 P / TO T t4

E E = E E / ^(i4)fai (t4M fa(t3)0j3 (t3)x

¿2,¿3=0 \j'i=P+i / ¿2,¿3=0 \ji=P+i t t

t3 t2 X 2

fa(¿2)fa (¿2^ fai(ti)^ii (ti)dtidt2dt3 dt4 ) = tt

p / o T r

= E E J Wt3)&3 (t3^ ^2(t2)0j2 (t2)x

¿2 ¿3=0 \j1=p+1 t t

t2 T \ 2

x J ^1(t1)hi1 (t1)dt^y ^4^4)^1 (t4)dt4dt2dt^ =

t t3

p / T t3

= E i 03)/ «t^ (t2)x

¿2 ,¿3 =0 \ t t

oo t2 T \ 2

x E y ^1(t1)0i1 (tOd^y (t4)dt4dt2dtH <

¿1=p+1 t t3 /

o / T t3

< E / (*)/ (t2)x

oo t2 T \ 2

x E / ^1(t1)hil (t1)dtW ^Mfe (t4)dt4dt2dt3 =

¿l=p+1 t t3 )

= I{t2<t3}^l (t3 )^|(t2)x

[t,T ]2

/ 00 t2 T \ 2

x E / ^1(t1)hil(t1)dt1 / ^4M^l(t4)dt^ dt2dt3 <

V-^11 t3 )

K

0 (2.737)

p2

if p ^ o, where constant K does not depend on p;

p / o \ 2 p / o T ^

E ( E «l) = E E J ^4 (t4M ^3 Mfe (t3)x

¿1 ,¿4=0 N¿2 =p+1 / ¿1 ,¿4=0 y2=p+1 t t

t3 t2 \ 2

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x y ^2(t2)0j2 (t2^ ^1(t1)hil (t1)dt1dt2dt3dt4 I =

tt

p / o T t4

= X 53 J Wt4)&4 (t4^ ^1(t1)hil (t1)x

¿1 ¿4=° y2=p+1 t t

t4 t4 \ 2

^2(t2)0j2 (t2^ ^3(t3)0j2 (t3)dt3dt2dt1dt^ =

t1 t2

p / T t4

= 53 i ^4(t4)0j4 M / ^1(t1)hil (t1)x

¿1j4=0 \ t t

o t4 t4 \ 2

x 53 ^202)^2(t2^«t^(t3)dt3dt2dt1dt^ <

¿2=p+1 tl t2 /

< E ( /^4(t4) / (tl^ (t1)x

¿1 ^ \i { o t4 t4 \ 2

x X ^^202)^2(t2^ «t3)&2(t3)dt3dt2dtldt4 I =

¿2=p+l tl t2 /

= 1{tl<t4}^42 (t4 )^2(tl)x

[t,T ]2

/ o t4 t4 \2

x J^MUnM) j«t^(t3)dt3dtJ dtldt4. (2.738)

V2 =p+l tl t2 /

Note that, by virtue of the additivity property of the integral, we have

o t4 t4

/ (t2M ^3(t3)0j-2(t3)dt3dt2 = (2.739)

¿2=p+1 tl l

= 53 J ^3)^2 (t3^ ^Mfe (t2)dt2dt3-

¿2=p+1 t t

o ll t3

- 53 / ^03)^2 (t3W ^Mfe (t2)dt2dt3-

¿2 =P+1 { {

t4 ti

TO 4 i

E / ^3(¿3)0^2(t3)dtW ^(¿2)^2(¿2)dt2. (2.740)

¿2=P+i ti t

However, all three series on the right-hand side of (2.740) have already been evaluated in (2.733) and (2.736). From (2.738) and (2.740) we finally obtain

P / TO \2 k

E E <-yz-£ 0 (2.741)

¿U4 =0 \72 =P+i /

if p fa to, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.

In complete analogy with (2.736), we have for (2.721)

P / TO \ 2 P / TO T t4

E E

= E E / ^4 M02 / ^3 ^^ (t3)x

¿i,¿3=0 N¿2 =P+1 / ¿i,¿3=0 \j'2=P+i t t

t3 t2 \ 2

^(¿2)072 (¿2^ ^i (¿i)0ji (tl)dtldt2dtзdt^ =

tt

P / TO T r

= E E I ^(¿^¿3 (¿3W «¿^ (¿2)x

¿1,¿3=0 y'2=P+i t t

t2 T \ 2

x J ^i (¿i)0ji (¿i)diidi^y fa (¿4^2 (t4)dt4dtп =

t t3

P / TO T r

= E E J ^(¿^¿3 (¿3^ fa^Ofa (¿i)x

¿U3=° y'2=P+i t t

t3 T \ 2

x^ ^2(¿2)0j'2 ^^d^ ^ fa^^ (t4)dt4dtз I =

ti t3

P / T t3

= E i ^(¿^¿3 (¿3^ ^i(ii)0ji (¿i)x

¿¿0 V t t

t3 T

o 3 r,

x ^ / ^2(t2)dt2dtW ^4(t4)0j2 (t4)dt4dt^ <

¿2=p+lti t3

oo ( T t3

< E / ^(*)/ (tl)hil (tl)x

¿1 J3=0 \ t t

t3 T x 2

x E / (t2 )dtW ^4^4)^2 (t4)dt4dtl dt3

¿2=p+lti t3

: J 1{tl<t3}^8 (t3M (tl)x

[t,T ]2

i oo t3 T \ 2

x( E y ^2(t2)hi2 (t2)dt^y ^4^4)^2 (t4)dt^ dtldt3 <

\j2=p+1 tl t3 /

K

< ^--> 0 (2.742)

" p2- v y

if p ^ o, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.

We have for (2.722)

p / o \ 2 p / ^

£ E = X E / ^4 (t4W ^3 (t3)x

¿1 ,¿2=0 \.?3=p+l / ¿1 ,¿2=0 y'3=p+l t t

t3 t2 \ 2

xj (t2^ ^(tl^l (tl)dtldt2dt3 dt^ =

tt

/ T T

p / o ^

= E E / ^l(tl)hil (tlW «t^fe (t2)x

j1,j2=0 y'3=p+l t tl

T t \ 2

x J (*)/ ^Mfc (t4)dt4dt3dt2dt^ =

t2 t3

, T T

P T

E (/^(¿i^i (¿i)/ «¿^ (¿2) x

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¿^,¿2=^ \ t ti

2

rsr, T T

00 „ „

x E / ^fe^a (¿3M fa^^ (t4)dt4dtзdt2dtl

<

¿3=P+i t2 t3

™ . T T

00

< E / ^ (¿i)0j'i (¿iW ^ (¿2)0j-2 (¿2) x

¿i ¿2=A i ti

TO T T \ 2

x E y ^3(¿3)0j'3(¿3) y fa^^(¿4)dt4dtзdt2¿¿i I

¿'3=P+i t2 t3 /

00

= J 1{ti<t2}^I (t1 M(t2) x

[t,T ]2

T T \ 2

x I E y ^(¿3^3(¿3)/ fa^fe(t4)dt4dtJ dt2dfa (2.743)

\¿3=p+1 t2 t3 /

It is easy to see that the integral (see (2.743))

T T

y fa(¿3)0J3 (*)/ ^4(¿4)0J3 (¿4)dt4dtз

t2 t3

is similar to the integral from the formula (2.739) if in the last integral we substitute ¿4 = T. Therefore, by analogy with (2.741), we obtain

P / TO V K

E E << ^ > 0 (2.744)

¿i ,¿2 =0 \73 =P+1 /

if p fa to, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.

Now consider (r2.723l)-(l'2.725l). We have for (12.72,3) (see Step 2 in the proof of Theorem 2.30)

TOTO \ 2 / p TO \ 2

^¿¿^¿i I = 1 C¿2¿1¿2 ¿i I <

¿2=P+1 ¿1 =P+1 / \7i =0 ¿2=P+1 /

p / o

< (p + 1)E E ¿^¿l) . (2.745)

¿1=0 N¿2 =p+1

Consider (2.721) and (2.742). We have

<

¿1=¿3

p / o \ 2 p / o

\ 53 ^C¿2¿1¿2¿1 J = I 53 ^¿^Jl ¿1=0 N¿2 =p+1 / ¿¿,¿3=0 N¿2 =p+1

p / o \2 K

£ ( V Cnhhh < -5-, (2.746)

¿¿,¿3=0 V?2=p+l /

where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p. Combining (2.745) and (2.746), we obtain

2

o o \ (p 1)K Kl

¿2=p+l ¿1 =p+l J

if p ^ o, where constant K1 does not depend on p. Similarly for (2.724) we have (see (2.720), (2.741))

oo \ 2 /p o \ 2

53 53 ^c¿l¿2¿2¿l J = i 53 C¿i¿2¿2¿l i <

¿2=p+l ¿1 =p+l / N¿1 =0 ¿2=p+l /

p o 2

< (p + 1)E E ^¿1 , (2.747)

¿¿=0 N¿2 =p+1 /

p o 2 p o 2

53 ( 53 ^c¿l¿2¿2¿l J = i 53

¿1=0 V?2=p+l / ¿¿,¿4=0 N¿2 =p+1

/ \ 2 p / o \ 2 K

<

¿¿=¿4

< £ £ < ^ (2-748)

¿¿,¿4=0 V?2=p+l /

where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p. Combining

2

(2.747) and (2.748), we obtain

oo oo

e eC

¿2=P+1 ¿1=P+1

¿1¿2¿2¿1 J — p2-£ — pi-£

if p fa to, where constant K1 does not depend on p. Consider (2.725). Using (2.662), we obtain

oo oo

C¿3¿3¿1¿1

¿3=P+1 ¿1=P+1

oo oo

TOP

E E

C¿3¿3¿1¿1

- E E

C¿3¿3¿1¿1

¿3=P+1 ¿1=0 ¿3=P+1 ¿1=0

1TO

2 y v ^ !/",/:■,/ ./

¿3 =P+1

TOP

EE

C¿з¿з¿l¿l;

¿¿l WO ¿3=p+1 ¿i=0

(2.749)

where (see (2.629))

C

¿3¿3¿1¿1

(¿¿iW-)

t3

T t4

= J ^4(¿4)0j3 (¿4^ y ^з(tз)0¿з (¿3)/ ^2(t2)^l(t2)dt2dtзdt4.

ttt

From the estimate (2.83) (polynomial case) and its analogue for the trigonometric case (see the proof of Lemma 2.2, Sect. 2.1.2) we get

^¿^'¿l

¿3=P+1

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where constant C is independent of p.

Further, we have (see (2.744))

\ 2

P TO 2

(¿¿iWO

<

c p:

(2.750)

PTO

53 53 ^¿s¿¿i ] <(p+1) i 53 ^¿^¿m

¿1=0 ¿3=P+1 / ¿1=0 V73=p+1

(p+1) E E

C¿3¿3¿2¿1

¿1 ,¿2 =0 \73=p+1

<

2

PTO

< (p +1) E E ^¿l | <

¿1 ,¿2=0 Y^P+1 < (p + l)K < K\

p2-£ p1-£

(2.751)

where constant K1 does not depend on p. Combining (2.749)-(2.751), we obtain

oo oo

C

¿3=P+1 ¿1 =P+1

¿^¿l / p1-£

K2

< --► 0

if p fa to, where constant K2 does not depend on p.

Let us prove (126)-(2Z28i). It is not difficult to see that the estimate (2.750) proves (2.726).

Using the integration order replacement, we obtain

Ey C¿з¿з¿l¿l ¿l=p+1

oo

T

t4

(¿373 WO t2

e / ^4 (¿4)^3 04) / fa^fai ^ / fa^Ofai (tl)dtldt2dt4 = ¿i=p+1 t t t

TO T / T \ t2

E i ^(¿2) / fa^fa^^ 0jl (¿2)/ fa^l^i (tl)dtldt2, ¿l=p+1 t \ t2 / t

(2.752)

00

Ey C¿1¿2¿2¿1 ¿l=p+1

OO

T

t4

(¿2¿2)^(•) t3

E J 04(¿4)0¿l (¿4^ y 0з(tз)02(tз^ fa^l^i (¿l)diid£3^£4 ¿i=p+1 t t t

T t4 t4

TO

E / (¿4)0¿ 1 (¿4) / fa^l^i (¿1) / 0з(tз)02(tз)dtзdtldt4 ¿l=p+11 t t1

2

oo

T

t4

t4 tiN

J (¿4) j (¿4^ (tl)

ji=P+1 t t

^3(t3)^2(t3)dt3dtidt4 =

tt

oo

T

t4

53 I I ^4(Î4W ^3)^3)^3 I j(Î4W Wti) j(ti)dtidt4-

ji=p+i t V t / t

(2.753)

00

T

t4

53 / ^4(¿4(t4)/ (Wti)/ ^3(t3)^2(t3)dt^ j(ti)dtidt4. (2.754)

ji=p+i t t

Applying the estimate (2.83) (polynomial case) and its analogue for the trigonometric case (see the proof of Lemma 2.2, Sect. 2.1.2) to the right-hand sides of (I2.752l)-(l'2.754l). we get

œ

53 Cj3j3jiji

j3=p+i (j3 J3WO

œ

53 Cj1j2j2j1

j1=p+i (j2j2W0

<

C P ''

c p:

(2.755)

(2.756)

where constant C is independent of p. The estimates (2.755), (2.756) prove (2.727), (2.728).

The relations (12.7171)-(l'2.728) are proved. Theorem 2.33 is proved.

t

1

2.13 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 5. The Case pi = ... = p5 ^ œ and Continuously Differentiable Weight Functions ^1(r), ..., ^5(r) (The Cases of Legendre Polynomials and Trigonometric Functions)

Theorem 2.34 [32], [37], [38], [63]. Suppose that (x)}=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in

the space L2([t,T]). Furthermore, let fa(t),...,fa(r) are continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of fifth multiplicity

* T * ¿2

J*[0(5)]T,t =/ fa(is) • fa^dw^ ...dw^ (2.757)

the following expansion

J*^E cj;1'..j

p^œ

jiv,j:=0

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that converges in the mean-square sense is valid, where i1,..., i5 = 0,1,..., m,

T t2

^...¿i = J fa^fa (t5) . . ./ fal(t1)0¿l (t1)dt1 . . .¿¿B

tt

and

T

cf = J 0 (s)dw<->

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wTi) = f(i) for i = 1,..., m and wT0) = t.

Proof. Note that in this proof we write k instead of 5 when this is true for an arbitrary k (k G N). As noted in Remark 2.4, Conditions 1 and 2 of Theorem 2.30 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space .faQ^T]). Let us verify Condition 3 of Theorem 2.30 for the iterated Stratonovich stochastic integral (2.757). Thus, we have to check the following conditions

2

P /TO \

= 0, (2.758)

P™, E E C'j: 'ji

jqi ,jq2 'j93 =0 \j31 =P+1

j31 j32

2

p / œ œ

) = 0, (2.759)

j31 =j32 'j33 =j34 1

2

P-imo 53 I 53 53 Cj: ''

jqi =0 \j31 =P+1 jS3 =P+!

P™ M ^ Cj:'' j

jqi =0 \jS3 =P+1

= 0, (2.760)

( j32 jg 1 W • ' ,js 1 = jS2 'j33 = jS4 =g1 + 1

where ({gi, {g3, }, {qi}) and ({gi, , {qi5q2,q3}) are partitions of the set {1, 2,..., 5} that is {gi,g2,g3,g4,qi} = {gi, g2, qi, q2, q3} = {1, 2,..., 5}; braces mean an unordered set, and parentheses mean an ordered set.

Let us find a representation for Cjk...j|. =. + that will be convenient

for further consideration.

Using the integration order replacement in Riemann integrals, we obtain

T ti+2 ii+1 tj t2

J hk (tk) ...J h/ (t/) J hi-i(ti-i) ...J hi(ti)dti...

t t t t t

... dt/—idt/dt/+i... dtk =

T t;+2 t;+i t;+i tj + i tj+i

= J hk (tk) ...J h/+i(ti+i^ hi(ti)y h2(t2) ...J h/-i(t/-i^ y hi (t/ )dt/ x

t t t ti tj—2 t j—1

xdt/-i... dt2dtidt/+i... dtk =

T tj+2 / t;+i \ t; + i t; + i tj+i

= J hk (tk) ...J h/+i(t/+i) I y h/ (t/)dt/ y hi(ti^ y h2(t2) ...J h/-i(t/-i) x

t t \t / t ti tj—2

xdt/-i... dt2dtidt/+i... dtk—

T tj+2 t; + i t; + i t;+i / tj —i \

— J hk (tk) ...J h/+i(t/+i^ y hi(ti^ y h2(t2) ...J h/—i(t/—iW y h/ (t/)dt/ x

t t t ti tj—2 \ t /

xdt/—i... dt2dtidt/+i... dtk =

T tj+2 / t;+i \ t;+i

= J hk (tk) ...J h/+i(t/+iW y h/ (t/)dt/ y h/—i(t/—i)... t t t t t2

...J hi(ti)dti... dt/—idt/+i... dtk—

t

T tl+2 t,+ i /tl-1 \ t, —i

- J fe)... J ^C^) f hl-l(tl-l) ( / h/(tl)dtl I / hl-2(tl-2) ■ • •

t t t \ t / t

¿2

■■■J h1(t1)dt1... dt/-2dt/-1dt/+1... itk, (2.761)

t

where 1 < l < k and h1(T),..., hk(T) are continuous functions on the interval [t,T]. By analogy with (2.761) we have for l = k

T t, t2

J h/(t/) J h/-1 (t/-1)... / h (t1)dt1... dt/-1dt/ =

t t t

TT T T

= J h1(t1^y h2(t2).. y h/-1(t/-1^ h/(t/)it/dt/-1 ...^2^1 =

t ti t, — 2 t,-i

T \ T T T

fa ft№)/Mo/ h2 ... / .. .dt2dt1-

t / t ti t, — 2

TT T /1,—1 \

- /M^jh2(t2)... J ft,-^-1) Jh(t/№ <ft(-1...it2it1=

t ti t, — 2 \ t /

T \ T t2

^ h/(t/)it/ I J h/_1 (t/_1)... / h1(t1)it1... it/-1

t t t

T / t, — 1 \ t, —i t2

-J h/-1(t/-1M J hi (t/)it/ J h/—2 (t/—2) .../ h1(t1)it1... it/—1. (2.762)

t \ t / t t

The formulas (2.761), (2.762) will be used further.

Our further proof will not fundamentally depend on the weight functions fa(T),... , fa(T). Therefore, sometimes in subsequent consideration we assume for simplicity that fa(T),..., fa (t) = 1.

Let us continue the proof. Applying (2761) to Cjk + j;-1...js+1 j; js-1j (more

precisely to hs(ts) = ^s(ts(ts)), we obtain for l + 1 < k, s — 1 > 1, l — 1 > s + 1

oo

Cjk •••j'i+1j'ij'i-1 -js + 1j'ijs-1."j'l = (2.763)

j=P+1

QQ T tl+2 ti+1 t(

53 / j (tk) -J j (tl ^ 0ji-1 (tl—1) ••• j; =p+11 t t t ts+2 ts+1 ts

■j ^js + 1 / M y 0js-1 (tS—1) - - -

t t t

t2

ft^ .■■«^■ ■ ■A ...«ft* =

t

QQ T ti+2 ^+1 t

53 / j (ik ) ■■■ J j (tl ^ 0ji-1 (il—1) ■■■ j; =p+11 t t t ts+2 / ts+1 \ ts+1

■ J 0js + 1 (tS + 0 I J (ts)dtj J 0js-1 (ts —1) ■■■ t \ t / t t2

■ ■■ J j (t1 )it1 ■ ■ ■ it.—1its+1 ■ ■ ■ itl—1itlitl+1 ■ ■ ■ itk —

t

qq T U+2 U + 1 t;

53 / j (tk ) ■■y^ji+1 j (tl ^ 0ji-1 (il—1) ■■■ j; =p+11 t t t ts+2 ts + 1 i ts-1 \ ts-1

■ ■ ■ J 0js+1 MO / 0js-1 (ts—0 I J (ts)dt J J 0js-2 (tS —2) ■ ■ ■ t t t t t2

■ ■ ■ J j (t 1 )it1 ■ ■ ■ its—2its —1its+1 ■ ■ ■ itl—1itiiti+1 ■ ■ ■ iik = t

to to

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= 53 Ajk...j'i + 1j'i j'i-1".js+1 j'i js-1...j1 ^53 Bjk •••j'í+1j'íj'í-1•••j's+1j'г js-1...j1 ■ j =p+1 j =p+1

Now we apply the formula (2.761) to the quantities A¿fc...¿г+1¿г¿г_1...¿s+1¿г¿s_1 ..¿^

and ^...j■í+lj■íj■í_l...js+lj■íjs_l...¿l (more precisely to h101) = fa(¿1 ^01)). Then we

have for l + 1 < k, s - 1 > 1, l - 1 > s + 1

TO

' n.

¿(+¿1 ¿i_i ...js+ljг js_l".jl

y ^ifc...

¿i=P+1

4

1, . . . , ¿s-1, ¿s+1, . . . , ¿1-1, ¿1+1, . . . , ¿k) x

[t,T]k_2 d=1

k

xn 0g (¿g ^ (¿g )dii... d^-id^i... dt/-ldt/+l... d£k =

s=l 3=I,s

44

EC *(d) = v^ c *(d)

¿fc ...¿I + ¿1 _ 1 ...¿S + ¿8 _ 1 "J 1 / v ¿fc ...¿q ...¿1

d=1 d=1

, (2.764)

q=1,s

where

1, . . . , ¿s—1, ¿s+1, . . . , ¿1-1, ¿1+1, . . . , ¿k) =

TO ts+1 Y

= 1{ti<...<ts_i<ts + i<...<ti_i<t;+i<...<tfc}

E / ^fafai (T)dW fa (T)fa (T)d^ ¿l=p+1 t t

(2.765)

Fp(2)0

1, . . . , ¿s—1, ¿s+1, . . . , ¿1-1, ¿1+1, . . . , ¿k) =

TO y y

1{tl<...<ts_i<ts+i<...<t;_i<t; + i<...<tfc }

E / ^(fafai(T)dW fa (T)fai(т)dт, ¿l=p+1 t t

(2.766)

Fp(3)0

1, . . . , ¿s—1, ¿s+1, . . . , ¿1-1, ¿1+1, . . . , ¿k) =

TO ts_1 ti + 1

1{ti<...<ts_i<ts + i<...<t;_i<t;+i<...<tfc }

E / 0s (T)fal (T)dW fa ^fai (т)dт, ¿l=p+1 t t

(2.767)

F(4)0

1, . . . , ¿s—1, ¿s+1, . . . , ¿1-1, ¿1+1, . . . , ¿k) =

to ts+1 U-1

ji =P+1 t t

(2.768)

By analogy with (2.764) we can consider the expressions

to

53 Cj;jk-1 .j;' (2.769)

j; =p+1

to

X Cjk...j-,+1 j;j;—1 j (l +1 < k), (2.770)

j'; =p+1

E Cj;jk—1 ."j's+1j'; js-1 j (S — 1 > 1)- (2.771)

j';=p+1

Then we have for (2.769)-(2.771) (see (2.761),

00 „ 2 k-1

TOTO n ¿j rv- _l

53 Cj;jk—1...j2j; = / 53Gpd)(t2,---,tk —^g (tg)j (tg)it2 - - -itk —1,

___, I 1 J j i --o

(2.772)

j';=p+1 rtTik-2 d=1 g=2

TO „2

EiCj-j-+1j-j-1..j2j; = J E^' (i2,..■,tl-1,tl+1,.■■,tk):

j'; =p+1 rt T] k — 2 d=1

j;+1j;j;-1...j2j; / / j^y 1^2, - - - , tl—1, tl+1, - - - , tk)x

[t,T ]k

k

x II^g(tg) j(tg)it2 - - -iti—1iti+1 ■■■itk, (2.773)

3=2

to „ 4

53 Cj';jk- 1-j's+1j';js-1 ...j'1 = 53 (t1, - - - , ts —1, ts+1, - - - , tk—1) x

7; =p+1 [t T]k-2 d=1

k—1

x n ^g(tg) j(tg)it1 - - - its—1^+1 - - - itk—1, (2.774)

g=1 g=s

where

TO T ^

Gy1)(t2,---,tk—1) = l{t2<...<tk-1} 53 / ^k (t )0j; (t )iW )0j; (t ^

j; =y+1 t

to tfc_1 ^

Gp2) 02, . . . , ¿k-1) = -1{t2<...<tk_i} E / ^k(t^¿i (t)dW 01(t^¿i (t)dт,

¿i =P+1 t t

£^(¿2, . . . ,¿1-1, ¿1+1, .. . , ¿k) =

to tl+1 ^

E / (t)0*(t)d T fa (t )0j■J (t )dT,

¿i =P+1 t t

E^fe . . . ,¿1-1, ¿1+1, .. . , ¿k) =

to y ^

1{t2<...<ti_i<ti+i<...<tfc} E / M0«(T)dT / ^lW0«(т)dт,

¿I =p+1 t

DP1) (t1, . . . , ¿s—1, ¿s+1, . . . , ¿k-1)

oo T ts+1

1 {t 1 <...<ts_ 1 <ts + i <...<tk_i }

E J 0k(t)0jl (t)dT I ^s(T)0j■í (T ^

¿I =p+1 t

DP2) (t1, . . . , ¿s—1, ¿s+1, . . . , ¿k-1) =

TO T V

1{ti<...<ts_i<ts+i<...<tfc_i} E / ^k(т)0jl(T)dW 0s(T)fai(т)dт,

¿l =p+1 t t

DP3) (t1, . . . , ¿s—1, ¿s+1, . . . , ¿k-1) =

TO y ts+1

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1{tl<...<ts_l<ts+l<...<tfc_l}

e / ^k(t)0jl(t)dT / ¿l =p+1 t t

DP4) (t1, . . . , ¿s—1, ¿s+1, . . . , ¿k-1) =

TO y ts_

1{tl<...<ts_l<ts+l<...<tfc_l}

E / ^k(т)0jl(T)dW ^s(ti0.(t)dT.

ys (T )0¿l(T )dT,

s_l<ts + l<----<tfc_l} / v I r '« \ /r./i\ /-- I r s V ) r¿l

¿l=P+1 t

Now let us consider the value Cj 7\ I. ,.. To do this, we will make

j .j1 js1 =jg2> g2=g1+1

the following transformations

T t;+2 t; + 1 t; t;-1 t2

J hk (tk) ■ ■■ J hi+1(ti+1^ hi (ti) J hi (ti—1) J hi—2(ti—2) ---J h1(t1)it1 --t t t t t t

- - - iti—2iti—iitiiti+1 - - - itk =

T t;+2 t; + 1 t;+1 t;+1

= J hk (tk) -J hi+1(ti+1^ h1(t1^ h2 (t2) ■ ■■ J hi—2(ti—2)x

t t t t1 t;-3

/ t; + 1 t;-2 \ / t; + 1 t;-1\

x I / * /J / * / J'.....*—........* =

T t;+2 / t; + 1 t;+1 \ t; + 1

= J hk (tk) - J hi+1(ti+1n J hi (ti )iti y hi (ti—1)iti—J J h1(t1)x

t t \ t t / t

t; + 1 t;+1

xJh2(t2) ■ ■ ■/ ^ft-^-2 ■„M^ -

t1 t;-3

T t;+2 / t; + 1 \ t;+1 t;+1

— J hk (tk) -J hi+1(ti+1H y hi (ti)iti y h1(t1^ /2 (t2) ■ ■■

t t t t t1

t; + 1 / t;-2 \

■■■/ hi-2(ti-2) (Jhi ^H ^^ —

t;-3 t

T t;+2 / t; + 1 t;-1 \ t;+1

— J hk (tk) ■ ■■J hi+1(ti+1) y hi(ti—1) J hi (ti )itiiti—1 I y h1(t1)x

t t \ t t / t

t;+1 t; + 1 xj h2(t2)■■^hl-2(il-2)itl-2■■■it2it1iil+1 ■■■^+

t1 t;-3

T t,+2 t,+ i t,+ i t, + i

+ /hk(tk) ... J h/+1(t/+1^ h1 (t^ h2(t2) ... J h/—2(t/—2)x

t t t ti t,—3

/1,—2 t,—i \

* 1 / hl"l_l)/h.....H ....................*.

T t,+2 / t, + i t,+ i \ t, + i

= /hk (tk)... j ^Om) J * M*/h/ft-^-1 f v^-,)* t t \ t t / t ti —2 t2 x J h/—3(t/—3)... J h1(t1)it1... it/—3it/—2it/+1... itk-

tt

T t,+2 /t, + i \ t,+ i

-J hk (tk) ... J h/+1(t/+1M J h/ (t/)it/I J h/—2(t/—2) x

t t t t / t, —2 \ t, —2 t2

* 1 / h/ W ...J h1(t1)it1... itl_зitl_2itl+l...itk —

\ t / t t

T t,+2 /t, + i t, —i \

-J hk (tk) ... J h/+1(t/+1M y h/ (t/—1) J h/ (t/)it/it/—1J x t t \ t t /

t, + i t, —2 t2

x / ^te-2) / hl-3(tl-3) ...j M^1... «^A^m...iitk+

t t t T t,+2 t,+ i / t,—2 t, —i \

+/hk(tk)... //v^) (/ h/c^) /h(tlx^ 1 *

t t t \ t t / t, —2 t2 x y h/—3(t/—3) ...J h1(t1)it1... it/—3it/—2it/+1 ...itk, (2.775)

tt

where l + 1 < k,l — 2 > 1, and h1(T),..., hk(t) are continuous functions on the interval [t,T]. The case l = k follows from (2.775) with t/+1 = T, and the case l = 2 is obvious.

Applying (12.7751) to Cjfc...jl+1jljljl-2......j, we obtain for l + 1 < k, l — 2 > 1

oo

•••jl+1jljl jl-2......j1

jl=P+1

» 4 k

n ^(tg) j(tg)x

xdt1 • • • dt/—2dt/+1 • • • dtk =

44

E.^**(d) _ \ A

jk •••jl +1 jl — 2 ••• jl = / v jk •••jq •••j

, (2.776)

q=/ —1,/

where

H(1)(t1, • • • , t/—2,t/+1, • • • ,tk) =

to t|+1 t|+1

i{ii<^<ii-2<ii+i<„xifc} 53 / (r)^j'(r)dr / 1(r)^ji(r)dr' (2.777)

ji=p+1 { t

Hj2)(t1, • • • , t/—2,t/+1, • • • ,tk) =

to t|+1 t|—2

i{t1<^<ti-2<ti+1<^<tfc} 53 / (r)^j'(r)dr / 1(r)^ji(r)dr' (2.778)

ji=p+1 { t

H3)(t 1, • • • , t/—2, t/+1, • • • , tk)

1, • • • , t/—2

to t|+1

1{t1<^<ti-2<ti+1<^<tfc} E / 1(r)<j(t) / (0)d0dr, (2.779)

j' =p+1 { t

• • • , t/—2,t/+1, • • • ,tk) =

p

t|-2

= 1{t1<^<t'-2<t'+1<^<tfc} E J M 1(t)<j(T)J (0)^ (2.780)

j' =p+1 t t

By analogy with (2.776) we can consider the expressions

to

53 Cjk•••j'+1j'j', (2.781)

ji=p+1

E ¿¿^...¿i. (2.782)

¿1 =p+1

Then we have for (2.781), (2.782) (see (2.775) and its analogue for ¿1+1 = T)

TO „ k

E C¿к ...¿l+l¿l¿l = J ^(¿3,..., ¿k )H fa (tg)0¿з (¿g )di3 ...dtk, (2.783)

[t,T

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¿l =P+1 [tT lfc_2 g=3

TO-

=P+ Cfa^^...¿1 = J ]T Mpd)(¿1,..., ¿k—2) n fag(tg)0¿з(¿g)dii ..^¿k-2,

(2.784)

¿1 =P+1 [tTlfc_2 d=1 g=1

where

t3 T

TO 3 „

Lpfe,... ^) = 1{t3<...<tfcJ (t)0,-i(T)J fa(tf)fai(^)d^dT,

¿i=P+1 t t

■m!^ (¿1,..., ¿k—2)

TO

1{tl<...<tk_2} E I (т)0jl (T)dW fak-1(T)fai (т)dт, ¿l =p+1 t t

■m!2) (¿1,..., ¿k—2) =

P

T T

TO

1{tl<...<tk_2} E I (t)0?1 (T)dW fak-1(T)fai (т)dт, ¿l=p+1 t t

■Ml3) (¿1,..., ¿k—2) =

P

T tfc_2

T

-1{ti<...<tk_2} E I fak-i(T)0¿l(t) / fa(0)d0dT, ¿1 =P+1 t t

■Mp4) (¿1,..., ¿k—2) =

TO tk_2 T

1{ti<...<tfc_2^ / fak-1 (fafa(t) / fa(0)d0dT.

11 <...<t k — 2 }

¿l =P+1 i

It is important to note that C*(d)„- , C**(d) .

(d = 1,..., 4) are , that is, we can use Parseval's equality

Fourier coefficients (see (2.764), in the further proof.

Combining the equalities (12.764)—(2.768) (the case > g1 + 1), using Parseval's equality and applying the estimates for integrals from basis functions that we used in the proof of Theorems 2.32, 2.33, we obtain for (2.764)

2

oo

E I E ••J i

J91 ,-J9k-2 =0 V^l =P+1

j's1 =j's2 ,g2>gl + 1

OO

E I E CJk•••j

= 0 \ jSi =p+1

9=31,32

jg1 =J32 ,g2>g1 + 1

E (ECfj j

= 0 \d=1

9=31,32

< V IV C *(d)

— ^^ 1 Z—/ Jk •••Jq •••J1

q=g1,g2/ j1,...,jq,...,jk = 0 \d=1

9=31,32

y / EFpd)(t1' • • • 'tff1-1'tff1 + 1' • • • '^52-1 'W^ • • • jtk)x

j9,'",jk =0 \ u mlk-2 d=1

J1- 9=31;32k=0 2 k

X n faq(tq) j(tq)dt1 • • • dtg1-1dtg1+1... dtg2-1dtg2+1... dtk

9=1

9=31,32

/

: J EFp(d) (t1' • • • 't31-1't31 + 1' • • • 't32-1't32 + 1' • • • ,tk) n faq(tq)

9=31,32

X

/

xdt1... dtg1-1dtg1+1... dtg2-1dtg2+1... dtk —

4

—4

d=1

[t,T ]

k-2

Fp(d)(t1 ' . . . 'tg1-1'tg1 + 1' . . . ,tg2-1'tg2 + 1' . . . jtk) (tq)

X

V

9=1 9=31,32

y

K

x dii... dtgi-idtgi+i... dtg2-idtg2+i... dtk < : —> 0 (2.785)

p

2

p

2

4

p

1

2

if p ^ to, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p. The cases (2.769)-(2.771) are considered analogously.

Absolutely similarly (see (12.785)) combining the equalities (I2.776l)-(l2.780l) (the case g2 = g1 + 1), using Parseval's equality and applying the estimates for integrals from basis functions that we used in the proof of Theorems 2.32, 2.33, we get for (2.776)

/ \ 2

P /to \

E I E Cjfc-j1

j91 ,••• ,jqk-2 =0 Vj^ =P+1 jg1 =jg2 ,g2=g1+1

E E Cjk-jl

=0 \jS1 =p+1

9=31,32

El V C **(d)

1 Z_^Cjk •••jq -j'l

j1>---,jq,---,jk =0 \d=1

9=31,32

j31 =jg2 ,g2=g1+1 4

oo

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< l c**(d)

< ^^ 1 Z-/ Cjk •••jq •••j1

q=g1,g2y

j1 >---,jq,---,jk =0 \d=1

9=31,32

q=g!,g2,

, tg1 —1, tg1+2, • • • ,tk)x

,j9v,jk = 0 \ r, mi k — 9 d=1

j1,"',jq,"',jk = ^ X [tT]k-2 9=31,32 \ ]

k

x n ^q(tq) j (tq)dt1 • • • dtgl — 1^+2 • • • dtk

9=1

9=31,31 + 1

/

: J ¿Hp^,...^—l,tgl+2,•••,tk) n ^q(tq)

[t,T]k-2 \ d=1 9=3<1=^11 +1

X

/

xdt1 • • • dtg1—1 dtg1+2......dtk <

4

< 4E

d=1

k

Hpd)(t1,...,tgl—1,tgl+2,...,tk) n ^q(tq)

[t,T ]

k-2

X

V

9=1

9=31,31 + 1

/

K

x dii... dtgi-idtg1+2 ... dtk < 0

p—

(2.786)

2

4

p

2

if p ^ to, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p. The cases (2.781), (2.782) are considered analogously.

From (2785), (2786) and their analogues for the cases (2769) i277TT). (2.781), (2.782) we obtain

E E Cjk.

j91 '•••'j9k-2 =0 \ j31 =p+1

j1

j31 =j32

I<

-

(2.787)

where constant K is independent of p. Thus the equality (2.758) is proved. Let us prove the equality (2.759). Consider the following cases

1. g2 > g1 + 1, g4 = g3 + 1, 2. g2 = g1 + 1, g4 > g3 + 1, 3. g2 > g1 + 1, g4 > g3 + 1, 4. g2 = g1 + 1, g4 = g3 + 1.

The proof for Cases 1-3 will be similar. Consider, for example, Case 2. Using (2.661), we obtain

2

oo oo

E E E

j91 =0 \j'31 =p+1 j33 =P+1

j31 =j32 ,j33 = j'34 ,g4>g3+1,g2=gl+1

to p

e i e e Cj5-jl

j91 =0 \j'31 =p+1 j33 =0

j'31 = j32 ,j33 = j34 ,g4 >g3+1 ,g2=gl+1

p / p to

E £ E

•••jl

j91 =0 \j33 =0 j31 =p+1

< (2.788)

j'31 = j32 ,j33 = j34 ,g4 >g3+1 ,g2=gl+1

pp

< (p+ E E Cj5..•j,

j91 =0 j33 =0 \ j31 =p+1

j31 = j32 ,j33 = j34 ,g4 >g3+1 ,g2=gl+1

2

pp

00

(p+ 1>E E E Cj5j

j91 =0 j33 ,j34 =^j31 =p+1

j31 = j32 ,g4>g3+1,g2=gl+1

<

j33 =j34 2

pp

< (p+ ^E E E Ci5-j.

j91 =0 j33 'j34 =0 \j31 =p+1

(2.789)

j31 =j32 ,g4>g3+1,g2=gl+1

2

p

2

p

2

It is easy to see that the expression (2.789) (without the multiplier p + 1) is a particular case (k = 5,g4 > g3 + 1,g2 = g1 + 1) of the left-hand side of (2.787). Combining (2.787) and (2.789), we have

oo oo

E E E ^...¿i

¿91 =0 N¿31 =P+1 ¿33 =P+1

¿3l =¿32 '¿33 =¿34 ,g4>g3+1,g2=gl + 1

<i1 + iW<KL^()

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p2-e p1-e

(2.790)

if p fa to, where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K1 does not depend on p.

Consider Case 4 (g2 = g1 + 1, g4 = g3 + 1). We have (see (2.662

\ 2

e e e

¿9i =0 Vfa =P+1 ¿33 =P+1 ¿31 =¿32 '¿33 =¿34

OO

TOP

E E E-E L

¿91 =0 N,¿31 =p+1 N¿33 =0 ¿33 =0

¿1

oo

E o E (

2

¿ql =0 N ¿3l =P+1

¿31 ¿32 '¿33 ¿34 PTO

e e ^...¿i

¿31 =¿32 ' ¿33¿33 •) ¿33 =0 ¿3l =p+1

2

<

¿31 ¿32 '¿33 ¿34

1

-2E E (¿■-¿'

¿91 =0 ^¿31 =P+1 pip TO

+2 E E E ^...¿i

¿91 =0 N¿33 =0 ¿31 =p+1

+

¿31 =¿32 ' ¿33¿33 •)

(2.791)

(2.792)

¿31 ¿32 '¿33 ¿34

An expression similar to us estimate (2.791). We have

was estimated (see

(2.790)). Let

oo

e e

¿91 =0 N¿31 =P+1

¿1

¿31 =¿32 ' ¿33¿33 •)

(T - ¿)£ E

¿91 =0 N¿31 =P+1

¿1

<

¿31 =¿32' ¿33 ¿33 W

2

P

2

P

2

P

2

P

2

p pi to

< (t—t) ££ E c,• ,*

j91 =0 j33 =0 \j31 =p+1

(2.793)

j31 =j32 , (j33 j33 W33

where the notations are the same as in the proof of Theorem 2.30.

The expression (2.793) without the multiplier T — t is an expression of type (2.717)-(2.722) before passing to the limit lim (the only difference is the

p—>-to

replacement of one of the weight functions ^1(r),... ,^4(r) in (E7I7)-(!2[7221) by the product ^/+1(t(t) (l = 1,..., 4). Therefore, for Case 4 (g2 = g1 + 1, g4 = g3 + 1), we obtain the estimate

2

oo

Cj5

j91 =0 \j'31 =p+1 j33 =P+1

j1

K

<--, (2.794)

j31=j32,j33=j34,g4=g3+1,g2=g1+1

p

where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K is independent of p.

The estimates (2.790), (2.794) prove (2.759).

Let us prove (2.760). By analogy with (2.793) we have

Cj5 j1

j91 =0 \j33 =p+1

(j'32 j'31) ^(•) ,j31 = j'32 ,j33 = j'34 .g2=gl+1

00

Cj5 j1

j91 =0 \j'33 =p+1

(j3l j'31 =j34 ,g2=gl+1

(t—t)E E Cj5 , , j,

j91 =0 \j33 =p+1

<

(j3l j'31 )^0,j'33 =j34 ,g2=gl+1

pp

00

< (T — t)EE E Cj5 • • jl

j91 =0 j'31 =0 \j33 =p+1

(j31 j'31 )^j'31 ,j33 =j34 ,g2=gl + 1

(2.795)

Thus, we obtain the estimate (see (2.793) and the proof of Theorem 2.33)

2

K

<__ (2.796)

(j32 j'31 W0j31 =j32 ,j33 =j34 ,g2=gl + 1

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00

E I E Cj5 • • jl

j91 =0 \j33 =p+1

p2

2

p

2

p

2

where £ is an arbitrary small positive real number for the polynomial case and £ = 0 for the trigonometric case, constant K does not depend on p.

The estimate (2.796) proves (2.760). Theorem 2.34 is proved.

2.14 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 6. The Case p = ... = p6 ^ œ and fa (t ),..., (t ) = 1 (The Cases of Legendre Polynomials and Trigonometric Functions)

Theorem 2.35 [32], [37], [38], [64]. Suppose that [fa(x)}fa0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of sixth multiplicity

* T * ¿2

= | dw<;'» ...dw'f (2.797)

t t

the following expansion

P

j;(;i-ie) = i.i.m. y c6 j... z(i6)

5je

¿i, . . . ,¿6=0

that converges in the mean-square sense is valid, where fa ..., i6 = 0,1,..., m,

T t2

c*. . ., = / *M-../*--dt6

t t

and

T

j = | & (s)dw<!>

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w^^ = f^ for i = 1,..., m and wT0) = t.

Proof. As noted in Remark 2.4, Conditions 1 and 2 of Theorem 2.30 are satisfied for complete orthonormal systems of Legendre polynomials and trigonometric functions in the space L2([t,T]). Let us verify Condition 3 of Theorem 2.30 for the iterated Stratonovich stochastic integral (2.797). Thus, we

have to check the following conditions

lim

p—TO

E I E Cj6 . ..jl

,jq2 >j93 'j94 0 \jS1 Î+1

= 0,

jS1 jg2

(2.798)

oo

He, E E E ■ ■ ji

p^œ

jq1 ,j'q2 =0 \jS1 =p+l jg3 =p+l

= 0,

jS1 jg2 'jS3 jg4

(2.799)

OO

Jim E E Cj6 ■ ■ j1

p^œ

j?1 ,j92 =0 \j'fl1 =P+1

= 0, (2.800)

(j's4 jg3 WO'j'31 =jg2 'jS3 =jg4 >04=g3 + 1

lim

p—TO

E E E Cj6 ■ ■ ■ j1

j =P+1 jg3 =P+1 jg5 =P+1

= 0, (2.801)

j31 jg2 'j33 jg4 'jS5 jS6

lim

p—TO

E E j

j =p+1 jg3 =P+1

(jS6 jg5 jg2 'jS3 jg4 'jS5 =jS6 ,g6=g5 + 1

OO

lim

p—TO

Cj6...j1

j =P+1

= 0,

(2.802)

2

= 0,

(j34 j33 j35 W0j31 =j32 ,j33 =j34 ,j35 =j36 ,g4=g3 + 1,g6=g5 + 1

' (2.803)

where the expressions ({g1, g2}, fe^L {g5,g6}}) , ({g1, g2}, {g3, g4}, {q1,q2}}) ,

({g1, g2}, {q1,q2, q3, q4}) are partitions of the set {1, 2,..., 6} that is {g1, g2, g3, g4,g5,g6} = {01,02,03,04,g1,tf2} = {g1 ,g2,q1,q2,q3,q4} = {1, 2, ..., 6}; braces mean an unordered set, and parentheses mean an ordered set.

The equalities (2.798), (2.800) were proved earlier (see the proof of equalities (2.787), (2.793)). The relation (2.803) follows from the estimate (2.83) for the

polynomial case and its analogue for the trigonometric case. It is easy to see that the equalities (2.799) and (2.802) are proved in complete analogy with the proof of (2.759), (2.793).

Thus, we have to prove the relation (2.801). The equality (2.801) is equivalent to the following equalities

oo oo

lim V V y C

p

p^œ •

j 1 =p+1 j2 =P+1 j3 =P+1

j3j2j1j3j2j1

= 0,

(2.804)

2

p

2

p

2

TO TO TO ^mm y y E Cjij3j2j3j2ji j1=P+1 j2=P+1 j3=P+1 = 0, (2.805)

TOTOTO plimTO E E E Cj3j2j3j1j2j1 j1=P+1 j2=P+1 j3=P+1 = 0, (2.806)

TOTOTO plimTO E E E Cj1j2j3j3j2j1 j1=P+1 j2=P+1 j3 =P+1 = 0, (2.807)

TOTOTO j|imTO ^ E E Cj1j2j2j3j3j1 j1=P+1 j2=P+1 j3 =P+1 = 0, (2.808)

TOTOTO lim Cj3j3j2j2j1j1 P^TO z-' z-' z-' j1=P+1 j2 =P+1 j3 =P+1 = 0, (2.809)

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TOTOTO lim Cj2j3j3j2j1j1 P^TO z-' z-' z-' j1=p+1 j2 =P+1 j3 =P+1 = 0, (2.810)

TOTOTO lim Cj3j2j3j2j1j1 P^TO z-' z-' z-' j1=p+1 j2 =P+1 j3 =P+1 = 0, (2.811)

TOTOTO PlimTO E E E Cj3j3j2j1j2j1 j1=P+1 j2 =P+1 j3 =P+1 = 0, (2.812)

TOTOTO PlimTO y y y Cj3j3j1j2j2j1 j1=P+1 j2 =P+1 j3 =P+1 = 0, (2.813)

TOTOTO PlimTO y y y Cj2j1j3j3j2j1 j1=P+1 j2 =P+1 j3 =P+1 = 0, (2.814)

TOTOTO PlimTO y y y Cj3j1j2j3j2j1 j1=P+1 j2 =P+1 j3 =P+1 = 0, (2.815)

TOTOTO Plimro y y y Cj2j3j1j3j2j1 j1=P+1 j2 =P+1 j3 =P+1 = 0, (2.816)

TOTOTO Plimro y y y Cj3j1j3j2j2j1 j1=P+1 j2 =P+1 j3 =P+1 = 0, (2.817)

oo oo oo

lim E E E Cj2j3j3jlj2jl = 0. (2.818)

p—TO z-' z-' z-'

jl=p+1 j2=p+1 j3=p+1

Consider in detail the case of Legendre polynomials (the case of trigonometric functions is considered in complete analogy).

First, we prove the following equality for the Fourier coefficients for the case ^(t), • • • ,^(T) = 1

Cj6j5j4j3j2j'l + Cj'lj2j3j4j5j6 = Cj6 Cj5j4j3j2j'l — Cj5j6 Cj4j3j2j'l +

+ Cj4j5j6Cj3j2j'l — Cj3j4j5j6Cj2j'l + Cj2j3j4j5j6Cjl • (2.819)

Using the integration order replacement, we have

C=

Cj6j5j4j3j2j1 =

T t6 t2

= J j M) j j (t5) . J j (t1)dt1 • • • dt5dt6 =

t t t T T t5 t2

= J j M) J j (t5^ j (t4) . J j (t1)dt1 • • • dt4dt5dt6—

t t t t T T t5 t2

— J j M) J j (t5^ j (t4) . J j (t1)dt1 • • • dt4dt5dt6 = t t6 t t

= C C

Cj6 Cj5j4j3j2j1

T T T t4 t2

— ^ j (t6^ j (t5^ j (t4^ j (t3) . J j (t1)dt1 • • • dt3dt4dt5dt6+

t t6 t t t

T T T t4 t2

+ J j (t6^ j (t5^ j (t4^ j (t3) . J j (t1)dt1 • • • dt3dt4dt5dt6 =

t t6 t5 t t

= C C

Cj6 Cj5j4j3j2j1

T T

— J j (t6) J j (t5)dt5dt6 Cj4j3j2jl + t t6

T T T t4 ¿2

+ J j (t6^ j j (t4^ j (t3) - J j (t1)dt1 . . . dMMM^ = t ¿6 ¿5 t t

= Cj6 Cj5j4j3j2j1 — Cj5j6 Cj4j3j2j1 +

T T T t4 t2

+ J j M J j (t5^ j (t4^ j (t3) -J j (t1)dt1 - - - dt3dt4dt5dt6 =

t t6 t5 t t

= Cj6 Cj5j4j3j2j1 — Cj5j6 Cj4j3j2j1 + Cj4j5j6 Cj3j2j1 — Cj3j4j5j6 Cj2j1 + Cj2j3j4j5j6 Cj1 —

T T T

- J j (t6) y j (t5) -J j (t1)dt1 - - - dt5dt6 =

t t6 t2

= Cj6 Cj5j4j3j2j1 — Cj5j6 Cj4j3j2j1 + Cj4j5j6 Cj3j2j1 —

— Cj3j4j5j6 Cj2j1 + Cj2j3j4j5j6 Cj1 — Cj1j2 j3j4j5j6 - (2.820)

The equality (2.820) completes the proof of the relation (2.819). Let us consider (2.804). From (2.655) we obtain

TO TO TO P P P

Cj3j2j1j3j2j1 •

Ey Ey Ey Cj3j2j1j3j2j1 = Ey Ey Ey Cj3j2j1j3j2j1 - (2.821) j1=P+1 j2=P+1 j3=P+1 j1=0 j2=0 j3=0

Applying (2.819), we get

Ey Cj3j2j1j3j2j1 ^ 5-y Cj1j2j3j1j2j3 = ^ 5-y C j1j2j3=0 j1j2j3=0 j1j2j3=0

P

Cj3j2j1j3j2j1

P /

;__n V

_ C- ■ C- ■ ■ ■ + C-.-C-.-_

j2j1j3j2j1 Cj2j3 Cj1j3j2j1 + Cj1j2j3 Cj3j2j1

j1 j2 J3=°

— Cj3j1j2j3 Cj2j1 + Cj2j3j1j2j3 Cj1 J- (2.822)

P

Note that

T-t

T t

Cj2j1 ^ y j (t) j j W^T = t t

l/V(2ji + l)(2ji + 3) if J2 = Ji + 1, ¿1 = 0,1,2,

l/VW^ if ¿2 =¿1-1, ¿1 = 1,2,

0

if ¿i = ¿2 = 0

otherwise

(2.823)

t (VT=t if¿i = 0

Cji = j (t)dT = s •

i 0 if ¿1 = 0

(2.824)

Moreover, the generalized Parseval equality gives

j1 J2,j3=0

T t3

j3j2j1

t2

pimœ ê j j(t3)y j(t2) y j(ti)dtidt2dt3x

j1,j2,j3=0 t t t

T t3 t2

xj j (*)/ j (t2^ j (ti)dti dt2dt3 =

ttt p T T T

pimœ E y j (*)/ j (t2^ j (ti)dtidt2dt3 X

j1,j2,j3 =0 t t3 t2

T t3 t2

xj j (*)/ j (t2^ j (ti)dti dt2dt3 =

ttt

1

p » 3

= pim S / 1{ts<t2<ti^ (tz)dtidÎ2dÎ3 X

j1j2,j3=0 [t t]3 1=1

3

X / 1{t1<t2<t3} JJ (t)dtidt2dt3 =

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[t,T]3 1=1

= I 1{t3<t2<ti}1{ti<t2<t3} dt1dt2dt3 = 0. (2-825)

[t,T ]3

Using the above arguments and also (2.655), (2.821), and (2.822), we get

oo oo oo

lim / V / V /v Cj3j2j1j3j2j1 = lim / v Cj3j2j1j3j2j1

p^œ z—* z—* z—* p^œ z—*

j1=p+1 j2=p+1 j3=p+1 j1 ,j2 ,j3=0

1 p /

2 «s. ^

... _ C ■ C ■ ■ ■ —

j2j1j3j2j1 Cj2j3 Cj1j3j2j1

j1 j2 ,j3=0

— Cj3j1j2j3 Cj2 j1 + Cj2j3j1j2j3 Cj1 J =

p /

lim E C*3 Cj

CC

P—>-TO z—'

j1 ,j2 ,j3=0

PP

= lim V^ ^j2jl0j2jl ~ / ^ ^ './:■,/./:./:■/ './:./ =

P—TO z-' P—TO z-'

j1 j2=0 j1 j2j3=0

P P TO

= VT-i lim E ' './:./ "./:./ • '"l" E E O,/ ./:./:A./ • (2"826)

P—TO z-' P—TO z-' z-'

j1 ,j2=0 j1,j2=0 j3=P+1

By analogy with the proof of (2.723) (see the proof of Theorem 2.33) we obtain

P TO TO

liP E Cj2j10j2j1 = lim E E Cj2j10j2j1 = 0 (2.827)

p—TO p—TO

j1,j2=0 j1=P+1 j2=P+1

where we used the following representation

Cj2j10j2j1 =

V

T ¿5 ¿4 t3 ¿2

-J MM J MM J J MM J MMdtidtoJUdUdh = t t t t t T t5 t4 t2 t4

^—-J MM J MM J MM J MMdh J dt.3dt2dt.4dt5 = t t t t t2 T t5 t4 t2

VtM,

t t t t T t5 t4 t2

+ ^—- J MM J M(^4) J MM{t - h) J MMdtidt2dt4dt5 = t t t t

def c + C

= Cj 2 j 1 j 2 j 1 + Cj 2 j 1 j 2 j 1 .

Further, we have (see (2.823))

p TO TO /

lim E E Cj3jlj2j3 Cj2jl = lim / , c00cj300j3+

p—TO z—' z—' p—TO z—' V

jl ,j2=0 j3 =p+1 j3=p+1

p p—1 \

+ y^ Cjl —1,jlCj3j'l,j'l —1,j3 + E Cjl + 1,jlCj3j'l,jl+1,j3 + C1,0Cj301j3 j • (2.828) 1 i /

Observe that

K

IQ1-U1I + \M+M < - (Jl = 1, • • • (2-829)

J1

|Cj300j3 1 + 1 Cj3 jl ,j 1 — 1 ,j31 + |Cj3jl,jl+1,j31 + |Cj301j31 < K1

c/3>p + 1), (2-830)

j3

where constants K, K1 do not depend on j, j3.

The estimate (2.829) follows from (2.823). At the same time, the estimate (2.830) can be obtained using the following reasoning. First note that the integration order replacement gives

T t4 t3 t2

Cj3j1j2j3 = / j (U)j j j (U)j j (t1 l^MMM^ =

tttt

T t3 / t2 \ / T

= f j (*)/ j (¿2) J j (t1)dtj dtJ J j (t4)dt4 I dt3- (2.831)

t t t t3

Applying the estimates (2.142), (2.143), and (2.159) to (2.831) gives the estimate (2.830).

Using (2.828), (2.829), and (2.830), we obtain

PTO

' c. . . . C

j2j1

Ey Ey Cj3j1j2j3Cj

j1 ,j2 =0 j3=p+1

ë ifl + e1! <

j3=p+1 j3 V j1 = 1 j1

TO / P \

<KJte ¡2 + f^ (2.832)

if p — to, where constant K is independent of p- Thus, the equality (2.804) is proved (see (2.826), (2.827), (2.832)).

The relation (2.805) is proved in complete analogy with the proof of equality (2.804). For (2.805) we have (see (2.819))

P P P

linl / v Cj1j3j2j3j2 j1 + Cj1j2j3j2j3j1 I = 2 lim / v Cj1j3j2j3j2j1 =

P—TO P—TO

V1 ,j2,j3=0 j1,j2,j3 =0 / j1,j2,j3=0

p

.71*2*3 =0 V

... _ C ■ C ■ ■ + C ■ ■ ■ C ■ ■ ■ —

j3j2 j3j2 j1 Cj3 j1 Cj2j3 j2j1 + Cj2j3 j1 Cj3j2 j1

Cj3j2j3j1 Cj2 j1 + Cj2j3j2j3j1 Cj1 J =

P

2 lim j \/Tr--7 y ( './:■,/:./',/: 11 ~~ / v

j2*3=0 j 1*2*3 =0

p

2 lim / V Cj2j1 Cj3j2j3j1"

p^œ

j1*2*3=0

To estimate the Fourier coefficient Cj***, we use the following (see the proof of (2.804) for more details)

T t4 t3 t2

Cj3j2j3j1 = j (t4M j (t3M j (t2M j (t1 )dt1dt2dt3dt4 =

T t4 t3 t3

= J j (t4^ y j ^ j (t1^ j (t2)dt2dt1dt3dt4 = t t t t1 T t4 / t3 \ t3

= y j (t4 )J j (t3M y j (t2)dt^ y j (t1)dt1 dt3dt4 — t t t t T t4 t3 / tl

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— J j (t4) y j (t3) J j (t1) ( y j (t2)dt^ dt1dt3dt4 = t t t t T / t3 \ t3 / T

= j j(t3M y j(t2)dt^ y^jl(t1 )dtM y j(t4)dt^ dt3—

t t t t3

T t3 / tl \ / T

— j j (t3) J j (t1M y j (t2)dt^ dtu y j (t4)dt4 I dt3.

t t t t3

Let us prove (2.806). From (2.655) we obtain

TO TO to p p p

Cj3j2j3jl j2jl •

53 53 Cj3j2 j3j'lj2j'l = EE 53 Cj3 j2j3j'l j2j'l • (2.833)

j'l=p+1 j2=p+1 j3=p+1 j1=0 j2=0 j3=0

Applying (2.819) and (2.833), we get (we replaced j3 by j4)

p p p

Cj4j2j4j1j2j1 + Cj1j2j1j4j2j4 = 2 Cj4j2j4j1j2j1

j'l J2,j4=0 j'l,j2,j4 =0 j'l,j2,j4=0

p ^

E (Cj4 cj

;„ n V

p

_ C- ■ C- ■ ■ ■ + C■ ■ ■ C■ ■ ■ —

y j2 j4 jl j2 jl Cj2j4 Cj4j1 j2 jl + Cj4j2 j4 Cj1j2j1

j'lj2,j4 =0

Cj'lj4j2j4 Cj2 j'l + Cj2j'lj4j2j4 Cj'l J =

p / 2C

j2 j1j4j2j4 Cj'l Cj'lj4 j2j4 Cj2j'l ) +

j'lj2 ,j4=0

+ E Cj4j2j4 Cj'lj2j'l • (2.834)

j'lj2,j4 =0

Further, we have (see

llm 53 Cj4j2j4 jj1 = llm 53 53 C

p^œ z—' p^œ z—' \ z—'

j1 ,j2 ,j4=0 j2=0 \j1=0

pp

Cj1j2j1

2

p œ 2

HeE E c^u =0, (2-835)

p^œ z

j2=0 \j1=p+1

where we applied the equality (2.697).

Furthermore, by analogy with the proof of (2.804), we have

lim y ( Cj2j1j4j2 j4 Cj1 — Cj1j4j2j4 Cj2 j1 ) = 0 (2.836)

p^œ z—' y /

j1,j2 ,j4=0

To estimate the Fourier coefficient Cj1j4j2j4 in (2.836), we use the following (see the proof of (2.804) for more details)

T t4 t3 / t2 \

Cj1j4j2j4 = J j j ^ j (t2M J j (t1)dt1 I dt2dt3dt4 = t t t \t /

T t4 / t2 \ t4

= J j (t^y j (¿2) ( J j (¿1)^1) J j (¿3)dt3dt2dt4 = t t t t2 T / t4 \ t4 / t2

= J j (¿4) ( J j(¿3)^3 I J j (t2) I J j (¿1)^1) dt2dt4-t t t t T t4 / t2 \ / t2

-J j (¿4 ) J j (¿2) ( y 0j4 (t3)dt3 II y 0j4 (¿1)^1 ) dt2 dt4.

t t t t

The relations (2833)-(2836) complete the proof of equality (2806). Let us prove (2.807). Using (2.655), we get

œ œ œ p p œ

Cj1j2 j3j3j2j1 •

53 53 Cj1j2j3j3j2j1 = 53 Cj1j2j3j3j2j1 • (2.837)

j1=p+1 j2=p+1 j3=p+1 j1=0 j2=0 j3=p+1

2

Applying (2.819) and (2.837), we obtain

pœ

Cj1j2j3j3j2j1

2 £ £ Cj

j1,j2 =0 j3=p+1

pœ

Ey ( Cj1 Cj2j3j3j2j1 — Cj2j1 Cj3j3j2j1 + (Cj3j2j1) —

j1,j2 =0 j3=p+1

C7o ^ j j C j j + C j jo jo j j C j )

j3j3j2j1 j2j1 j2j3j3j2j1 j1

pœ

2 ^ ^ Ey ( Cj1 Cj2j3j3j2j1 — Cj2j1 Cj3j3j2j1 j + j1,j2=0 j3=p+1 ^

pœ

+ E E (Cj3j2j1 )2. (2.838)

j1,j2=0 j3=p+1

Using the estimate (1.217), we get

pœ

lim E E (Cj3j2j1 )2 = 0. (2.839)

p^œ z—' z—'

j1,j2=0 j3=p+1

By analogy with the proof of (2.804), we have

p œ , x

^^ Ey (Cj1 Cj2j3j3j2j1 — Cj2j1 Cj3j3j2j1 j = 0, (2.840)

j1,j2=0 j3 =p+1 ^

where we applied the equality (2.724). To estimate the Fourier coefficient Cj3j3j2j1 in (2.840), we used the following (see the proof of (2.804) for more details)

T t4 t3 t2

Cj3j3j2j1 = J j (t4^ y &3 M/ j (t2^ y j (t1 )dt1dt2dt3dt4 = tttt

T T T T

= y j (t1^ j (t2)J j to)J j (t4)dt4dt3dt2dt1 =

t t1 t2 t3

T T / T \ 2

= \J fe(ii) J <f>h(t2) J o,JI,)<ll, dtodh. (2.841)

t t1 t2

Combining the equalities (I28371)-(28m we obtain (2807)).

Let us prove (2.808) (we replace j2 by j4 and j3 by j2 in (2.808)). As noted in Remark 2.4, the sequential order of the series

TOTOTO

ees

j 1 =P+1 j2 =P+1 j4 =P+1

is not important. This follows directly from the formulas (2.662) and (2.655). Applying the mentioned property and (2.655), we get

TO TO TO P TO TO

Ey Ey Ey Cj1 j4j4 j2 j2 j1 = — y^ Cj1j4j4j2j2j1 - (2.842)

j1=P+1 j2=P+1 j4=P+1 j1=0 j2=P+1 j4=P+1

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Observe that (see the above reasoning)

TO TO TO TO

Ey 5-y Cj1j4j4j2j2j1 = y ^ Ey Cj1j4j4j2j2j1 - (2.843)

j2 =P+1 j4 =P+1 j4 =P+1 j2 =P+1

Using (2.819) and (2.843), we obtain

P TO TO P TO TO

yi y^ y^ ( Cj1j4j4j2j2j1 + Cj1j2j2j4j4j1 j = 2 ^^ ^^ ^y Cj1 j4j4j2j2j1 =

j1=0 j2=P+1 j4 =P+ 1 ' j1=0 j2=P+1 j4=P+1

P TO TO

= y ^ ^^ Ey ( Cj1 Cj4j4j2j2j1 — Cj4j1 Cj4j2j2j1 + Cj4j4j1 Cj2j2j1 —

j1=0 j2=P+1 j4=P+1

— Cj2j4j4j1 Cj2j1 + Cj2j2j4j4j1 Cj1 ^ =

P TO TO

TOTO

y^ y^ y^ ( Cj1 Cj4j4j2j2j1 — Cj4j1 Cj

j1=0 j2=P+1 j4=P+1

OO

j4j2j2j1 Cj2j4j4j1 Cj2j1 + Cj2j2j4j4j1 Cj1 J +

+ El E ) - (2.844)

j1=0 \j2=P+1

2

The equality

pœ

p&E E «j = 0 (2.845)

p^œ

j1=0 \j2=p+1

follows from the relation (2.696).

By analogy with the proof of equality (2.804) we obtain

p TO TO

p œ œ

p&E £ £ (C'-1 C

—n .;„—nxi ___nxi \

— C C

._. ,_. ,_. , _ y j4 j4j2 j2 j1 Cj4j1 Cj4j2j2j1

p^œ z—' z—' z—'

j1 =0 j2=p+1 j4=p+1

Cj2j4j4j1 Cj2j1 + Cj2j2 j4j4j1 Cj1 J = 0, (2.846)

where we applied the equality (2.725). To estimate the Fourier coefficient Cjjj in (2.846), we used the following (see the proof of (2.804) for more details)

T t4 t3 t2

Cj2j4j4j1 = J j (t4^ y j ^ j (t2^ y j (t1 )dt1dt2dt3dt4 = tttt

T t4 t4 t4

= y j (¿4) y j (¿1 )J j (¿2^ j (¿3)dt3dt2dt1 dt4 =

t t t1 t2

T t4 / t4 X 2

= 2 / fe^) / <fei(*l) I I (pjAMdto ) =

t t t

T / t4 \ 2 t4

2 /' ^2(^4) I I MiMdto ) I t t t T t4 / t1 ^ 2

y ^2(^4) y ^(ii) jy (f>j4{h)dt2 ) dt.1dt.4-t t t t / t4 \ t4 / t1

-j j (¿4 ) ( y^j4 (¿2)^2 I y j (¿1) I y 0j4 (¿2)^2 ) dt1 dt4.

t t t t

The relation (2808) follows from (E842), (E844D-(E

Consider (2.809). Using the integration order replacement, we obtain

C=

Cj3 j3j2j2j1j1

T te t5 t4 / t3 X 2

= ^ j OjJIc,) j OjJI:,) J (f>j2{U) J (f)j2{h) i J faitfadti I dt3dt4dt5dt6 = t t t t t t / t3 \2T T T

= 2 / fe^) ( I (f>h{ti)dti ) j (f)j2{U) I Oj.Jh,) j (f)j3{te)dt6dt5dUdt3 =

t / t3 \2t / T

t t t3 t4 t5 1 J 4>nih) I y ^(ijrfiij

t3 t4

4 j fete) I I I j (f>h(U) ( j fa,(h)dh I «3- (2.847)

Applying the estimates (2.142), (2.143), and (2.159) to (2.847) gives the following estimate

I ^3,3,2,2,ml < O'l•./:', > 0, Jo > 0), (2.848)

where constant K does not depend on j2, j3. Further, we get (see (2.662

œ œ œ œ œ œ

e e e

Cj3j3j2j2j1j1

E E E

Cj3j3j2j2j1j1

j 1 =p+1 j2 =p+1 j3 =p+1 j 1 =p+1 j3 =p+1 j2 =p+1

OO OO

1 ^ ^ p œ œ

2 Ey ^ '/•../••./../.././ ~~ Ey ^ '/•./ •../../. ././ • (2-849) ' - (j2j2)^(') j2=0 j1=p+1 j3=p+1

j1 =p+1 j3=p+1

where

C

Cj3j3j2j2j1j1

(j2j2)^(')

T t6 t5 t4 t2

= J j (t6^ j(t5^ J j fa) J j (t1)dÎ1dÎ2dt4dt5dt6 =

t t t t t T t6 t5 t2 t5

= J j fa) J j (t5)J j fa) j j (t1)dt^y dt4dt2dt5dt6 = t t t t t2

2

T t6 t5 t2

= J j (t6)J j (t5) (t5 — t) J j (t2^ j (t1)dtxdt2dt5dt6+ t t t t T t6 t5 t2

+ J j (t6^ j(tB)/ j (t2)(t — t2) y j (t1)dt1 dt2dt5dt6 d=f

t t t t

= Cj'3j3j'ljl + Cj//3j3jljl • (2.850)

Let us substitute (2.850) into (2.849)

TO TO TO 1 TO TO

E E E ( '.IM.l-.i r = 2 E E ( './:■■./:',/ ./ j l =p+1 j2 =p+1 j3 =p+1 j 1 =p+1 j3 =p+1

1 TO TO p TO TO

+ 2 E E ^3.73.71.71 ~ E E ^hhhhhh- (2.851)

j 1 =p+1 j3 =p+1 j2 =0 j 1 =p+1 j3 =p+1

The relation (2.725) implies that

TO TO TO TO

Jim £ £ //. = 0 p-£ £ j/ = (2 852)

p^œ ' ^ ' j^^^1 p^œ

j 1 =p+1 =p+1 j 1 =p+1 j3 =p+1

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From the estimate (2.848) we get

p œ œ

Cj3j3j2j2j1j1

£ £ £ Cj

j2=0 j1=p+1 j3=p+1

œœ

<A> + 1) £ ? £

j1=p+1 j1 j3=p+1 j3

2 2

<^- 0 (2.853)

p

if p ^ œ, where constant K is independent of p.

The relations (ESSHHESSS) complete the proof of (28m

Let us prove (2.810). Using the integration order replacement, we get

C=

Cj2 j3j3 j2 j1 j1

T te t5 t4 / t3 X 2

= 2 / ^2^6) / Oj.JI:,) / / <fe2(*3) I I (f)n{ti)dti ) (ll'/ll\(ll:,(llv, =

T / t3

T T T

fe te) / fe (t1)dtu / fe te) / j te) / fe (t6)dt6dt5dt4dt3 =

tt T / t3

t3 t4 t5

T T

t5

fe te) / fe (t1)dtu / fe te) / j (ta)dt^ / j (t4)dt4dt5dt3 =

tt t / t3

t3 t5

t / T

t3

t5

= o / fete) / fete№ / o;, (/.-,) / <f>h{t6)dt6 / o:Jt\)(H I x

tt

T / t3

t3 t5

xdt5dt3—

t3

t / T

2

fe (t3M / fe (t1)dt1 / fe (t4)dt4 / j (t5M / j (t6)dt6 x

tt

t t3 t5

x dt5dt3. (2.854)

Applying (2.655) and (2.662), we obtain

oo oo oo

E E E

Cj2j3j3j2j1j1 =

—

Cj2j3j3j2j1j1

j 1 =p+1 j2 =p+1 j3 =p+1 j 1 =p+1 j3 =p+1 j2 =p+1

p TO TO

EE E

Cj2j3j3j2j1j1

j2 =0 j 1 =p+1 j3 =p+1

p TO

j2=0 j'l=p+1

p TO

p p TO

EE E

Cj2j3j3j2j1j1

(j3j3 WO j2=0 j3=0 j1=p+1

2 53 53 ^ '/ ./•./••./ ./ / j2=0 j'l=p+1

TO

C0000j1j1 (j3j3W0 j'1=p+1

p TO p TO

E E

C0j3j30j'lj'l

—

Cj200j2j'lj'l

j3 = 1 j'l=p+1 j2 = 1 j'l=p+1

p p TO

EE E

Cj2j3j3j2j'l j'l •

j2 = 1 j3 = 1 j'l=p+1

(2.855)

2

1

2

1

2

1

The equality

1 p to li:u - £ £

j2=0 j1=p+1

j2j3j3j2j1j1

= 0 (2.856)

(*3*3W • )

follows from the inequality similar to (2.751) (see the proof of Theorem 2.33), where we used the following representation

C......

Cj2j3j3j2j1j1

(j3j3W • )

T ¿6 ¿4 ¿3 ¿2

= J j (to^ J j fa) j j (t2^ j (11)dtidt2dt3dt4dt6 =

t t t t t T t6 t3 t2 t6

= J j fa)j j fa)j j fa) j j (ti)dtidt^y dt4dt3 dto = t t t t t3 T t6 t3 t2

+ J 0j2(to)(to - t) y j(t3)J j(t2) y j (ti)dtidt2dt3dto+ t t t t T t6 t3 t2

+ j j fa)j j(t3)(t - *)/ j (t2^ j (ti)dtidt2dt3dto d=f

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t t t t

= Cj2j2j1 j1 + Cj2!j2j1j1 * (2.857)

Applying the estimates (2.142), (2.143), (2.159), and (2.711) (e = 1/2) to (2.854) gives the following estimates

K

j2,j3 > 0), (2.858)

Jlj2j3

K

1^200^1 <72- (J1,J2> 0), (2.859)

Ji J2

K

^ '"*:*:'."* * <727 Ol ••/:'.> 0). (2.860)

Ji J3 K

|Coooo,m|<72 (J1>0). (2.861)

Using the estimate (2.858), we have

p p œ

j2j3j3j2j1j1

EE E Cj

j2 = 1 j3 = 1 j1=p+1

œ 1 p 1 p 1

^^ j2 ^ -3/4 j1 =p+1J1 j2 = 1 2 j3 = 1 j3

œ / p \ i p

< tf I % M + / ^II |1 +

X2

X

(te \ < Kl_±lnp ^ 0 (2862)

X3/4

p

3/4

if p — to, where constants K, K1 do not depend on p. Similarly we get (see (2859) (2861))

p TO

E E

œ

y C0000j1j1 j1=p+1

+

pœ

EE

C0j3j30j1j1

j3=1 j1=p+1

+

j2 = 1 j1=p+1

^ 0 (2.863)

if p ^ œ.

The relations (2.855), (2.856), (2.862), (2.863) prove (2.810). Consider (2.811). Using the integration order replacement, we get

T

te

t5

C=

t4

t3

2

j(¿6M j(¿5) / j(¿4M j(¿3) / j (¿1)^1 dMMM^ =

T

t3

T

j(¿3) / j (¿1)^1 / j(¿4M j(¿5) / j(¿6)dÎ6dÎ5dÎ4dÎ3 =

t3

t4

t5

T

t3

T

t5

j (¿3) / j (¿1)^1 / j (¿5) / j / j (¿4)dMMÎ3 =

t

T

t

t3

T

t5

t3

t5

T

j (^3 M / j fa^ / j (^5 M / j ^4^4 / j ^6^6 X

t3 t

xdi5di3-

t5

T

t3

T

-Ô / fete) /

2

j ^^ / j (¿5^ ( / j te^ X

t3

t5

2

1

2

1

2

1

2

1

2

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2

x dt5dt3- (2.864)

Applying (2.655), we obtain

TO TO TO TO TO TO

Ey Ey Ey Cjsj2j3j2j1j1 = ^^ yi Cj3j2j3j2j1j1 =

j 1 =P+1 j2 =P+1 j3 =P+1 j 1 =P+1 j3 =P+1 j2 =P+1

P TO TO

= — ^y Ey Ey Cj3j2j3j2j1j1 - (2.865)

j2=0 j1=P+1 j3=P+1

Further proof of the equality (2.811) is based on the relations (2.864), (2.865) and is similar to the proof of the formula (2.810).

Let us prove (2.812). Applying the integration order replacement, we obtain

C=

Cj3j3j2j1j2j1 =

T ¿6 ¿5 ¿4 ¿3 ¿2

= J fe (t6) y fe (t5) y fe (t4) y fe (*)/ fe (^2^ y fe (t 1 )dtidt2dt3dt4dt5dt6 =

i i i i i i TTTTTT

= / fe (t1^ fe (t2^ fe &)/ fe (t4^ fe te^ fe (t6)dt6dt5dt4dt3dt2dt1 =

i i1 i2 i3 i4 i5

T t T T / T \ 2

= ^ I <f>ji(tl) /fete) /fete) /fe(*4) I I Oj.J I:,)(!/:, ) (¿MMM^! =

T / T \ 2 ¿4 ¿3 ¿2

i i1 i2 i3 i4

i \i4 / i i i

i \i4 / i i ¿2

1

2J fete) I J Oj.JI:,)(!/:, I I fete) I fete) I fete)dMMMi4 =

i i4

T / T \ 2 i4 ¿2 ¿4

2y fete) jy Oj.JI:,)(ll:, J j' fete) j (f)n{ti)dti j (f)n(U)dUdtodtA =

i i4

T / T \ 2 / ¿4 \ ¿4 / ¿2

2 y fete) [ J 4>n(h)dh J Or(l,)dl, J 4>n(to) J faitfadt! I X i4 i i i xdt2dt4-

T / T \ 2 t4 / t2

-\ J MM J MMdh J MM J MMdti I x t t4 t t

x dMÎ4. (2.866)

Using (2.655), we get

œ œ œ œ œ œ

53 53 Cj3j3j2j1j2j1 = 53 Cj3j3j2j1j2j1 =

j 1 =p+1 j2 =p+1 j3 =p+1 j 1 =p+1 j3 =p+1 j2 =p+1

p œ œ

= — y y 53 Cj3j3j2j1j2j1 • (2.867)

j2=0 j1=p+1 j3=p+1

Further proof of the equality (2.812) is based on the relations (2.866), (2.867) and is similar to the proof of the relations (2.810), (2.811).

Consider (2.813). Using the integration order replacement, we have

C=

T te t5 ^

= J M (¿6) y M (¿5) y M (¿4) y j (¿3) y M (¿2) y M (Î1)dMÎ2dÎ3dMMÎ6 =

tttttt TTTTTT

= J M (¿1^ m (¿2) y M (¿3) y M (¿4) y M(¿5^ 0j3(t6)dÍ6dt5dÍ4dtзdÍ2dtl =

t t1 t2 t3 t4 t5

T T T T /T \ 2

= \ ^JiM / fete) / fete) / fete) I / Oj.Jh,)(lh, ) ûMMMil =

t t1 t2 t3 t4

1

T / T \ 2 t4 t3 t2

2J fete) I / Oj.JI:,)(ll:, ) / (fe2(i3) / fete) / ^(¿î^MMM^ = t t4 t t t T / T \ 2 t4 t2 t4

= IJ MM ij Oj.JI:,)(ll:, I y ^2(i2) y y MMdhdtodU =

t t4 t t t2

1Tf (tÎ X T/ V/ r/ X

= 2 J ^^ J J MMdh J MM J MMdh x

t t4 t t t

xdt2dt4—

T / T \ 2 i4 / ¿2 \ / ¿2

\ I 4>n(U) j 4>n(h)dh j 4>n(U) I faitfadh I o,V,)dl, ) x i i4 i i i

x dt2dt4- (2.868)

Applying (2.655) and (2.662), we obtain

TO TO TO TO TO TO

Ey Ey Ey Cj3j3j1j2j2j1 = — y^ Cj2j3j1j2j2j1 =

j 1 =P+1 j2 =P+1 j3 =P+1 j2 =P+1 j3 =P+1 j 1 =P+1

P TO TO P TO TO

C

y^ y^ y^ Cj2j3j1j2j2j1 = ^^ y^ y^ Cj2j3j1j:

j1=0 j2=P+1 j3=P+1 j1=0 j3=P+1 j2=P+1

PTO

2

j1=0 j3=P+1

P P TO

o>; ey ^ '/••-/••-/./././ ey ey ey ^ '/••-/•././././ • (2.869)

(j2j2)^(') ^=0 j2=0 j3=P+1

The equality

1 P TO

P—TO2

j1=0 j3=P+1

j3j3j1j2j2j1

= 0 (2.870)

(j2j2)^(')

follows from the inequality (2.751), where we proceed similarly to the proof of equality (2.856) (see (2.857)).

The relation

P P TO

lim E E E Cj j2j2j1 = 0 (2.871)

p—TO

j1=0 j2=0 j3=P+1

is proved on the basis of (2.868) and similarly with the proof of (2.810). The equalities (2.869)-(2.871) prove (2.813).

Let us prove (2.814). Using (2.655) and (2.662), we get

TO TO TO TO p

Cj2j1j3j3j2j1 = Cj2j1j3j3j2j1 =

j 1 =P+1 j2 =P+1 j3 =P+1 j3 =P+1 j 1 ,j2 =0

P

2

j1,j2=0

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P

ey ^ '/:./ ./::./:;./:./ ey ^ '/:./ ./:;./:;./:./ • (2.872)

(j3j3)^( 0 j1,j2,j3=0

Using the equality (2.723) we have

1P

lim - V C

p—>-to 2 ' j1 ,j2=0

j2j1j3j3j2j1

= 0, (2.873)

(j3j3W-)

where we proceed similarly to the proof of equality (2.856) (see (2.857)).

Further, we will prove the following relation

P

Hm Y Cj3j-1 = 0 (2.874)

P—TO z-'

j1,j2,j3 =0

using the equality (2.819). From (2.819) we have

P 1 P ( \

2

j1,j2,j3 =0 j1 ,j2 ,j3=0

1p

2 / V V - ./2 ~ j1 j3j3 j2 j1 — C j1 j2 C j3 j3j2 j1 + Cj3j1j2 C j3 j2 j1 — j1 ,j2 ,j3=0

E (Cj2 Cj

;„ ;__n V

Cj3j3j1j2 Cj2j1 + Cj2j3j3j1j2 Cj1 J =

Ey ( Cj2j3j3j1j2 Cj1 Cj3j3j1 j2 Cj2j1 j + j1 ,j2 ,j3=0

1p

+ 2 Ey ^ './'■./ ^ './:',/:./ ' (2.875)

j1 ,j2 ,j3=0

The generalized Parseval equality gives (by analogy with (2.825))

1p

^O E 0. (2.876)

P—to 2 z—'

j1 ,j2 ,j3=0

Let us prove the following equality

P-i—TO Ey (Cj2j3j3j1 j2 Cj1 Cj3j3j1j2 Cj2.71) 0- (2.877)

j1,j2 ,j3=0

»

The relation

p

lim V Cj2j3j3jl/2 Cj1 = 0 (2.878)

rt—Vm * *

p—TO

j'l,j2,j3 =0

is proved by the same methods as in the proof of equality (2.804) and also using Theorem 2.33 and (2.662).

Further, we have (see (2.662))

y V ^ './:■,/:■,/•./:• ~~ ^ './",/:',/ ./: ^ ^ ^ './:',/",/ ./: • (2.879)

(j3j3W0 j3 =p+1

2

j3=0

Moreover,

C

Cj3j3j1j2

(j3 j3 WO

T t3 t2

= /1 j ^Z * (t1)dt1 dt2dt3 =

t t t T t2 T

= J j (t2^ / (t1 )dt^ y dt3dt2 =

t t t2 T t2

= J(T — t2)j (t2)J j(t1)dt1dt2

tt T T

= y j (t1) J(T — t2)j (t2)dt2dt1

t t1 T T

= f j (t2) J(T — t1)0jl (t 1)dt 1 dt2

t t2

def

(T — t1)1{t2<tl}0jl (t1)0j2 (t2)dt1dt2 =

[t,T

= j • (2.880) http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal. http://diffjournal.spbu.ru/ A.568

Using (2.879), (2.880), and the generalized Parseval equality, we obtain

p 1 p

lim y Cj3j3j1 j2Cj2j1 = — lim \ Cj2jlCj2jl —

p—TO z—* 2 p—to z—*

j'l J2,j3=0 j'l,j2=0

p TO p TO

— lim /v Cj3j3j'lj2 Cj2j'l = — lim / . Cj3j3j'lj2 Cj2j'l • (2.881) p—TO z-' z-' p—TO z-' z-'

j' 1 ,j2 =0 j3 =p+1 j 1 ,j2 =0 j3 =p+1

We have (see (2.841))

T T / T \ 2

' W./, =lj J O ,.('■>) J 0,„(l,)dl, dtodtl. (2.SS2) t tl \t2 /

By analogy with (2.832) and also using (2.882), we get

p TO

lim zJ zJ Cj3j3j1j2 Cj2j1 = 0. (2.883)

p—TO z-' z-'

j'l ,j2=0 j3 =p+1

Combining (2.881) and (2.883), we obtain

p

p—œ

lim £ MM2 Cj2j1 = 0. (2.884)

)—vœ ' *

j1,j2,j3=0

The relation (2877) follows from (2878) and (2884). From (2875)-(2877) we get (27874). The equalities ^2872)-^2874P complete the proof of ^2833).

For the proof of (2.815)-(2.818) we will use a new idea. More precisely, we will consider the sums of expressions (2.815)-(2.818) with the expressions already studied throughout this proof.

Let us begin from (2.815). Applying the integration order replacement, we obtain

Cj3j1 j2 j3 j2 j1 + Cj3j1j2j3j1 j2 = T te t5 t4 / t3 \ / t3

= J M M j M (¿5) J M M J M fcO I J M ^2^2 I I J M (¿1)^1 ) X t t t t t t

xdi3di4di5 di6 =

T t6 t5 / t3 \ / t3 \ t5

= J 03 M y j (t5^ y j (t3H y j (t2)dt2 y j (ti)dt i y j (t4)dt4X t t t t t t3

xdt3dt5dt6 =

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T t6 / t5 \ t5 / t3

= J j W J j (t5M y j (t4)dt4 I y j (t3) ( y j (t2)dt2 j x t t t t t

r/ \

x / j (ti)dtM dt3dt5dt6-

T t6 t5 / t3 \ 2 / t3

-f 03 fa) f j (t5^ j (t3) y j (t2)dtj y 01 (ti)dti | X t t t t t

xdt3dt5dt6 =

T / t5 \ t5 / t3

= J j Ml y j (t4)dt4 I y j (t3M y j (t2)dt2 j x t t t t

Ci \ ( T

x / (ti)dti dtn / 0j3(t6)dt^ dt5-

T t5 / t3 \ 2 / t3

- J j C^ y j (t3M y j (t2)dt2 J I y (ti)dti ) dt3x

t t t t

x ^y j(t6)dt6^ dt5. (2.885)

Using (2.655), we get

to to to / \

Ey ( Cj3j1j2j3j2j1 + Cj3j1j2j3j1j2 j =

j1 =p+i j2 =P+i j3=P+i

P P TO X N

= ^ ^ ^^ Ey ( Cj3j1j2j3j2j1 + Cj3j1j2j3j1 j2 j • (2.886)

j1=0 j3=0 j2 =P+i ^

Further, by analogy with the proof of equality (2.810) and using (2.885), we obtain

p p œ

p^œ

iï111^ V ^ y y y jlj2j3j2jl + Cj3jlj2j3jlj2 j — 0- (2.887)

jl =0 j3=0 j2=p+1

From (2.886) and (2.887) we get

œ œ œ , N

j™ ^ ) V y V (Cj3j'lj2j3j2jl + Cj3jlj2j3jlj2 j — 0- (2.888)

jl=p+1 j2=p+1 j3=p+1

Moreover (see (2.804)),

oo oo oo

^m E E E Cj3jlj2j3jlj2 = 0. (2.889)

j l =p+1 j2 =p+1 j3 =p+1

Combining (2.888) and (2.889), we have

oo oo oo

1 ""EE E Cj3jlj2j3j2jl — 0-

p^œ

j l =p+1 j2 =p+1 j3 =p+1

The equality (2.815) is proved.

Consider (2.816). Using the integration order rep lacement, we have

Cj2j3jlj3j2jl + Cj2j3jlj3jl j2 —

T te t5 t4 / t3 \ / t3

— J 02 M y 03 (t5) y j y 03 (t3M y j (t2)dt2 I I y 0jl (t1)dt1 ) X t t t t \t / \t

xdt3dt4dt5 —

T te t5 / t3 \ / t3 \ t5

— J 02 (t6) y 03 (t5^ y j Ml y 02 (t2)dt2l I y 0l (t1 )dt1 I y j (t4)dt4 x

t t t t t t3

xdt3dt5 —

T te / t5 \ t5 / t3

— J 02 M y 03 M y j (t4)dt4 y 03 (t3 H y 02 (t2)dt2 ) x t t t t t

is

x [ J j (ti)dt^ dt3dt5dt6-

T t6 t5 /is \ / is X 2

- J j (t6 ^ j M J j (t3M J j (t2)dt2l I J j (t1)dt1 ) x i i i i i

xdt3dt5dt6 =

T / t5 \ i5 /is

= J j (t5 W J j (t4 )dt4 J j (t3M J j (t2)dt2 | X i i i i

x (/^j1 (ti)dti^ j (tß)dt^ dt5-

T i5 /is \ / is \ 2

-J j M J j (t3M y j (t2)dt2l I y j (t1)dtll dt3X

X ^y j (to)dtoj dt5. (2.890)

Using (2.655), we obtain

to to to ✓ \

— ^ v ^ v I Cj2jsjljsj2 jl + Cj2jsj1jsj1 j2 ) =

jl=p+1 j2=P+1 js =P+1

p to to

53 53 53 ( Cj2js jl js j2 jl + Cj2jsj1jsj1j2 j • (2.891)

js =0 jl=P+1 j2=P+1

By analogy with the proof of (2.810) and applying (2.890), we get

p TO to / \

Hm ) : ) : 53 (Cj2jsj1jsj2jl + Cj2jsj1jsj1j2 j = 0. (2.892)

js=0 jl=P+1 j2=P+1

From (2.891) and (2.892) we have

TO TO TO / \

^ ) V 53 (Cj2jsj1 jsj2jl + Cj2jsj1jsj1j2 j = 0. (2.893)

jl =p+1 j2 =p+1 js=P+1

Moreover (see (2.805)),

œœœ

I™ £ £ £ Cj2j3jlj3jlj2 — 0. (2.894)

p^œ z—* z—* z—*

jl=p+1 j2=p+1 j3=p+1

Combining (2.893) and (2.894), we finally obtain

oo oo oo

„1im- £ £ £ Cj2j3jlj3j2jl — 0-

p^œ

jl=p+1 j2=p+1 j3=p+1

The equality (2.816) is proved.

Now consider (2.817). Using the integration order replacement, we obtain

Cj3jlj3j2 j2 jl + Cj3jlj3j2jl j2 —

T te t5 t4 / t3 \ / t3 \

— J 0j3 0jl (t5) J 0j3 (t4^ 0j2 (t3M J 0j2 (t2)dt2 J 0jl (t1)dtH x

t t t t t t

xdt3dt4dt5 —

T te t5 / t3 \ / t3 \ t5

— y 0j3 0jl (t5^ y 0j2 (t3M y 0j2 (t2)dt2 y 0jl (t1)dtH y 03 (t4)dt4 x

t t t t t t3

xdt3dt5 —

T te / t5 \ t5 / t3

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— J 0j3 0jl (t5M y 0j3 (t4)dt4 I y 02 (t3 ) I y 0j2 (t2)dt2 ) x

t t t t t

T/ \

x / 0jl (t1)dtM dt3dt5

T te t5 / t3 \ / t3

-f 03 (to) y 0l (t5) y 02 te) y 0j2 te)dt2 y 0jl (¿1)^1 ) x t t t t t

fr

x / 0j3 (t4)dt4 J dt3dt5dtö —

T / t5 \ t5 / t3

= J / (t5H J j (t4)dt4 J j (t3) i J j (t2)dt2 | x t t t t

x (/^jl(t1)dt1 ^ dt3 *(t6)dt6 | dt5-

T t5 / t3 \ / t3

— J j (t5) / j (t3H J j (t2)dt2 J j (t1)dt^ x

t t t t

x ^y* j (t4)dt4^ dt3 j (t6)^6 ) dt5.

(2.895)

Applying (2.655) and (2.662), we obtain

oo oo oo

53 53 ( Cj3j'lj3j2j2j'l + Cj3j'lj3j2j'lj2 j

j'l =p+1 j2 =p+1 j3=p+1

p TO TO / \

53 53 53 ( Cj3j1 j3j2j2j'l + Cj3j'lj3j2j'lj2 j

j'l =0 j3=p+1 j2=p+1

p p TO

EEEc

+ Cj

j3j1j3j2j2j1 j3j1j3j2jlj2

j1=0 j2=0 j3=p+1

1 p TO

■

2 53 53 ^ */••-/ ./•../ ./ ./

jl=0 j3 =p+1

(j2j2)^(^)

The equality

1

p TO

pij^ 2 53 53 ^ '/ •-/ ./::./: ./: ./

j1=0 j3=p+1

=0

(j2j2)^(^)

(2.896)

(2.897)

follows from the equality (2.723), where we proceed similarly to the proof of equality (2.856) (see (2.857)).

By analogy with the proof of (2.810) and applying (2.895), we get

p p TO X N

p—m>:>: 53 ( Cj3j1 j3j2j2j'l + Cj3j'lj3j2j'lj2 j = 0 (2.898)

j1=0 j2=0 j3=p+1

From (E396HE39S]) we have

oo oo oo

lim ^fa y y y y ( Cj3jlj3j2j2jl + ^janjjjlj^ ) — 0- (2.899)

p^œ z—' z—' z—' V /

jl =p+1 j2 =p+1 j3 =p+1

Moreover (see (2.806)),

oo oo oo

lim E E E Cj3jlj3j2jlj2 — 0. (2.900)

p^œ

jl=p+1 j2=p+1 j3=p+1

Combining (2.899) and (2.900), we finally obtain

oo oo oo

■ lim- y y E Cj3jlj3j2j2jl — 0-

p^œ

jl=p+1 j2=p+1 j3=p+1

The equality (2.817) is proved.

Finally consider (2.818). Using the integration order replacement, we have

Cj2j3j3jlj2jl + Cj2j3j3jljl j2 — T te t5 t4 / t3 \ / t3 \

— J 0j2 (t6^ 0j3 (t5) J 0j3 (t4^ 0jl (t3M J 0j2 (t2)dt2 I I J 0jl (t1)dt H x

t t t t t t

xdt3dt4dt5 dtö —

T te t5 / t3 \ / t3 \ t5

— J 0j2 (t6^ 0j3 M y 0jl (t3M J 0j2 (t2)dt2 J 0jl (t1)dt 1 J 0j3 (t4)dt4 x

t t t t t t3

xdt3dt5 dtö —

T te / t5 \ t5 / t3

— J 0j2 M y 0j3 M y 0j3 (t4)dt4 y 0jl (t3 H y 0j2 (t2)dt2 | x t t t t t

r/ \

x / 0jl (t1)dtM dt3dt5dtß—

T te t5 / t3 \ / t3

-J 0j2 (t6^ 0j3 (t5 ) y 0jl M y 0j2 (¿2^2 y 0jl (t1)dt^ x t t t t t

t3

x | J j (t4)dt4 J dt3dt5dt6 =

T / t5 \ t5 / t3

= J j (t5 ) I J j (t4 )dt4 I J j (t3M J j (t2)dt2 j x t t t t

x (/ ^jl(t1)dt1J dt3 ^y* /(t6)dt^ dt5-

T t5 / t3 \ / t3

— J j (h)J j (t3H y j (t2)dt2 y j (t1)dt^ x

t t t t

x 1 y j (t4)dt4j dt3 ^ j (t6)^6 ) d^5-

(2.901)

Using (2.655) and (2.662), we get

TO TO TO / \

53 53 ( Cj2j3j3j'lj2j'l + Cj2j3j3j'lj'lj2 j

-m I 1 /1 ^-ivi I 1 /»^v-m_Ll ^ '

j'l =p+1 j2 =p+1 j3=p+1

oo oo

2 53 53 (^ttj-ij-ijmji j'l=p+1 j2 =p+1

(j3j3W-)

+ Cj2j3j3j'lj'lj2

(j3 j^W")

p TO TO /

53 53 53 ( Cj2j3j3j'lj2j'l + Cj2j3j3j'lj'lj2

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j3=0 j'l=p+1 j2=p+1

OO oo

2 53 53 (Cjrjwrjrji j'l=p+1 j2=p+1

+ C

(j3j3)^(-)

j2j3j3jljlj2

+

(j3j3)^(-)/

p p TO X N

+ ^ ^ ^3 53 ( Cj2j3j3j'lj2j'l + Cj2j3j3j'lj'lj2 j —

n;—n .;„—„xi V /

/1=0 j3=0 j2=p+1

1 p TO

2 53 53 ^ '/: ./"■./:',/ ./ ./•/

j3=0 j2 =p+1

(2.902)

C/'lj'lWO

The equalities

TO /

" + C

1 TO TO

E E [C—

j1=p+1 j2=P+1

j2j3j3j1j1j2 (j3j3W-)

= 0, (2.903)

(j3j3)^(')/

1 P TO

JifTO 2 E E ^nhhjijm

j3=0 j2=P+1

(j1j1)^(')

1TO

P—to 4 z—' j2=P+1

1 TO TO

53 ^hhhjijm j3=P+1 j2=P+1

= 0 (2.904)

CmW ■ )

follows from the equalities (2.723), (2.724), where we used the same technique as in (2.857). When proving (2.904), we also applied (2.662) and (2.83).

By analogy with the proof of (2.810) and applying (2.901), we obtain

P P TO

lim y v y v I Cj2j3j3j1j2j1 + Cj2j3j3j1j1j2 ) = 0 (2.905)

p—TO

j1 =0 j3=0 j2=P+1

From (2.902)-(2.905) we have

TO TO TO

lim 53 y ^ ( Cj2j3j3j1 j2j1 + Cj2j3j3j1jlj2 ) = 0- (2.906)

p—TO

j1=p+1 j2=P+1 j3 =P+1

Furthermore (see (2.808)),

TO TO TO

lim E E E CCj2j3j3j 1 j 1 j2 = 0. (2.907)

p—TO

j 1 =p+1 j2 =P+1 j3 =P+1

Combining (2.906) and (2.907), we finally obtain

TO TO TO

„lim- E E E Cj2j3j3j1j2j1 = 0

p—TO

j 1 =p+1 j2 =P+1 j3 =P+1

The equality (2.818) is proved. Theorem 2.35 is proved.

2.15 Estimates for the Mean-Square Approximation Error of Iterated Stratonovich Stochastic Integrals of Multiplicity k in Theorems 2.30, 2.31

In this section, we estimate the mean-square approximation error for iterated Stratonovich stochastic integrals of multiplicity k (k £ N) in Theorems 2.30, 2.31.

Theorem 2.36 [32], [37],

. Suppose that every ^ (t) (/ = 1,..., k) is a continuously differentiate nonrandom function at the interval [t,T]. Furthermore, let (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then the following estimates

m

J *[^(k)](i1-ik)

JT,t

53 il j

ji,-jfc=0 l=1

(ii) ji

<

< K1

/

1

[k/2]

p + 53

r=1 ({{gi,S2}>--->{S2r-1>S2r }},{91>--->9fc-2r }) \ {S1,S2>'">S2r-1>S2r>91>'">9fc-2r } = {1>2>--->k}

M< R

)(p)r,g1,g2,---,g2r-1,g2r

/

(2.908)

m

J *[^(k)]

p

k

j1,-,jfc=0 1=1

(ii)

<

< K2(s)

P

[k/2] + E

E

mR

V

r=1 ({{S1,S2 }>--->{S2r-1>S2r }},{91>--->9fc-2r }) {S1,S2 .■■■>S2r-1>S2r>91>--->9fc-2r }={1>2>--->k}

)(p)r,g1,g2,---,g2r-1,g2r

^s.t

/

(2.909)

hold, where s £ (t, T] (s is fixed), i1,..., ik = 1,..., m,

R(p)r,g1 ,g2,---,g2r-1,g2r = R(p)r,g1,g2,---,g2r-1,g2r Rs,t = RT,t

T=s

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2

k

p

2

1

2

R(rp]r'gi'g2'""g2r-1'g2r is defined by (2676), J*^]^0 and J*^]^"'*) are iterated Stratonovich stochastic integrals (12.6,36) and (12.6851). Cj^.j and Cj^.j(s) are Fourier coefficients (2.628) and (2.683), constants K and K2(s) are independent of p; another notations are the same as in Theorems 1.1, 2.30, 2.31.

Proof. Note that Conditions 1 and 2 of Theorems 2.30, 2.31 are satisfied under the conditions of Theorem 2.36 (see Remark 2.4). From the proof of Theorem 2.30 it follows that the expression (2.681) (¿i,..., ik = 1,..., m) before passing to the limit l.i.m. has the form

p^œ

p k

E Cj-jill zf = J[*k)i?P)p+ j1,",jfc =0 1=1

[k/2] / .

r=:L \ 2 (sr ,...,Sl)GÀfc,r

+ E RTP]r'gi'g2'...'g2r-1'g2^ (2.910)

({{31>32}>'">{32r-1>32r }},{91 .■■■>9fc-2r }) /

{si,S2>'">S2r-1>S2r>91>'">9fc-2r } = {i>2>'">k}

w. p. 1, where

is the approximation (11.222) of the iterated Itô stochastic integral (12.643), I[^W^^81-1*8^2...^-1isr+2-ik)p is the approximation obtained using (1222) for the iterated Itô stochastic integral J[0(k)]yt" 'si (see (2.682)). '

Using (2.910) and Theorem 2.12, we have

pk

E cv.jiii zf = j

ji,."jfc=0 i=i

[k/2]

r=1 2 (sr,...,si)eAfc,r

+ ( J№(k)]TÏ"ifc)p - J[0(k)]iVfc) ) +

[k/2]

1 I T r / (kh(i1...isi-1«si+2...«sr-1«sr+2...«fc)p

+E E yUhnïr"1"1"1"

r=1 (sr ,...,si)eAfc

r

_I [^(k)](i1-is1-1is1 + 2-iSr-1isr + 2-"ik ) j +

[k/2]

)(p)r,g1,g2,---,g2r-1,g2r tT,t

r=1 ({{S1 ,32}>--->{32r-1>32r }},{91>--->9fc-2r }) {31,32>--->32r-1>32r>91v>9k-2r } = {1>2>--->k}

+E E R

= j ^(k)](rik) + j ^(k)](rik )p _ j [^(k)]Ti;t'"ik^+ [k/2]

+E E ¿1'

r=1 (sr ,...,s1)eAfc,r

_I [^(k)](i1---is1-1is1 + 2---iSr-1isr + 2---ik ) j +

[k/2]

+ ^ ^ RTp)r,g1,g2,...,g2r-1,g2r (2.911)

r=1 ({{S1 ,S2}>'",{S2r-1>S2r }}>{91>--->9fc-2r }) {S1,S2>'">S2r-1>S2r>91 >'">9k-2r } = {1,2>'">k}

w. p. 1, where we denote J[^(k)]f,t-'s1 as I[^(k)^•••M-1^2--^-1isr+2-ik). Applying (1.223) (see Remark 1.7), we obtain the following estimates

N 21 C

(i1-ifc)p 7L/,(kh(i1--ik) » 1 -

m { ( - -W Mt ) } < (2.912)

m ^ | I [^^J](i1'"is1-1is1+2'"isr-1isr +2 •••ik)p _ I [^(k)] (i1"'is1-1is1 + 2- "isr-1V + 2 •••ik ) | ^ <

< (2.913)

P

where constant C does not depend on p.

From (I2.911l)-(l2.913l) and the elementary inequality

(a1 + a2 + ... + an)2 < n (a^ + a| + ... + On) , n £ N

2

we obtain (2.908). The estimate (2.909) is obtained similarly to the estimate (2.908) using Theorems 1.11, 2.31 and (1.249) (see Remark 1.12). Theorem 2.36 is proved.

2.16 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 3-5 in Theorems 2.32-2.34

In this section, we consider the rate of convergence of approximations of iterated Stratonovich stochastic integrals in Theorems 2.32-2.34. It is easy to see that in Theorems 2.32-2.34 the second term in parentheses on the right-hand side of (2.908) is estimated for k = 3, 4, 5. Combining these results with Theorem 2.36, we obtain the following theorems.

Theorem 2.37 [32], [37], [38], [63]. Suppose that [fa(x)}j=0 is a com-

plete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let 0i(t),^2(t),^3(t) are continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

J* [0(3)]T,i = / «ts)/ «*2)/ 0l(il)dfi(il)dfi(;2)dfi(sis) t t t

the following estimate

M j - ± Q^'cfcfVUf

[ V jl ,j2,js=0 /J 1

is fulfilled, where i1, i2, i3 = 1,..., m, constant C is independent of p,

T ¿3 ¿2

Cjsj2ji = J 03(t3)fajs (t3)/ ^(¿2)02 (t2) y ^l(tl)faji (tl)dtidt2dt3

t t t

and

T

j = / fa

t

are independent standard Gaussian random variables for various i or j. http://doi.org/l0.2l638/ll70l/spbu35.2023.ll0 Electronic Journal. http://diffjournal.spbu.ru/ A.581

Theorem 2.38

, [37],

. Let (x)}|=0 be a complete orthonor-

mal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^1(t),... , ^4(t) be continuously differentiable non-random functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

*T ^ £4 ^ £3 ^¿2

J*

T,£ = ^4)/ ^3(i3)/ ^fe)/ ^faf dft2

(i1)df (i2 ) df (i3)df (i4)

^3

¿4

£ £ the following estimate

m

J*[^(4)]T,t _ £

Cj

Z(i1)Z(i2) Z(i3)Z(i4)

Sjl S72 S7q Sj

j4j3j2j^j1 Sj2 Sj3 Sj4

j1,j2,j3,j4=0

< —

holds, where i1,i2,i3,i4 = 1,... ,m, constant C does not depend on p, £ is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space L2([t, T]) and £ = 0 for the case of complete orthonormal system of trigonometric functions in the space L2([t,T]),

T £4 £3 £2

Cj4j3j2j1 = J ^4(t4)0j4 (t4) y ^3(t3)0j3 M/ ^2(t2)0j2 (t2^ ^1(t1)^j1 (t1)dt1x £ £ £ £

xdt2dt3dt4;

another notations are the same as in Theorem 2.37.

Note that Theorem 2.26 is an analog of Theorem 2.38. At that £ = 0, ^1(t),..., ^4(t) = 1, and i1,..., ik = 0,1,..., m in Theorem 2.26.

[37], [38], [63]. Assume that {^ (x) j 0 is a complete

Theorem 2.39

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orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t, T]) and ^1(t),..., ^5(t) are continuously differentiable nonrandom functions on [t, T]. Then, for the iterated Stratonovich stochastic integral of fifth multiplicity

±t

£2

J*

T,£ =

(is) ... ^1(t1)dft

(i1) £1

df

(i5)

£5

the following estimate

m

E

j1vj5 =0

CZ

Z(i1) . . . Z(i5) S j . . . S

j5

< —

is valid, where ii,..., = 1,...,m, constant C is independent of p, £ is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space L2([t, T]) and £ = 0 for the case of complete orthonormal system of trigonometric functions in the space L2([t,T]),

t t2

Cj5...j1 = J ^(¿5(t5) - J (ii)fai (ti)dti... dts; t t

another notations are the same as in Theorem 2.37, 2.38.

2.17 Generalization of Theorems 2.4—2.8. The Case p, p2, p3 ^ ro and Continuously Differetiable Weight Functions (The Cases of Legendre Polynomials and Trigonometric Functions). Proof of Hypothesis 2.3 for the Case k = 3

This section is devoted to the following theorem.

Theorem 2.40 [32], [37], [38]. Suppose that {fa(x)}=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let fa(t),^2(t),^3(t) are continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

T * is * ¿2

(n*^) _ I „/,_ (t_ ) / „/,„ (t. W „/,. (t. )dw(;1)dw(;2)dw(;S)

1

t t t

J*i^(3)iTr _ I ut2)i ^i(ti)dw(ii°H2'dw<s

the following expansion

Pi P2 PS

J•[^(»ijij'2'3' _ l.i.m ££ Wj' (2.914)

ji=0 j2=0 js =0

that converges in the mean-square sense is valid, where i1, i2, _ 0,1,..., m,

T is ¿2

Cj _ /«t3) jW /^2(t2)^j2M /WOfci(tl)dtldt2dt3

i i t

and

T

Sj(i) = / j (s)dw<i>

£

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[i) = f(i) for i = 1,..., m and wT0) = t.

Proof. Let us consider the case of Legendre polynomials (the trigonometric case is simpler and can be considered similarly). Applying (2.650), we obtain

p1 p2 p3

V^ V^ V^ C Z(i1)Z(i2)Z(i3) = p[K ](i1i2i3) I

Cj3j2j1 j Zj2 Zj3 = J [KP^P2P3]T,t +

j1=0 j2=0 j3=0

P3 min{p1,p2}

+ 1{i1=i2=0} 53 53 Cj3j1j1 J'[jj3](Zi,3t +

j3=0 j1=0 P1 min{p2,p3}

+1 {i2 = i3=0} Y 53 Cj3j3j1 J' [ ] T^ + j1=0 j3=0

p2 min{p1,p3}

+ 1{i1=i3=0^ 53 Cj J'[j ] Zi2t) (2.915)

j2=0 j1=0

w. p. 1, where notations are the same as in (2.650).

Using (2.382), Theorem 1.1 (see (1.43)), Theorem 2.12 (see (2.372)) as well as (2.667) (see the derivation of (2.667)) and (2.662), we get

T £3

= +ii{il=i2^} fuu) f

t t T t3

= l.i.m. J [Kp1p2p3]Zг,t ) +

p1,p2,p3^^ '

1

P3

P3—to 2 z—'

j3 =0

(j2 j'iWOji =j2

J '[j ^+

1

Pi

ji=0

(j3j2W0,j2=j3

j =

= l.i.m. J [Kpip2P3]T,t ) +

Pi,P2,P3 — TO

P3 TO

+ l{ii=i2=0}l.i.m^ V] Cj3jiji J'[ j]Ti,3t) +

P3—TO ' ' '

j3=0 ji =0

Pi TO

+ 1{i2 = i3=0}l.i-m^ E Cj3j3ji J'[j]T,t

pi—>-TO z-' z-' '

ji=0 j3 =0

w. p. 1.

Using (2.915), (2.916) and the elementary inequality

(a + b + c + d)2 < 4 (a2 + b2 + c2 + d2) ,

we obtain

pi p2 p3

m< | j-^Tr3'-£££Cj3j2jicji-'W

<

ji=0 j2=0 j3=0

(2.916)

< 4m

(:i:2:3) _ t/[k ](:i:2:3M I +

T,t J [KPiP2P3]T,t f +

X

+4 • 1{:i=:2=0}X

P3 min{pi,p2}

(:3) V^ n 1(:3)

P3 TO

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m{ (P3—to eeCj3jijiJ'[j] Tt e e Cj3jijij [j]

j3=0 ji=0 j3=0 ji=0

T,t

+

xm

+4 • 1{:2=:3=0}X

Pi TO

pi — TO

ji=0 j3=0

pi min{p2,p3}

|(:i) V^ V^ n V\J, ](:i)

]T,t

ji=0 j3=0

l.i-m. E ECj3j3jiJ/[^jiW " E E Cj3j3jiJ/[0i](

+

2

p2 min{p1,p3} " 2

+ 4 • 1{i1=i3=0}m { ]T Cj1j2j1 J'[j^

j2=0 j1=0

= 4Ap1p2p3 + 4 • 1{i1 = i2=0}Bp1p2p3 + 4 • 1{i2=i3=0}Cp1p2p3 + 4 ^ 1 {i1=i3=0} Dp1p2p3 .

(2.917)

Theorem 1.1 gives (see (1.43))

lim Ap1p2p3 = 0. (2.918)

p1,p2,p3^ro

Further, in complete analogy with (2.715) and using (2.655), we obtain

Dp1p2p3 =

p2 imin{p1,p3} \ 2 p2 / TO \ 2

= E I E Cj1j2j J = E I E Cj1j2j J <

j2=0 \ j1=0 J j2=0 \j1=min{p1,p3} + 1 J

TO / TO \ 2 K

^ E E Cnnn < -T7\> 0 (2.919)

V-1 =min1;r,p3}+1 J (min{p1,p3})

if p1,p2,p3 ^ to, where £ is an arbitrary small positive real number, constant K is independent of p.

We have

Bp1p2p3 = p3 TO p3 TO

= m l^EE cj J'fe ft _ E E C?3j1j1 j'[ j ft+

l \ V j3=0 j1=0 j3=0 j1=0 J

/ p3 TO p3 min{p1,p2} \ \ 2

+ (E ECj3j1j1 J'[jj3]Ti3t _E E Cj3j1j1 J'[jj3]Ti3t M f <

\J3=0 j1=0 j3=0 j1=0 JJ

< 2Ep3 + 2Fp1p2p3, (2.920)

where

E =

Ep3

m

p3 TO

^jm. 53 53 Cj3jiji j/[^j3 ]T1,3t) - 53 53 Cj3jijiJ /[j ]ST3

p3 TO

P3—TO

I (13) ,t

j3=0 ji=0

m

p3 TO

F

1(13)

pip2p3

P3 min{pi,p2}

y^ Cj3jiji J [^j3 ^ Cj3jiji J [^j3 ^

j3=0 ji=0 j3=0 ji=0

p3

m E E Cj3jijiJ [j]

j3=0 ji=min{pi,p2}+i

(:3)

T,t

p3

00

53 ( 53 Cj3j^i

j3=^ \ji=min{pi,p2}+i

(2.921)

By analogy with (2.702) we get

p3

00

531 53 Cj3jiji 1 <

j3=0 \ji=min{pi,p2}+i

j3jiji I <

<C

j3=0 \ji=min{pi,p2}+i

K

<-* 0

(min{pi,p2})

2

if pi,p2,p3 — to, where constant K does not depend on p. Moreover,

lim EP3 = lim EP3 = 0.

P3 — TO pi,p2,l3 — TO

Combining (E32DH2323]), we obtain

lim BPiP2P3 0' Pi,P2,P3 — TO

(2.922)

(2.923)

(2.924)

Consider CPiP2P3. We have

C

pip2p3

2

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2

p1 TO p1 TO

mip.—m EEc?3j3j1 j[jft_ E Ec?3j3j1 j[jf))+

j1=0 j3=0

p1 to p1 min{p2,p3} \ \ 2

+ ( 53 Cj3j3j1J [jj1 ]T,t _ 53 53 Cj3j3j1J [jj1 ]2i,t

j1=0 j3=0 j1=0 j3=0

<

< 2Gp1 + 2Hp1p2p3;

(2.925)

where

m

p1 TO

r* — Gp1 =

Li\m. E E Cj3j3j1J[ j ft _ E E jjJ'[jj1]

p1 TO

p1—>-TO

l(i1) ]T,t

j1=0 j3=0

m

p1 TO

H

(i1)

p1p2p3 =

p1 min{p2,p3}

(i1)

53 53 Cj3j3j1J [jj1 ]2i,t E 53 Cj3j3j1J [jj1 ]:T1

j1=0 j3=0

p1

M Cj3j3j1 J [jj1]

j1=0 j3=min{p2 ,p3 }+1

(i1) T,t

p1

Cj

53 ( 53 ^333331

j 1 =0 \j3==min {p2 ,p3 }■+1

(2.926)

By analogy with (2.706) we get

p1

C

j1=0 V3=mm{p2,p3}+1

j3j3j1 <

TOTO

<C

j 1 j3 =min{p2 ,p3 }+1

K

<-* 0

(min{p2,p3})

2

(2.927)

if p1,p2,p3 — to, where constant K does not depend on p.

2

Moreover,

lim Gp1 = lim Gp1 = 0. (2.928)

pi^œ pi,p2,p3^œ

Combining (2.925)-(2.928), we obtain

lim Cp1p2p3 = 0. (2.929)

pi,p2,p3^œ

The relations (2.917)-(2.919), (2.924), (2.929) complete the proof of Theorem 2.40. Theorem 2.40 is proved.

2.18 Generalization of Theorem 2.30 for Complete Orthonormal Systems of Functions in L2([t,T]) and (t ),..., (t ) G L2([t,T]) such that Condition 3 of Theorem 2.30 is Satisfied

In this section, we generalize Theorem 2.30 to the case of complete orthonormal systems of functions in the space L2([t,T]) and ^1(t), ..., (t) G L2([t,T]) such that Condition 3 of Theorem 2.30 is satisfied.

def

Let (Q, F, p) be a complete probability space and let f (t,w) = ft : [0,T] x Q — R1 be the standard Wiener process defined on the probability space

(Q, F, p).

Let us consider the family of a-algebras {Ft, t G [0, T]} defined on the probability space (Q, F, p) and connected with the Wiener process ft in such a way that

1. Fs C Ft c F for s < t.

2. The Wiener process ft is Ft-measurable for all t G [0,T].

3. The process ft+A — ft for all t > 0, A > 0 is independent with the events of a-algebra Ft.

Let ^(t, w) = : [0, T] x Q — R1 be some random process, which is measurable with respect to the pair of variables (t, w) and satisfies to the following condition

T

j |£T|dT < to w. p. 1 (t > 0).

t

Let t(N), j = 0,1,..., N be a partition of the interval [t, T], t > 0 such that

)

t = rk"' < ' < .

_(N )

< T

(N )

N

= T,

max

0<j<N-1

T(N) _ _(N)

Further, for simplicity, we write Tj instead of rjN'.

Consider the definition of the Stratonovich stochastic integral, which differs from the definition given in Sect. 2.1.1.

The mean-square limit (if it exists)

^ 0 if N ^ œ.

(2.930)

N1

l.i.m.

1

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rj+i

T

6ds (j - /j) = I £r ◦ d/

j=o Tj+i - T

(2.931)

is called [128], [129] the Stratonovich stochastic integral of the process , t g [t, T], where Tj, j = 0,1,..., N is a partition of the interval [t, T] satisfying the condition (2.930).

We also denote by

6 ◦ d/s

the Stratonovich stochastic integral like (12.9,31) (if it exists) of 1{se[t,T]} for t e [t,T],t > 0.

It is known [129] (Lemma A.2) that the following iterated Stratonovich stochastic integral

t2

js [^(k) ](n-ik ) = ^ (tk ) ... (ti ) ◦ dw

(ii) ti

dw

(ik)

(2.932)

t t

exists, where t e [t,T], ^i(t),... (t) e L2([t,T]), ii,... ,ik = 0,1,... ,m, w^' = f^ for i = 1,..., m and wT0) = t, f^' (i = 1,..., m) are independent standard Wiener processes defined as above in this section.

In [130] (2021) an analogue of Theorem 2.12 (1997) is proved for the case of a scalar standard Wiener process and (t),..., (t) e L2([t, T]).

Using Theorem 2.12 together with Proposition 3.1 [130] and the proof of Lemma A.2 [129], we can write

js [^(k) ]Ti1t-ifc) = J

(ii---¿fe ) T,t

[k/2]

1

53 2 r

=1 (sr ,---,si)GAk

J

Trt---'si w. p. 1, (2.933)

r

where ^1(t),...,(t) G L2([t,T]), is supposed to be equal to zero;

0

J[^)]Trfc) is the iterated Ito stochastic integral

t t2

J [^(k)]Trfc) = / ^ (tk) ..J ^i(ti)dwt(;i)... dw(:k); (2.934) tt

another notations are the same as in Theorem 2.12.

Further, by analogy with (26391), (12646)-(2648) and using (1313) (see Theorem 1.17) instead of (1.257) (see Theorem 1.13) without passing to the limit l.i.m. and for $(t1,..., tk) = Kpi...pk(t1,..., tk) (see (2.640)), we obtain

PlvvPfc —TO „„,_,,„

the following generalization of (2.648) to the case of an arbitrary complete orthonormal system of functions in the space L2([t, T]) and ^1(t), ..., (t) G L2([i,T ])

p k p

£ Cfk^IIj' = E Cfcjj'[j...j]Tr'+

ji,-jfc=0 1=1 ji,...,jfc=0

[k/2]

+ E Cjk. ..jiE E II1{;g2s-i = ;S2s =0}x

jiv-jfc =0 r=1 ({{si,S2>,-,{S2r-i,S2r }},{9i,-,9fc-2r}) S = 1

{3i,32v>32r-i>32r>9i>--->9fc-2r } = {i>2>--->k}

1

{j

g2s-T

2s } J . . . 09fc-2r ]

(;

qi ...;9fc-2r >

lT,t

w.

p. 1,

(2.935)

where J'[j ... j^V^), J'[j ... 2 ](n<ti qk 2r) are multiple Wiener stochastic integrals defined by the formula (1.299).

Using the equalities (1.309) and (1.312), we can reformulate Theorem 1.16 as follows

pi pk

J [V(k>]Trk) = l.i-m. E ... £ Cjk.ji J [ j ... j tfa w. p. 1, (2.936)

pi,...,pfc —TO Z-' Z-'

ji=0 j=0

where J' [j ... j ]T;it..;k) is the multiple Wiener stochastic integral defined by (1.299); another notations are the same as in Theorem 1.16.

Passing to the limit l.i.m. in (2.935) and using the equality (2.936), we get

p—TO

w. p. 1

lpL- E 6V--;iCj;:i)...j> = J>+

ji,---jk=0

[k/2] r

+ E E nii;g2s-i = ¿32S =01X

r=1 ({{3i>32}>--->{32r-i>32r }}>{9i>--->9fc-2r}) s = 1 {Si>S2>--->S2r-i>S2r>9i>--->9fc-2r } = {i>2>--->k}

P

P r

Et—r f ( ¿q^ ¿q )

^ Cjk -- -ji 1{jS2s-i = jg2s } J [^j9i . . . ^j9fc-2r ]l,t , (2.937) ji, - - - Jk =0 s = 1

where J

.. .^jqk 2 - - ¿qk 2r ) is the multiple Wiener stochastic integral dei (;i - - -¿k) •

fined by (Om J[^^f- - ¿k) is the iterated Itô stochastic integral (2934).

Suppose that (x)}°=0 is an arbitrary complete orthonormal system of functions in L2([t,T]) and $1(r), $2(r) G L2([t,T]). Then we have

œ

E

j=0

T

(t)$1(r)dr / (t)$2(t)dr

<

œ

-

(il

T

1{r<s}0j (t)$1(r)dT + / 1{T>s}0j (t)$2(t)d

T

V

j=0

i.e. the series

2

!

< œ,

(2.938)

T

œ

£ J h(T)$1(T)dry h(r)$2(r)dr

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j =0 _

converges absolutely.

By interpreting the integrals in (12.656)—(2.659) as Lebesgue integrals, using Fubini's theorem in (2.656) and Lebesgue's Dominated Convergence Theorem in (2.658), we obtain (2.654) (see (2.659), (2.938)) for the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^i(T),..., ^(t) e L2([t,T]).

Using the equality (2.113) for the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^1(t),... (t) e L2([t,T]) as

s

2

s

well as absolute convergence of the series on the left-hand side of (2.113) for this case (see Theorem 2 in [107]), we obtain the generalization of (2.662) for

the case of an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^(t),...,^(t) G L2([t,T]).

Repeating the steps of the proof of Theorem 2.30 below the formula (2.663) using (2.937), we obtain for complete orthonormal systems {fa(x)}°=0 (fa(x) = 1/faT^t) in the space L2{[t,T]) and fa{r),...,fa{r) e L2{[t,T]) (for which Condition 3 of Theorem 2.30 is satisfied) the following equality

p k

(ii)

Kim. £ 6V..;,n Z

j'iv-Jfc=0 1=1

[k/2] .

= .Jc"[;:r + E w{k)]trai = jvk)trik) (2-939)

r=1 (sr ,...,si)GAfc,r

w. p. 1, where notations in (2.939) are the same as in Theorem 2.12 and is the iterated Stratonovich stochastic integral defined by (2.932). Thus the following theorem is proved.

Theorem 2.41. Assume that the complete orthonormal system {fa(x)}°=0 {(f)o{x) = 1/faT - t) in the space L2{[t,T]) and fa{r),..., 'ifafa) G L2{[t, T}) are such that Condition 3 of Theorem 2.30 or a weaker condition (2.687) are satisfied. Then, for the iterated Stratonovich stochastic integral (2.932) of arbitrary multiplicity k

t t2

JS [fa(k) ]Trifc) = / fa (tk ) ...J ^i(ti ) ◦ dwt(;i)... ◦ dwt(:k ) t t

the following expansion

p k

JS [fa<k>]Tr > = l.i.m. É Cjk^n Zj: ' (2.940)

j'i,...,jfc=0 1=1

that converges in the mean-square sense is valid, where

T t2

Cjfc ...ji =

J fa (tk )fafc (tk ) ...J fa (t0fai (t1)dt1 ...dtk tt

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, ..., ik = 0,1,..., m,

T

cf = | h (T )dw«

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[;) = fT;) for i = 1,..., m and w[0) = t.

In this section, it is also appropriate to mention the so-called multiple Stratonovich stochastic integral [128], [129] (also see [124]).

The mean-square limit (if it exists)

N —1 N—1 1 ,,

l.i.m. V ... V --—- / K(tu ..., tk)dh ... dtk Aw^ ... Awfl)

N^œ f=0 f=L ATli... ATlk J Tli Tlk

;i=0 1k=0 Ar;i X---XAnk

= JS [K]^-ifc} (2.941)

is called [128], [129] the multiple Stratonovich stochastic integral of the function

K(t1,...,tk) G L([t,T]k), where Awj = wîj+ — wj (i = 0,1,...,m), t, (j = 0,1,...,N) is a partition of the interval [t,T] satisfying the condition (2.930), i1,...,ik = 0,1,..., m, wT;) = fi;) for i = 1,...,m and w(0) = t , fi;) (i = 1, . . . , m) are independent standard Wiener processes defined as above in this section.

Note that in [129] the case i1 = ... = ik = 0 was considered. We also denote

by JS[K]Sit 'ifc) the multiple Stratonovich stochastic integral (2.941) (if it exists) of the function K(ti,...,tk)1{(t1,...,tk)G[i)S]*}, where K(ti,...,tk) G L2([t,T]k), s G [t,T], t > 0.

Let the function K(t1,..., tk) be chosen as follows

^i(ti) ...^k (tk), ti < ... < tk K (ti,...,tk )=< , (2.942)

0, otherwise

where ^(t),...,(t) G L2([t,T]), ti,...,tk G [t,T] (k > 2) and K(ti) = ^i(ti) for ti G [t,T].

We will denote the multiple Stratonovich stochastic integral (2.941) of the function (2.942) as JS[^(k)ffc).

It is known [129] (Lemma A.2) that the Stratonovich stochastic integrals JS[fa^-^ and JS]Ti;t-ifc) exist (also see the equality (2.933)). Moreover, JS= JS[fa^'^) w. p. 1 [129] (Lemma A.2).

Recall that an expansion similar to (2.637), (2.940) was obtained in [127 for the multiple Stratonovich stochastic integral (2.941) under the condition of convergence of trace series (see Remarks 2.4, 2.7 for details).

Recently, another approach to the expansion of integrals (2.941) has been proposed, where multiple Fourier-Walsh and Fourier-Haar series (k £ N) have been applied [200]. The convergence was proved with respect to the special

subsequence (p = 2m, m ^ to in a formula similar to (2.i

2.19 Algorithm of the Proof of Hypothesis 2.2

Let us make some remarks about the development of the approach based on Theorem 2.30 and describe the algorithm of the proof of Hypothesis 2.2 (see Sect. 2.5). First, consider the case k = 2n +1, n = 3, 4,... (k is the multiplicity of the iterated Stratonovich stochastic integral (2.636)). Let Conditions 1 and 2 of Theorem 2.30 be satisfied. Consider the equality (2.686). The right-hand side of (2.686) has the form

53 • • -53

j31 =

j32

r-1"

j31 j32 '•••'j32r-1 j32r

2r

ni

i=i

{g2l =g2i-1 + 1}Cjfc"j1

(jg2 jg1 )^(0"-(jg2r jg2r-1 = jg2 vJg.

1 ->«2''"'•'i'2r-1 jg2r

Iterated application of the formulas the values

I, (2.762), (2.775) separately to

53 • • - 53 •••j1

j31 =0 jS2

r-1"

jS1 jS2 '•••'jS2r-1 jS2r

and

2 r ZT ^

{g2i=g2i-1 + 1}Cjfc •••j1

1=1

(jS2 jS1 )^() ••• (jg2r jS2r-1 WOj^ = jS2 '•••'j32r-1 = jS2r

(gl,g2, - - - ,g2r-1,g2r as in

representation

I, r = 1, 2,..., [k/2], 2r < k) gives the following

0

1

0

p p

E E-Ec

\jsi =0 jg2r_1 =0

9=31>32v>32r-_1>32r

j31 j32 '•"'jS2r_1 j32r

1

2T

1{g2i=g2i_1 + 1}Cjfc •••j1

1=1

<

(j32 j31 ••• (j32r j32r_1 = j32 '•••'jS2r_1 = j32r .

pp

<

...

C

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jk •••j1

= 0 \jS1 =0 jg2r_1 =0 9=31,32>'">S2r_1>S2r

j31 j32 '•••'j32r_1 j32r

IT ^

{g2l =g2i_1 + 1}Cjfc •••j1

1=1

(j32 j31 )^(•)•••(jS2r j32r_1 WOj^ = j32 '•••'jS2r_1 = j32r .

oo

E

j10 \ [t TI 9=31,32 '■■■'32r_ 1,32r \ [t'T ]

Rp (t1 5 • • • 5 tg1-1) tg1 + 1) • • • ) tg2r-1' + 1 5 * * * 5 ^ ) X

k_2r

\

x

^q(tq)j (tq) dt1 • • • dtg1-1dtg1 + 1 • • • dtg2r-1^tg2r + 1 • • • dtfc

9=1

9=31,32v>32r_1>32r-

y

(2.943)

where

Rp (t1; • • •5 tg1-15 tg1+15 • • •5 tg2r-1' tg2r+1' • • •5 tk) x

[t,T ]k_2r k

x H ^q (tq) j (tq) dt1 • • • dtg1-1dtg1+1 • • • dt^+1 • • • dtk

9=1

9=31,32 .■■■>32r_1>32r

is the Fourier coefficient of

Rp (t1 5 • • • 5 tg1-1 5 tg1 + 1 5 • • • 5tg2r 1 5 tg2r + 15 • • • 5 tk) =

p

2

r

2

Rp (tb * * * ^-^W^ * * * ^r-l^r + l,***,tk) fa (tq ^

9=1

9=31,32>'">32r-1>32r

where

Rp(tl, . . . ,tg1-l,tg1 + l, . . . ,tff2r-l,tff2r + l, * * * ) =

4r

E^P^fe * * * ^-^W^ * * * ^-l^^b * * * )-

d=l

2r

(t^ • • • ,tg1-l• • • ,tg2r-l^r+l, • • • ,tk) e L2([t,T]k-2r)

d=l

and some of the functions (tl, • • •,tg1-l,tg1+l, * * * , tg2r-l,tg2r+l, * * * ,tk) and (tl, • • •,tg1-l,tg1+l, • • • ,tg2r-l,tg2r+l, * * * ,tk) can be identically equal to zero. Obviously, we could use another representation for the function

Rp(tb * * * , tg1-1, tg1+l, * * * , tg2r-l, +1, * * * , ^) (2.944)

based on the left-hand side of the equality (2.686) and (2.761), (2.762), (2.775) (see Sect. 2.13 for details). In Sect. 2.13, we considered the function (2.944) in detail for the case k > 5, r = 1*

Parseval's equality gives

oo I

£

j1, -,j9, -,ife=0 1 r, T]fc-2r 9=31,32>--->32r-1>32r \ J

9=1

9=31,32>'">32r-1>32r

Rp (tb * * * , tg1-1, tg1+l, * * * , tg2r-1, tg2r+1, * * * , ^k) x

x II ^q (tq ) j (t

q) dtl * * * dtg1-ldtg1 + l * * * dtg2r-ldtg2r +1 * * * dtk

. \ 2 -Rp^b * * * , tg1-1, tg1+l, * * * , tg2r-b +1, * * * , tk) J dtl * * * dtg1-ldtg1 + l

[t,T Jk-2r

2

* * * dtg2r-ldtg2r + 1 * * * dtk || Rp || L2([t,TJfc-2r(2.945)

Combining (2.943) and (2.945), we obtain

E I E . . . E Cjk---ji

Ji,-,Jq,-,Jk=0 \jSi =0 jg2r-i =0 9=Si.S2>'">S2r-i>S2r

jgi jS2 '---'jS2r-i j32r

ni

1=1

jg2i=g2i-i + 1}Cjk---ji

<

(j32 jSi )^0)---(jS2r jS2r-i W-)^ = jS2 '---'jS2r-i = jS2r .

< llil

'plL2([i,T ]k-2r ).

(2.946)

Assume that we have succeeded in proving the following equality

II ^ l|2

Applying (2.946) and (2.947), we get (compare with (2.687))

(2.947)

lim

p^œ

E I E . . . E Cjk---ji

Ji,-,jq,-,Jk=0 Vgi =0 jS2r-i = 9=Si.S2>'">S2r-i>S2r

jSi jS2 '---'jS2r-i jS2r

2' n ^

{g2i=g2i-i + 1}Cjk ---ji

1=1

(jS2 jSi )^(0---(j32r jS2r-i = jS2 '---'jS2r-i = jS2r .

= 0.

(2.948)

As noted in Sect. 2.10, Condition 3 of Theorem 2.30 can be replaced by a weaker condition (2.687) (or (2.948)). Also Condition 3 of Theorem 2.30 can be replaced by (2.947). From (2.948) we obviously obtain

lim .

p—t>00 ^—'

jgi =0

E

jS2r-i =0

C

Cjk---ji

jsi j32 '---'jS2r-i jS2r

2?- IT

1=1

(jg2 jsi )^(0---(jg2r jg2r-i = jg2 >--->j32r-i = js2r

. (2.949)

According to (2.686), the equality (2.949) will be satisfied if

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lim Si,S12 ...Si^ C

(p)

p^œ

jk---jq---ji

= 0,

(2.950)

q=gi,g2,---,g2r-i,g2r

2

1

2

r

2

0

2

p

r

where gi,g2,..., g2r-i, g2r as in

l, l1,l2,...,ld such that l1,l2,...,ld G

{1, 2,..., r}, l1 > l2 > ... > ld, d = 0,1, 2,..., r - 1, r = 1, 2,.[k/2],

SZl ... C

(P)

d | jk -jq -jl

def £f(p)

q=gl,g2,-,g2r—l,g2r

jk -jq •••jl

q=gl,g2,-,g2r—l,g2r

for d = 0, where

c (p)

Cjk •••jq •••jl

q=0l,02v,g2r-l,g2r

s ) C (P)

Cjk •••jq •••jl

q=gl,g2,-,g2r—l,g2r

are defined by (2.631), (2.632), l = 1, 2,...,r (see Sect. 2.10 for details).

Let us make some remarks about the function (2.944) for the case k > 5, r = 2. In this case, using the left-hand side of the equality (2.686) and (2.761), (2.762), (2.775), we represent the function (2.944) as the sum of several functions. In particular, among these functions will be the following functions

Qp(t1, . . . , ¿s-b ¿s+b . . . , ¿Z-b ¿Z+b . . . , ¿q-1, ¿q+1, . . . , ¿g-b ¿g+b . . . , ¿k) =

1

{tl^Xtg—l^s + l^Xt; —l^+l^Xtq—^tq+l^^tg—^tg+l^Xtk }

X

OO

ts + 1

il-1

x E J ^(T)0j!(T)dT y ^z(T)0j!(T)dTX jl =p+1 t t

rq+l

ig—1

x E / ^q(t) j(r)dW ^g(t) j(r)d^

jq =P+1 { {

(2.951)

(Qp(i1, . . . , ¿Z-2, ¿Z+3, . . . ) = 1 {tl < ••• <ti—2 <ti+3 < •••<tk }

X

tl — 2

0

x E I J (0) / ^z(u)dud^ x

ji=p+1 V t t

TO /tl—2 0

jq =p+1 \ t t

(2.952)

. . . ,£/-2,^+3, • • • ) — l{ii<...<i,_2<ii+3<...<ifc}x

^ ^ tj+3 / T 0 x E E / ^/+i(T) j(t) / № / (u)<j(u)dud^ x

jl=P+1 jq =P+1 { \t t

x j ^/+2(u)0jq(u)dudr, (2.953)

t

QQp(tl, • • • , t/-1, t/+2, • • • , tq-l, tq+2, • • • , tk) —

— l{tl<...<t,_1<t;+2<...<tq_1<tq+2<...<tfc} X

c» c» /tJ+2 0

x E E / (0) I )x

=p+1 jl+1=p+1 \t t

tq+2 0

x I i ^q+1Wji+i (0)J ^(u)0j! (u)dudtf I • (2.954)

tt

Note that the pairs (g1;g2), (g3,g4) for the functions (2.952) and (2.953) have the property: g2 — g1 + 1, g4 — g3 + 1, g3 — g2 + 1 At the same time, the pairs (g1,g2), (g3,g4) for the function (2.951) have the following property: g2 > g1 + 1, g4 > g3 + 1, g3 > g2 + 1- For the function (2.954), the pairs (g1,g2), (g3,g4) chosen as follows: g2 > g1 + 1, g4 > g3 + 1, g4 — g2 + 1, g3 — g1 + 1 Generally speaking, all possible pairs (g1,g2), (g3,g4) must be considered. We consider the functions (l2.951l)-(2.954) only as an example.

Suppose that s + 1 — l — 1, l + 1 — q — 1, q + 1 — g — 1 in (2.951). Let us show that (we consider the case of Legendre polynomials; the trigonometric

case is simpler and can be considered similarly)

2

— 0, (2.955)

2

^^¿2(1^-4) — 0, (2.956)

2

p^^QQP^L2([t,T]k_4) — 0, (2.957)

p ^L2([t,T ]k—4)=0.

(2.958)

First consider the proof of (2.955). We have (s + 1 = l - 1, l + 1 = q - 1,

q + 1= g - 1)

2

(Qp(*1, . . . , ¿Z-3, ¿Z-1, ¿Z+1, ¿Z+3, ¿Z+5, . . . , ¿k)) =

1

{^••Xti —3<ti — l<ti+l<ti+3<ti+5<^<tk }

x

ti—1

ti—l

X ( E J ^Z-2(T)0ji (T)d^ ^Z(T)0ji (T)dTx j =p+1 t t

oo

ti+3

x E J ^ + 2(t) j (T)d^ ^Z + 4(T)0jq (T)d

jq =p+1 t t

Using the estimate (2.711), we obtain

¡V (r )dr

t

(2.959)

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K

<

j W2(1 - z2(s))1/4-e/4'

(2.960)

where j G N, s G (¿,T), z(s) is defined by (2.20), £ G (0,1), constant K does not depend on j, {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials in the space L2([£, T]), ^(t) is a continuously differentiable nonrandom function on [¿, T].

Applying (2.960) and (2.714) (we take £ instead of £/2 in (2.714)), we get

ti—l

E J ^Z-2(T)fai (T)d^ ^Z(T)0ji (T)dTx .ji=p+1 t t

oo

ti+3

x E J ^Z + 2(T) j (T)d^ ^Z+4(T) j (T)d^ <

jq =p+ 1 t t

<

K1

p4(1-^)(1 - Z2(¿Z-1))1-£(1 - z2(¿Z+3))1-£'

(2.961)

2

where £/—1,t/+3 £ (t,T), constant K1 is independent of p^ Combining (2.959) and (2.961), we have (2.955).

Let us prove (2.956). Applying the estimate (2.710) in (2.622) and taking into account the boundedness of the functions ^1(r), ^2(r) and their derivatives, we obtain

+C2 E

E Cjj(s)

j=m+1

/z(s)

A2- £

j=m+1

< C

1

+

1

z(s)

dx

n1—£ m1—£

1

(1 — X2 )1/2—£/2

z(s)

+

dy

V

+

1

(1 — y2 )1/2—£/2 (1 — z 2 (s))1/4—(1 — y2 )1/4—£/4

z (s)

z(s)

dx

1

(1 — y2)1/4—(1 — x2)1/4—£/4

—1 \

dy

/

where

Cjj(s) — Mr(r) / (tf)dtfdr,

(2.962)

s £ (t,T), constants C1, C2 do not depend on n and m. From (2.962) we have

oo

E Cjj (s)

j=m+1

oo

Ki V — I 1 1

—+ V2Jr+1 ^ I + (i-^(S))1/4-£/1,

m1—£

, (2.963)

where s £ (£,T), constants K1, K2 do not depend on m.

Applying (2.714) (we take £ instead of e/2 in (2.714)) in (2.96i

E Cjj (s)

j=m+1

<

K

m1—£ (1 — z2 (s))

2 1/4— /4

, we get (2.964)

where s £ (t,T), constant K is independent of m. Using the estimate (2.964), we obtain (see

_ 2

(QQp(t1, • • • ,£/—2, £/+3, • • • ^ — l{ti<...<t,_2<ti+3<...

.<tk}

X

n

1

s

OO ' * — 2 0

x y =p+ y J ^Z-1(Wji (0) J fa (u)0ji (u)dud0j x

TO / * — 2 0 \ \ 2

x E / ^(0) j(0) / faZ+2(u) j(u)dud0 <

jq=p+1 t t

Ki

- ,,»' (1 .^i//,))1 • (2-965)

where ¿Z-2 G (¿,T), constant K1 is independent of p. The inequality (12.965 completes the proof of (2.956).

Let us prove (2.957). Applying (2.621) in (2.953), we get

(Qp(t1, . . . , tZ-2, tZ+3, . . . , ¿k)) <

oo oo *+3 / T 0

< (E E / faZ+1(T) j(T) / faZ-1(0)0ji(0) / fa(u)dud0 ) x

,ji=p+1 jq =p+1 t \t t

2

x faZ+2(u)0jq (u)dudT =

jq

t

1 TO *+3 / T 0

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2 E / I I ^i-iWfoAO) I faifafatifadudO ) fa+2(r)dT-ji =p+1 {

p ti+3 to / T 0

TO

e / faZ+1(T)0jq (t) e / faZ-1(0)0ji (0) / fa (u)dud0 I x

jq=0 t ji=p+^{ {

2

x fa+2(u) j (u)dudT =

jq

t

= (a - b)2 < 2(|a|2 + |b|2). (2.966)

Further, we have

tl+3

M <l[ №+l(T)|

oo

Y J —1(0)<j (0)J ^ (u)0j! (u)dud0

j'=P+1 t t

^/+2(r)|

(2.967)

tl+3

|b|<£ / l^/+1(r)j(t)|

jq=0 t

oo

/ —1(0)^ (0) / ^(u)0j! (u)dud0 ji=P+1 t t

X

^/+2(u) j (u)du

dr^

(2.968)

Combining (2.964) and (2.967), we obtain

, , C

a < 1 1 " p1—£

(2.969)

where constant C is independent of p^

Separating in (2.968) the term with the number jq — 0 and then applying (2.142), (2.278), (2.964), we obtain

|b|<

tl+3

K dt

tl+3

~ p!-£

jq -jq=1 t

dt

(1 — z2(r))1/2—£/4 (1 — z2(r))3/4—£/4

<

<KLI

p

K1 (2 + Inp)

p

1- £

0

(2.970)

if p ^ ^ The estimates (2.966), (2.969), (2.970) complete the proof of (2.957).

0

p

Finally, consider the proof of (2.958). Using the elementary inequality | ab| < (a2 + b2)/2 and Parseval's equality, we have

Qp(^1, . . . , ¿Z-1, ¿Z+2, . . . , ¿q-1, ¿q+2, . . . , ¿k ) ) <

TOTO <

v j'i=p+1 j'i+l=p+1

*+2 0

J faZ+1(0)j(0) J faZ(u)dud0 tt

X

tq+2 0

/ faq+1(0) jl (0) / faq (u)0ji (u)dud0

<

1

< -

- 4

*t +2 *0

V

E E I J faZ+1(0)0ji+l(0^ fa(u)dud0 | + j'i=p+1 j'i+1 =p+1 \t t

oo oo

iq+2

*0 2 + £ £ / faVn(0)jl (0W fa (u)^j'i ('«)d«d0

j'i =p+1 j'i+l=p+1 \t t / y

<

/0000/ *+2

<

4

E E I J faZ+1(0) jl(0^ fa(u)dud0 | +

yji =p+1 ji+l=0 \t t

TO TO /*+2 0 \V

+ E E I / faq+1(0) jl(0) J fa(u)^ji(u)dud0j

j i =p+1 j i+1=0 \ t

<

1

< -

- 4

/ OO *+2

E J fa?+1(0H J fa(u)0ji(u)du ) d0+

yji =p+1 t \t

ttq+2

+ E I fa?+1(0M I fa(u)^ji(u)du d0 ji =p+1 t Xt J J

(2.971)

2

0

2

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0

2

From (2.971) and (2.25), (2.278) we obtain

. \ 2 Qp^b • • • , £/—1, t/+2, • • • , tq—1, £q+2, • • • , ) J <

K

<— o

p2

if p ^ oo, where constant K does not depend on p^ Thus the equalities (2.955)-(2.958) are proved.

Recall that the function (2.944) (this function is defined using the left-hand side of the equality (2.686)) for the case k > 5, r — 2 is represented as the sum of several functions. Four of them, namely Qp, QQp, Qp, QQp (these functions correspond to the particular case of choosing the pairs (g1, g2), (g3, g4); generally speaking, all possible pairs (g1,g2), (g3,g4) must be considered), have been studied above. Absolutely similarly, we can consider the remaining functions (for all possible pairs (g1, g2), (g3, g4)) whose sum is the function (2.944) for the

case k> 5, r — 2^ As a result, we will have

2

pl^0^R?p^L2([t,T]k_2r) — 0 (k> 5, r — 2)^

After that, we can go to the function (2.944) for the case k > 5, r — 3, 2r < k (this function is defined using the left-hand side of the equality (2.

and follow the same steps as above. This will lead us to the following equality

2

^Pp IL2([t,T]k_2r) — 0 (k > 5, r — 3, 2r < k)

Then we can move on to the next step and so on. As a result, we get the equality (2.947) (r — 1, 2, • • •, [k/2]) and thus prove Hypothesis 2.2 for the case k — 2n + 1, n — 3,4, • • • (see Sect. 2.5).

For the case k — 2n, n — 3,4, • • • we follow the above steps for r — 1, 2, • • •, [k/2] — 1 (2r < k — 2). For 2r — k we use the same technique as in the proof of the equalities (2Z23)-(2.725). Recall that we used (2.655), (12.6621) and Parseval's equality in the proof of (2Z23)-(2.725).

The obvious disadvantage of the proposed algorithm is the drastic increase of complexity of the proof when moving from r — 1 to r — 2, r — 2 to r — 3 and so on.

The proofs of Theorems 2.33 and 2.34 contain a rather simple trick of passing from r — 1 to r — 2^ Unfortunately, this procedure cannot be applied

already at the transition from r = 2 to r = 3.

Note that the case k = 6, r = 3 was successfully considered in Theorem 2.35 under the following simplifying assumption: fa1(T),..., fa6(T) = 1.

Nevertheless, the results obtained in the previous sections of Chapter 2 are quite sufficient for practical needs (see Chapters 4 and 5 for details).

2.20 Theorems 2.1-2.9, 2.32-2.35, 2.40 on Expansion of Iterated Stratonovich Stochastic Integrals from Point of View of the Wong-Zakai Approximation

The iterated Ito stochastic integrals and solutions of Ito SDEs are complex and important functionals from the independent components fs(i), i = 1,... ,m of the multidimensional Wiener process fs, s G [0,T]. Let fs(i)p, p G N be some approximation of fs(i), i = 1,... ,m. Suppose that fs(i)p converges to fs(i), i = 1,..., m ifp —y to in some sense and has differentiable sample trajectories.

A natural question arises: if we replace fs(i) by fs(i)p, i = 1,..., m in the functionals mentioned above, will the resulting functionals converge to the original functionals from the components fs(i), i = 1,...,m of the multidimentional Wiener process fs?

The answere to this question is negative in the general case. However, in the pioneering works of Wong E. and Zakai M. [68], [69], it was shown that under the special conditions and for some types of approximations of the Wiener process the answere is affirmative with one peculiarity: the convergence takes place to the iterated Stratonovich stochastic integrals and solutions of Stratonovich SDEs and not to the iterated Ito stochastic integrals and solutions of Ito SDEs.

The piecewise linear approximation as well as the regularization by convolution [68]-[70] relate to the mentioned types of approximations of the Wiener

process. The above approximation of stochastic integrals and solutions of SDEs is often called the Wong-Zakai approximation.

Let fs, s G [0,T] be an m-dimensional standard Wiener process with independent components fs(i), i = 1,... ,m. It is well known that the following

representation takes place [114], [115] (also see Sect. 6.1 of this book for detail)

T t

TO T T

f«'> - f<"> = £ I to(s)ds cj", j fa jWdfj", (2.972)

j=0

where r £ [t,T], t > 0, (x)}o=0 is an arbitrary complete orthonormal system

of functions in the space L2([t,T]), and j are independent standard Gaussian random variables for various i or j. Moreover, the series (2.972) converges for any r £ [t, T] in the mean-square sense.

Let fT")p — ft(")p be the mean-square approximation of the process f(i) — ft(i), which has the following form

p T

j =0

From (2.973) we obtain

f«p — f<")p — £ J fr(s)dsCf. (2.973)

p

dfi")p — £ (r Xfdr. (2.974)

j=0

Consider the following iterated Riemann-Stieltjes integral

T t2

Vk (tk)• • • / ^1(t1)dwt("l)pi • • • dwt("fc)pfc, (2.975)

where p1, • • • ,pk £ N, i1, • • •, ik — 0,1, • • •, m,

!dfT")p for i — 1, • • •, m

, p £ N, (2.976)

dr for i — 0

and dfT")p in defined by the relation Let us substitute (2.976) into

/T ^ pi pk k Vk (tk)... V (t1 )dw<"i)pi .. • dw^ — £ , • • £ Cjk...ji n j>, (2.977)

t { ji=0 jk =0 /=1

where p1,... ,pk £ N,

T

j — J h (s)dwS") t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w(i) = fs(i) for i = 1,..., m and w(0) = s,

t t2

cjfc •••ji = J fa (tk ) j (tk) ■■■J fa (ti)0ji (t1)dt1 t t

is the Fourier coefficient.

To best of our knowledge [68]-[70] the approximations of the Wiener process

in the Wong-Zakai approximation must satisfy fairly strong restrictions [70 (see Definition 7.1, pp. 480-481). Moreover, approximations of the Wiener process that are similar to (2.973) were not considered in [68], [69] (also see

70], Theorems 7.1, 7.2). Therefore, the proof of analogs of Theorems 7.1 and

7.2 [70] for approximations of the Wiener process based on its series expansion (2.972) (also see (6.16)) should be carried out separately. Thus, the mean-square convergence of the right-hand side of (2.977) to the iterated Stratonovich stochastic integral (2.6) does not follow from the results of the papers [68], [69 (also see [70], Theorems 7.1, 7.2) even for the case p1 = ... = pk = p.

From the other hand, Theorems 1.1, 2.1-2.9, 2.32-2.35, 2.40 from this monograph can be considered as the proof of the Wong-Zakai approximation based on the iterated Riemann-Stieltjes integrals (2.975) of multiplicities 1 to 6 and the Wiener process approximation (2.973) on the base of its series expansion. At that, the mentioned Riemann-Stieltjes integrals converge (according to Theorems 1.1, 2.1-2.9, 2.32-2.35, 2.40) to the appropriate Stratonovich stochastic integrals (2.6). Recall that {fa (x)}TO=0 (see (2.972), (2.973), and Theorems 1.1, 2.1, 2.2, 2.4-2.9, 2.32-2.35, 2.40) is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([i,T]).

To illustrate the above reasoning, consider two examples for the case k = 2, fa(s), fa(s) = 1; i1 ,i2 = 1,... ,m.

The first example relates to the piecewise linear approximation of the multidimensional Wiener process (these approximations were considered in [68]-[70]).

Let b^(i), f G [0,T] be the piecewise linear approximation of the ith component ft(i) of the multidimensional standard Wiener process ft, £ G [0,T] with independent components ft(i), i = 1,..., m, i.e.

/')(/) - f(0 |

a \l) ~ k,A + ^ /ca>

lkA = f(k+1)A fkA

where Af« = f« - f(A, t G [fcA, (k + 1)A), k = 0,- 1.

Note that w. p. 1

db

(") A

dt

(t) —

Af

(")

kA

A

t £ [kA, (k + 1)A), k — 0,1,..., N — 1. (2.978)

Consider the following iterated Riemann-Stieltjes integral

T s

J y^ dbAAi)(r )dbAA2)(s), i1,i2 — 1,...,m. 00

Using (2.978) and additive property of Riemann-Stieltjes integrals, we can

write w. p. 1

T s

J dbAAi)(r)dbA2)(s) — J J

db

("i) A

db

("2) A

dr

ds

00

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n _1 C+;>A/ ,_1 (q+1>AAf (,i)

E E

/=0 /A N- 1 /-1

q=0 qA

A

s (" ) ^

[ M'uLdr

A

Af("2) AtlA -ds =

/A

J

(/+1)A s

A

N—1 /—1 . N—1 s

EE^a^a+^E^a^a / drds =

/=0 q=0 /=0 /A /A

N—1 /—1 1 N—1

/=0 q=0 2 /=0

(2.979)

Using (2.979), it is not difficult to show (see Lemma 1.1, Remark 1.2, and (2.8)) that

T

Ni.m. y J dbA°(r)dbA2)(s) — J J

di[H)diM + llu

T

2

{"i="2}

ds —

00

T ^ s

J J df("i)dfs("2), tt

(2.980)

where A ^ 0 if N ^ o (NA — T).

s

Obviously, (2.980) agrees with Theorem 7.1 (see [70], p. 486).

The next example relates to the approximation (2.973) of the Wiener process based on its series expansion (2.972), where t = 0 and {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([0,T]).

Consider the following iterated Riemann-Stieltjes integral

T s

ff= i....,m. (2981)

00

where dfT;)p is defined by the relation (12.974).

Let us substitute (2.974) into (2.981)

T s p

J ydffrfp = Y j jj, (2.982)

0 0 j1'j2=0

T s

Cj2j1 = j j (s) J j (t)dTdS 00

is the Fourier coefficient; another notations are the same as in (2.977).

As we noted above, approximations of the Wiener process that are similar to (2.973) were not considered in [68], [69] (also see Theorems 7.1, 7.2 in [70]). Furthermore, the extension of the results of Theorems 7.1 and 7.2 [70] to the case under consideration is not obvious.

On the other hand, we can apply the theory built in Chapters 1 and 2 of this book. More precisely, using Theorems 2.1, 2.2 we obtain from (2.982) the desired result

T s p *T *s

where

lpLm | yfl)pdfs(i2)p = Cj2jicj;i)cj:2) = J J dfiil)dfs( i2).

0 0 jiJ2=° t t

(2.983)

From the other hand, by Theorem 1.1 (see (1.46)) for the case k = 2 we

obtain from (2.982) the following relation

T s p

«îjpdf(i2)p _ i î m c . zz(i2)

j2

0 0 ' j1 'j2=°

l.i . m . i i f 1)pdf(i2)p _ l.i . m . J2 Cj2jicjn)zji

p^œ z—'

ji ,j2=0

p

r. . (Z(ii)z(i2) _ 1 . 1 r.

rj2j1 I j Z72 ±{«1=«2}±{j1 =

j2|) + !{i1= œ =Î2> Y] Cj1j1

j1 =0

œ

• Y] Cj1j1.

j1=0

T s

J J f f) + I{i1=i2^ j. (2.984)

00

Since

E^m = (jhMdr] = \ (T[Mr)dr\ = \ Ids, ji=0 ji=0 \o J \o /0

then from (2.8) and (2.984) we obtain (2.983).

2.21 Wong—Zakai Type Theorems for Iterated Stratonovich Stochastic Integrals. The Case of Approximation of the Multidimensional Wiener Process Based on its Series Expansion Using Legendre Polynomials and Trigonometric Functions

As we mentioned above, there exists a lot of publications on the subject of

Wong-Zakai approximation of stochastic integrals and SDEs [68]-[70] (also see

116]-[123]). However, these works did not consider the approximation of iter-

ated stochastic integrals and SDEs for the case of approximation of the multidimensional Wiener process based on its series expansions. Usually, as an approximation of the Wiener process in the theorems of the Wong-Zakai type, the authors [68]-[70] (also see [116 -

choose a piecewise linear approximation or an approximation based on the regularization by convolution.

The Wong-Zakai approximation is widely used to approximate stochastic integrals and SDEs. In particular, the Wong-Zakai approximation can be used to approximate the iterated Stratonovich stochastic integrals in the context of

numerical integration of Ito SDEs in the framework of the approach based on the Taylor-Stratonovich expansion [79], [80] (see Chapter 4). It should be noted that the authors of the works [78] (pp. 438-439), [79] (Sect. 5.8, pp. 202-204), 80] (pp. 82-84), [88] (pp. 263-264) mention the Wong-Zakai approximation

68]-[70] within the frames of approximation of iterated Stratonovich stochastic integrals based on the Karhunen-Loeve expansion of the Brownian bridge process (see Sect. 6.2). However, in these works there is no rigorous proof of convergence for approximations of the mentioned stochastic integrals of milti-plicity 3 and higher (see discussion in Sect. 6.2).

From the other hand, the theory constructed in Chapters 1 and 2 of this

monograph (also see [14]-[16]) can be considered as the proof of the Wong-Zakai

approximation for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 based on the Wiener process series expansion using Legendre polynomials and trigonometric functions.

The subject of this section is to reformulate the main results of Chapter 2 of this book in the form of theorems on convergence of iterated Riemann-Stiltjes integrals to iterated Stratonovich stochastic integrals.

Let us reformulate Theorems 2.2, 2.4-2.10, 2.14, 2.17, 2.30, 2.32-2.35, 2.40 and Hypotheses 2.1-2.3 of this monograph as statements on the convergence of the iterated Riemann-Stiltjes integrals (12.975) to the iterated Stratonovich stochastic integrals (2.356).

Theorem 2.42 [38] (reformulation of Theorem 2.2). Suppose that the following conditions are fulfilled:

1. Every fa(r) (l = 1, 2) is a continuously differentiate function at the interval [t,T].

2. {fa (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]).

Then, for the iterated Stratonovich stochastic integral of second multiplicity

* T * ¿2

J*[fa2)]T,t =/ fa(*2)/ fatiOdw^dw^ (M2 = 0,1,..., m)

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the following formula

t t2

(2)]T,t = l.i.m. / fa(t2) / fa(ti)dwt(;i)pidwt(;2)p2

is valid.

Theorem 2.43 [38] (reformulation of Theorems 2.4 and 2.6). Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

J J J <»ff (M2,i3 = 1,...,m)

i t t

the following formula

*T * is * ¿2 T ts t2

JJJ dfi(;i)dfi;2)dfi(;s) = p u.m.^ J J J fii)pif(;2)p2f;s)ps t t t t t t

is valid.

Theorem 2.44 [38] (reformulation of Theorem 2.5). Let (x)}°=0 be a

complete orthonormal system of Legendre polynomials in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

ffi' = / (t -(t -(t - ii)'1 f^ff

t t t

the following formula

T ¿3 ¿2

V _ i )'l

' = l.i-m. / (i - i3)'s / (t - t2)'W (t - ii)'1 df((il)pidftÜ2)P2dff»,

12st,Î / / / i2s

t

where ii, i2, i3 = 1,..., m, is valid for each of the following cases

1. ii = ¿2, ¿2 = i3, ii = ¿3 and li, /2,/3 = 0,1, 2,...

2. ii = ¿2 = i3 and l1 = l2 = l3 and l1, l2, l3 = 0,1, 2,...

3. ii = i2 = i3 and li = l2 = l3 and li, l2, l3 = 0,1, 2,...

4. ii,i2,i3 = 1,..., m; li = l2 = l3 = l and l = 0,1, 2,...

Theorem 2.45 [38] (reformulation of Theorem 2.7). Let (x)}°=0 be a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and ^(t) (/ = 1, 2,3) are continuously differentiate

functions at the interval [t, T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

# # t where il5 i2, i3 = 1,..., m, the following formula

T ¿3 ¿2

t t t is valid for each of the following cases

1. ii = ¿2, ¿2 = ¿3, ¿1 = ¿3,

2. ¿1 = ¿2 = ¿3 and fa(t) = "fa(t),

3. ¿i = ¿2 = ¿3 and fa(t) = fa(T),

4. ¿1, ¿2, ¿3 = 1,..., m and fa(T) = fa2(T) = fa3(T).

Theorem 2.46 [38] (reformulation of Theorem 2.8). Let {fa(x)}°=0 be a

complete orthonormal system of Legendre polynomials or trigonomertic functions in the space L2([t,T]). Furthermore, let the function fafa) is continuously differentiate at the interval [t,T] and the functions fa(T), fa3(T) are twice continuously differentiate at the interval [t, T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

J*[fa3)fc = / «to/ )/ fa (ti)dfi;i)f^fs),

where il5 i2, i3 = 1,..., m, the following formula

T ¿3 ¿2

j*[#>]„ = i.i.m. f utfaut?) f Mf^f*

is valid.

Theorem 2.47 [38] (reformulation of Theorem 2.9). Let {fa(x)}°=0 be a

complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic in-

tegral of fourth multiplicity

*T ^£4 ^ t3

= J J J J dw<;')dwi;2»dw<33)dw«:4»,

t t t t

where ii, i2, i3, i4 = 0,1,..., m, the following formula

T t4 t3 t2

/;J;ii2i3i4) = LLm. JJ J J dw( dwt(22)pdwt(33)pdwt(44)p

tttt

is valid.

Theorem 2.48 [38] (reformulation of Theorems 2.10 and 2.14). Suppose that every ^(t) (l = 1,..., k) is a continuously differentiate function at the interval [t,T] and (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral

* T * t2

J* № (k)]T,t =f (tk)...[ ^i(ti)dwtll) ...dw£k), (2.985)

where ii,..., ik = 0,1,..., m, the following formulas

2n

IT,,-. W TTi""^

lim lim ... lim M < J'hr'b, - J'Wk) Sr'p* > = 0.

^TO pfc_i^TO pi^TO

2n'

lim lim ... lim M <M J* - J*^^-^ 1 }> = 0

are valid, where

T t2

J *[^(k)]T1f"Pfc = ^ (tk)... ^i (ti)dwt;i)pi... dw£k , (2.986)

n G N, and lim means lim sup.

Theorem 2.49 (reformulation of Theorem 2.17). Suppose that every ^(t) (l = 1,2,3) is a continuously differentiate function at the interval [t,T]

and {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integrals J*[fa2)]T,t and J*[fa3)]T,i (¿1 ,¿2,23 = 1,...,m) defined by (2.985) the following formulas

lim lim m < ( J *

pi^œ p2^œ

T,t

-J *

P1,P2 T,t

= 0,

lim lim m < ( J *

P2^œ pi^œ

T,t

-J *

P1,P2 T,t

= 0,

lim lim lim m

pi^œ p2^œ p3^œ

J*[fa(3)]T,t - J*[fa(3)]T?2'P3

lim lim lim M{ | J*[fa(3)]T,t - J*[fa(3)]

p3^œ p2^œ p1^œ '

Pl,P2,P3

T,t

= 0,

=0

TT2 and J *

pi,p2,p3 are defined by (2.986).

are valid, where J-JT t u^u, ^ -JTt

Theorem 2.50 (reformulation of Theorem 2.30). Assume that the continuously differentiate functions fa(t) (l = ) at the interval [t,T] and the complete orthonormal system {fa(x)}°=0 of continuous functions (fa(x) = 1/faT — t) in the space Lo{[t,T}) are such that the following conditions are satisfied:

1. The equality

t2

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- I $i(ii)$2(ii)dii = E / ^2(i2)fa(i2) I ^ifafafafadtfaU t j=0 t t

holds for all s £ (t,T], where the nonrandom functions $1(r), $2(r) are continuously differentiate on [t,T] and the series on the right-hand side of the above equality converges absolutely.

2. The estimates

fa (t )$x(t )dt

<

ll(s)

,?'l/2+a

T

fa (t )$2(t )dT

<

ttl(s)

,yl/2+CK

2

s

£ / $2(T)0j (T) / (0)d0d

j=P+! i t

T

<

p

£

hold for all s G (t,T) and for some a,^ > 0, where (t), $2(t) are continuously differentiate nonrandom functions on [t,T], G N, and

T T

J ^2(t)dT < œ, y*|^2(T)| dT < œ.

î Î

3. The condition

lim

p^TO

E

(p)

j* -j'q -j'l

jlv,jqvj'k = 0

9=S1>S2>'">S2r-1>S2r

= 0

q=gi,02v,g2r—i,g2r

holds for all possible gi, g2,..., g2r-i, g2r (see (12.6271)) and li, l2,..., ld such that li, l2,..., ld G {1, 2,... ,r}, li > l2 > ... > ld, d = 0,1, 2,... ,r - 1, where r = 1, 2,..., [k/2] and

Sl2 • • • C

(p)

d | jk •••jq •••jl

def £(p)

q=0l,02,-,02r—l,g2r

jk •••jq •••jl

q=gl,g2v>g2r—l,g2r

for d = 0.

Then, for the iterated Stratonovich stochastic integral of arbitrary multiplicity k

T

t2

J '[v/*']^

^ ' = v* (tk) ■■■ Vi(ii)dw<;l) ...dw';*1,

tt where ii,..., ik = 0,1,..., m, the following formula

T

t2

j*[v(k)]Trik) = i.i.m. I v*(t*)••• Vi(ti)dwi;l)p• • • dwi;*)p

p^œ

oo

j (x)}j =0

is valid.

Theorem 2.51 (reformulation of Theorem 2.32). Suppose that (x)} is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let ^i(s),^2(s),^3(s) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated

s

2

Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

J*[fa3)]T,t = / fa(t3)/ fa(*2)| ^l(ti)dwi;i)dwii2)dwt(Ss), t t t

where ¿1, ¿3, ¿3 = 0,1,..., m, the following formula

T ¿3 ¿2

J*[fa (3)]T,i = fa(t3)/ «t2)/ fai(ti)dwi;i)pdwi;2)pdwt(:s)p

t t t

is valid.

Theorem 2.52 (reformulation of Theorem 2.33). Suppose that {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let fa(s),... ,fa(s) are continuously differentiable nonrandom functions on [t, T]. Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

* T * i4 * "s * ^

J*[fa(4)]T,t = j MU)J Mt3)J Mt2)f fai(ti)dw(;i)dw(;2)dw(:s)dw(;4), t t t t

where ..., i4 = 0,1,..., m, the following formula

T t4 ts t2

J*[fa(4)]T,t = 1^ / fa^)/ fa^)/ fa^)/ fa(ti)dwt(;i)px

tttt

xdwt(i2)pdwt(3s)pdwt(44)p

is valid.

Theorem 2.53 (reformulation of Theorem 2.34). Suppose that {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let fa(s),... ,fa(s) are continuously differentiable nonrandom functions on [t, T]. Then, for the iterated Stratonovich stochastic integral of fifth multiplicity

* T * "£5 * t4 * ¿S * t2

J*[fa(5)]T,t = J fa^)/ Mt4)j fa^)/ fa^)/ fai(ti)dwt(;i)x t t t t t

xdwf2)dwfs)dwf4) dwt(i5),

where ¿i,..., ¿5 = 0,1,..., m, the following formula

T t5 t4 ts t2

J*[fa(5)]T,t = lp;i;m^ fa^)/ fa^)/ fa^)/ fa^)/ fai (ti x

ttttt

x dwt(;2)pdwt(is)pdwt(i4)pdwt(i5)p

is valid.

Theorem 2.54 (reformulation of Theorem 2.35). Suppose that {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of sixth multiplicity

*T * ¿6 * t5 * t4 * "s * t2

j,(,:,.,6)=J J J J J J dw<:i»dw«:2>dw«;s>dw<44»dw«;5>dw<:G),

t t t t t t

where ii,..., = 0,1,..., m, the following formula

T ¿6 ¿5 ¿4 ¿3 ¿2

J*1' ■'6) = lA.m. JJJJJ J dwfadwfadwfadwfadwfadwfa

i i i i i i

is valid.

Theorem 2.55 (reformulation of Theorem 2.40). Suppose that {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Furthermore, let fa(s),fa(s),fa(s) are continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * ¿3 * ¿2

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(3)]t,; = / (ts)J ^^(iOdw^w^w^, ¿¿¿

where ii, i2, i3 = 0,1,..., m, the following formula

T ¿3 ¿2

J*[^ (3)]t,; = l.i.m. i UU)i Mt2) [ ^Ы^1dw£2)p2dwi33)p3

¿¿¿

is valid.

Hypothesis 2.4

(reformulation of Hypothesis 2.1). Let (x)}°=0 be

a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then, for the iterated Stratonovich stochastic integral of kth multiplicity

.T

<Î2

dw!n)... dw

)

tfc

(ii,..., ik = 0,1,... ,m)

the following formula

* T * Î2

dw

(il)

T

dw<:k ) =

l.i.m.

p^œ

t2

dw

(i1)P

. dw

(:fc )P tfc

is valid.

Hypothesis 2.5 [38] (reformulation of Hypothesis 2.2). Let (x)}°=0 be a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, every ^(t) (l = 1,...,k) is an enough smooth nonrandom function on [t,T]. Then, for the iterated Stratonovich stochastic integral of kth multiplicity

T

J *

T,t = / ^k (tk )

= Î2

^i(ti)dw(;i)

. . dw

(ik) tfc

(ii,...,ik = 0,1,... ,m)

the following formula

J *

T,t =

T

t2

l.i.m.

p^œ

^ (tk )... MiMï» ...dw«:"p

38] (reformulation of Hypothesis 2.3). Let (x)}

œ j =0

is valid.

Hypothesis 2.6

be a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, every ^(t) (l = 1,...,k) is an enough smooth nonrandom function on [t,T]. Then, for the iterated Stratonovich stochastic integral of kth multiplicity

T

■M

J *

T,t = ^k (tk )

^i(ti)dw(:i)

. . dw

(:fc)

(ii,... ,ik = 0,1,... ,m)

i

the following formula

T t2

J*[^(k)]T,t = l.i.m^ I ^ (tk )... ^i(ti)dwi:i)pi ... dwt:k )pk

t t

is valid.

Note that Hypothesis 2.6 is valid under weaker conditions if ii,..., are pairwise different and ii,...= 0,1,... ,m (see (1.44)).

2.22 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k. The Case i1 = ... = ik = 0 and Different Weight Functions ^1(t),... (t)

In this section, we generalize the approach considered in Sect. 2.1.2 to the case i1 = ... = = 0 and different weight functions ^i(t),... (t) (k > 2). Let us formulate the following theorem.

Theorem 2.56 [33]. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, ^1(t),...,(t) (k > 2) are continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral

J*[#k)]T,t = / ^k (tk ).../ ^i (ti)dft(i:i)... dft(k:i) (ii = 1,...,m)

t

the following equality

p p \ 2n

ton m J*[^(k)]T,t - £ ... £ Cjk. jiCj;i)... =0

[ V ji=0 j=0 J )

is valid, where n G N,

T t2

Cjk -ji = J ^k (tk ) j (tk) .J ^i(ti)0ji (ti)dti ...dtk tt is the Fourier coefficient and

T

j) = / h (T )df(:i) (ii = 1,...,m) t

are independent standard Gaussian random variables for various j.

Proof. The case k = 2 is proved in Theorem 2.16. Consider the case k > 2. First, consider the case k = 3 in detail. Define the auxiliary function

K\tlMM = \{ 6

>i(ti)fa(t2)fa(t3 fai(ti)fa(t3)fa(t2 fai(t2)fa(ti)fa(t3 fai(t2)fa(t3)fa(ti fai(t3)fa(t2)fa(ti >(t3)fa(ti)fa(t2

, ti < t2 < t3

, ti < t3 < t2

, t2 < ti < t3

, t2 < t3 < ti

, t3 < t2 < ti

, t3 < ti < t2

ti,t2,t3 £ [t,T].

Using Lemma 1.1, Remark 1.1 (see Sect. 1.1.3), and (2.382), we obtain w. p. 1

N-i N-i N-i

J[K']T3i = Ni.m. E E E K(Tii,T2,71sfff =

is=012=0 ii=0 i 2 3

/N-i 1s-i 12-i

= lNi.m. EE EK (Tii ^ ,Tis) AfT:ii) A fT:2i) Af(:si) +

U=0 12=0 1i=0

N-i 1s — i 1i-i

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+ E E E K fa .12. Tis f M^ffa

/s=0 1i=0 /2=0

N-i l2-i ii-i

+ E E E K (Tii, TJ2, Tis f AfS2i> Afft)+

/2=0 1i=0 /s=0

N-i 12-i 1s-i

+ E E E K' (Ti • T2, T/s f^ffa

/2=0 /s=0 /i=0

N-i /i-i /2-1

+ E E E K (t/i - ■t/2 , t/s f f

/i=0 /2=0 /s=0

N-1 li-l I3 —1

+ £ £ £ K'to • ,713

11=0 /3=0 /2=0

N-1 /2-1 2

+ ££ K' (T/1 -T/2 >T/1 )(Afi^)

/2=0 /i=0 i 2 N-1 /3-1 2

+ ££ K' (T/i ,T/3 ,T/3 )(Af<^) Af^

/3=0 /1=0

N-1 /i-1 2

+ ££ K' (t/i ,712 ,T,2 ^f) f +

/1=0 /2=0

N-1 /3-1 2

+ ££ K' (T/3,712,713 Iff) AfT:> +

/3 = 0 /2=0

N-1 /3-1 2

+ ££ K'(712,71b,T/3)(f) AfC» +

/3 = 0 /2=0

N-1 /2-1 2

+ ££ K' (7/2,712.713 }(fj>) AfTi1)

/2=0 /3=0 2 3

/ T t3 t2

= U J Uh) J Mh) J Mt^df^dth \t t t T t2 t1

+ / Ut2)f Mh)J ^1(t3)dft(3;1)dft(1;1)dft(2;1)+ ttt T t2 t3

+ / Mh)J Mt3)J ^1(t1)dft(1;1)dft(3;1)dft(2;1)+ ttt T t3 t1

+ / Mt3)J ^fa)/ ^1(t2)dft(2;1)dft(1;1)dft(3;1) + ttt T t1 t2

+ / ^fa)/ «*2)/ ^1(t3)dft(3;1)dft(2;1)dft(1;1) + ttt

+ fa3(ti) / fa2(t3) / fai(t2)dfi(2:i)dfi(s:i)dft(;i) +

+ fajfe) / fa2(ti)fai(ti)dtidft(2:i) + fa3(tiW fa2(t2)fai(t2)dt2dft(i:i) +

T ¿i ¿s

ti

ttt T ¿2 T ti

■(:i) I I J, (+ \ I J, (+ \J, (+ \rH- ^7-P(:i)

t t t t T is T is

+ / ^3(t3^ fa2(ti)fai(ti)dtidfi(s:i) +J fa(t3)fa(t3) j fai(ti)dfi;i)dt3 + t t t t T is T ¿2

+ J fa3(t3)fa2(t3)J fai(t2)dft(2:i)dt3 + J Ih(t2)*h(t2) J fai(t3)dfi(s:i)dt2 t t t t T is ¿2

= / fajfe)/ fa^)/ fa (ti)dft(:i)dft(2:i)dft(s:i) +

ttt T is T is

yW3) j UhyUtfadhd^+ j Uh)Uh) j Uh)dtl)dh = t t t t * T * ¿3 * ¿2

= / fa^)/ fa(t2)/ fai(ti)dft(i:i)dft(2:i)dft(s:i) =f J*[fa(3)]T,t, (2.987)

where the multiple stochastic integral J[K']^" is defined by (11.161) and {rj}N=0 is a partition of [t,T], which satisfies the condition (1.9).

Using Proposition 2.2 for n = 3 (see Sect. 2.1.2) and generalizing the Fou-rier-Legendre expansion (12.57) for the function K'(t1,t2,t3), we obtain

p p p 1 ,

K'{tl,to,h) = lim E E E 6 ( f + f './:■■./ ./: + './:./ ./:■. '

^ °°ji=0 j2=0 js=0 ^ + Cj2j3ji + Cjij2js + Cjij3j2^ j (t0 j (t2)0js (t3) , (2.988)

where the multiple Fourier series (2.988) converges to the function K'(t1,t2,t3) in (t,T)3. Moreover, the series ('2.988) is bounded on [t,T]3; {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). For the trigonomertic case, the above statement

follows from Proposition 2.2 (the proof that the function K'(t1,t2,t3) belongs to the Holder class with parameter 1 in [t, T]3 is omitted and can be carried out in the same way as for the function K '(t1,t2) in the two-dimensional case (see Sect. 2.1.2)). The proof of generalization of the Fourier-Legendre expansion (2.57) to the three-dimensional case (see (2.988)) is omitted as well as the proof of the boundedness of the Fourier-Legendre series (12.988) on the boundary of the cube [t,T]3.

Denote

g \ ~ J3J2J1 + ^ './:■../ ./: + ^ './: ./ ./:■.

p p p 1 /

KpiMM,h) = K'ihMM) "EEEer-

¿1=0 ¿2=0 ¿3=0 ^ + Cj2j3¿1 + Cj1j2¿3 + Cj'1j3j2^ j (t1)^j2 (t2)0j3 (t3 ).

Using Lemma 1.3 and (2.987), we get w. p. 1

p p p 1 /

¿1=0 ¿2=0 ¿3=0 ^

+ ^'¿^¿^¿1 + ^¿'¿2¿3 + C¿1¿3¿2^ C¿; (¿2 £¿3 + J [Rppp]r,t = ppp

= Z( ;1)Z( ;1)Z( ;1) + t[R' 1 ( 3)

= Z^ Z^ Z^ ¿¿¿1 SJ1 ^¿2 ¿J + J [Rppp]T,t. ¿1=0 ¿2=0 ¿3 =0

Then

p p p ^ 2n

m j J [Rj?>n = m j *[^(3)iT,f - ee e ¿¿1 ¿¿'c^'

1 [ V ¿1 =0 ¿2=0 ¿3=0

where n G N.

Applying (we mean here the passage to the limit lim ) the Lebesgue's Dom-

p—>-n

inated Convergence Theorem to the integrals on the right-hand side of (2.433)

for k = 3 and Rppp(t1,t2,t3) instead of Rp1p2p3(t1 ,t2,t3), we obtain

p—n Mj J[flJpplS3^)2"} = 0.

Theorem 2.56 is proved for the case k = 3.

To prove Theorem 2.56 for the case k > 3, consider the auxiliary function

fai(ti).. .fak(tk), ti < ... < tk

fai(tgi) ...fak (tgfc), tgi < ... < tgfc , ti,...,tk £ [t,TL

v fai(tk).. .fak(ti), tk < ... < ti

(2.989)

where {g1,..., gk} = {1,..., k} and we take into account all possible permutations (g1,..., gk) on the right-hand side of the formula (2.989).

Further, we have w. p. 1

[k/2]

1

2r

=1 (sr ,...,si)eAk

sr (...,s i

T,t ,

(2.990)

where the function K '(t1,...,tk) is defined by (2.989); another notations are the same as in (2.370) and Theorem 2.12 (i1 = ... = ik = 0 in (2.370)).

From (2.990) and Theorem 2.12 we obtain w. p. 1

J*

T,t = J [K']Tki.

(2.991)

Generalizing the above reasoning to the case k > 3 and taking into account (2.991), we get w. p. 1

p p

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J*

1

T,t =

£ * * * £ Ti I £ (1 • • • ^ + J h>>i,..r r.

k! ,

ji =0 jfc =0 \(ji,...,jfc)

?' i(k)

\t

pp

C Z(:i) Z(:i) + T[ R' ](k) ... Cjfc...jij ... j + J [Rp...p]T,t,

ji=0 jk =0

where

R' (ti,...,tk) = K' (ti,...,tk)-

p... p

r

p p 1 / \

E or(h)...on(i,). ¿1=0 ¿fc=0 ' \(ji,...,jfc) )

the expression

£

Ciiv-jfc)

means the sum with respect to all possible permutations (j,..., ). Further,

m {(j [R,,®)2n} = m | J -i^fe - E •■■ E c...* cj;i).. -c^)2" J,

where n € N.

Applying (we mean here the passage to the limit lim ) the Lebesgue's Dominated Convergence Theorem to the integrals on the right-hand side of (2.433) for Rp...p(ti,... , tk) instead of Rpi...:Pk(ti,... , tk), we obtain

Urn m{ (J=0.

Theorem 2.56 is proved.

2.23 Comparison of Theorems 2.2 and 2.7 with the Representations of Iterated Stratonovich Stochastic Integrals With Respect to the Scalar Standard Wiener Process

Note that the correctness of the formulas (2.35) and (2.245) can be verified in the following way. If i1 = i2 = i3 = i = 1,...,m and (t),^2(t),^3(t) = ^(t), then we can derive from (2.35) and (2.245) the well known equalities (see Sect. 6.7)

* T * t2 / t

J MMf^df^ = I y ^(r)dfW

t t

ST !jc t3 ¡|St2

3

w. p. 1, where ) is a continuous nonrandom function at the interval [t,T].

From (2.35) (under the above assumptions and p1 = p2 = p) we have (see (2.424) and (1.59))

p

J*[fa(2,]T.i = ^ E Cj2jijj =

ji,j2=0

p ji-1 / \ p 0N

(:)A:) , /7«

p ji-1 / \ p

^00 , £ £ I Cj2ji + Cjij2 j jj + £ Cjiji (Zi:

= l.i.m.

ji=0 j2=0 ji=0

p ji-1 -i p ji=0 j2=0 ji=0

p p ^EfafaciM' + ^Efaici:1)"

, jij2 = 0 ji=0 J

\ ji=j2 /

X 2 / T x 2

p \ 2 /

= lis- 5 (E faci, J = 2i (/ I <2-992»

w. p. 1. Note that the last step in (2.992) is performed by analogy with (1.55). From (2.245) (under the above assumptions) we obtain (see (2.425) and

(00H02]))

p

-(iMiMi) -

J*[fa(3)]T,t = y£. £ Cj3j2jicj

ji,j2,js =0

p ji 1 j2 1 / \ = ^ X ^ ( Cjsj2ji + Cj'sj'ij2 + Cj2jijs + Cj2jsji + Cj'ij2jS + CjijSj2 j x

P ji=0 j2=0 js=0^ '

xc':)c!:)c':)+

ji j2 j

p ji 1

+EE (j js+j+j (cjs')2 cj:>+

ji =0 js=0

p ji-1 / \ 2 p A

+EE j + Cj, j,js + j (Zj'iO2 j+E Cjijij, j3 =

ji=0 js=0V J ji=0 /

p ji-1 ¿2-1

■is- (EEEc*jc!2)<j3)+

p—>-to

*i=0 ¿2=0 ¿3=0

1

p ¿i-1

2 i- 2

¿i=0 ¿3=0

1

p

3 Z (i) 6 ~ * Vzji *i=0

¿i=0 ¿3=0 3

1- I 1 = l.i.m. -

p—to \ 6

p

E c*2 c* zj;)4:)zj:)+

jij2>j3 = 0 ji=j2 j2=j3 ji=j3

1

p ¿i-1

2 i- 2

¿i=0 ¿3=0

1

p

3 Z (:) 6 ~ * V Zji ¿i=0

¿i=0 ¿3=0 3

1

p

p—TO \ 6 p ¿i-1

='is- (* e c^mm.

p *i-1

ii:*2:*3=0 2

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/ p ¿i-1

3 E E C* C*i j+3££ c* C*3 (cj;y C«3'+

V ji=0 ¿3=0

+E C

ji=0

3 (z(:)

p ji-1

¿i=0 ¿3=0 3

+

p ¿i-1

.(;)\2 >(;)

¿i=0 ¿3=0

(OV Ai) , ^ V^ V^ ri2 r i fV) \

ji Vsia / A- 2 Z^Z^ ./'• IV / V ^

1

p

3 Mi) 6 ^^ VZji ji=0

ji=0 ¿3=0 3

1

3

= l.i.m.

p—TO 6

Ec* z

(:)

i

3!

(2.993)

(r)df^

,ji=0 "v Vt

w. p. 1. Note that the last step in (2.993) is performed by analogy with (1.58).

3

2.24 One Result on the Expansion of Multiple Stratonovich Stochastic Integrals of Multiplicity k. The

Case ¿i = ... = ik = 1,..., m

Let us consider the multiple stochastic integral (1.16)

N-1 k

l.i.m. £ $(3 ,...,rA) IIAw j ' = J WTp(2.994)

N^ . ^ 77 j 1

JlvJk =0 1=1

where we assume that $(t^..., tk) : [t, T]k ^ R1 is a continuous nonrandom function on [t,T]k. Moreover, {t3}N _0 is a partition of [t,T], which satisfies the condition (1.9) and ¿1,..., ¿k = 0,1,..., m.

The stochastic integral with respect to the scalar standard Wiener process (i1 = ... = ik = 0) and similar to (2.994) (the function $(t1,..., tk) is assumed to be symmetric on the hypercube [t,T]k) has been considered in literature (see, for example, Remark 1.5.7 [124]). The integral (2.994) is sometimes called the multiple Stratonovich stochastic integral. This is due to the fact that the following rule of the classical integral calculus holds for this integral (see Lemma

1.3)

J[$]&'= Jfafe'... Jw. p. 1, where $((1,... ,tk) = ^(tO .. .^t(it) and

T

J [^]Tt = / ^ (s)dwSi' (l = 1-----k).

It is not difficult to see that for the case i1 = ... = ik = 0 we have w. p. 1

N-1

l.i.m. Y $(rji '...'r3k )Aw{jl'... Awj =

Ji,-Jk =0

E h(E .....

i.e.

k k J [$]Tif.i1' = J [$]Ti,1...il' w. p. 1, (2.995)

where

<h(ti t.A =

k\

is the symmetrization of the function ..., ); the expression

Z

(ai,...,afc)

means the sum with respect to all possible permutations (al5..., ak).

Due to (2.995) the condition of symmetry of the function ..., ) need not be required in the case i1 = ... = ik = 0.

Definition 2.1 [125]. Let ..., ) G L2([t,T]k) is a symmetric function and q = 1, 2,... [k/2] (q is fixed). Suppose that for every complete orthonormal

system of functions (x)}°=0 in the space L2 ([t,T]) the following sum

p p

J2 J2 J . . . ) j (t1 ) j (t2) . . . j (t2q-1)0jq (t2q)X

ilv-Jq=0 j2q+1,...,jfc =0 [t T

X 0j2q+1 (t2q+1) . . . j (tk)dt1 . . . • 0j2q+1 (t2q+1) . . . j (tk) (2.996)

converges in L2([t, T]k-2q) if p ^ to to a limit, which is independent of the choice of the complete orthonormal system of functions (x)}°=0 in the space L2([t, T]). Then we say that the qth limiting trace for (t1,..., ) exists, which by definition is the limit of the sum (2.996) and is denoted as Tr9Moreover,

—def -x-

Tr $ = $.

Consider the following Theorem using our notations.

Theorem 2.57 [125]. Let ) G L2([t,T]k) is a symmetric non-

4, . . . , tk) G L2(

random function. Furthermore, let all limiting traces for <T(t1,..., tk) (see Definition 2.1) exist. Then the following expansion

rw<ip?) = £ Ckjzjii)..j

ji ,...jfc=o

that converges in the mean-square sense is valid, where

k

j o[$]Titi-ii )

is the multiple Stratonovich stochastic integral defined as in [126] (1993) (also see [125], pp. 910-911),

C —

/k

)n j (ti)dti ...dtk

[t,T]k

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, i1 — 1,..., m,

T

Cjfl) — J to (s)df<>1>

t

are independent standard Gaussian random variables for various j.

In addition to the conditions of Theorem 2.57, we assume that the function $(t1,... ,tk) is continuous on [t,T]k. Then [124]

k k J0^-*1 ) — J[S]^1'"*1 ) w. p. 1, (2.997)

where the multiple Stratonovich stochastic integral

k

j [^Tp)

is defined by (2.994). As a result, we get the following expansion

J [*]T;r ) — l^m. E Cjk...ji Cf ...Cf. (2."8)

ji,-"jk =0

It should be noted that the expansion (2.998) is valid provided that for the function $(t1,... ,tk) there exist all limiting traces that do not depend on the choice of the complete orthonormal system of functions {fa (x)}°=0 in the space L2([t,T]). The last condition is essential for the proof of the equality (2.997) (this proof follows from Theorem 1.5.3, Remark 1.5.7, and Propositions 2.2.3, 2.2.5, 4.1.2 [124]). More precisely, in [124], to prove Proposition 4.1.2 (p. 65)

a special basis {fa(x)}°=0 was used. This means that the existence of a limit of the sum (12.996) for the function $(t1,... ,tk) in the case when {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) requires a separate proof. Such a proof in the simplest case is given in Sect. 2.1.2 (see Lemma 2.2).

It is not difficult to show that (see (2.991))

k k

) = J[K]^1-'1) w. p. 1, (2.999)

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where

k

*r„/,Wl( H-H )

J ,.„ ,

is the iterated Stratonovich stochastic integral (2.356) (¿i = ... = = 0),

k

j [K ^)

is the multiple Stratonovich stochastic integral (2.994) (i1 = ... = = 0) for the continuous function K'(t1,..., ) defined by (2.989).

If we assume that the limiting traces from Theorem 2.57 exist, then we can write (see (2.

k k p

1=1=um. v ( ', v cÄ...,)cj:i)...cj:)

ji,-jk=° \ (ji,...,jk)

p

= 'pL^j- £ Cjk...jiC«!1).j (2.1000)

ji,-jk=0

where

T t2

Cjk ...j! = J 1 (tk ) j (tk) ...J li(ti ) j (ti )dti.. .dtk (2.1001) t t is the Fourier coefficient,

E

(jiv-jfc )

means the sum with respect to all possible permutations (j,..., jk); another notations are the same as in Theorem 2.57.

The equality (2.1000) agrees with Hypothesis 2.2 (see Sect. 2.5) for the particular case ii = ... = ik = 0.

From the other hand, the following expansion (see (1.44))

pi pk

.ni(k)]Tr')=pJiifa E. ■ ■ £ Ckjzf ■ ■. j) (2.1002)

ji=0 j'fc=0

is valid, where Cjk...ji has the form (12.10011). the numbers i1,...,ik are pair-wise different (i1,..., ik = 0,1,..., m); another notations are the same as in Theorem 1.1.

The equality (2.1002) corresponds to Hypothesis 2.3 (see Sect. 2.5) for the pairwise different numbers i1,..., ik = 0,1,..., m.

2.25 A Different Look at Hypotheses 2.1-2.3 on the Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k

In the previous section, we saw that in a number of papers (see, for example, 124]-[126]) the conditions of theorems related to multiple stochastic integrals

(see Theorem 2.57 in Sect. 2.24) are formulated in terms of limiting traces (see Definition 2.1). In addition to limiting traces, the concept of Hilbert space valued traces (integral traces) is introduced in [124]. The concepts of traces considered in [124]-[126] are close to some expressions that we used in this

chapter. For example, the following integral (see (2.10))

T T

i J MtiyhMdh = J k* (t-i, t\ )dt\ t t

is an example of a trace introduced in [124] (Definition 2.2.1). However, the function K*(t1,t2) defined by (2.95) is not symmetric compared with [124] (De-

finition 2.2.1). In addition, the expression (see (2.10))

œ œ „

ECm = E / K(Î1,t2)j(t1)j(t2)dt1dt

_A _A J

ji=0 ji=0 UT12

is an example of a limiting trace (see Definition 2.1) for the function K(t1,t2), which is not symmetric (see (2.

As noted in the previous section, the proof of the existence of limiting traces is a nontrivial problem. Therefore, in this section, we reformulate Hypothesis 2.3 from Sect. 2.5 (Hypotheses 2.1 and 2.2 are special cases of Hypothesis 2.3) in the form of Theorem 2.58 (see below). Moreover, the conditions of this theorem will be formulated using some expressions, which can be conditionally considered as "stochastic analogs" of limiting traces from Definition 2.1.

Theorem 2.58. Assume that the continuously differentiate nonrandom functions ^(t),..., Ak(t) : [t,T] ^ R and complete orthonormal system of functions (x)}°=0 in the space L2([t,T]) are such that the following equality is fulfilled

[k/2] .

E^ E J^{k)]sA;rsl =

r=1 (sr ,...,s1)eAfc,r P1 P: k

= l-i-m £ -.^Cj-k ...j1 l.i.m. E IK <T|, )Aw<;») (2.1003)

w. p. 1, where

l

is, ,...,S1 def

isp =isp+1

p=1

j[A^T;-'1 =f n i{„p=„p+1=0, X

T ts,+3 is, + 2

X J Ak (tk) ...J As,+2 (ts, +2) J A s, (ts,+1 +1(ts,+1) X t t t

ts, + 1 ts1+3 ts1 + 2

X J As,-1(ts,-1) ...J As^sM J (ts1+1)^s1+1(ts1+1 )x t t t

ts1 + 1 t2

X y As1-1(ts1-1) ...J A1(t1)dwt(i1) . . . dWt(;S--11)dts1 + 1dWt(;S+^2) . . . tt

... dw;;-1' dts,+Kiw;::++- ...dw>:k \

T t2

y ^k (tk ) j (tk ) ...J (t1)0j1 (t1)dt1 ...dtk tt

is the Fourier coefficient,

Gk = Hk\Lk, Hk = {(/1,...,/k) : l1,...,1k = 0, 1,...,N - 1}, Lk = {(l1,...,1k): I1 ,...,1k = 0, 1,...,N -1; lg = lr (g = r); g,r = 1,..., k}.

Ak,i = {(s/ s/ > si-1 + 1,...,S2 > si + 1, s/ ,...,si = 1,..., k - 1},

(s/,..., s1) E Ak;/, l = 1,..., [k/2] , i1,..., ik = 0,1,..., m, [x] is an integer part of a real number x, l.i.m. is a limit in the mean-square sense, 1A is the indicator of the set A, AwTj = wTj++1 — wTj (i = 0,1,...,m), {Tj}N=o is a partition of [t,T], which satisfies the condition (1.9). Then, for the iterated Stratonovich stochastic integral of multiplicity k

* T * ¿2

J*[fa(k) ]T,t = / fa (tk) ..J fa (ti)dwt(;i)... dwi:k) (2.1004)

the following expansion

Pi Pk k

J*^'l" = Jiï^ £IIZ^ (2.ioo5)

ji=0 jfc =0 1=1

that converges in the mean-square sense is valid, where

T

C7(;) = i fa (t)dwT;)

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[:) = f^ for i = 1,..., m and w[0) = t.

The proof of Theorem 2.58 immediately follows from Theorems 1.1 and 2.12 (see (1.10) and (2.372)).

It should be noted that a significant part of Chapter 2 is devoted to the proof of Theorem 2.58 (see (12.1005)) for various special cases (Theorems 2.1-2.9, 2.30, 2.32-2.35, 2.40). More precisely, in Theorems 2.1, 2.2, 2.4-2.9, 2.32-2.35, 2.40 we assume that {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). The above systems of functions are most suitable for the expansion of iterated stochastic integrals from the Taylor-Ito and Taylor-Stratonovich expansions (see Chapter 5).

Note that Theorems 2.1 and 2.2 prove the expansion (2.1005) for the case k = 2 (p1,p2 ^ to). At that fa2(T) is a continuously differentiable nonrandom function on [t, T] and fa1(T) is twice continuously differentiable nonrandom function on [t,T] (Theorem 2.1). In Theorem 2.2, the functions fa1(T) and fa2(T) are assumed to be continuously differentiable only one time on [t, T]. Theorems 2.42.8, 2.32, 2.40 prove the case k = 3 of (2.1005). In Theorems 2.4 and 2.6, the

case Ai(r),A2(t), A3(t) = 1 (PbP2,P3 ^ to) is considered. Theorem 2.8 proves the expansion (2.1005) for the case when A2(t) is a continuously differentiable nonrandom function on [t,T] and A1(r), A3(t) are twice continuously differentiable nonrandom functions on [t,T] (p1 = p2 = p3 = p ^ to). Theorem 2.32 is an analogue of Theorem 2.8 for continuously differentiable functions A1(r), A2(t), A3(r) (p1 = P2 = P3 = P ^ to). Theorem 2.40 proves the expansion (2.1005) for the case when A1(r),A2(t),A3(t) are continuously differentiable nonrandom functions on [t,T] (p^p2,p3 ^ to). In Theorems 2.5 and 2.7, we consider narrower particular cases of the functions ^(t), A2(r), A3(r)• The cases k = 4 and k = 5 of (2.1005) are considered in Theorems 2.9, 2.33, 2.34. Moreover, the functions ^(t), • • •, As(t) are continuously differentiable on [t,T] in these theorems (p1 = ... = p5 = p ^ to). Theorem 2.35 proves the case k = 6 of (2.1005). At that ^(t), • • •, A6(t) = 1 and p1 = ... = p6 = p ^ to in Theorem 2.35.

Remark 2.5. The equalities (1.10) and (1.54) imply that the condition (2.1003) can be replaced by the following condition

[k/2]

1

Pi

Pk

[k/2]

E£ E

r=1 (sr ,...,si)GAfc,r

?rsi = - LLm

Pi,...,Pk ^to

E • • • E c.-* E (-!)rx

ji=0 jk=0

r=1

X

E

ni

{i

g2s-1 "32.

W =0}1{j

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g2s-1

jg2s }

({{3i ,32 },•••,{32r-i,32r }},{9i,---,9fc-2r }) S=1 {Si,32 ,...,32r-i,S2r,9i,...,9fc-2r }={i,2v,k}

k-2r

nc

1=1

(iq )

(2.1006)

w. p. 1, where notations are the same as in Theorems 1.2 and 2.12.

Remark 2.6. Applying Theorems 1.14, 1.16, we can reformulate the equality (2.1006) as follows

[k/2]

1

2 r ^

r=1 (sr ,...,si)eAfc,r

Pi Pk

5n 'si = Ji-TO £••• £c-i

ji=0 jk=0

X

c

(ii)

j) - n

ji

1=1

V

1{ml =0} + 1{ml>0} <

V

n ^ (j)), if ii = 0

s=1

i=1 j

s=1 v

ns,i

if ii = 0

/

k

w. p. 1, where notations are the same as in Theorems 1.14, 1.16 and 2.12 (Hn(x) is the Hermite polynomial (1.263)).

Remark 2.7. Recently, in [127], an approach to the proof of expansion similar to (2.492) was proposed. In particular, this approach uses the representation of the multiple Stratonovich stochastic integral (2.941) as the sum of some constant value and multiple Wiener stochastic integrals of multiplicities not exceeding k. Note that a similar representation in a different form is defined by the formula (2.646).

It should be noted that an expansion similar to (2.492) was considered in 127] for an arbitrary k. The system of basis functions {fa(x)}°=0 in the space

L2([t,T]) can also be arbitrary. However, in [127], the condition on convergence of trace series is used as a sufficient condition for the validity of expansion similar to (2.492) (see [127] for details). Note that the verification of the above condition for the kernel (1.6) is a separate problem.

In Theorems 2.24-2.29, 2.37-2.39 the rate of mean-square convergence of expansions of iterated Stratonovich stochastic integrals is found. Determining the rate of mean-square convergence in the approach [127] is an open problem.

2.26 Invariance of Expansions of Iterated Itô and Stratonovich Stochastic Integrals from Theorems 1.1 and 2.58

In this section, we consider the invariance of expansions of iterated Ito and Stratonovich stochastic integrals from Theorems 1.1 and 2.58.

Consider the multiple Wiener stochastic integral J'fa^ ... fakl^V^' defined by (11.23) ($(t1,...,tk ) = faj (ti) •••fajk (tk )), where {faj (x) j=0 is a complete orthonormal system of functions in the space L2([t,T]), each function faj(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7).

Taking into account the equalities (1.35) and (1.43), we obtain

fajkfr;k) w. p. 1, (2.1007)

where J^Kly^"^' and J'[faj1... fajkly^"^' are multiple Wiener stochastic integrals defined by (1.23), the function K(t1,... ,tk) has the form (1.6).

Pi Pk

j[K]Trk) = l.i-m^ Ê■■.£cjWiJ'[faji...

ji=0 jk =0

On the other hand, the expansion (2.1005) can be written as follows (see Lemma 1.3 and (2.384))

pi pk

J[K*].iir,k> = l.i.m. £ ... £ C,,..ji J [j... jjTil^-'*» w. p. 1, (2.1008)

PivvPfc ^^ Z-' Z-'

ji =0 jk =0

where J [K *]T!1t"ik) and J [ j ... j j k ) are multiple Stratonovich stochastic integrals defined by (2.994), the function K*(ii,... ,tk) has the form (2.362).

It is not difficult to see that w. p. 1

J [Kfr" > = J' [K >.

Therefore, the expansions (2.1007) and (2.1008) have the same form. At that the expansion (2.1007) is formulated using multiple Wiener stochastic integrals and the expansion (2.1008) is formulated using multiple Stratonovich stochastic integrals.

The expansions (2.1007) and (2.1008) can be written in a slightly different way. Using (1.39), (1.42), and (1.85), we obtain

pi pfc

J [^k> ■ -!k) = l . i.m. £ . ■ ■„ x

ji=0 jk=0

T t2

x £ J j (tk) ...y j (ii)dw};i) ...dwtik) w. p. 1, (2.1009) (ji,--,jk) t t

where J[fa^]^"^) is the iterated Ito stochastic integral (11.51),

L

(jivjk )

means the sum with respect to all possible permutations (j,...,jk). At the same time if jr swapped with j in the permutation j..., jk), then ir swapped with in the permutation (i1,... ,ik); another notations are the same as in Theorem 1.1.

The iterated Stratonovich stochastic integrals

* T * t2

i j(tk).../ j(ti)dw<:i)...dw<:k)

satisfy the following equality

* T * ¿2

4;,)• ••j' = E J j(tk)•••/ j(t1)dwi;i).••dw<;k> w.p. 1,

(jiv,jk) t t

(2.1010)

where

T

j = J ^ (t)dw(;) (i = 0,1, • • •, m, j = 0,1,...) t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), the expression

E

(jiv-jk)

has the same meaning as in (2.1009).

For the case i1 = ... = ik = 0 we obtain from (2.1010) the following well known formula from the classical integral calculus (see (1.

T t2

y j (¿1) • • • j (tk)dt1... dtk = ^ y j (tk) • • • y j (t1)dt1 • • • dtk =

[t,T]k (ji,.",jk) t t

T t2

= E /••^^ji (¿1) ...^jk (tk)dt1 ...dtk, (2.1011)

(t i ,...,tk ) t t

where

E

(ji,.",jk)

means the sum with respect to all possible permutations j..., jk) and permutations (t1, • • • ,tk) when summing

E

(ti,...,tk)

(see (2.1011)) are performed only in the values dt1... dtk (at the same time the indices near upper limits of integration in the iterated integrals are changed correspondently).

Let us check the formula (2.1010) for the cases k = 2 and k = 3. Using (1.46), (2.381), and (2.1009) (k = 2), we have

c T * t2

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E / j&)/ j (ii)dwj;,)dw<22)

(jlj2) t t

*T * t2 *T * t2

J jj(ti)dwt2l)dwt(22) + J j(h)J j(ti)dw( 22)dw( 2l) =

t t t t

T t2 T t2

= / jfe)/ j (tl)dw<;')dw<22) + / j fe)/ j(ii)dw<:2)dw<2'» +

t t t t

T

+ 1{«1=Î2=0W j (t1)0j2 (t1)dt

i=

_ /(ilV(Î2) 1 1 I "1 1 _

= Zjl zj2 - i{»1=»2=0}1{ji=j2| + i{2l=22=0}1{ji=j2| =

= c(2l)C(22) w. p. 1. Sjl Sj2 F

Applying (1.47), (2.382), (2.1009) (k = 3), and the integration order replacement technique for Ito stochastic integrals (see Chapter 3) or Ito's formula, we obtain w. p. 1

* T * ts * t2

E f j(*)/ j(¿2)/ (il)dw<;l)dw|22)dw<Ss) =

/ ... \ U \J \J

(jl,j2,jS) t t t

T ts t2

E /j(*)/jte)/j(ii)dw<;l)dwi22)dw<ss)+

(jl,j2,j's) t t t

/ T ts

+ 1{*l=*2=0} j j («3) j j («i)Ajl fa^M^^ tt T tl

+ f j («i) j («i)/ j(Î3)dwt(ss)d«^ + tt

/ T t2

+ 1{:i=:3=0} I J j № j j (t1)0,3 (t1 +

tt T ti \

+/ 0,3 (t1)0ji , (t2)dw(:2)dt j + tt / T ti

+ 1{:2=*3=0} J 0ji (t1^ 0,2 (t3)0j3 (t3)dt3dwt:i) + tt T t3

+ f 0,3(t3)0j2(¿3)/ 0ji (t1)dwt(:i)dt3 tt T t3 t2

L f 0,3 (¿3) / 0,2 fe) / 0ji (i1)dw<:i)dwi;2)dw<33) +

(ji,j2,j3) t t t

/ T t3

+ 1{:i=2=0} J 0,3 ^ 0,2 (t1)0,i (t1)dt1dwt(33) + tt T T \

+ / 0,3 (t3^ 0,2 (t1)0,i (t1)dt1dwt(33M +

t t3

/ T t2

+ 1{:i=i3=0 J J 0,2 (t2^ 0,i (t1)0,3 (t1)dt1dwt(:2) + tt T T \

+ f 0,2 (¿2)/0,i (11)0,3 (t1)dt1dwt(:2M + t t2

/ T ti

+ 1{:2=3=0} I J 0,i (t1^ 0,2 (t3)0,3 +

tt T T

+ f 0,i(¿1)/ 0,2 (¿3)0,3(t3)dt3dwt(;i} t ti

T t3 t2

= E / 0,3 (*)/ 0,2 (¿2^ 0,i (¿1 )dw(:i)dwt(:2)dwt(:3)+

(,i ',2 7,3) t t t

T T

+ 1{:i=:2=0^ 0,3 ^ 0,2 (t1)0jl (¿1^1 ^t^

tt T T

+ 1{:i=:3=0^ 0,2 0,i ^O0, ^t^

tt T T

+ 1{:2=:3=0^ 0,i (¿1 ^ 0,2 i^^^0, =

tt T t3 t2

= E / 0,3(«/ 0,20,i (¿1 )dwi;i)dw<«dw«:3>+

O'l^^ t t t

+ 1{i1 = i2=0}1{j1=j2}Cj(33) + 1{i1=i3=0}1{j1=jз}Cj(2г2) + C^ =

= Z(:1)Z(:2) Z(:3)_ ZJ1 Z ,2 ZJ3

— 1 1 Z^ _ i i z(:2) _1 1 Z(:i) +

Mn^^}1^^}^ 'M»^^^} 'Mil^?^} 1 {«2 = ^3=0} 1{,2 ^MS, +

+ 1{i1 = i2=0}1{j1=j2}Cj(33) + 1{i1=i3=0}1{j1=jз}Cj(2г2) + C^ =

= z (:i)z (:2 )z (:3)

Using (2.1010), we can write the expansion (2.1005) as follows

pi pk

j^Tr' = l-i-m E.-Ec»..* x

pi,. . . ,pk^^ ^^ ,1=0 ,k=0

T t2

x E / 0,k (¿k) ..J 0,1 (¿1)dwt(:i) ...dwt(:k) w.p. 1, (2.1012)

Cilr-Jk) t t

where J*[^(k)]T:it ■ ■ :k) is the iterated Stratonovich stochastic integral (2.1004); another notations are the same as in (2.1009).

Obviously, the expansions (2.1009) and (2.1012) have the same form. At that the expansion (2.1009) is formulated using iterated ItO stochastic integrals and the expansion (2.1012) is formulated using iterated Stratonovich stochastic integrals.

Taking into account the expansions (2.1009) and (2.1012), we can reformulate the condition (2.1003) as follows

pi pk

./•[V'(t)]T,r!'1 - J W^lr"' = l.i-m. Ë ... Ë Ckj X

pi, ..-,pfc^—' ^—' ji=0 j'fc=0

/ * T * Î2

x E/ j(tk)...J j(ii)dwi;11 ...dwikk1 -

(ji,• • • ,jfc ' \ t t

T t2 \

-/ j (tk ) j (ti )dwt;i) ...dwi:k) tt

w. p. 1.

2.27 Expansion of Multiple Stratonovich Stochastic Integrals of Arbitrary Multiplicity k. The case of a multidimensional Wiener process and a smooth function $(¿1, ..., tk)

As we have seen in this chapter, one of the main difficulties in obtaining expansions of iterated Stratonovich stochastic integrals is related to the properties of the kernel (1.6). The kernel (1.6) is discontinuous, which causes difficulties in applying the theory of multiple Fourier series converging pointwise. Moreover, the Volterra integral operator with the kernel

1, ¿2 <ti K(ti,t2)H , ¿1, ¿2 G [0, 1]

0, otherwise

is not a trace class operator [131 .

Let us assume that the function ..., ) : [t, T]k ^ R satisfies sufficient

conditions for its expansion into a multiple Fourier series

p k

lim E Cjk ...^n (ti ) (2-1013)

ji =0 1=1

converging pointwise in (t,T)h to the function ... , th) and converging on the boundary of the hypercube [t,T]h. Here {fa(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and

/h

$(ti,...,tk)n(ti)dti ...dtk (2.1014)

/—i

[t,T ]k

is the Fourier coefficient.

The mentioned conditions for k = 1 and k > 2 (trigonometric case) are given in Sect. 2.1.1, 2.1.2.

Consider the multiple Stratonovich stochastic integral (1.16) (also see (2.994))

N-1

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Um. £ $(t, = Jf^r',

j1,---Jfc =0 1=1

N

where we suppose that the function ... ,th) is the same as above, {rj}j—0 is a partition of [t,T], which satisfies the condition (1.9) and i1,...,ih = 0,1,..., m.

Denote

p k

Rp(ti,...,th ) = $(ti,...,th) - £ Cj ...j^ (t/), (2.1015)

j1,",jfc =0 / = i

where Cjk...j has the form (12.1014).

Applying (we mean here the passage to the limit lim ) the Lebesgue's Dominated Convergence Theorem to the integrals on the right-hand side of (2.433) for Rp(ti,... ,th) (see (2.10151)) instead of Rpi...,pk(ti,... ,th) (see (24351)), we obtain

p k ^ 2n'

pKm ^ (J' - £ Cik-^n z

ji ,---,j'fc=0 i=i

k

= Hm m j J[R]^"^)2" j = 0, (2.1016)

where n G N and

T

Cf = / h (T)dwTi) t

are independent standard Gaussian random variables for various i or j (in the case when i = 0).

Note that the equality (2.1016) will also be satisfied if the multiple Fourier series (2.1013) converges to the function $(t1,... ,tk) almost everywhere on the hypercubes [t,T]k-r (r = 0,1,..., [k/2]) that are domains of integration for the integrals on the right-hand side of the inequality (2.433) and if the indicated multiple Fourier series is bounded on the hypercube [t,T]k.

Chapter 3

Integration Order Replacement Technique for Iterated Ito Stochastic Integrals and Iterated Stochastic Integrals with Respect to Martingales

This chapter is devoted to the integration order replacement technique for iterated Ito stochastic integrals and iterated stochastic integrals with respect to martingales. We consider the class of iterated Ito stochastic integrals, for which with probability 1 the formulas on integration order replacement corresponding to the rules of classical integral calculus are correct. The theorems on integration order replacement for the class of iterated Ito stochastic integrals are proved. Many examples of these theorems usage have been considered. The mentioned results are generalized for the class of iterated stochastic integrals with respect to martingales.

3.1 Introduction

In this chapter we performed rather laborious work connected with the theorems on integration order replacement for iterated Ito stochastic integrals. However, there may appear a question about a practical usefulness of this theory, since the significant part of its conclusions directly follows from the Ito formula.

It is not difficult to see that to obtain various relations for iterated Ito stochastic integrals (see, for example, Sect. 3.6) using the Ito formula, first

of all these relations should be guessed. Then it is necessary to introduce corresponding Ito processes and afterwards to use the Ito formula. It is clear that this process requires intellectual expenses and it is not always trivial.

On the other hand, the technique on integration order replacement introduced in this chapter is formally comply with the similar technique for Riemann integrals, although it is related to Ito integrals, and it provides a possibility to perform transformations naturally (as with Riemann integrals) with iterated Ito stochastic integrals and to obtain various relations for them.

So, in order to implementation of transformations of the specific class of Ito processes, which is represented by iterated Ito stochastic integrals, it is more naturally and easier to use the theorems on integration order replacement, than the Ito formula.

Many examples of these theorems usage are presented in Sect. 3.6.

Note that in Chapters 1,2, and 4 the integration order replacement technique for iterated Ito stochastic integrals has been successfully applied for the proof and development of the method of approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series (see Chapters 1 and 2) as well as for the construction of the so-called unified Taylor-Ito and Taylor-Stratonovich expansions (see Chapter 4).

Let (Q, F, p) be a complete probability space and let f (t,w) : [0,T] x Q ^ R1 be the standard Wiener process defined on the probability space (Q, F, p).

Further, we will use the following notation: f (t,w) =f ft.

Let us consider the family of a-algebras {Ft, t G [0, T]} defined on the probability space (Q, F, p) and connected with the Wiener process ft in such a way that

1. Fs C Ft C F for s < t.

2. The Wiener process ft is Ft-measurable for all t G [0,T].

3. The process ft+A — ft for all t > 0, A > 0 is independent with the events of a-algebra Ft.

Let us recall that the class M2([0,T]) (see Sect. 1.1.2) consists of functions £ : [0,T] x Q ^ R1, which satisfy the conditions:

1. The function £(t,w) is measurable with respect to the pair of variables (t,w).

2. The function £(t,w) is Ft-measurable for all t G [0,T] and £(t, w) is independent with increments ft+A — ft for t > t, A > 0.

3. The following relation is fulfilled

T

J m {(£(t,u))2} dt < oo.

0

4. m {(£(t,u))2} < o for all t G [0,T].

Let us recall (see Sect. 1.1.2) that the stochastic integrals

T T

J £тdfT and J £тdr, (3.1)

00

where £t G M2([0, T]) and the first integral in (3.1) is the Ito stochastic integral, can be defined in the mean-square sense by the relations (1.2) and (1.4).

We will introduce the class S2([0, T]) of functions £ : [0, T] x Q ^ R1, which satisfy the conditions:

1. £(т,и) G M2([0,T]).

2. £(т, u) is the mean-square continuous random process at the interval [0,T ].

As we noted above, the Ito stochastic integral exists in the mean-square sense (see (1.2)), if the random process £(т, u) G M2([0,T]), i.e., perhaps this process does not satisfy the property of the mean-square continuity on the interval [0,T]. In this chapter we will formulate and prove the theorems on integration order replacement for the special class of iterated Ito stochastic integrals. At the same time, the condition of the mean-square continuity of integrand in the innermost stochastic integral will be significant.

Let us introduce the following class of iterated stochastic integrals

T ifc-i tfc

J [ф, fa(k)b = J fa (ti) ...J fa (tk) J фт dw^Wf... dw^, t t t

where ф(т, u) == фт, фт G S2([t,T]), every fa/(т) (l = 1,...,k) is a continous nonrandom function at the interval [t,T], here and further wTl) = /т or wTl) = т for т G [t,T] (l = 1,...,k + 1), (fai,...,fak) d=f fa(k),fa(1)d=f fai.

We will call the stochastic integral J^,fa(k)]Tjt as the iterated Ito stochastic integral.

It is well known that for the iterated Riemann integral in the case of specific conditions the formula on integration order replacement is correct. In particular, if the nonrandom functions f (x) and g(x) are continuous at the interval [a, b], then

6 x 6 6

J f (x) y g(y)dydx = J g(y) y f (x)dxdy. (3.2)

a a ay

If we suppose that for the Ito stochastic integral

T s

J[^kt = J Ws^ Atdw^dw^ tt

the formula on integration order replacement, which is similar to (3.2), is valid, then we will have

T s T T

y A1(s)y Atdw(2)dwS1) = J At y A1(s)dwS1}dw[2). (3.3)

t t t t

If, in addition wS1), wS2) = fs (s G [t, T]) in (13.3), then the stochastic process

T

nT = At J ^1(s)dwS1)

t

does not belong to the class M2([t,T]), and, consequently, for the Ito stochastic integral

T

t

nT dwT2)

on the right-hand side of (3.3) the conditions of its existence are not fulfilled. At the same time

T T T T

JdfsJ ds = y (s — t)dfs ^(fs — ft)ds w. p. 1, (3.4)

t t t t

and we can obtain this equality, for example, using the Ito formula, but (3.4) can be considered as a result of integration order replacement (see below).

Actually, we can demonstrate that

T

T T

J (fs - /t)ds = y J d/T-ds = y J dsd/T w. p. 1.

t t t t T

Then

T

T t

T T

J (s — t)dfs + J (fs — ft)ds = J J dsdfT + J J dsdfT = J dfs J ds w. p. 1.

t t t t t T t t

The aim of this chapter is to establish the strict mathematical sense of the formula (13.3) for the case w(1), w(2) = fs (s G [t,T]) as well as its analogue corresponding to the iterated Ito stochastic integral J[h, A(k)]T,t, k > 2. At that, we will use the definition of the Ito stochastic integral which is more general than (1.2).

Let us consider the partition t(N), j = 0,1,..., N of the interval [t, T] such

that

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_(N)

t = Tn" ' < T1"'' < . . . < T

(N) = T N = T

max

0<j<N-1

T(N) T(N) Tj+1 Tj

0 if N oo.

(3.5)

In [105] Stratonovich R.L. introduced the definition of the so-called combined stochastic integral for the specific class of integrated processes. Taking this definition as a foundation, let us consider the following construction of stochastic integral

N-1

H.m. Y M (fTj+1 j =0

T

= 0t d/r 0T ,

(3.6)

where At, ^ G S2([t,T]), {t^}n 0 is the partition of the interval [t,T], which satisfies the condition (3.5) (here and sometimes further for simplicity we write Tj instead of t,-N)).

Further, we will prove existence of the integral (3.6) for 0T G S2([t,T]) and from a little bit narrower class of processes than S2([t,T]). In addition, the integral defined by (3.6) will be used for the formulation and proof of the theorem on integration order replacement for the iterated Ito stochastic integrals

J[0,A(k)]T,t, k > 1.

s

Note that under the appropriate conditions the following properties of stochastic integrals defined by the formula (3.6) can be proved

T T

J fadf g(T) = J fag(T)d/T w. p. 1,

t t

where g(r) is a continuous nonrandom function at the interval [t,T],

T T T

f («0T + №) dfT* = af ^dfT* + p / fadfT* w. p. 1,

t t t T T T

/<MfT+ №) = a/f + w. p. 1,

t t t where a, p G R1.

At that, we suppose that the stochastic processes fa, , and faT are such that the integrals included in the mentioned properties exist.

3.2 Formulation of the Theorem on Integration Order Replacement for Iterated Ito Stochastic Integrals of Multiplicity k (k G N)

Let us define the stochastic integrals /[fa(k)]TjS, k > 1 of the form

T T T

%(k)]T,s = J fa (tk fa-1 (tk-i)dwt-11) ..J fa (ti)dwt(1)

S tfc t2

in accordance with the definition (3.6) by the following recurrence relation

N-1

/[fa(k)]T,t = l.i.m. £ fak(t)Aw(f)/[fa(k-1)]T,Ti+i, (3.7)

1=0

where k > 1, I[fa(0)]T,S = 1, [s,T] C [t,T], here and further Awi? = 1 -wi*), i = 1,.. .,k + 1, l = 0,1,. ..,N - 1.

Then, we will define the iterated stochastic integral J[fafa(k)]Tjt, k > 1

T

J[0,fa(k)]T,t = i fadw(k+1)I[fa(k)]t,s

similarly in accordance with the definition (3.6)

N-1

= l.i.m. £ A<+1)/[^(k)]T:

._. . - - Jl+1-

Let us formulate the theorem on integration order replacement for iterated Ito stochastic integrals.

Theorem 3.1 [111] (1997), (also see [1]-[16], [72], [112]). Suppose that G

S2([t,T]) and every A(t) (l = 1,..., k) is a continuous nonrandom function at the interval [t,T]. Then, the stochastic integral J[A, A(k)]T,t, k > 1 exists and

J[0,#fe>kt = J[0,#fe>kt w. p. 1.

3.3 Proof of Theorem 3.1 for the Case of Iterated Ito Stochastic Integrals of Multiplicity 2

At first, let us prove Theorem 3.1 for the case k = 1. We have

N —1 T

J[A,A1]T,t d=f l.i.m. V ^(TOAwW / AtdwT2) =

t

N—1 1-1 Tj+1

l.i.m. V ^Am^V / dwT2>, (3.8)

N^œ 1=0 j=o 7

Tj

N-1 T

J[0,^1]r>t = l.i.m. V Aw<2> / A1(s)dw(1> N^œ ^ 3 J

j=0 T

7 Tj+1

N -1 N -1 Tl+1

l.i.m. ^ ^ Aw(2> ^ / A1(s)dw(1> = N^œ 3 3 J

j=0 1=7+1 Tl

N-1 Tl+1 1-1

l.i.m. ^ / 0T3Aw(2>. (3.9)

N=œ f-' J ^ 33

1=0 Ti j=0

It is clear that if the difference ¿N of prelimit expressions on the right-hand sides of (3.8) and (3.9) tends to zero when N fa to in the mean-square sense, then the stochastic integral J [fa exists and

J[0,fai]r,t = J[0,fai]r,t w. p. 1.

The difference ¿N can be represented in the form ¿N = £N + £N , where

£N =

N-1 1-1 Tj+X

£>(t)Aw(1 ^ / (fa - faj) dwT2); 1=0 j=0 i

£ N =

N-1 Tl+1 1-1

E / (^1(ti) - ^1(s)) dwi1^ faj AwTJ)

i=o T j=o

We will demonstrate that w. p. 1

l.i.m. £N = 0.

N faTO

In order to do it we will analyze four cases:

1. (2) wT = /, AwT1) = a/TI

2. (2) wT = T, AwT1) = = a/ti .

3. (2) wT = /t , AwT1 = at .

4. (2) wT = t, AwT1) = = at .

Let us recall the well known standard moment properties of stochastic integrals

m

T

M{ Kt |2} dT,

m

T

£r dT

T

< (T - t) / M{ |£T|2} dT,

(3.10)

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where e M2([t,T]).

For Case 1 using standard moment properties for the Ito stochastic integral as well as mean-square continuity (which means uniform mean-square continu-

2

ity) of the process far on the interval [t, T], we obtain

N-1 k-1 Tj+1

M{ ^|2} = £ ^2(Tk)Ar^ / m{ |far - fa, I2} 7. n ;_n J

dr <

k=0

j=0

N- 1 k-1

2 (T-f) fc--—

2

< C2e Y Ark E ArJ < C k=0 j =0

i.e. m j|£N|2} fa 0 when N fa to. Here Arj < J(e), j = 0,1,... ,N - 1

(£(e) > 0 exists for any £ > 0 and it does not depend on r), |fa1(r)| < C.

Let us consider Case 2. Using the Minkowski inequality, uniform mean-square continuity of the process far as well as the estimate (3.10) for the stochastic integral, we have

2

m

N-1 / k-1 ^ ^

{|£n|2} = E ^2(Tk)ATkM I J2 / (faT - far,)dr

k=0 V=0 rj >/

s <

N-1

< E fa2(rk)Ark k=0

k-1

E

j=0

/ f / rJ+1 \ 2 \

m ( far - far, )dr >

V J Vi J / /

/

\ I/2 \2

/ /

<

N1

k1

2 (T-f);

< C2£ Y Ark E < ^

k=0 \j=0 J

i.e. m {|£n|2} fa 0 when N fa to. Here At, < ¿(¿), j = 0,1,...,N - 1

(^(¿) > 0 exists for any £ > 0 and it does not depend on t), |^1(t)| < C.

For Case 3 using the Minkowski inequality, standard moment properties for the Ito stochastic integral as well as uniform mean-square continuity of the process faT, we find

2

m

{|£n |2}

/ N-1 / r / k-1 y ^ 2 \ 1/2

< E!fa1(Tk)| Ark m E / (far - farj )dfr >

k=0 V \ / / /

2

I k-1

N-1 k-1 'jt1

(Tk)|ATk Y / m{I^T - |2} dT

ivk=0 v=0 j

\ 1/2 \

/

<

/

N1

k1

1/2

<C2e £ At* £ At,

'k i / k=0 \j =0

<6 £ -

i.e. m{|£N|2} ^ 0 when N ^ to. Here At, < J(e), j = 0,1,...,N - 1 (£(e) > 0 exists for any £ > 0 and it does not depend on t), |^(t)| < C.

Finally, for Case 4 using the Minkowski inequality, uniform mean-square continuity of the process 0T as well as the estimate (3.10) for the stochastic integral, we obtain

2

m

{k~N I2}

N-1 k-1 / r ('j+1 \ r 2 \ 1/2

< EEl^1(Tk )|ATk m < / (0T - 0Tj )dT >

k=0 j=0 \ V U ) / / /

<

N 1 k 1

< C2e ^ ATk^ At, <C

2 (T-f)

4

. k=0 j=0

i.e. m {|£N|2} ^ 0 when N ^ to. Here At} < j = 0,1,...,N - 1 (£(e) > 0 exists for any £ > 0 and it does not depend on t), |^i(t)| < C. Thus, we have proved that w. p. 1

l.i.m. £N = 0.

N=to

Analogously, taking into account the uniform continuity of the function ^i(t) on the interval [t, T], we can demonstrate that w. p. 1

l.i.m. £n = 0.

N

Consequently,

l.i.m. eN = 0 w. p. 1.

N

Theorem 3.1 is proved for the case k = 1.

2

Remark 3.1. Proving Theorem 3.1, we used the fact that if the stochastic process fat is mean-square continuous at the interval [t, T], then it is uniformly mean-square continuous at this interval, i.e. V £ > 0 3 > 0 such that for all ti , t2 ^ [t, T] satisfying the condition 111 — t21 < the inequality

m

{l0ti — fa*2 I2} < £

is fulfilled (here 5(e) does not depend on ti and t2).

Proof. Suppose that the stochastic process fat is mean-square continuous at the interval [t, T], but not uniformly mean-square continuous at this interval. Then for some e > 0 and V 5(e) > 0 3 ti,t2 £ [t,T] such that |ti — t2| < 5(e), but

m{ Ifati — fat2|2} > e.

Consequently, for 5 = 5n = 1/n (n £ N) 3 tin), t2n) £ [t, T] such that

t(n) _ t(n) ti — t2

<

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n

but

m

fa,(n) — fa.

(n)

>

The sequence tin) (n £ N) is bounded, consequently, according to the Bolza-no-Weierstrass Theorem, we can choose from it the subsequence tikn) (n £ N) that converges to a certain number t (it is simple to demonstrate that t £ [t,T]). Similarly to it and in virtue of the inequality

t(n) _ t(n) ti — t2

<

n

we have t^ fa t when n fa to.

According to the mean-square continuity of the process fat at the moment t and the elementary inequality (a + b)2 < 2(a2 + b2), we obtain

0m

fa.(fcn) — fa,(kn)

<

<2 m

when n fa oo. Then

fat(kn) — fat

+ m

fat(kn ) — fat ¿2

lim m

n—T^OO

fa.(fcn) — fa,(kn)

= 0.

0

1

2

1

2

It is impossible by virtue of the fact that

m

0,(fc„) — 0t

kn) — 0. (fcn) l2

2

> £ > 0.

The obtained contradiction proves the required statement.

3.4 Proof of Theorem 3.1 for the Case of Iterated Itô Stochastic Integrals of Multiplicity k (k G N)

Let us prove Theorem 3.1 for the case k > 1. In order to do it we will introduce the following notations

0 tr

I T+% = J (ti ).. ■ /',+r (WiMwtr'+T'... dwtf,

s s

0 tr tr + 1

j[0,^ks =f j(ti)...j ^(w)y dw(q+r+i)dwt(q++ir)...dwtf,

s s s

n—1 jq-1 jq+r-1 —1 r+q

g^'u =£ E - £ W№k•

jq=mjq+i=m jq+r=m l=q

(^„...,^,+r ) = ^<r+1), <> = ^,

(V>i,...,Vv+i) =f^1r+1), ^1r+1)= ^(r+1).

Note that according to notations introduced above, we have

s

IMs,* = i Mi(T)dwTl).

To prove Theorem 3.1 for k > 1 it is enough to show that

J[0, A(k)iT,t = l.i.m. S[0, M(k)]N = J[0, A(k)iT,t w. p. 1, (3.11)

N

where

jk —1

0T1 Aw(k+1),

1=0

where AwT-+i) = w(-f++i) — +i).

At first, let us prove the right equality in (3.11). We have

N — i

J[fa,fa(-)]T,t = l.i.m. ^ faTl Aw^/^]^. (3.12)

._. . - - Ti+1"

On the basis of the inductive hypothesis we obtain that

I= %(-)]T,Ti+i w. p. 1, (3.13)

where /[fa(-)]T,s is defined in accordance with (3.7) and

T tfc-2 tfc-1

I [fa(-)]T,s = J fai(ti) ...J fa-—i(t-—i) | fa- (tk )dwt(k)dwt(-;ii).. .dw(i).

s s s

Let us note that when k > 4 (for k = 2, 3 the arguments are similar) due to additivity of the Ito stochastic integral the following equalities are correct

N — i j+1 ^

I[fa(-)]T,Ti+1 = £ / fai(ti) / fa2(t2)1 [fa3-—2)]t2,Ti+1 dwt(22)dwt(i) =

Ti+1

Ni

TJ1 + 1 / 1 TJ2 + 1 tA

fai(ti) E +

■■■ \j2=1+i 7 7 /

\ '72 '71 /

j1=1 + i ,

'71

fa2(t2)I [fa3- 2)]t2,Ti+1

= ... = G[fa(-)] N,i+i + H[fa( )]n,i+i w. p. 1, (3.14)

where

N—i Tj'1+1 s H[fa(-)]N,1+i = £ f fai(S^ fa2(T)I [fa3-—2)]t,ti+1 dw(2)dwii) +

k—2 jr-1 —i Tj'r-+1 s

+ £ G[fa(r—%,1+i £ / far (s) / far+i (t )I [fai+—r—1)]t,ti+1 dw(r+i)dwir) + r=2 jr =1+i / /

jfc-2 — i

+ G[fa<k—2>]AW £ I [faí2-l]тJl_l+l,тJl_l. (3.15)

jfc-1=1+i

1

Next, substitute (3.14) into (3.13) and (3.13) into (3.12). Then w. p. 1

N-i f .

J[0, ^(k)]T,t = l.i.m. V 0Tl AwTf+1) + H[^(k)ki+1 . (3.16)

Since

N-1 ji-1 jfc-i-1 N-1 N-1 N-1

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EE---E = E E ••• E aji-jk, (3.17)

ji=0 j2=0 jfc=0 jk=0 jfc-1=jfc+1 ji=j2+1

where aji...jk are scalars, then

N-1 N-1 k

G[<A(k)km = £ ... £ ni№k+1.T„• (3.18)

jk = 1+1 ji=j2 + 1 1 = 1

Let us substitute (3.18) into

N1

N,1+1

1=0

and use again the formula (3.17). Then

N-1

E Am^^km = S[0, . (3.19)

1=0

Suppose that the limit

l.i.m. S[0,#feV (3.20)

N

exists (its existence will be proved further).

Then from (3.19) and (3.16) it follows that for proof of the right equality in (3.11) we have to demonstrate that w. p. 1

N1

l.i.m. E AwTf+1)H[^(k)]N,i+1 = 0. (3.21)

1=0

Analyzing the second moment of the prelimit expression on the left-hand side of (13/211) and taking into account (1.3.15), the independence of , Aw(f+1), and H[^(k)]N,1+1 as well as the standard estimates for second moments of

stochastic integrals and the Minkowski inequality, we find that (3.21) is correct. Thus, by the assumption of existence of the limit (3.20) we obtain that the right equality in (3.11) is fulfilled.

Let us demonstrate that the left equality in (3.11) is also fulfilled.

We have

N— 1

J[ф,/(к)Ь = l.i.m. E ^i(ti}AwT1)J[0,4k-1)kt. (3.22)

1=0

Let us use for the integral J[0, ^ in (13.22) the same arguments,

which resulted to the relation (3.14) for the integral I. After that let us substitute the expression obtained for the integral J[0, 02k-1)]Tt into (3221). Further, using the Minkowski inequality and standard estimates for second moments of stochastic integrals it is easy to obtain that

J[0,kk)]Tt = l.i.m. R[0,/(k)]N w. p. 1, (3.23)

N

where

N —1 jW Tl+1

Я[0,/(к)к = E Wj)А<)С[/2*—1°koE / Фтdwlk+1). ji=0 1=0 J

We will demonstrate that

l.i.m. R[0,/(fe)k = l.i.m. S[0,/(fe)k w. p. 1. (3.24)

It is easy to see that

R[0,/(k)k = U[0,/(k)k + V[0,/(k)k + S[0,/(k)k w. p. 1, (3.25)

where

N—1 jk —1

U[Ф,кк)к = £ «j)Awji)G[/2*—11)]ji,^ I[Аф]тг+1,тг,

ji=o 1=0

N—1 jfc — 1

V[Ф,/(% = £ I[А^1]Тл+ьТлС[</?—1)к0 E Фт,Aw<f+1»,

ji=o 1=0

тЛ + 1

I[A/J^^ = J (/1(Tji) — /1(т))dw(1),

тл

Ti+1

I[AfaWi = J (faT — fa'i)dw(k+i). 'i

Using the Minkowski inequality, standard estimates for second moments of stochastic integrals, the condition that the process fa' belongs to the class S2([t, T]) as well as continuity (which means uniform continuity) of the function fai(r), we obtain that

l.i.m. V[fa,fa(-)]N = l.i.m. U[fa,fa(-)]N = 0 w. p. 1.

Nito Nito

Then, considering (3.25), we obtain (3.24). From (3.24) and (3.23) it follows that the left equality in (3.11) is fulfilled.

Note that the limit (3.20) exists because it is equal to the stochastic integral J [fa, fa(-)]T,t, which exists under the conditions of Theorem 3.1. So, the chain of equalities (3.11) is proved. Theorem 3.1 is proved.

3.5 Corollaries and Generalizations of Theorem 3.1

Assume that = {(ti,... ,tk) : t < ti < ... < tk < T} and the following conditions are fulfilled:

AI. £ S2([t,T]).

AII. $(ti,..., tk—i) is a continuous nonrandom function in the closed domain i (recall that we use the same symbol i to denote the open and closed domains corresponding to the domain i).

Let us define the following stochastic integrals

T T T

JC, = /itkdwii' > ...J dw^y $(ti,t2,. .. ,it—i)dwii1) =f

t is t2

N—i T T T

d=f Nim. ? ^Ti AwT;fc) J dwt:--1) ..J ¿w^y ^^ ,...,t-—i)dwt ;1)

■=0 Ti+1 ts t2

for k > 3 and

T T

•fa = / & dwt22^ $(ii)dw<11'=f

t t2

N-1 T

= l.i.m. £ £„Aw<;2> / $(i1)dw«;')

Nfaœ ~~~~ J

■=0 T

Ti+1

for k = 2. Here w';) = f';) for i = 1,..., m and w'0) = t, f';) (i = 1,..., m) are FT-measurable for all t £ [0,T] independent standard Wiener processes, 0 < t < T, ii,..., ik = 0,1,.. .,m.

Let us denote

T tfc-1

J[C, = / .../^(ti-----tk—i^dw^ ...dwt(;1), k > 2, (3.26)

tt

where the right-hand side of (3.26) is the iterated Ito stochastic integral. Let us introduce the following iterated stochastic integrals

T T T

./[$lTt—1) = | dwfc'1 ..J dwg2'/ $(t1,t2,...,ik_1)dwi

t ¿3 ¿2

T T T

N —1

= l.i.m A (;: ')

/ v ' i

> 7_n

Tl + 1 ¿3 ¿2

N —1 T 11 d=fi^.m. £Aw[;:-'^dwt(;:-22)...y^w^y$(t1,t2,...,tk—1)dwt(;i),

T ¿:-2

J/[$]T71) = /... i $(t1,...,tk—1)dwt(;-^i)...dwt(;i), k > 2.

Similarly to the proof of Theorem 3.1 it is easy to demonstrate that under the condition AII the stochastic integral •/[^]T-:—i) exists and

J[$]T-—i) = -/MTV1' w. p. 1. (3.27)

Moreover, using (3.27) the following generalization of Theorem 3.1 can be proved similarly to the proof of Theorem 3.1.

Theorem 3.2 [111] (1997) (also see [1]-[16], [72], [112]). Suppose that the

conditions AI, AII of this section are fulfilled. Then, the stochastic integral J[£, exists and for k > 2

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jk, = jk, w.p. i.

(k)

Let us consider the following stochastic integrals

T T T ¿2

i = i fi2) i j = , ,

T T ¿2

f2)|$1(t1,t2)dfi;i), j = J |$2(t1,t2}dfi;i)df(;2).

t t2

If we consider

T

$l(il,i2)f

(ii) i

t2

as the integrand of I and

t2

$2(Î1,Î2)<H)

as the integrand of J, then, due to independence of these integrands we may mistakenly think that m{1J} = 0. But it is not the fact. Actually, using the integration order replacement technique in the stochastic integral I, we have w. p. 1

T ti T t2

/ = / yVcti.t^'fi1' = / |$1(i2,ii)dfi(r)dft;i).

t t t t

So, using the standard properties of the Ito stochastic integral [95], we get

T ¿2

m{1J} = l{ii=i2} / I $l(i2,il)$2(il,i2)dMi2,

¿¿

where 1A is the indicator of the set A.

Let us consider the following statement.

Theorem 3.3 [ill] (1997) (also see [1]-[16], [72], [112]). Let the conditions of Theorem 3.1 are fulfilled and h(r) is a continuous nonrandom function at the interval [t,T]. Then

T T

T;T = h(r)dw( + )1 [0( )]t;T w. p. 1, (3.28) ¿¿

where stochastic integrals on the left-hand side of (3.28) as well as on the right-hand side of (3.28) exist.

Proof. According to Theorem 3.1, the iterated stochastic integral on the right-hand side of (3.28) exists. In addition

T T

/ fah(r)dw( + ^/[/fa )]t,t =

N-1

N->oo

l.i.m. £ fa, Ah(r1)Aw(.f+1)/[fa(k)J T,T; + i W. p. 1,

1=0

where Ah(r/) = h(r/+i) - h(r/).

Using the arguments which resulted to the right equality in (3.11), we obtain

1=0

N-1

H.m Y faTiAh(Ti L'fa 'JT,n+i

jk -i

l.i.m. fanAh(n)Aw(f+1) w. p. 1. (3.29)

1=0

Using the Minkowski inequality, standard estimates for second moments of stochastic integrals as well as continuity of the function h(r), we obtain that the second moment of the prelimit expression on the right-hand side of (3.29) tends to zero when N fa to. Theorem 3.3 is proved.

Let us consider one corollary of Theorem 3.1.

Theorem 3.4 [111] (1997) (also see [1]-[16], [72], [112]). Under the condi-

tions of Theorem 3.3 the following equality

T ti

J h(ti) I fadwTk+2)dwif+1)/ fa 'Jr>ii = t t

T T

= J fadwTk+2) y h(ti)dwt(f+1)/[fa(k)]T,t1 w. p. 1 (3.30)

t T

is fulfilled. Moreover, the stochastic integrals in (3.30) exist.

Proof. Using Theorem 3.1 two times, we obtain

T T

i 0tdw^ / h(t1)dwt(f+1)/ [0^,

T ifc-i tfc

= J 01 (ti) ■■■J 0k (tk ) J Pt dwTk+1)dwt(k)... dwti1) = t t t T T T

: J Pt dwTk+1) y 0k (tk )dwifck) ..J ^(tOdw^ w. p. 1,

t T t2

where

T

Pt = h(T) / 0sdwSk+2).

t

Theorem 3.4 is proved.

3.6 Examples of Integration Order Replacement Technique for the Concrete Iterated Ito Stochastic Integrals

As we mentioned above, the formulas from this section could be obtained using the Ito formula. However, the method based on Theorem 3.1 is more simple and familiar, since it deals with usual rules of the integration order replacement for Riemann integrals.

Using the integration order replacement technique for iterated Ito stochastic integrals (Theorem 3.1), we obtain the following equalities which are fulfilled w. p. 1

T t2 T

J J dfti dt2 = J (T - t1)dfti, t t t T t2 T

i cos(t2 - T) / dftidt2 = i sin(T - t1 )dfti,

1

T ¿2 T

J sin(t2 - T) J d/tldt2 = J (cos(T - ti) - 1) d/tl,

t t t T ¿2 T

J ea(t2"T) J dftldh = i J (l - e^1-^) d/tl5 a ^ 0,

t t t T ¿2 T

J(t2-Trj dMi2 = -_l- J(ti-T)a+1dftl, a^-1, t t t

T T

^(ioo)T,t =2 J(T ~ hfdftn J(oio)T,t = J{t\ - t)(T - ti)dftl, tt T ¿2

J(ii0)T,t = J(T - t2) J dftidft2, (3.31)

tt T ¿2 T ¿3 ¿2

J(101)T,t = J j(t2 - ii )dfti¿/¿2 , ^(1011)^ = J J J(t2 - ii )d/ti^/¿2^/¿3 , t t t t t

T ¿3 ¿2

^(1101)^ = J j(t3 - t2) J dftid/t2d/t3, ¿ ¿ ¿

T ¿3 ¿2 T ¿2

J(ino)T,t = J(T-h) J J dftldft2dft3, J(uoo)T,t=2 J(T — t2)2 J dftldft2, t t t t t

T ¿2

^(1010)^ = J(T - t2) J(t2 - t1 )d/ii¿/¿2, (3.32)

¿¿

T ¿2 T ¿2

^(iooi)T,t =2 J J to- tl )2<tftidft2, J(ouo)T,t = J (T -t2) J {t\ - t)dftldft2, ¿ ¿ ¿ ¿

T ¿2

^(0101)^ = J J (t2 - t1)(t1 - t)d/tld/t2, ¿¿

T T

J{ooio)T,t =2 J(T — t\)(ti - t)2dftl, J(oioo)T,i =2 J(T ~ f)2fa - t)dftl.

tt

T

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J{iooo)T,t = ^ J(T — ti)3dftl, t

T

^- i IT — i, r-Mf^.

J(1 0...0 = 71—TTT (T - ti)k

(k - 1)!

k-i t

T t2

~ (fc - 2)! 2

ti1-

k-2 t t T

Ji^0)T,t = J (T - t1) J(s1^)ti,td/ti'

k-i t k-2 T t2

^i^o^TO = ^ _ 2)! J Jfa ~ ff^dftstft^

k-2 t t T ts t2

J(10 1...1 )T,t = J ... J J (t2 - t1)dftidft2 . . . d/tfc-i, k-2 t t t

T tk-i tk-2 t2

Jl^ 01)T,t = J J (tk-1 - ifc-2^ ...J dfti . . . dftk-s dftk-2 dftk-i > k-2 t t t t

J(10)T,t + J(01)T,t = (T — t) J(1)T,t5 J(110)T,t + J(101)T,t + J(011)T,t = (T — t) J(11)T,t5

f +/ +/ (T - *)2 r

J(1100)T,t + J(1010)T,t + J(1001)T,t + J(0110)T,t +

+ / +1 (T -7

+ ^(0101 )T,t ~r 'J(OOll)T,t — -~-J(ll

/ + 7 + 7 + 7 (T-t)37

^(1000)T,i + ^(0100)T,i + ^(0010)T,i + ^(0001)T,i — -gj-J(i)i>

J(1110)T,t + J(1101)T,t + J(1011)T,t + J(0111)T,t = (T - 0^(111)^,

k 1 k-l_

£ = ^ _ - 1

1=1 l-i fc-i k

£ 0^^t = (T - 0 Jl^T^

1 = 1 l-i fc-i k-i

_^ (T _ t)k-m

ii+...+ik=m

ijejo, i}, ¿=i,...,fc

(A: — m)!

where

T ¿2

J(1i...1k)T,t = J ...J ^t^ . . . dwlk), ¿¿

/j = 1 when w(i) = /ti and /j = 0 when w(i) = t (i = 1,..., k), /T is a standard Wiener process.

Let us consider two examples and show explicitly the technique on integration order replacement for iterated Ito stochastic integrals.

Example 3.1. Let us prove the equality (3.31). Using Theorems 3.1 and 3.3, we obtain:

T ¿3 ¿2 T T T

J(110)T,t = J J J d/ti d/t2 dt3 = J d/ti y d/t2 y dt3 = ¿ ¿ ¿ ¿ ¿i ¿2

T T T T

= J d/t^ d/t2 (T - t2) = y d/t^(T - t2)d/t2 =

¿ ¿i ¿ ¿i T ¿2

= J(T - t2)J d/tid/t2 w. p. 1. ¿¿

Example 3.2. Let us prove the equality (3.32). Using Theorems 3.1 and 3.3, we obtain

T t4 t3 t2 T T T T

J(ioio)T,t = ^ | J d/t1 dÎ2d/i3dt4 = J d/tx J dt2 J f J dt4 =

t t t t t t1 t2 t3 T ^p ^n ^n ^n ^n

= J d/ti y dt2 J d/t3 (T - ta) = J dfh J dt2 J(T - t3)d/t3 =

t t1 t2 t t1 t2 T t3 t2 T / t3 t2

= I (T - ta) / / d/ti dt2dft3 = y (T - ta) N J dfa dt^ dft3 =

t t t t \t t T / t3 t3

= J(T - ta) y d/t^ dt^ dft3 = t t t1 T / t3

= /(T - ta) y d/ti (ta - tiH dft3 = tt T / t3

= I(T - ta) J (ta - ti)d/tj d/t3 = tt T t2

^ y (T - t2) J (t2 - ti)dfti dft2 w. p. 1. tt

3.7 Integration Order Replacement Technique for Iterated Stochastic Integrals with Respect to Martingale

In this section, we will generalize the theorems on integration order replacement for iterated Ito stochastic integrals to the class of iterated stochastic integrals with respect to martingale.

Let (Œ, F, p) be a complete probability space and let {Ft,t G [0, T]} be a nondecreasing family of a-algebras defined on the probability space (Œ, F, p). Suppose that Mt, t G [0,T] is an Ft-measurable martingale for all t G [0,T], which satisfies the condition m {|Mt|} < to. Moreover, for all t G [0,T] there

exists an Ft- measurable and nonnegative w. p. 1 stochastic process pt, t G [0, T] such that

m

{(Ms - Mt)2 | Ft} = Ml / Pt dr

Ft w. p. 1,

where 0 < t < s < T.

Let us consider the class H2(p, [0,T]) of stochastic processes t G [0,T], which are Ft-measurable for all t G [0, T] and satisfy the condition

m | j ^?ptdt| < to. For any partition t(+), j = 0,1,..., N of the interval [0, T] such that

0 = t0N) < t<+) < ... < t++) = T, max

0 i N 0<j<+-1

T(+) _ T(+)

fa 0 if N fa to (3.33)

we will define the sequence of step functions

^t,w) = (w) w. p. 1 for t G

T(+) T(+)

where ^(N)(t,w) G H (p, [0,T]), j = 0,1,. ..,N - 1, N = 1, 2,...

Let us define the stochastic integral with respect to martingale for ^(t, w) G H2(p, [0, T]) as the following mean-square limit

Ni

um-e „<+> (t<+>,u) (m(t<+n^) - m(t<+>, j=0

u

T

=f / dMT

where ^(N)(t,w) is any step function from the class H2(p, [0,T]), which converges to the function ^(t,w) in the following sense

T

lim m

NfaTO J 0

)(t,u) - <^(t,u) }> ptdt = 0

It is well known [95] that the stochastic integral

T

J dMT 0

exists and it does not depend on the selection of sequence )(t,w).

Let HH2(p, [0, T]) be the class of stochastic processes , t G [0, T], which are mean-square continuous for all t G [0,T] and belong to the class H2(p, [0,T]).

Let us consider the following iterated stochastic integrals

T tk-i tfc

S[0,0( )]T,t = J 01 (t1).. .J 0k (tk) J 0r dMik+1)dMt(kk)... dM^, (3.34) t t t

T tk-i

S [0(k)]T,t = J 01 (t1).. .J 0k (tk )dMt(kk)... dMt(11). (3.35)

t t

Here 0T G HH2(p, [t,T]) and 01(t),... ,0k(t) are continuous nonrandom functions at the interval [t,T], M^ = MT or M^ = t if t g [t,T], / = 1,... ,k + 1, MT is the martingale defined above.

Let us define the iterated stochastic integral S[0(k)]T,s, 0 < t < s < T, k > 1 with respect to martingale

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T T

S [0(k)]T,s = J 0k (tk )dMt(kk) ..J 01(t1)dMt(1)

S ¿2

by the following recurrence relation

N-1

S[0(k)]T,t d=f l.i.m. £ 0k(ti)aMT^S[0(k-1)]T,Ti+i, (3.36)

1=0

where k > 1, S[0(0)]T,S =f 1, [s,T] C [t,T], here and further amT? = mT^ -M() , i = 1,..., k + 1,/ = 0,1,..., N - 1, {t1}N0 is the partition of the interval [t,T], which satisfies the condition similar to (3.33), another notations are the same as in (3.34), (3.35).

Further, let us define the iterated stochastic integral S[0,0(k)]T,t, k > 1 of the form

T

S[0,0(k)]T,t = J 0sdMs(k+1)S[0(k)]T,s t

by the equality

N-1

S[0,0(k)]T,t d=f l.i.m. £ 0TI AM^S[0(k)]T,Ti+i,

1=0

where the sense of notations included in (>.341) (137361) is saved.

Let us formulate the theorem on integration order replacement for the iterated stochastic integrals with respect to martingale, which is the generalization of Theorem 3.1.

Theorem 3.5 [132] (1999) (also see [1]-[16], [112]). Let fa G ^(p, [t,T]),

every fa (t) (l = 1,..., k) is a continuous nonrandom function at the interval [t,T], and |pT| < K < oo w. p. 1 for all t G [t,T]. Then, the stochastic integral fa(k)]T,t exists and

S= S[fa#fe)]T,t w. p. 1.

The proof of Theorem 3.5 is similar to the proof of Theorem 3.1.

Remark 3.2. Let us note that we can propose another variant of the conditions in Theorem 3.5. For example, if we not require the boundedness of the process pT, then it is necessary to require the fulfillment of the following additional conditions:

1. m{|pT|} < oo for all t e [t,T].

2. The process pT is independent with the processes fa and MT.

Remark 3.3. Note that it is well known the construction of stochastic integral with respect to the Wiener process with integrable process, which is not an FT-measurable stochastic process — the so-called Stratonovich stochastic integral [105].

The stochastic integral S[fa, is also the stochastic integral with in-

tegrable process, which is not an FT-measurable stochastic process. However, under the conditions of Theorem 3.5

S= w. p. 1,

where Sis a usual iterated stochastic integral with respect to martingale. If, for example, MT, t e [t,T] is the Wiener process, then the question on connection between stochastic integral S[fa, and Stratonovich stochastic

integral is solving as a standard question on connection between Stratonovich and Ito stochastic integrals [105 .

Let us consider several statements, which are the generalizations of theorems formulated in the previous sections.

Assume that = {(t1,..., tk) : t < t1 < ... < tk < T} and the following conditions are fulfilled:

BI. ^T G H2(p, [t, T]).

BII. $(t1,..., tk-1) is a continuous nonrandom function in the closed domain Dk-1 (recall that we use the same symbol Dk-1 to denote the open and closed domains corresponding to the domain Dk-1).

Let us define the following stochastic integrals with respect to martingale

T T T

S K, = / & dM«k> ..J dM^ j $(i1, t2,..., tk-1)dM«1> =

t t3 t2

N— 1 T T T

d=f l.i.m. £ ^ AMf i dM^ ... / dMf i ^(t1, t2,..., tk-1 )d^^t(]1) '1—0 J J J

for k 3 and

Tl + i ¿3 ¿2

T T

SK, ^]T2t = / 4dMt(22) J $(t1)dMii1) d=f

t t2

T

N-1 »

=f l.i.m. £ i„ AM<2) / $(i1)dM<1)

Tl + i

for k = 2, where the sense of notations included in (I3.34j)-(I3361) is saved. Moreover, the stochastic process £T, t G [t,T] belongs to the class H2(p, [t,T]).

In addition, let

T tk-i

S[e, ^]Tkt = / -J ^(t1,...,tk-1)etkdMt(kk) ...dMt(1), k > 2, (3.37) tt

where the right-hand side of (3.37) is the iterated stochastic integral with respect to martingale.

Let us introduce the following iterated stochastic integrals with respect to martingale

T T T

SWTV11 = / dAit;1' -J dM® j $(i1, ,..., tk_l)dM«1» =f

t ¿3 ¿2

def

N-1 T

l.i.m. V AM?-1) / dM 1=0 1 /

T

(k-2)

tfc-2

dM

(2)

t2

$(ti,t2 ,...,tk-i)dMt(i1),

is

t2

T

tfc-2

S 1)

$(ii,... ,tk-i)dMt(k-;1).. .dMt;j, k > 2.

(1)

It is easy to demonstrate similarly to the proof of Theorem 3.5 that under the condition BII the stochastic integral S^y-^ exists and

(k-1) T,t

w. p. 1.

In its turn, using this fact we can prove the following theorem similarly to the proof of Theorem 3.5.

Theorem 3.6 [132] (1999) (also see [1]-[16], [112]). Let the conditions BI, BII of this section are fulfilled and |pT| < K < to w. p. 1 for all t G [t,T]. Then, the stochastic integral £[£, exists and for k > 2

SK, = s[i, *]£ w. p. 1.

Theorem 3.6 is the generalization of Theorem 3.2 for the case of iterated stochastic integrals with respect to martingale.

Let us consider two statements.

Theorem 3.7 [132] (1999) (also see [1]-[16], [112]). Let the conditions of Theorem 3.5 are fulfilled and h(r) is a continuous nonrandom function at the interval [t,T]. Then

T T

J fadMÎk+1)h(r)S[fa(k)]t,t = J fah(T)dMÎk+1)S[fa(k)]T,r w. p. 1, (3.38) ii

where the stochastic integrals in (3.38) exist.

Theorem 3.8 [132] (1999) (also see [1]-[16], [112]). Under the conditions of Theorem 3.5

t ti

J h(t1) I fadM(k+2)dMt(f+1)5[fa(k)]T,ti = ii

(k)

T T

= J 0tdMik+2^ h(t1)dMt(ik+1)S[0(k)]T,ti w. p. 1, (3.39)

t T

where the stochastic integrals in (3.39) exist.

The proofs of Theorems 3.7 and 3.8 are similar to the proofs of Theorems 3.3 and 3.4 correspondingly.

Remark 3.4. The integration order replacement technique for iterated Ito stochastic integrals (Theorems 3.1-3.4) has been successfully applied for construction of the so-called unified Taylor-Ito and Taylor-Stratonovich expansions (see Chapter 4) as well as for proof and development of the mean-square approximation method for iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series (see Chapters 1 and 2).

Chapter 4

Four New Forms of the Taylor—Ito and Taylor—Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations

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The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed time moment is considered in this chapter. The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions are transformed to four new representations, which include the minimal sets of different types of iterated It o and Stratonovich stochastic integrals. Therefore, these representations (the so-called unified Taylor-Ito and Taylor-Stratonovich expansions) are more convenient for constructing of the high-order strong numerical methods for Ito SDEs. Explicit one-step strong numerical schemes with the convergence orders 1.0, 1.5, 2.0, 2.5, and 3.0 based on the unified Taylor-Ito and Taylor-Stratonovich expansions are derived.

4.1 Introduction

Let F, p) be a complete probability space, let {Ft,t G [0,T]} be a non-decreasing right-continuous family of a-algebras of F, and let ft be a standard m-dimensional Wiener process, which is Ft-measurable for any t G [0,T]. We assume that the components ft(i) (i = 1,..., m) of this process are independent.

Consider an Ito SDE in the integral form

t t

xt = x0 + J a(xT, t)dr + J B(xT, t)dfT, x0 = x(0,w).

(4.1)

Here xt is some n-dimensional stochastic process satisfying to the Ito SDE (4.1). The nonrandom functions a : Rn x [0,T] ^ Rn, B : Rn x [0,T] ^ Rnxm guarantee the existence and uniqueness (up to stochastic equivalence) of a solution to the equation (4.1) [95]. The second integral on the right-hand side of (4.1) is interpreted as an Ito stochastic integral. Let x0 be an n-dimensional random variable, which is F0-measurable and m{|x01 ) < to. Also we assume that x0 and ft — f0 are independent when t > 0.

It is well known [79], [80], [88], [133], [134] (also see [13]) that Ito SDEs are

adequate mathematical models of dynamic systems of different physical nature that are affected by random perturbations. For example, Ito SDEs are used as mathematical models in stochastic mathematical finance, hydrology, seismology, geophysics, chemical kinetics, population dynamics, electrodynamics, medicine and other fields [79], [80], [88], [133], [134] (also see [13]).

Numerical integration of Ito SDEs based on the strong convergence criterion of approximations [79] is widely used for the numerical simulation of sample trajectories of solutions to Ito SDEs (which is required for constructing new mathematical models on the basis of such equations and for the numerical solution of different mathematical problems connected with Ito SDEs). Among these problems, we note the following: filtering of signals under influence of random noises in various statements (linear Kalman-Bucy filtering, nonlinear optimal filtering, filtering of continuous time Markov chains with a finite space of states, etc.), optimal stochastic control (including incomplete data control), testing estimation procedures of parameters of stochastic systems, stochastic stability and bifurcations analysis [77], [79], [80], [87], [88], [114], [135]-[139].

Exact solutions of Ito SDEs are known in rather rare cases. For this reason it is necessary to construct numerical procedures for solving these equations.

In this chapter, a promising approach [77], [79], [80], [87], [88] to the numerical integration of Ito SDEs based on the stochastic analogues of the Taylor formula (Taylor-Ito and Taylor-Stratonovich expansions) [140], [141] (also see [51], [72], [142]-[

is used. This approach uses a finite discretization of the time variable and the numerical simulation of the solution to the Ito SDE at discrete time moments using the stochastic analogues of the Taylor formula

mentioned above. A number of works (e.g., [77]-[80], [87], [88]) describe nu-

merical schemes with the strong convergence orders 1.5, 2.0, 2.5, and 3.0 for Ito SDEs; however, they do not contain efficient procedures of the mean-square approximation of the iterated stochastic integrals for the case of multidimensional nonadditive noise.

In this chapter, we consider the unified Taylor-Itô and Taylor-Stratonovich expansions [142], [144] (also see [51], [72]) which makes it possible (in contrast

with its classical analogues [79], [140]) to use the minimal sets of iterated Ito and Stratonovich stochastic integrals; this is a simplifying factor for the numerical methods implementation. We prove the unified Taylor-Ito expansion [142] with

using of the slightly different approach (which is taken from [144]) in comparison with the approach from [142]. Moreover, we obtain another (second) version

of the unified Taylor-Ito expansion [76], [145]. In addition we construct two

new forms of the Taylor-Stratonovich expansion (the so-called unified Taylor-Stratonovich expansions [144]).

It should be noted that in Chapter 5 on the base of the results of Chapters 1, 2 we study methods of numerical simulation of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1, 2, 3, 4, 5, and 6 from the Taylor-Ito and Taylor-Stratonovich expansions. These stochastic integrals are used in the strong numerical methods for Ito SDEs [77], [79], [80], [87] (also see [13]). To approximate the iterated Ito and Stratonovich stochastic integrals appearing in the numerical schemes with the strong convergence orders 1.0,1.5, 2.0, 2.5, and 3.0, the method of generalized multiple Fourier series (see Chapter 1) and especially method of multiple Fourier-Legendre series will be applied in Chapter 5. It is important that the method of generalized multiple Fourier series (Theorem 1.1) does not lead to the partitioning of the integration interval of the iterated Ito and Stratonovich stochastic integrals under consideration; this interval length is the integration step of the numerical methods used to solve Ito SDEs; therefore, it is already fairly small and does not need to be partitioned. Computational experiments [1] show that the numerical simulation for iterated stochastic integrals (in which the interval of integration is partitioned) leads to unacceptably high computational cost and accumulation of computation errors. Also note that the Legendre polynomials have essential advantage over the trigonomentric functions (see Chapter 5) constructing the mean-square approximations of iterated Ito and Stratonovich stochastic integrals in the framework of the method of generalized multiple Fourier series (Theorem 1.1).

Let us consider the following iterated Ito and Stratonovich stochastic integrals:

t2

J[0(k)]T,t = / 0k(tk). . . I 0i(ti)dwt(;i)... dw(:k), (4.2)

J*[0(k)kt = J 0k (tk) ...J 0i(ti)dwt(;i)... dw£k), (4.3)

t t

where every 0/(t) (l = 1,...,k) is a continuous nonrandom function at the interval [t, T], w[i) = f(i) for i = 1,..., m and wT0) = t, i1,..., ik = 0,1,..., m.

It should be noted that one of the main problems when constructing the high-order strong numerical methods for Ito SDEs on the base of the Taylor-Ito and Taylor-Stratonovich expansions is the mean-square approximation of the iterated Ito and Stratonovich stochastic integrals (4.2) and (4.3). Obviously, in the absence of procedures for the numerical simulation of stochastic integrals, the mentioned numerical methods are unrealizable in practice. For this reason, in Chapter 5 we give the extensive practical material on expansions and mean-square approximations of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions. In Chapter 5, the main focus is on approximations based on multiple Fourier-Legendre series. Such approximations is more effective in comparison with the trigonometric approximations (see Sect. 5.2) at least for the numerical methods with the strong convergence order 1.5 and higher

The rest of this Chapter is organized as follows. In Sect. 4.1 (below) we consider a brief review of publications on the problem of construction of the Taylor-Ito and Taylor-Stratonovich expansions for the solutions of Ito SDEs. Sect. 4.2 is devoted to some auxiliary lemmas. In Sect. 4.3 we consider the classical Taylor-Ito expansion while Sect. 4.4 and Sect. 4.5 are devoted to first and second forms of the so-called unified Taylor-Ito expansion correspondingly. The classical Taylor-Stratonovich expansion is considered in Sect. 4.6. First and second forms of the unified Taylor-Stratonovich expansion are derived in Sect. 4.7 and Sect. 4.8. In Sect. 4.9 we give a comparative analysis of the unified Taylor-Ito and Taylor-Stratonovich expansions with the classical Taylor-Ito and Taylor-Stratonovich expansions. Application of the first form of the unified Taylor-Ito expansion to the high-order strong numerical methods for Ito SDEs is considered in Sect. 4.10. In Sect. 4.11 we construct the high-order strong numerical methods for It o SDEs on the base of the first form of the unified

Taylor-Stratonovich expansion.

Let us give a brief review of publications on the problem of construction of the Taylor-Ito and Taylor-Stratonovich expansions for the solutions of Ito SDEs. A few variants of a stochastic analog of the Taylor formula have been obtained in [140], [141] (also see [77], [79]) for the stochastic processes in the

form R(xs,s), s G [0,T], where xs is a solution of the Ito SDE (4.1) and R : Rn x [0,T] fa R1 is a sufficiently smooth nonrandom function.

The first result in this direction called the Ito-Taylor expansion has been obtained in [141] (also see [140]). This result gives an expansion of the process R(xs,s), s G [0,T] into a series such that every term (if k > 0) contains the iterated Ito stochastic integral

dw

(ii)

. dw

(ik) tk

(4.4)

as a multiplier factor, where ii,..., ik = 0,1,..., m. Obviously, the iterated Ito stochastic integral (4.4) is a particular case of (4.2) for fa1(r),... , (t) = 1.

In [140] another expansion of the stochastic process R(xs, s), s G [0,T] into a series has been derived. The iterated Stratonovich stochastic integrals

(ii)

. . dw

(ik) tk

(4.5)

were used instead of the iterated Ito stochastic integrals; the corresponding expansion was called the Stratonovich-Taylor expansion. In the formula (14.5) the indices i1,..., take values 0,1,..., m.

In [142] the Ito-Taylor expansion [140] is reduced to the interesting and unexpected form (the so-called unified Taylor-Ito expansion) by special transformations (see Chapter 3). Every term of this expansion (if k > 0) contains the iterated Ito stochastic integral

(s - tk)

(s - t1)

11 dfti1

(ii)

.dft

(ik)

'tfc

(4.6)

where l1,..., lk = 0,1, 2,... and i1,..., ik = 1,..., m.

i

t

i

s

k

It is worth to mention another form of the unified Taylor-Ito expansion 76], [145] (also see [1]-[16]). Terms of the latter expansion contain iterated Ito

stochastic integrals of the form

(t - tk )1k... I (t - t^1 ..f(4.7)

where /i,..., = 0,1, 2,... and ii,..., = 1,..., m.

Obviously that some of the iterated Ito stochastic integrals (4.4) or (4.5) are connected by linear relations, while this is not the case for integrals defined by (4.6), (4.7). In this sense, the total quantity of stochastic integrals defined by (4.6) or (4.7) is minimal. Futhermore, in this chapter we construct two new forms of the Taylor-Stratonovich expansion (the so-called unified Taylor-Stratonovich expansions) [146] (also see [144]) such that every term (if k > 0)

contains as a multiplier the iterated Stratonovich stochastic integral of one of two types

j * ¿2

(t-tk)1;.../ (t-ti)11 dft(;i}..f(4.8)

t t : S * ¿2

(s-tk)1;.../ (s-tikdfi;1}...f;k}, (4.9)

t

where 11,..., 1k = 0,1, 2,..., i1,..., ik = 1,..., m, and k = 1, 2,...

It is not difficult to see that for the sets of iterated Stratonovich stochastic integrals (4.8) and (4.9) the property of minimality (see above) also holds as for the sets of iterated Ito stochastic integrals (4.6), (4.7).

As we noted above, the main problem in implementation of high-order strong numerical methods for Ito SDEs is the mean-square approximation of the iterated stochastic integrals (4.4) (491). Obviously, these stochastic integrals are particular cases of the stochastic integrals (4.2), (4.3).

Taking into account the results of Chapters 1, 2, 3, 5 and the minimality of the sets of stochastic integrals (4.6)) (479), we conclude that the unified Taylor-Ito and Taylor-Stratonovich expansions based on the iterated stochastic integrals (4.6)) (O)) can be useful for constructing of high-order strong numerical methods with the convergence orders 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, ... for Ito SDEs.

s

4.2 Auxiliary Lemmas

def

Let (Q, F, p) be a complete probability space and let f (t, w) = ft : [0, T] x Q fa R1 be the standard Wiener process defined on the probability space (Q, F, p).

Let us consider the family of a-algebras {Ft, t G [0, T]} defined on the probability space (Q, F, p) and connected with the Wiener process ft in such a way that

1. Fs C Ft c F for s < t.

2. The Wiener process ft is Ft-measurable for all t G [0,T].

3. The process ft+A — ft for all t > 0, A > 0 is independent with the events of a-algebra Ft.

def

Let us consider the class M2([0,T]) of random functions £(t,w) = £t : [0,T] x Q fa R1 (see Sect. 1.1.2).

Let us recall (see Sect. 2.1.1) that the class Qm([0,T]) consists of Ito processes , t G [0,T] of the form

T T

nr = no + J asds + J bsdfs, (4.10)

o o

where (ar)m, (br)m G M2([0,T]) and

lim m{ |bs — bT|4} =0 for all t G [0,T].

The second integral on the right-hand side of (4.10) is the Ito stochastic integral.

Also note that the definition of the Stratonovich stochastic integral in the mean-square sense is given by (2.3) (Sect. 2.1.1) and the relation between Stratonovich and Ito stochastic integrals (see Sect. 2.1.1) has the following form [105] (also see [79])

* T t T

J F(rjT,r)dfT = J F(i]T,r)dfT + ^J^(ilT,r)bTdr w. p. 1. (4.11) t t t

If the Wiener processes in (4.10) and (4.11) are independent, then

* T T

i F(nr,T)dfr = / F(nr, t)dfr w. p. 1. (4.12)

Recall that a possible variant of conditions providing the correctness of the formulas (4.11) and (4.12) consists of the following conditions: nT G Q4([t,T]), F(nT,t) G M2([t,T]), F(x,t) G C2'1(R1 x [0,T]), where C2'1(R1 x [0,T]) is the space of functions F(x, t) : R1 x [0, T] ^ R1 with the following property: these functions are twice differentiable in x and have one derivative in t. Moreover, all these derivatives are bounded.

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Remark 4.1. Note that if F(x,t) = F1(x)F2(t), then the smoothness condition F(x,t) G C2'1(R1 x [0,T]) can be weakened. Namely, it suffices to replace the condition with respect to t by continuity with respect to this variable.

Also remind that S2([0,T]) is a subset of M2([0,T]) and S2([0,T]) consists of the mean-square continuous random functions (see Sect. 3.1).

Let us apply Theorem 3.1 (see Sect. 3.2) to derive one property for Ito stochastic integrals.

Lemma 4.1 14

51]. Let h(r), g(t), G(t) : [t, s] ^ R1 be continuous nonrandom functions at the interval [t, s] and let G(t) be a antiderivative of the function g(t). Furthermore, let G S2([t,s]). Then

s T 9 s 9

i g(t) i fc(0) / ^d^fW = i(G(s) - G(0))h(0) /&ff}

w. p. 1, where = 1, 2 and fk, fT2) are independent standard Wiener processes that are FT-measurable for all t G [t, s].

Proof. Applying Theorem 3.1 two times and Theorem 3.3, we get the following relations

s t 9 s s s

J g (t ) J h(0) y & dtff W = y & df« y h(^)df9j) J g (t )dT =

t t t tu 9

s s s s

= G(s) / ^d^ / h(^)df9j) - / ^d^ / G(^)h(^)df9j) =

s

é»

G(sW h(0) / ^udfUi)df9j) - G(0)h(0) / &dtff)

uu

t t t

s

é»

(G(s) - G(0))h(0) / ) w. p. 1.

6»

s

The proof of Lemma 4.1 is completed.

Let us consider an analogue of Lemma 4.1 for Stratonovich stochastic integrals.

Lemma 4.2 [144] (also see [1]-[5], [12]-[16], [51]). Let h(r), д(т), G(t) : [t, s] fa R1 be continuous nonrandom functions at the interval [t, s] and let G(t) be a antiderivative of the function д(т). Furthermore, let ^ £ Q4([t,s]) and

T T

= J audu + J b„dfU°, l = 1, 2.

t t

Then

S * T ^ о ^ S ^ о

Jg(t)J dfjdT = y (G(s)-G(£))h(0)| ^df«df^j) (4.13)

t t t t t

w. p. 1, where l = 1, 2 and f(1), fT2) are independent standard Wiener processes that are FT-measurable for all t £ [t, s].

Proof. Under the conditions of Lemma 4.2, we can apply the equalities (4.11) and (4.12) with F(ж,0) = жВД and

* о

По = J eU/)dfUi), t

since the function xh(0) is sufficiently smooth (see Remark 4.1) and the following obvious inclusions hold: n0 £ Q4([t, s]), n0h(0) £ M2([t, s]).

Thus, we have the equalities

т ^ о т * о т

I h(9)j = J h(9) J edfMdfW+±l{i=j} J hmfde, (4.14)

t t t t t

* о о о

J ^dW = J + il{i=i} J Kdu (4.15)

t t t w. p. 1, where 1A is the indicator of the set A.

Substituting the formulas (4.14) and (4.15) into the left-hand side of the equality (4.13) and applying Theorem 3.1 twice and Theorem 3.3, we get the

following relations

s * 1

0

g(T) I h(0) / ^ffW =

u

t t t

s s s

= J ^^ I h(0)df9j) | g(T)dT+

tu

s s s

IT. r ......jW , _ 1

/ Kdu / / g{r)dr + -!{,=?} / /

2 — {(<—t/ i - «— i v / —-9 1 j \ j — ■ 2

t u t

s

( / f h(e)df^ + ii{i=j} f h{o)$)M+

u

s s

+il{/=i} f budu f h(0)df®

2

tu

s

CiPdî^ I G(0)h(0)dft1J) + ±l{i=j} /

2

s s

1

+-l{/=i} / budu /

2 _ 11—t ri u

t 9

= G(S) J h(9) J ^dl^dî^ + il{i=j} J h(0)$)d£>+

u 'dig' + 2

tt

s 9

J h(9) J bududï{0j)

tt 9 s

J ^d^df® + ±i{i=j} J G(e)h(e)$)M+

tt s 9

[ h{6)G{6) [ bududî{d]) ) (4.16)

s

w. p. 1. Applying successively the formulas (4.14), (4.15) together with the formula (4.14) in which h(9) replaced by G(9)h(9) as well as the relation (4.16), we obtain the equality (4.13). The proof of Lemma 4.2 is completed.

4.3 The Taylor—ItO Expansion

In this section, we use the Taylor-Ito expansion [140] and introduce some necessary notations. At that we will use the original notations introduced by the author of this book.

Let C2,1 (Rn x [0, T]) = L be the space of functions R(x,t) : Rn x[0,T] fa R1 with the following property: these functions are twice continuously differentiate in x and have one continuous derivative in t. We consider the following operators on the space L

dR

i=1

d x(i)

j=1 1,i=1

d2R dxWdxW

n (x,^),

(4.17)

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G^x, t) = Y B{31)(x> ^TIT) (x> > i = 1 > • • • >

dR

j=1

d x(j)

(4.18)

where x(j) is the jth component of x, aj)(x,t) is the jth component of a(x,t), and B(j)(x,t) is the ijth element of B(x,t).

By the Ito formula, we have the equality

s s

/m „

LR(xT, t)dr + £ / Gid)R(xT ,t)df(i) (4.19) , ¿=1 .

w. p. 1, where 0 < t < s < T. In the formula (4.19) it is assumed that the functions a(x,t), B(x,t), and R(x,t) satisfy the following condition: LR(xT,t), G0)R(xT, t ) G M2([0,T]) for i = 1,...,m.

Introduce the following notations

(k)A =

■■■ik)

mi ... mk

ii=1,...,ifc=1

m1,..., mk > 1,

(4.20)

n

(k+i)^ l (i)b(k) =

m, m,

^ ... A(ii--ifc+;)b(ii-i') ¿1 = 1 i; = 1

m; + 1 ... m; + fc

i ;+i=1,...,i;+fc=1

for k > 1

mi m

^ ... A(ii...i)B(ii...¿;)

ii=1 i =1

mi ... mfc

for k = 0

Ak+1Dkik )Ak ...A2 ^ A1R(x,t)

(ii)

= (k)Ak+1Dk Ak ...A2 D1A1R(x,t),

ii=1,...,ifc=1

i (4.21)

where Ap and D^9) are operators defined on the space L for p = 1,..., k + 1, q = 1,..., k, and iq = 1,..., mq. It is assumed that the left-hand side of (4.21)

exists. The symbol 0 is treated as the usual multiplication. If m/ = 0 in (4.20) for some l G {1,..., k}, then the right-hand side of (4.20) is treated as

A(ii...i ;-ii ;+i-.ifc)

mi ... m;-i m;+i ... mfc

ii=1,...,i ; -i=1,i;+i=1,...,ifc=1

(shortly, (k-1)A).

We also introduce the following notations

mAi ... mA;

= »Qa, ... Qa,R(x,t),

qA;»...QAl1»R(x,t)

ii=Ai,...,i =A

(pk)J

(Afc...Ai)s,i

J

(ifc...ii)

(Afc ...A,)s,t

mA, ... mAk

ii=Ai,...,ifc=Ak

Mk ^(Ak,...,A1): Ai = 1 or Ai = 0; l = 1,...,kl, k > 1,

S t2

J

(ifc ...ii) (Afc ...A,)s,t

dwt(ik»...dwjk1', k > 1,

(ii)

tfc

t t

where Ai = 1 or Ai = 0, qA;» = L and ii = 0 for Ai = 0, qA,;) ii = 1,..., m for Ai = 1,

G0i;» and

pi = ^^ Aj for l = 1,..., r + 1, r G N,

j=1

i

wT;) (i = 1,... ,m) are FT-measurable for all t G [0,T] independent standard Wiener processes and wT°' = t.

Applying (4.19) to the process R(xs, s) repeatedly, we obtain the following Taylor-Ito expansion [140

R(xs, s) = R(xt, t) + ^ £ (pk)QAfc... QAlR(xt, t) ? (pk) J(Afc...Al)s,t+

k=1 (Afc ,...,Ai)GMfc

+ (Dr+1 )s,t (4.22)

w. p. 1, where

(Dr+1)s,t =

= ^ [...( j (Pr+l)QAr+i ...QAl R(xti ,t1) Ar-+1 dwtJ ... Al dwtr+i.

(4.23)

It is assumed that the right-hand sides of (4.22), (4.23) exist.

A possible variant of the conditions, under which the right-hand sides of (4.22), (4.23) exist is as follows

(i) Q^ ... R(x, t) G L for all (A/,..., Ax) G U Mg;

g=i

r+i

(ii) Q^j'1... Q^1 'r(xt, t) G M2([0, T]) for all (A/,..., Ai) G U Mg.

g=i

Let us rewrite the expansion (4.22) in the another form

r mA i mAfc

R(x„s)=R(x„t)+£ e E £qA:1 .■■QA':)R(xf,t)j(t;A1))S,(+

k=1 (Afc,...,Ai)GMfc ii=Ai ik=Ak

+ (Dr+i)s,t w. p. 1.

Denote

Grk = j(Ak,..., Ai) : r + 1 < 2k - Ai - ... - Ak < 2r J,

Eqk = ^(Afc,..., Ai) : 2k - Ai - ... - Ak = gj, where A/ = 1 or A/ = 0 (l = 1,..., k).

r

The Taylor-Ito expansion ordered according to the order of smallness (in the mean-square sense when s 1t) of its terms has the form

R(Xs, s) =

r mAi mAk

= R(xf, t) + E £ E ■ ■ • E Q:1 ■ ■ • QA!>(xt, t) Jti'w+

q,k—1 (A:,...,Ai)GEqk ¿1—Ai ¿k—Afc

+ (Hr+1 )s,t w. p. 1, (4.24)

where

r mA1 mAk

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(Hr+1 )s, =E E E • • • E qA:1 • • • Qiii)R(xt, t)+(Dr+1 )„(.

k—1 (A:,...,Ai)eGrk ¿1—Ai ¿k—A:

4.4 The First Form of the Unified Taylor—Ito Expansion

In this section, we transform the right-hand side of (4.22) by Theorem 3.1 and Lemma 4.1 to a representation including the iterated Ito stochastic integrals (4.7).

Denote

I,

(n-.-ifc )

S ¿2

= y(t-tk... J(t-ti)l1 dft(;i)...) for k>i

(4.25)

and

I,

,(il-,-ik ) = 1 for k = 0,

where i1, • • •, = 1,..., m. Moreover, let

(k)1

l1---lfcs,i

I,

(¿1---«fe)

l1---lfcs,i

«1,---,ife = 1

= i (G^L - LG^) , p=l,2,..., i = 1,..., m, (4.26)

where L and G/», i = 1,... ,m are determined by the equalities (14.17), (14.18). Denote

Aq = <j (k,j,1i,...,1k ) : k + j + ^ = q; k, j, li,..., l* = 0,1,... f,

p=i

m

k

G^1... G((jk'LjR(x, t)

d=f (k)G/i... G/kLjR(x, t),

ii,...,ik=i

Lj R(x, t) = ^

L..^LR{K,t) forj>l

R(x, t)

for j = 0

Theorem 4.1. Let conditions (i), (ii) be satisfied. Then for any s,t G [0,T] such that s > t and for any positive integer r, the following expansion takes place w. p. 1

R(xs, s) =

R(xt ,t) + E E

j!

£ gî;11 ...G(;k 1 LjR(xt,t)/;;xk;+

(k,j,/i,...,/k)GAq ^ ;i,...,;k=i

+ (Dr+i),t ,

(4.27)

where (Dr+i)st is defined by (14.2,3). Proof. We claim that

(Pq )

E (Pq 'QAq ...QAi R(xt,t)"q (Pq 'j(Aq ...Al)s,t

(Aq ,...,Al)GMq

E

(k,j',/1,...,/k)GAq

(« - ty fl

E gî;1'...G;k'LRK^.1.-; (4.28)

(il ...ik )

;i,...,ik=i

w. p. 1. The equality (4.28) is valid for q = 1. Assume that (4.28) is valid for some q > 1. In this case, using the induction hypothesis, we obtain

E (Pq+l)QAi . . . QAq+iR(xt,t) '+1 (Pq+l) J(Ai...Aq+i)s,t

(Aq+l ,...,Al )GMq+i

E / E i(Pq+l)QAi ...QAq+iR(xt,t)

^q+lGji, °} t (Aq,..,Al)GMq\

Pq

q (PqJ

(Al ...Aq)0,t

Aq+i ,

• dw,9 =

E / E

Aq+l Gji, °} t (k,j,/l,...,/k )GAq

j!

x

m

,

x I >G ... GQ R(x(,t) k №/,,,] V dw„ =

s

+ ^^...G^Go^x^)' / ..,fcJ -dfj (4.29)

J!

\ t w. p. 1.

Using Lemma 4.1, we obtain

s

1 , (s - t)j+1 for k = 0

(j + 1)! ■

■(s - t)j+1 ■ (k>/ii..,fcsi - (-1)j+1 ■ (k>/ii..,fc-i IkWsi for k> 0

(4.30)

w. p. 1. In addition (see (4.25)) we get

s

...ifcifc+1> (A Ol \

J j I '/•••// ~ j I '/•••//./-., 1 1

t

in the notations just introduced. Substitute (4.30) and (4.31) into the formula (4.29). Grouping summands in the obtained expression with equal lower indices at iterated Ito stochastic integrals and using (4.26) and the equality

1 P !

G«fi(x, f) = - £(-1)^L^L'-m^, t), c; = ^¿zr^y (4.32)

(this equality follows from (4.26)), we note that the obtained expression equals to

(kj,l1,...,Ifc>eAq+1

w. p. 1. Summing the equalities (4.28) for q = 1, 2,...,r and applying the formula (4.22), we obtain the expression (4.27). The proof is completed.

Let us order terms of the expansion (4.27) according to their smallness orders as s 1t in the mean-square sense

R(xs, s) =

r__( _ ,)j m

+ (Hr+i)a>i w. p. 1, (4.33)

where

( _ ,)j m

(fcj,/i,...,/fc)eUr ii,...,ifc=1

+ (Dr+1)s,t >

k

Dq = < (k,j,/i,...,/k) : k + 2 ( j + > ^ I = q; k,j,/i,...,/k = 0,1,...J>,

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(k, j, li,..., Ik) : k + 2 +

q _

\ \ p=i /

(4.34)

k

Ur = < (k, j, li,... ,1k) : k + j + ^ < r,

I p=I

k + 2 + ^ > r + 1; k,j,/i,...,/k = 0,1,...J, (4.35)

and (Dr+i)st is defined by (14.23). Note that the remainder term (Hr+i)st in (4.33) has a higher order of smallness in the mean-square sense as s | t than the terms of the main part of the expansion (4.33).

4.5 The Second Form of the Unified Taylor—Ito Expansion

Consider iterated Ito stochastic integrals of the form

S ¿2

J/,.' v) = /(* - tk)/k - J(s - ti)1'f1... <W for k > 1

and

J

(¿i...ik)

= 1 for k = 0,

where i1, . . . , ik = 1, . . . , m.

The additive property of stochastic integrals and the Newton binomial formula imply the following equality

i&t!=£ • • ■ £ IK (t - s)'1^-*--* « w. p. 1 (4.36)

'k k

r(«i---«k )

ji—0 j:—0 g—1

where

ck =

l!

k!(1 - k)!

is the binomial coefficient. Thus, the Taylor-Ito expansion of the process ns = R(xs,s), s G [0,T] can be constructed either using the iterated stochastic integrals I^1.^ similarly to the previous section or using the iterated stochastic

integrals J'(г-1-/•гk)• This is the main subject of this section.

Denote

J

(¿1—¿k)

' 1k

d=f (k) j

'1-'kSlt '

«1,...,«:—1 m

Lj G^î11 •••G((;k 1 R(x,t)

(»k ' :

= (k)LjG'1 ■ ■ ■ G':R(x,^

îl,---,îk—1

Theorem 4.2. Let conditions (i), (ii) be satisfied. Then for any s,t G [0,T] such that s > t and for any positive integer r, the following expansion is valid w. p. 1

R(xs, s) =

R(xt,t) + E E

(s-ty j!

£ LjG((lîl) ■ ■ ■ G^'R(x,t)j£.tk) +

q=1 (k,j,ii,...,ifc )GAq i1,...,ifc = 1

+ (Dr+1 )s,t ,

where (Dr+1 )st is defined by (14.23).

Proof. To prove the theorem, we check the equalities

(s - t)j m

(4.37)

£

(k,j,'1 v"?': )g

j !

ll,---,lk—l

1

m

( _ j m

= E far- E w.p.l (4.38)

(fc,j,/i,...,zfc)eAq ' «i,---,«fe=i

for q = 1, 2,...,r. To check (4.38), substitute the expression (4.36) into the right-hand side of (4.38) and then use the formulas (4.26), (4.32).

Let us order terms of the expansion (4.37) according to their smallness orders as s 1t in the mean-square sense

R(xs, s) =

r__( _ ,)j m

= /?(x„i) + E E far1 £ L'G^K..G^R(Xt,t)4:fa+

q=i (k,j,Zi,---,zfc)eDq ii,---,ifc=i

+ (Hr+1)s,t W. p. 1,

where

( _ ,)j m

№+i),f = E far- E

(k,j,Zi,---,Zfc)GUr ii,---,ifc=1

+ (Dr+1 )s,t .

The remainder term (Dr+1 )st is defined by (14.23); the sets Dq and Ur are defined by (4.34) and (4.35), respectively. Finally, we note that the convergence w. p. 1 of the truncated Taylor-Ito expansion (14.22) (without the remainder term (Dr+1)st) to the process R(xs,s) as r fa to for all s,t G [0,T] such that s > t and T < to has been proved in [79] (Proposition 5.9.2). Since the expansions (4.27) and (4.37) are obtained from the Taylor-Ito expansion (422) without any additional conditions, the truncated expansions (4.27) and (4.37) (without the reminder term (Dr+1 )s t) under the conditions of Proposition 5.9.2 [79] converge to the process R(xs,s) w. p. 1 as r fa to for all s,t G [0,T] such that s > t and T < .

4.6 The Taylor—Stratonovich Expansion

In this section, we use the Taylor-Stratonovich expansion [140] and introduce some necessary notations. At that we will use the original notations introduced by the author of this book.

Assume that LR(xT,t), G0i}R(xT,t) e M2([0,T]) for i = consider the Ito formula (4.19).

In addition, suppose that the function G0i)R(x,t) (i = 1,. that the formulas (4.11) and (4.12) can be applied. Then

1,... ,m and .., m) is such

J GfR{KT,r)di^ = j J Gq^Gq^ R(x.T, r)dr

(4.39)

w. p. 1, where i = 1,..., m.

Using the relation (4.39), let us write (4.19) in the following form

/110 p

lR(xT,r)dr + ^ / R(xT, t)df(i) , i=1 ,

w. p. 1, (4.40)

where

LR{x,t) = LR{x,t) ~ TGJ Go

2

(i)G(i) 0 G0

¿=1

Introduce the following notations

mAi ... mA|

d£ > ...DA;i)R(x,i)

(Pfc) J *

J(Afc ...Ai)s,t

=f (pi)Dai ...DaiR(x,t),

¿i=Ai,...,i =A;

J,

;(ifc...ii)

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(Afc ...Ai)s,t

mAi ... mAk

¿1—Ai ,...,ik=Ak

(4.41)

Mk ^(Afc,...,Ai): A/ = 1 or A/ = 0; l = 1,...,kL k > 1,

<¿2

J

= (ifc ...ii) (Afc ...Ai )s,t

dwt(ik) ...dwj;u, k > 1,

r(ii)

t t

where A/ = 1 or A/ = 0, D^) = l and i/ = 0 for A/ = 0, D^0 i/ = 1,..., m for A/ = 1,

Goil) and

p/ = ^^ Aj for l = 1,..., r + 1, r e N, j=i

s

m

s

/

wT:) (i = 1,... ,m) are FT-measurable for all t G [0,T] independent standard Wiener processes and w[0) = t.

Applying the formula (4.40) to the process R(xs, s) repeatedly, we obtain the following Taylor-Stratonovich expansion [140]

R(xs, s) = R(xt, t) + £ Y )DAk... DAlR(xt, t) ? (pk' J(*Afc...Al)s , +

k=1 (Afc,... , Ai)GMfc

w. p. 1, where

S J

(Ar+i,... , Ai)GMr+i "£

+ (Dr+1 )s ,t

(Dr+1)s,t =

(4.42)

<Î2

(Pr+l)DAr+i ...DaiR(xti,ti) r+1 dw

• dwt

r + i •

(4.43)

It is assumed that the right-hand sides of (4.42), (4.43) exist.

A possible variant of the conditions under which the right-hand sides of (4.42), (4.43) exist is as follows

(i*) Q^ ... Qi;i}R(x, t) G L for all (A,,..., Ax) G (J Mg;

g=x

r+1

(ii*) for all x, y G Rn, t, s G [0, T], (A,,..., Ax) G U Mg, and for some v > 0

g=x

Qa;}...QC;}R(x,t) - Qiil}... Qi;1 }R(y,t) < K|x - y|, (4.44)

qA:,)•••QA;i)R(x,t) <k(i+|x|),

(4.45)

and

qA,0 ... QAii)R(x,t) - QAr • • • QArR(x, s) < K|t - sr(1 + |x|),

)(:,) r»(ii)

where K < to is a constant, QA^ = L and i/ = 0 for A/ = 0, QA^ = G/1) and i = 1,..., m for A/ = 1;

(iii*) the functions a(x,t) and B(x, t) are measurable with respect to all variables and satisfy the conditions (4.44) and (4.45);

(iv*) x0 is F0-measurable and m {|x0|8} <

.

r

s

1

Let us rewrite the expansion (4.42) in another form

R(xs, s) =

r mA i mAfc

= R(xt, t) + E E £... £ D:>... dA:»R(x(, t)j(*A:':.a!;))s,(+

k=1 (Afc,...,Ai)GM: ii=Ai i: =A:

+ (Dr+i )s,t w- P. 1.

Denote

Grk = |(Ak,..., Ai) : r + 1 < 2k - Ai - ... - Ak < 2r J,

Eqk = j(Ak,..., Ai) : 2k - Ai - ... - Ak = gj,

where A/ = 1 or A/ = 0 (l = 1,..., k).

Let us order terms of the Taylor-Stratonovich expansion according to their smallness orders as s 1t in the mean-square sense

R(xs, s) =

r mA i mA:

= R(xt, t) + e e e... e dA: )... DAi')R(xi, t) ,,+

q,k=1 (A:,...,A 1 )eEq: : i=A i ::=A:

+ (Hr+i)s,t w. p. 1, (4.46)

where

r mA i mA:

(Hr+i),, = £ £ £ ■.. £ D:)... D^Rix,. t)+

k=i (A:,...,Ai)eGr: ii=A i i: =A:

+ (Dr+i )s,, .

4.7 The First Form of the Unified Taylor—Stratonovich Expansion

In this section, we transform the right-hand side of (4.42) by Theorem 3.1 and Lemma 4.2 to a representation including the iterated Stratonovich stochastic integrals (4.8).

Denote

SÎ2

I

li---lks,t

= (t - tk )l

(t - tl)11 df(;i)

.. dff) for k > 1 (4.47)

and

where i i,..., = 1, Moreover, let

I *(n...ifc )

Ji.-Vt

= 1 for k = 0,

m.

(k) T

^ I

h-hs,t

I

■•(ii—ik) h---lks,t

ii,...,ik=1

= i (G^iL - LG^i) , p=l,2,..., i = 1,..., m, (4.48)

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s

k

m

where G0i) == G0i), i = 1,... ,m. The operators l and Gq;, i = 1,... ,m are determined by the equalities (4.17), (4.18), and (4.41).

Denote

(i)

Aq = <j (k,j,1i,...,1k) : k + j + = q; k,j, 1i,...,1k = 0,1,

p=i

(G^ ...(5 ((ik )Lj R(x,t)

=f (k)(5ii ...(5ikLjR(x,t),

il,...,ik = 1

Lj R(x,t) =f {

. .^LR{x,t) forj>l

R(x,t)

for j = 0

Theorem 4.3

(also see [1]-[16], [51], [146]). Let conditions (i*)-(iv*)

be satisfied. Then for any s,t G [0,T] such that s > t and for any positive integer r, the following expansion takes place w. p. 1

k

m

R(xs, s) =

«<*.*)+£ E -far1 E

+ (Dr+i)a,t, (4.49)

where (Dr+1)st is defined by (14.4,3). Proof. We claim that

£ (pq D ...Da 1 R(xt,t) (pq 'JA, ...A 1 )s,t = (A, ,...,A 1 )eMq

( _ ,)j m

= E iixL E fa-'-favR^utfax:' <4-5°)

(k,j,/l,...,/fc)GAg ' i 1,...,ifc = 1

w. p. 1. The equality (4.50) is valid for q = 1. Assume that (4.50) is valid for some q > 1. In this case using the induction hypothesis we obtain

E 1 'Da 1... Da,+1 R(xf, t) 1 <pq+1 >J,*a 1...v 1 w =

(A,+ 1 ,...,A 1 )GM,+ 1

E f E i(P'+1,DA1.. ■ Da,+1 R(x„i)«"■>J(V.,, JA-

Vq+1G{1, 0} i (A,,...,A1)eMq\ J

Aq+1 ,

aw, =

E / E

Aq+1G{1, 0} t (kj,/1,...,/fc)eA,

-n—x

J!

x [ (k+A,+1 'G/1 ■ ■ ■ g/kLDa,+1 R(xt, t) k (k)//UsJ Aq+1 dw, =

E • • • GlkP+1R(*ut) k j

■. s

+ ... G\^G0B(x(, t) ' [ faHm* 1 1

-------- — - j j\ («I)

w. p. 1.

s

Using Lemma 4.2, we obtain

(.e-ty

■{k)I* d0 =

j!

t

j ! n---lke,t

(s - t)j+1 for k = 0

(j + 1)!

(s - t)3+1 • (k)lUst - (-l)j+1 • ik+i+i. t for k > 0

(4.52)

w. p. 1. In addition (see (4.47)) we get

f (^ — t.y J*(i1...ik) 1) _ (—1 )J J*(h...ikik+1) ggX

J j\ h---he,t 9 j\ h--lkjs,t ^ ' '

t

in the notations just introduced. Substitute (4.52) and (4.53) into the formula (4.51). Grouping summands in the obtained expression with equal lower indices at iterated Stratonovich stochastic integrals and using (4.48) and the equality

1 P !

G«fi(x,i) = ^£(-1 refill"R(x,t), c; = ^¿zr^y (4.54)

(this equality follows from (4.48)), we note that the obtained expression equals to 3

j

(k,j,l1,...,lk)eAq+1

w. p. 1. Summing the equalities (4.50) for q = 1, 2,...,r and applying the formula (4.42), we obtain the expression (4.49). The proof is completed.

Let us order terms of the expansion (4.49) according to their smallness orders as s 11 in the mean-square sense

R(xs, s) =

r __( _ t)j m

q=l (h,j,h,...,lk )GDq ii,...,ih = 1

+ (Hr+i)st w. p. 1, (4.55)

s

1

where

( _ j m

№+ !)„,= E -far1 E

(k,j,/i,...,/fc )eUr ii,...,ifc=i

(k, j, li,..., Ik) : k + 2 + ^ Zp^

k

Dq = <J (k, j, Zi,.. ., Zk) : k + 2 I j + V Zp I = q; k, j, Zi,.. ., Zk = 0,1,.. . ¡> ,

(4.56)

k

Ur = < (k, j, Zi,..., Zk) : k + j + Y Zp < r,

i p=i

k + 2 + Zp^ > r + 1; k,j, Zi,...,Zk = 0,1,...J, (4.57)

and (Dr+i)st is defined by (14.43). Note that the remainder term (Hr+i)st in (4.55) has a higher order of smallness in the mean-square sense as s | t than the terms of the main part of the expansion (4.55).

4.8 The Second Form of the Unified Taylor—Stratonovich Expansion

Consider iterated Stratonovich stochastic integrals of the form

j/ir:*=/ (s-tk)/k... j (s-ti)11 <;i)...for k>i

t t

and

j;Vfc) = i for k = 0,

where ii,..., ik = 1,..., m.

The additive property of stochastic integrals and the Newton binomial formula imply the following equality

/1 /: k

i/S =£ • •. £ IK* (t - s)/i+-■ ■+"-ji-■ ■ j j ic> w. p. l, (4.58)

ji=0 j: =0 g=i

where

nk — 1 ~ k\(l-k)\

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is the binomial coefficient. Thus, the Taylor-Stratonovich expansion of the process = R(xs,s), s G [0,T] can be constructed either using the iterated stochastic integrals I/*^1/.^ similarly to the previous section or using the iterated

stochastic integrals J^/k^. This is the main subject of this section.

Denote

J

= ( il...ik)

/1-/ks,t

=f (k) j

/1 ■■■/kst ;

il,...,ifc = l

Lj 5 (il) ...(5 (ik )R(x,t)

def

(k)Lj (5/1 ...(5/k R(x,t).

ii,...,ik=i

Theorem 4.4 [144] (also see [1]-[16], [51], [146]). Let conditions (i*)-(iv*) be satisfied. Then for any s,t G [0,T] such that s > t and for any positive integer r, the following expansion is valid w. p. 1

R(xs, s) =

R(xt ,t) + E E

(s-ty j!

£ Lj (5 ((i1)...(( ((ik)R(xt,t)J/

(ik)

r*(i1...ik)

/1.../k

+

q=l (k,j,h,-,h)eAq ii,--,ifc=1

+ (Dr+1 )s,t 5

where (Dr+1 )st is defined by (14.4,3).

Proof. To prove the theorem, we check the equalities

(s - t)j m

(4.59)

E

j !

^ Ljo¡^... ai;k)R(xt, t)j/;(i^ik)

(k,j,/1,...,/k )GAq i1,...,ik = 1

(s - i)j ^

E

E <5il0..-<5ikk)LjR(xt.t)/^:1 w.p.1

(k,j',/1,...,/k)GAq

j !

v(ik) Lj lk

(i1...ik) 1...lks,t

i1,...,ik = 1

(4.60)

for q = 1, 2,...,r. To check (4.60), substitute the expression (4.58) into the right-hand side of (4.60) and then use the formulas (4.48), (4.54).

Let us order terms of the expansion (4.59) according to their smallness orders as s 11 in the mean-square sense

m

,i) + E E

R(xs, s) =

(s - t)j m

j !

E L G

i i,...,ifc = 1 + (Hr+l)s,i w. p. 1,

where

(Hr+1)s,i =

E

(k,j,/ i,...,/fc )eU,

(s - t)

E L G

i i,...,ifc=1

(i l ) /

G((ik )R(xt ,t)j/;(i-.ik )

l1.../fcs,i

+

+ (Dr+i )

s,t '

The remainder term (Dr+i)st is defined by (14.43): the sets Dq and Ur are defined by (4.56) and (4.57), respectively. Finally, we note that the convergence w. p. 1 of the truncated Taylor-Stratonovich expansion (14.42) (without the remainder term (Dr+1)st) to the process R(xs,s) as r fa to for all s,t G [0,T] such that s > t and T < to has been proved in [79] (Proposition 5.10.2). Since the expansions (14.49) and (14.59) are obtained from the Taylor-Stratonovich expansion (4.42) without any additional conditions, the truncated expansions (4.49) and (4.59) (without the reminder term (Dr+1)s t) under the conditions of Proposition 5.10.2 [79] converge to the process R(xs,s) w. p. 1 as r fa to for all s, t G [0, T] such that s > t and T < to.

r

m

4.9 Comparison of the Unified Taylor—Itô and Taylor-Stratonovich Expansions with the Classical Taylor— Ito and Taylor-Stratonovich Expansions

Note that the truncated unified Taylor-Ito and Taylor-Stratonovich expansions contain the less number of various iterated Ito and Stratonovich stochastic integrals (moreover, their major part will have less multiplicity) in comparison with the classical Taylor-Ito and Taylor-Stratonovich expansions [140 .

It is easy to notice that the stochastic integrals from the sets (4.4), (4.5) are connected by linear relations. However, the stochastic integrals from the sets (4.6), (4.7) cannot be connected by linear relations. This also holds for the stochastic integrals from the sets (4.8), (4.9). Therefore, we will call the sets (4.6) (479) as the stochastic bases.

Let us call the numbers rankA(r) and rankD(r) of various iterated Itô and Stratonovich stochastic integrals, which are included in the sets (14.6)-(1479) as the ranks of stochastic bases when summation in the stochastic expansions is performed using the sets Aq (q = 1,..., r) and Dq (q = 1,..., r) correspondingly. Here r is a fixed natural number.

At the beginning, let us analyze several examples related to the Taylor-Ito expansions (obviously, the same conclusions will hold for the Taylor-Stratonovich expansions).

Assume that the summation in the unified Taylor-Ito expansions is performed using the sets Dq (q = 1,... ,r). It is easy to see that the truncated unified Taylor-Ito expansion (4.33), where the summation is performed using the sets Dq when r = 3 includes 4 (rankD (3) = 4) various iterated Itô stochastic integrals

I (i 1 ) I (i 1Ï2) I (i 1 ) I (i 1Ï2Ï3) J0s,t , J00s,t , J1s,t , J000s,t •

The same truncated classical Taylor-Ito expansion (4.24) [79] contains 5 various iterated Ito stochastic integrals

t(h) t (Ï1Ï2) t(H0) t(0Ï1) t (Ï1Ï2Ï3) J(1)s,t' J(11)s,t' J(10)s,t' J(01)s,t ' J(111)s,f

For r = 4 we have 7 (rankD(4) = 7) stochastic integrals

I(i1) r(i1i2) /-(Ù) I(i1i2i3) r(i1i2) I(Ï1Ï2) I(i1i2i3i4) J0Sji, J00Sji , J1Sji, J000Sji , J01M , J10Sji , J0000Sji

against 9 stochastic integrals

T (i1 ) T (i1i2) T (i 10) T (0i 1 ) t (Ï1Ï2Ï3) j (i10i3) j (i1i20) r(0i1i2) t (i 1 i2i3i4 )

J(1)s,t ' J(11)s,t ' J(10)s,t ' J(01)s,t ' J(111)s,t ' J(101)s,t ' J(110)s,t ' J(011)s,t ' J(1111)s,t '

For r = 5 (rankD(5) = 12) we get 12 integrals against 17 integrals and for r = 6 and r = 7 we have 20 against 29 and 33 against 50 correspondingly.

We will obtain the same results when compare the unified Taylor-Stratonovich expansions [144] (also see [1]-[16], [51], [146]) with their classical analogues 79], [140] (see previous sections).

Note that the summation with respect to the sets Dq is usually used while constructing strong numerical methods (built according to the mean-square criterion of convergence) for Ito SDEs [77], [79] (also see [13]). The summation with respect to the sets Aq is usually used when building weak numerical methods (built in accordance with the weak criterion of convergence) for Ito SDEs [77], [79]. For example, rankA(4) = 15 while the total number of various iterated Ito stochastic integrals (included in the classical Taylor-Ito expansion [79] when r = 4) equals to 26.

Let us show that [3]-[16], [51]

rankA(r) = 2r — 1.

Let (/l5..., lk) be an ordered set such that l1,..., lk = 0,1,... and k =

def

1, 2,... Consider S(k) = l1 + ... + lk = p (p is a fixed natural number or zero). Let N(k,p) be a number of all ordered combinations (l1,...,lk) such that l1,..., lk = 0,1,..., k = 1, 2,..., and S(k) = p. First, let us show that

N (k,p) = C-kU

where

n !

_

n

m!(n — m)!

is a binomial coefficient.

It is not difficult to see that

N (1,p) = 1 = Cp—J—1, N (2,p) = p +1 = Cp—1—1,

Moreover,

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N(k + 1,p) = £ N(k, l) = £ C/k+—k— 1 = Ck+k,

/=0 /=0

where we used the induction assumption and the well known property of binomial coefficients.

Then

rankA(r) =

= N(1,0) + (N(1,1) + N(2, 0)) + (N(1, 2) + N(2,1) + N(3, 0)) + ... ... + (N(1, r — 1) + N(2, r — 2) + ... + N(r, 0)) =

= C + ( c0 + C1 ) + ( C20 + C2 + C2 ) +...

... + (Cr_ 1 + Cr _ 1 + Cr_ 1 + ... + Cr—1 ) = = 20 + 2 1 + 22 + ... + 2r—1 = 2r — 1.

p

Let nM(r) be the total number of various iterated stochastic integrals included in the classical Taylor-Ito expansion (14.22) [79], where summation is performed with respect to the set

r

UMk.

k=1

If we exclude from the consideration the integrals, which are equal to

(s - i)j

then

j !

ПМ (r) =

= (21 - 1) + (22 - 1) + (23 - 1) + ... + (2r - 1) =

= 2(1 + 2 + 22 + ... + 2r-1 ) - r = 2(2r - 1) - r

It means that

lim ^L = 2.

r

rankA(r) Numbers

rankA(r), nM(r), f (r) = nM(r)/rankA (r)

for various values r are shown in Table 4.1. Let us show that [3]-[16], [51]

rankD(r) =

fr-1 (r-1)/2+[s/2]

E

s=0

E C

for r = 1, 3, 5,

/=s

=

(4.61)

r-1 r/2-1+[(s+1)/2]

E E

s=0 /=s

C/s

for r = 2, 4, 6, . . .

where [x] is an integer part of a real number x and C^ is a binomial coefficient. For the proof of (4.61) we rewrite the condition

k + 2(j + S(k)) < r,

def

where S(k) = 11 + ... + (k, j, 11,..., = 0,1,...) in the form

j + S(k) < (r - k)/2

Table 4.1: Numb er s rank a (r), ??.m (r), / (r) = ??.m (r) / r ankA (r)

r 1 2 3 4 5 6 7 8 9 10

rankA(r) 1 3 7 15 31 63 127 255 511 1023

nM(r) 1 4 11 26 57 120 247 502 1013 2036

/(r) 1 1.3333 1.5714 1.7333 1.8387 1.9048 1.9449 1.9686 1.9824 1.9902

and perform the consideration of all possible combinations with respect to k = 1,..., r. Moreover, we take into account the above reasoning.

Let us calculate the number nE (r) of all different iterated Ito stochastic integrals from the classical Taylor-Ito expansion (4.24) [79] if the summation in this expansion is performed with respect to the set

r

U

q,k=l

The summation condition can be rewritten in this case in the form

0 < p + 2q < r,

where q is a total number of integrations with respect to time while p is a total number of integrations with respect to the Wiener processes in the selected iterated stochastic integral from the Taylor-Ito expansion (4.24) [79]. At that the multiplicity of the mentioned stochastic integral equals to p + q and it is not more than r. Let us rewrite the above condition (0 < p + 2q < r) in the form: 0 < q < (r — p)/2 ^ 0 < q < [(r — p)/2], where [x] means an integer part of a real number x. Then, performing the consideration of all possible combinations with respect to p = 1,..., r and using the combinatorial reasoning, we come to the formula

r [(r—s)/2]

nE (r) = £ ^ Cf(r—s)/2]+s—l, (4.62)

s=1 l=0

where [x] means an integer part of a real number x.

Numbers

rankD(r), nE(r), g(r) = nE(r)/rankD(r) for various values r are shown in Table 4.2.

Table 4.2: Numbers rankü(r), nfar), g{r) = 'rtE(V)/rankD(r)

r1 2 3 4 5 6 7 8 9 10

rankD(r) 1 2 4 7 12 20 33 54 88 143

nE(r) 1 2 5 9 17 29 50 83 138 261

g(r) 1 1 1.2500 1.2857 1.4167 1.4500 1.5152 1.5370 1.5682 1.8252

4.10 Application of First Form of the Unified Taylor— Ito Expansion to the High-Order Strong Numerical Methods for Ito SDEs

Let us rewrite (4.33) for all s,t G [0,T] such that s > t in the following form

R(xs, s) =

r ( _ t)j m

= *(*.*) + £ E far1 E

q=l (k,j,h,...,lk)GDq ii,-..,ik = 1

(s _ t)(r+l)/2

+ 1 wW('(r +'1)/2), L^R{x,,t) + (Hr+1)it w. p. 1, (4.63)

where

_ (s — t) (r+1)/2

(H>'+l)s,t = (Hr+l)s,t - 1{r=2d-l,dGN} + L{r+1)/2R{xt,t).

Consider the partition {tp}N=0 of the interval [0,T] such that

0 = t0 < t1 < ... < tn = T, An = max |rj+1 — Tj | .

0<j<N—1 J J

From (4.63) for s = Tp+1, t = Tp we obtain the following representation of explicit one-step strong numerical scheme for the Ito SDE (4.1), which is based on first form of the unified Taylor-Ito expansion

q=l (k,j,h,...,lk)eDq il,...,ifc = 1

I -I r {Tp+1 -Tp)(,+1)/2 (r+1)/2 (AKA\

+ l{r=2d-l,dGN} l)/2)! ^ ^

r

where /(i1 •/¿k) is an approximation of iterated Ito stochastic integral

il • • • lkTp+i,Tp

1 • i¿k' of the form

11 • • • lkTp+1,Tp

S ¿2

•£.') = A* - tk )lk...( (t - ii)'1 f;1'.. .<ik

Note that we understand the equality (4.64) componentwise with respect to the components yP;) of the column yp. Also for simplicity we put Tp = pA,

A = T/N, T = TN, p = 0,1,..., N.

It is known [79] that under the appropriate conditions the numerical scheme

(4.64) has strong order of convergence r/2 (r G N).

Let Bj (x, t) is the jth column of the matrix function B(x, t).

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Below we consider particular cases of the numerical scheme (4.64) for r = 2,3,4, 5, and 6, i.e. explicit one-step strong numerical schemes for the Ito SDE (4.1) with the convergence orders 1.0, 1.5, 2.0, 2.5, and 3.0. At that for simplicity we will write a, La, B^, G^Bj, ... instead of a(yp,Tp), La(yp, Tp), Bi(yp,Tp), G0i)Bj(yp,tp), ... correspondingly. Moreover, the operators L and i = 1,... ,m are determined by the equalities (4.17), (14.18).

Scheme with strong order 1.0 (Milstein Scheme)

yp+1 = yp + £ B.14:p+1,Tp + Aa + £ G«>2)B,11'-1',Tp. (465)

¿1 = 1 ¿1,«2 = 1

Scheme with strong order 1.5

m

yp+1 = yp + £ B„/«:p+1,Tp + Aa + £ G«>2)B,1 /«-^p +

¿1 = 1 ¿1,i2 = 1

+ V [G^a (A/«'"1' + /(i1) ) - LB;1 if'

L 0 V Urp+1'Tp 1tp+1,tp / 1 1rp+1,Tp_

+

I Y^ G(;3)G(i2 )R / (i3i2i1) , + G° G° ^¿1 J°0°rp+1,rp +

¿1,i2,i3 = 1

m

A2 + ~2~

(4.66)

Scheme with strong order 2.0

yp+i

yp + £ B,itL + Aa + £ g«'2»b>1 ^+

¿1=1

¿1,«2 = 1

+ £ [G<'l)a (A/0

i1=1

(il) + /1(il) ) - LBil/(il) Tp+1,Tp 1rp+1>Tp/ 1 1rp+1>Tp_

+

A2

I \ rii't-V riin)-T) f{'<3'<2n; I — T ^ I

+ Z^ 0 0 £*ii'/oooT + —

¿1,«2,«3 = 1

TP+1-Tp 2

m

+ £ [g0'2)lb,1 (/

(¿2H) 10

(¿2*1)

¿1,«2 = 1

'p+1>TP

01

'p+1>Tp

T ri(i2) U T (i2i1) 1 - LG0 Bi1 J1a +

p+1,Tp

+G0i2)G0i1)a f/S12i1) + A/((02i1)

0 0 v 01tp+1,tp 00tp+1,tp

+

V^ G(i4)G(i3)G(i2) B f(i4«3i2«1) + G0 G0 G0 Bi11 0000Tp+1,Tp •

i1,i2,i3,i4 = 1

(4.67)

Scheme with strong order 2.5

yP+1 =

yp + £ b„ + Aa + £ G0i2)B,1 /S0r;+)1.„+

i1=1

i1,i2=1

+ ^ [G0i1)a (A /T +/

i1=1

0;i) +/1il) ) - LBi1/iil)

0tp+1.tp 1rp+1.r^ 1 1rp+1.rp

+

+ £ G0i3)G0i2)Bi1/(

i1,i2,i3 = 1

A2

00Vi^P + T

m

+ £ [Gf »LB. (/

(i2i1) 10

- /,

( i 2 i 1)

i1,i2=1

'p+1>Tp

01

Tp+1.Tp

Tn(i2)U T (i2i1) I — LG0 Bi11 10 +

tp+1.tp

I rf(i2)G(i1)„ ( T(i2i1)

+G0 G0 a 01T .T

0 0 v 01tp+1,tp

+ A /0i2i1)

00

Tp+1.Tp

+

m

(i4)^f(i3)^f(i2) R f(i4i3i2il) | 0000T +

i R f + Z^ G0 Go Go Bii h

il,i2,i3,i4 = 1

G^Lsl (

+£

il=1

+\llbul

1

(il)

JTp+1>Tp A2

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(n) +Af(<i) i +

______—LG^ali^

2 2Tp+i_Tp 0 ^ 2Tp+1,Tp

-r^(Ti)

2tp i 1 .tp 1 T~p+1}T"p

+ A/}

+

+ £

i1,i2,i3 = 1

r^(i3) r r^(i2) n ( f (i3i2i1)

G0 LG0 Bi1 |v/100tp+1,TP

f (i3i2i1) . I — h010 ) +

tp+1.tp

I /^(i3)^(i2) r D ( f (i3i2i1) f (i3i2i1) \ ,

+G0 G0 LBi1 l/010Tp+1,Tp — /001tp+1,tJ +

I ^feW^WuK / A f(i3i2i1) I f(i3i2i1)

+G0 G0 G0 a I A/000T , _ Tp + h001

l000

Tp+1.Tp

■r(i3Wi2) R f(i3i2i1)

_ /-/^(i3)^(i2) D f LG0 G0 Bi1 J100-

"p+1>Tp_

+

\ " g^G^G^G^) R f(i5i4i3i2i1) I + G0 G0 G0 G0 Ri1 J00000 +

¿1,i2,«3,i4,«5 = 1

Tp+1.Tp

A3

H--LLsl.

6

(4.68)

Scheme with strong order 3.0

yp+1=yp +£ Bi1 /aTp+„Tp + Aa + ^ Ga'2)Bi1 /0aT;;+„Tp+

i1=1

i1,i2 = 1

+ £ [G0"»a (A/0

i1=1

(i1) + f1(i1) ) — LBi 71(i1)

Tp+1.Tp 1tp+1.T^/ 1 1tp+1.tp

+

+ E G0i3)Gai2)Bi, /,

i1,i2,i3 = 1

A2

(«3)^2) R f(i3«2H) 1 ^ 7-0,

000 "r -^«."r

000tp+1.tp 2

m

+ £ [Gf »LB« (/

(i2i1) 10

-l

( i 2 i 1)

i1,i2 = 1

Tp+1.Tp

01

Tp+1.Tp

rG(i2)R f (i2i1) 1 — LG0 Ri1 h10 +

Tp+1.Tp

m

where

+G0i2)G0i1)a (/(i:2i1) + A /(02i1)

0 0 V tp+1.tp 00tp+1,tp

+

+ e G0i4)G0i3)G0i2)Bi1 tet+%+1,p+rp+1,p, f^.^

ii,i2,«3,i4 = 1

qp+1,p

E

i1=1

(i1) A2 (i1)

GMLa. I - ffl} + Air' + J +

0 1 2 2tp:1,tp 1tp:1,tp 2 0tp:1,tp /

+-LLBiJiil) -LG^'alPo

2 i1 2rp:1.rp 0 V 2Tp:1.rp

(i1) (i1)

+A

(i1)

1tp:1,tp

+

i1,i2,i3 = 1

( T(i3i2i1) G0 LG0 Bi1 ^1 100rp:1,rp

/(i3i2i1) . I - 010 +

tp:1.tp

I G(i3)G(i2) T R f / (i3i2i1) / (i3i2i1) \ I

+G0 G0 LBi1 I 1 010rp:1,rp - 1 001rp:1,rJ +

I G(i3^(i2^(i1) _ f A / (i3i2i1) 1 / (i3i2i1) +G0 G0 G0 a 1000rp:1,rp + 1 001rp:1,rp

Y(i3)G(i2) D T (i3i2i1)

_ T G(;3)G(:2) R T

LG0 G0 Bi1J 100

Tp:1>Tpi

+

and

1 ri(i5)ri(i4)ri(i3)ri(i2) U T

+ G0 G0 G0 G0 Bi1

i1,i2,i3,i4,i5 = 1

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A3

H--LLa,

6

(i5i4i3i2i1) + 00000rp:1,rp +

r

p+1,p

E

i1,i2=1

r _ I 1 f(«2H) Go Go 2 o\+1,rP

+ A 7((i2i1)

01-p+i,rp + 2 0(WP 1

_l rrr^^R 7-(*2H) 1 2 0 1 20tp:1,tp

+G0i2)LG0i1)a (7|12i1)

(i2i1)

>:1.Tp

02

p:1.Tp

+ A (/i02i1)

-1,

(i2 i1)

>:1.Tp

01

>:1.Tp

+

+LG0i2)LBi1 (/112i1)

0 1 v 11tp:1,tp

- 1202i1) ) + 20tp:1,tp/

m

+dS')LLBil ( „ + „ - ¡fa

1)

- LGWa A/<;?"» T. + /¡if'

Jrp+1,TP -LJ-Tp+l>TP

+

»1,i2,i3,«4=l

^mW^W^W^K / A f(i4i3i2i1) i T(i4i3i2i1) \ ,

Go Go Go Go a 1 ooooTp+1,Tp + 1 oooiT.+1iTp I +

I g(Ï4)G(Ï3) LG(Î2) R ^ T (¿4«3i241) _ T (¿4«3i241)

+Go Go LGo RH ^1 oiooTp+1iTp 1 ooioTp+1iTp

_ L G(i4)G(i3)G(i2) R T (i4i3i2i1) +

LGo Go Go Ri11 ioooTp+1iT. +

+ G(Ï4) LG^G^) R T (i4i3i2i1 ) _ T (M^Ù) \ +

+Go LGo Go Ri1 ^1 ioooTp+1,Tp 1 oiooTp+1,T l +

+ G(i4)G(i3)G(i2) LR ^ T (¿4Î3«2«1) _ T (i4«3«2«1)

+Go Go Go LRn ^1 ooioTp+1,Tp 1 oooiTp+1,Tp

+

I \ A G^^^G^G^G^) D T (i^M^Ù)

+ Go Go Go Go Go Ri11 (

¿1,i2,i3,«4,i5,«6=i

1)

o Go Go Go Go Ri11 ooooooT1,T. •

It is well known [79] that under the standard conditions the numerical schemes (I4.65I)-(I4691) have strong orders of convergence 1.0, 1.5, 2.0, 2.5, and 3.0 correspondingly. Among these conditions we consider only the condition for approximations of iterated Ito stochastic integrals from the numerical schemes (4.65)-(4.69) [79] (also see [13])

M{( if 1 • ;• ik) - 1-;• ik) ^ 1 < C Ar+i, (4.70)

| l ;1.../fcrp+1,Tp / V '

where constant C is independent of A and r/2 is the strong convergence orders for the numerical schemes (4.65I)—(4769), i.e. r/2 = 1.0,1.5, 2.0, 2.5, and 3.0.

As we mentioned above, the numerical schemes (4.65) (4769) are unrealizable in practice without procedures for the numerical simulation of iterated Ito stochastic integrals from

In Chapter 5 we give an extensive material on the mean-square approximation of specific iterated Ito stochastic integrals from the numerical schemes (I4.65)-(I4jB9). The mentioned material based on the results of Chapter 1.

m

4.11 Application of First Form of the Unified Taylor— Stratonovich Expansion to the High-Order Strong Numerical Methods for Ito SDEs

Let us rewrite (4.55) for all s,t G [0,T] such that s > t in the following from

R(xs, s) =

r__( _ t)j m

+ 1

(s _ t)(r+1)/2

1 ; wo, ¿'r+1)/2^> + w.p.1, (4.71)

where

_ (s __ £)(r+1)/2

Consider the partition {tp}N=0 of the interval [0,T] such that

0 = t0 < ti < ... < tn = T, AN = max |t?-+1 — t?- | .

0<j<N—1 7 7

From (4.71) for

s = Tp+i, t = Tp we obtain the following representation of explicit one-step strong numerical scheme for the Ito SDE (4.1), which is based on first form of the unified Taylor-Stratonovich expansion

r / xj m

yP+i=y,+E E ^p1 E ■ ■ ■ <vyP +

q=1 (kj,/i,...,zfc )eDq ii,...,ifc=1

-Li (Tp+1 ~rp)('+1)/2 r(r+l)/2,r M 79\

+ l{r=2rf-m<en} ((r + 1)/2)! ^ ^

where 1'z*L(ilz*fe'ife) is an approximation of iterated Stratonovich stochastic integral /;(Vifc) of the form

* S ^ ¿2

(H..ifc) _ / /, , Nlfc / (. . 7.p(i1) Jf(ifc)

err = I (t-tk)/k... / (t-ti)11 <$;»...f

Note that we understand the equality (4.72) componentwise with respect to the components y^ of the column yp. Also for simplicity we put Tp = pA,

A = T/N, T = tn, p = 0,1,..., N.

It is known [79] that under the appropriate conditions the numerical scheme

(4.72) has strong order of convergence r/2 (r G N). Denote

m

j=i

where Bj(x,t) is the jth column of the matrix function B(x,t). It is not difficult to show that (see (4.41))

o n n o n

Zfl(x,t) = ^(x,t) + £&U>(x,t)^(x,t), (4.73)

j=l

where a(j)(x,t) is the j th component of the vector function a(x,t).

Below we consider particular cases of the numerical scheme (4.72) for r = 2, 3, 4, 5, and 6, i.e. explicit one-step strong numerical schemes for the Ito SDE (4.1) with the convergence orders 1.0, 1.5, 2.0, 2.5, and 3.0. At that, for simplicity we will write a, La, La, B^, Gg^Bj, ... instead of a(yp, tp), La(yp, tp), La(yp,Tp), Bj(yp,Tp), G0)i)Bj(yp,Tp), ... correspondingly. Moreover, the operators L and g)^, i = 1,... ,m are determined by the equalities (I4.17), (I4.18), and (4.73).

Scheme with strong order 1.0

yp+i = y + £ EHî«^ + Aa + £ G^Ri! jo«1),,. (4.74)

i1 = i i 1 ,i2 = i

m

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m

Scheme with strong order 1.5

yp+i = yp + £ Bi1+ Aa + £ G«i2»s!1 +

ii=i

i1,i2=i

+ £ |G«i1)a(A +1

(i1)

i1=i

;(i1) )- l

+

m

I G(i3)G(i2)R T ;(i3i2i1) ,

+ G0 G0 000Tp+1iTp +

¿1,«2,«3 = 1

A2

+ —La. (4.75)

Scheme with strong order 2.0

yp+1 = yp + £ Bi1 ^ + Aa+ £ G«'2»B,1 i**1^ +

¿1 = 1 ¿1,«2 = 1

m

+£ [G4i1'&(A ^ + — L b,1/;(i1)

¿1=1

+

0 * 0rp+1.rp 1tp+1>tp/ i1 1rp+1.rp_

m a 2

I Y^ G(«3)rf(«2)R f*(* 3«2«l) , ^ ,

+ £MoooTp+1,Tp + "2"^+

i 1 ,¿2 ,¿3 = 1

m

+ £ [g0,2)ZBi1 (tO:2^ — ^p;),) — LG<;2>B., tO^+

¿1,«2 = 1

+GOi2 )cOi1)af /O;(i2i1) +A /O;Oi2i1) 1

0 o * o1tp+1.tp

+

+ GO GO GO Bi11 OOOOTp+1iTp. (4.76)

¿1,«2,i3,«4 = 1

Scheme with strong order 2.5

m

yp+1 = yp + £ Bi1 + Aa + £ g0'2»B,1 +

¿1 = 1 ¿1,«2 = 1

m

+ V [GOn)a (A TO^ + ) — L

«1=1

+

m A 2

+ £MoooTp+1,Tp + ~2 a+

¿15*25*3=1

m

+ £ [gq'2)lbn (/**<;

¿1,«2 = 1

:('2'l) _ J*('2'l) \ _ Lg('2)B' /*('2'l) +

"p+l.Tp 01Tp+1,r^ 0 '1 10Tp+i,Tp

+G«'2)G<'l)a (/olT2'1»TP +A for')

p+1, TP

p+1,TP / _

+

('4^('3^('2^ f*('4'3'2'l) + 0000T +

I Y^ G('4 R f

+ G0 G0 G0 B'lJ (

'1,'2,'3,'4 = 1

Tp+1, TP

+

'1=1

Gq^Lbl (ll<n)

+A/;(il) +—/o^ ) +

2 2tp+1 >tp 1tp+1 >tp 2 0tp+i 'tp )

A2

K'l)

2 1 2rf+i'Tf

- LG)'l)af/*('l) + A/;^'l)T

0 \ 2Tp+l,Tp 1TP+1,TP

+

m

+ G0 LG0 B'l ^ 100Tp+l,Tp 010td+i,tJ +

'1,'2,'3 = 1

°TP+1.TP /

,g('3^('2) T r Z' f *('3'2'1) M'3'2'0 \ ,

+G0 G0 LB'l ^ 1 010Tp+l,Tp - 1 001tp+i,tJ +

iG^W^W'l) T ^ A f*('3'2'l) I fK'3'2'0 \

+G0 G0 G0 a I A 1000T , _ Tp + 1 001T , _ Tp J

TP+1>TP

TP+1>TP /

y('3^('2^ f*('3'2'l)

('3)G('2) D f : LG0 G0 B'lJ 100-

P+1>TP_

+

I V^ G('5)r,('4)r,('3)r'('2) R I

+ / J G0 G0 G0 G0 B'11 Q0000 +

'1,'2,'3,'4,'5 = 1

JTp+1>TP

A3

H--LLSL.

6

(4.77)

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Scheme with strong order 3.0

yP+1

yp + E B,l/0'Tp:»Tp + Aa+ £ G«'2)B'lI*)'

'1 = 1

Tp+1'TP

('2) 0

K'2'l)

' 1 ,'2 = 1

"p+1>TP

+

+ £ [G<'l)a(A f

'1 = 1

('l)

+

= ('l)

K'l)

1 i — L B'! 1-1

Tp+1,TP 1Tp+1>TP/ 1 1tp+1>tP

+

m

where

and

, V^ G(«3W«2) R f*(«3«2H) , — /-=,

+ Go Go BnJoooT + —

A2,

«1,«2,¿3 = 1

Tp+1,TP

2

m

+ £ [g0'2)LB,1 (;;<

:(¿2¿l) — ^(^¿l) ^ — LG^R. f^) +

¿1,42 = 1

Tp+1>TP

Tp+1>TP /

'O 1O

Tp+1>TP

+g0'2 )G<1!l»a( №•'>+a TOri.)

Tp+l > T2 Tp+l> TP /

(¿4) ^^(¿3) ^^(¿2) R T ;(¿4¿3«2¿l)

+

I r^MW^WM R T

+ GO GO GO ^¿l1 (

«1,*2,*3,*4 = 1

OOOOTp+l , Tp + qp+1,p + rp+1,p,

(4.78)

Op+bp =

¿1=1

G^La (

1

rO

2 p+l, tP

+ A T^0

a2 t=

<¿1)

+izZB,1/*<")

— L GO")a( /* (,l) +A T^1'

1Tp+l,Tp

+

m

+ GO L GO ^¿l^ 1OOTp+l,Tp — 1 O1OT2+l,TJ +

¿1,«2,¿3 = 1

°Tp+l.Tp /

+GO GO I 1 O1O — 1 OO1 I +

¿11 JO1O

Tp+1>TP

cOO1

Tp+1>TP /

I _ / A f*(i3i2i1) I T^^^¿l^) \

+GO GO GO a I A 1OOO + 1 OO1 I

OOVl,rp ^ OO1Tp+ljTp y

_ _ G(i3)G(i2) R T ^^^¿l^)

LGO GO RHJ 1OOrp+l,TP

+

I Y^ r^W^W^WM R T + GO GO GO GO ^l1 (

*1,*2,¿3,¿4,¿5 = 1

A3--+—LLa, 6

;(¿5¿4¿3¿2¿l) +

OOOOOT +

Tp+1>TP

r

p+1,p

£

(^Wil) f = I 1 f*(*2«l)

Go Go 2

+ A /;(i2il)

a 2

, ^ f*(«2«l) I _ Jn

01-P+1.-P ^ 2 JQ(WP '+

m

1

+-LLG[;2)BUL

i1J 2o

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^ ( i 2 i 1 ) Tp+1,Tp

+

+Goi2)L Goi1)af Tf2i1)

(i2i1)

>+1>Tp

-îoT1; +A Tio o2Tp+1,Tp v io

(i2i1)

-I,

=(i2i1)

+LGf)L5,1 (- T.

T.+1,Tp (i2i1)

0i

>+1>TP

+

1 Mioil) 1

Tp+1,Tp 2 Tp+1'rp

-I

(i2i1)

ii

1) ,+1,Tp

+

-L Goi2)Goi1)a(A Tfa 2H

(i2i1)

T.+1,Tp

+

(i2i1)

ii

T.+1,Tp

+

i1,i2,i3,i4 = i

G(i4)G(i3)G(i2)G(i1)^ ( A f*(i4i3i2i1) , f*(i4i3i2i1) I ,

Go Go Go Go a I A 1 ooooT , + 1 ooou I +

p+1,Tp

Tp+1,Tp

I g(Î4)G(Î3) T G(,2) R ^ f*(i4i3i2i1) _ T*(i4i3i2i1)

+GQ Go LGo R^ I 1 omn 1 n

0 i1 0i00

Jrp+1>Tp

cooio

Tp+1,Tp

_TG (i4^(i3^(i2^ î *(i4Î3i2Ï1) I

LGo Go Go Ri11 iooo +

Tp+1,Tp

_i_g(Î4) rG(Ï3)^(Ï2)p / î *(i4i3i2i1) î*(i4i3i2i1) i ,

+ Go LGo Go Ri1 ^1 ioooT_1T. 1 oiooT.+1,T. I +

0 0 i1 i000

_l_G(i4)G(i3)G(i2) T R It

+Go Go Go LRii I 1 (

(i4i3i2i1) (i4i3i2i1)

— J<

i1 00i0

Tp+1,Tp

oooi

Tp+1,Tp

+

E

'(Ï6)g(Ï5)G(Ï4)G(Ï3)G(Ï2) R T*(i6i5i4i3i2i1)

o Go Go Go Go RH1 ooooooT

i1,i2,i3,i4,i5,i6=i

Tp+1,Tp

It is well known [79] that under the standard conditions the numerical schemes (4.74)—have strong orders of convergence 1.0, 1.5, 2.0, 2.5, and 3.0 correspondingly. Among these conditions we consider only the condition for approximations of iterated Stratonovich stochastic integrals from the numerical schemes (EUH™) [79] (also see [13])

m

(Ï1...ifc ) 11 —Ifc

- L

= (i1...ifc)

Tp+1,Tp

11 —1fe

Tp+1,Tp

< CAr+i,

where constant C is independent of A and r/2 is the strong convergence orders for the numerical schemes (474)-(B~7K); i.e. r/2 = 1.0,1.5, 2.0, 2.5, and 3.0.

As we mentioned above, the numerical schemes (14.741)-(И~7Я1) are unrealizable in practice without procedures for the numerical simulation of iterated Stratonovich stochastic integrals from (4.71).

In Chapter 5 we give an extensive material on the mean-square approximation of specific iterated Ito and Stratonovich stochastic integrals from the numerical schemes (14.651)-(14ТШ). (I4.741)-(4~TK1). The mentioned material based on the results of Chapters 1 and 2.

Chapter 5

Mean-Square Approximation of Specific Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 6 from the Taylor—Ito and Taylor—Stratonovich Expansions Based on Theorems From Chapters 1 and 2

5.1 Mean-Square Approximation of Specific Iterated Ito and Stratonovich Stochastic Integrals of multiplicities 1 to 6 Based on Legendre Polynomials

This section is devoted to the extensive practical material on expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 on the base of Theorems 1.1, 2.1-2.9, 2.30, 2.32-2.35, 2.40 and multiple Fourier-Legendre series. The considered iterated Ito and Stratonovich stochastic integrals are part of the Taylor-Ito and Taylor-Stratonovich expansions. Therefore, the results of this section can be useful for the numerical solution of Ito SDEs with non-commutative noise.

Consider the following iterated Ito and Stratonovich stochastic integrals

t t2

J = / (tk)^i(ti)dwt(il) ...dwi:k), (5.1)

t t

j]T,t=J ^(tk)... / ^i(ti)dwt:i)... dw(:k), (5.2) tt

where every (t) (l = 1,..., k) is a continuous nonrandom function on [t, T],

w

(i)

= f(i) for i = 1,... ,m and w^ = t, (i = 1,... ,m) are independent standard Wiener processes; ii,..., ik = 0,1,..., m.

As we saw in Chapter 4, ^(t) = 1 (l = 1,..., k) and i1,..., ik = 0,1,..., m in (5.1), (5.2) if we consider the iterated stochastic integrals from the classical Taylor-Ito and Taylor-Stratonovich expansions [79]. At the same time ^(t) = (t — t)qi (/ = 1,..., k, q1,..., = 0,1, 2,...) and i1,..., ik = 1,..., m for the iterated stochastic integrals from the unified Taylor-Ito and Taylor-Stratonovich expansions [1]-[16], [51], [142], [144.

(i)

Thus, in this section, we will consider the following collections of iterated Ito and Stratonovich stochastic integrals

T

r(ii---ifc )

1 (Zi...Zfc )T,t

= / (t — tk)Z

t2

(t — t1)Z1 dft(i1) ...df

(ik) tk '

(5.3)

ST

p

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= (ii...ifc ) (Zi...Zfc)T,t

= (t — tk )Z

(t — ti)Zl dft(il)

..f

(ik )

tfc

(5.4)

where i1,..., ik = 1,..., m, 11,..., 1k = 0,1,...

The complete orthonormal system of Legendre polynomials in the space L2([t,T]) looks as follows

(x) =

T-t 3

x —

T + t:

2

Tt

j = 0,1, 2,

(5.5)

where

P (x) =

1 dJ 2 JjldxJ

(x2 — 1):

(5.6)

is the Legendre polynomial.

Let us recall some properties of Legendre polynomials [109] (see Sect. 2.1.2)

Pj(1) = 1, Pj+1(—1) = —Pj(—1), j = 0,1, 2,

dPj+i{x) _ = (2j + l^rr),

dx

jv ; 2j + l J

k

J xkPj(x)dx = 0, k = 0,1,...,j - 1, -i

1 r 0 if k = j

/ Pk(x)Pj(x)dx = ^ --i 2/(2j + 1) if k = j

m

Pn(x)Pm(x) ^ ^ Km,n,kPn+m-2k(x)

k=o

where

flm-kakfln-k 2n + 2m - 4k + 1 (2k - 1)!!

an a- =-----, =-—.-, m < n.

' ' am+n-k 2n + 2m - 2k + 1 k!

Applying the above properties of the Legendre polynomial system (5.5) and Theorems 1.1, 2.1-2.9, 2.30, 2.32-2.35, 2.40 we obtain the following expansions of iterated Ito and Stratonovich stochastic integrals from the sets (5.3), (5.4)

= fa^fa, (5-7)

= + ^CÎ"1) , (5.8)

T*(i m) _ T — t I Ji^Jio,) V"^ ( Ah) Ah) _ Ah)Ai2)\ \ (K

J(00)T,t~ 9 [ SO SO + / , ^.Q _ ^ l^i-lSi Si C.i-1 ; I , (O.IU)

T — t / j;-, ) u(i„) 1

r(H«2) _1 I AfaAfa i V^ 1 f AfaAfa A'li)Afa\ i 1

J(00)T,i — 9 | So So / ' . ^ _ 1 ^ ^>»-1J A{»1=»2} j '

i=i

(5.11)

T-*(n«2) _ T — t J^ixn) ~ | ^ I

J(0i)T,i - 2 4 I ^

, v (i^^rc, (i-ikUA _ Ç \ \ ( 2)

¿¿1 V(2* + l)(2i + 5)(2i + 3) (2i — l)(2i + 3) i i '

i

j*{nh) _ _t_j*{nh) (T t) I 1 Jj2)Mi)

(io)T,i 2 i00)^ 4

, • (/ • . dn)dt2) | | f5131

^ \J(2i + l)(2i + 5)(2i + 3) (2i — l)(2i + 3) ' ' j

¿=0

or

where

p

Wn^) _lim V^ C01 ZV(i2) 1 (oi)T,t _ Cj2jij zj2 '

ji,j2=0 p

1*(i1i2) _lim V^ C10 z (i1 )z (i2) 1 (io)T,t _ Cj2ji j zj2 '

ji,j2=0

yi2il ~~ g ^ 3231 '

+ + (5.14)

1 y

Cjj _ -J (1+ v)j (v) / Pji (x)dxdy, -1 -1 i y

10

j _ - Pj2(v) / (1+ x)Pji(x)dxdv;

j2 ji

— 1 -1

I

■(ilia) _ /-»(ilia) _i_ /71 _ 1

(io)T,i ~~ J(io)T,t "T" 4J-{»i=»2}^ ^ w. p. ±

I

■(ilia) _ r*(iiia) , ^-t /t-t_1

(01)T,i ~~ J(01)T,t "T" ^J-{ii=ia}l-£ W. p. l

r(iiia) _ T ~ 1 r(iiia) (T ~ ¿)2 / 1 ^(ii)^(i2) , (oi)T,i - 2 t00^ 4 I y^ 0 ^

tA \/ (2i + l)(2i + 5)(2i + 3) (2i — l)(2i + 3) j

or

r(ili 2) _ T T{hh) (T ~ tf I 1 ,

(io)T,i 2 i00)^ 4

, ^(u • • dn)cf2) 11 ,51(n

til \/(2i + l)(2i + 5)(2i + 3) (2i — l)(2i + 3) ' ' j

p

1('1'2) =lim V^ C 01 iz ('l)Z ('2) _ 1 r ,l1r.

1 (01)T,t = Z^ Cj2jl[ j zj2 1{'l='2}1{jl=j2}

j1,j2=0 p

r('l'2) = l • V^ C10 I z('l)z('2) _ 1 1

1 (10)T,t = Z^ Cj2j^ j zj2 1{'1='2}1{j1=j2}

j1,j2=0

p

t *('1'2'3) = lim V^ C Z ('l)z ('2)z ('3) (5 17)

1(ooo)T,t =lj;1;m^ Cj3j2jlzjl zj2 zj3 , (5.17)

jl,j2,j3=0

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p

= ^i1^. S Cj3j2j1 I jl,j2,j3=0 \

1('1'2'3) _lim V^ C- ■ I z ('1)z ('2) z ^ - 1r • ilr- -^z ('3)-

1 (ooo)T,t = l:!;m. Cj3j2jl I zjl zj2 zj3 1{'l='2}1{jl=j2}zj3

^('l) 1 1 A'2)

1{'2='3} 1 {j2 =j3} zjl1 1{'l = '3} 1 {j 1 =j3}zj22 , (5.18)

3

& = fa~t)3/2( Ct>) w.p.l, (5.19)

where

V/(2j1 + 1)(2J2 + 1)(2J3 + 1) 3/2

° j3j2.il — g ^3332311 l^O.ZUJ

1 z y

jj = y pj3(z) J pj2 (y) J pjl (x)dxdydz; (5.21) 111

here and further in this section

T

zf _/ h (s)f(i) (i _ 1,..., m, j _0,1,...) t

are independent standard Gaussian random variables for various i or j;

Ahhh) _ j^nhh)

J(000)T,t ~~ J(000)T,t 2 L«i=«2^0}J(l)T,t

7(is) _

-1

2

{¿2 = «3=0}

_t) 1 (ii) + 1 (ii)

t)J(0)T,t + J(1)T,t

w. p. 1 ,

I,

(T -t)2 J- *(ii*2)

(02)T,t

T»(¿ii2) _ /rri _ y \ r»(iii2) ,

J(00)T,t (T t)J(01)T.t +

(ii¿2) , (T - t)3

(01)T,t

8

2 Z^M^-L So

oo

i,«.>,<«> I 1 a + 2)(i + sjdSci"' - (»

"TqSo SO "t"

3 1=0 V a/ (2i + 1)(2i + 7)(2i

(¿2)Z(¿i) i Zi+3

3 \ a / ('¿% + i ) ('¿% —i— /) t -¿i, + ) (-¿i

2 I qA^^M^ (¿2 \ OA iu(i2V(ii)'

+ ^ - 3)Q+iC - (j + - Q+i

V/(2i + l)(2i + 3)(2i - l)(2i + 5)

(5.22)

I,

(iii2) (T -1)2 J- *(iii2)

(20)T,t

T»(iii2) _ /rri _ /\ r*(iii2) ,

J(00)T,t (T t)J(10)T.t +

(ii¿2) , (T - t)3

(10)T,t

8

2 Z^)/-^!) _l

"T^so S2

1,«.),<«> I / (i + !)(» + 2)d+ic,'"' - (j + 2)(» + 3)C,('2'd+i

' .,/>11 So I --—-

i=0

(i2 + 3i - l)Cit2iCfl} - (j2 + j - 3)Cr;Cm

i- l)(2i + 5)

(i2) (ii)

(5.23)

(iii2) (T -t)2 (iii2) (T -1) fr (iii2) (iii2)

(11 )T,t~ 4 J(00)T,t 2 V i10)^ (01)T'V

(T -1)

Jci(il)cf2) + E(

~ /(i + 1)(i +

(i2)z(ii) z(i2)z(ii) i+3 zi zi Zi+3

f

+

(* + i)2 (dgcfl}-cf2)c^)

\J(2i + l)(2i + 3)(2i - l)(2i + 5)

(5.24)

or

/Ji^t = l.i.m. V C02 j1 )z(

(02)T,t j2jl^ jl j

(02)T,t K'1'2)

.m

02 z('1 )z('2) j2 ,

j1,j2=0 p

1 20)T,t = ^im. S Cj2j'l zj'1 j

(20)T,t K'1'2)

20 z('1 )z('2) j2 ,

j1,j2=0 p

r *('1'2) = l i m V^ C11 z('l)z( 1 (11)T,t = lPlIS . Cj2jl zjl zj'

11 Z('1) Z('2) j2 ,

j1,j2=0

where

C02 = Cj2j1 =

V/(2j1 + 1)(2J2 + 1)

16

(T — t)3CT

3 T 02

j2j1

C20 = Cj2j1 =

V/(2j1 + 1)(2J2 + 1)

16

(T — t)3CT;

3 T 20

j2j1

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n V/(2j1 + 1)(2J2 + 1)

j = I Pj2 (y)(y + 1)^ Pjl (x)dxdy,

2

—1 —1

1 y

jl = Pj2(y) / Pjl(x)(x + 1)2dxdy,

—1 —1 1 y

jl = / Pj2(y)(y + 1W Pjl(x)(x + 1)dxdy,

1

r*(nn) An) , w n 1

i(ii)T,i - 2 V P' '

I

(il'h) _ j*\W2) _ Jll /71 _ -i

(02)T,t (02)T,t gJ-{«i=i2}^ W. p. 1

K'1'2)

1

(5.25)

p

1

y

I,

(n%^ — T*[H'l2) - -I r- .,(T — t\6 W -n 1 (20)T,t (20)T,t g-L{»i=»2}l-£ W. p. 1,

(iii2)

(5.26)

I

(U)T,t ~ ^(11)T,t ~ - ^ w. p. 1,

(iii2)

6

I

(ii¿2) (02)T,t

(T -1)2 (iii2) (iii2) (T -1)3

~ [1 ~ t)lQ1 +

(00)T,t

8

2 Ah)Ah)_1 z2 z0 +

00

(ii)z(i2) 0 z0

(i + 2)(i + 3)c;;23ci':i) - (»+ 1)(«

i=0

(i2)z(ii) i Zi+3

(i2 + i - 3)C-t2iCfl} ~ (¿2 + 3i - i)Cr;Cm

i — l)(2i + 5)

(i2) (ii)

241{ii=i2}(T ~ O3'

(5.27)

I,

(iii2) (20)T,t

(T - t)2 r(iii2)

- (T - *)/,

(iii2) (10)T,t

+

(71 - tf 8

-(i2)Aii)

^ An)An)_1 "T^so S2

(i2) (ii)

_l v / « + 1)« + 2)c™cr" - « + 2)(i + 3)c; c+3

"TgSO SO ^

i=0

V/(2i + l)(2i + 3)(2i - l)(2i + 5)

(i2) (ii)

(5.28)

f

I,

(ii¿2) (11)T,t

(T -tf

(00)T,t

^_^ ( r(«l«2) 1 r(«l«2) , I

2 y (io)T,i (oi)T,t J

(T -1)3

ziii)zii2 )+E

'(i +1)(i +

(i2) Z(¿i) _ Z(i2) Z(¿1 )

i+3 Si ci ci+3

i=0

V/(2i + l)(2i + 7)(2i + 3)(2i + 5)

or

\J(2i + l)(2i + 3)(2i - l)(2i + 5)J

il{il=i2}(T-i)3 (5.29)

p

r('1'2) = l i m C02 Iz('l)z('2) _ 1 r

1 (02)T,t = lpim . Z^ j'll zj1 z j2 1{'1='2} 1{j1=j2}

jl,j2=0 \ p

r('l'2) = l i m V^ c20 i z('l)z('2) _ i i

1 (20)T,t = lpim . Z^ CJ2j'M zjl z j2 1{'1='2}1{j1=j2}

jl,j2=0 \ p

r('l'2) =l i m V^ C11 z('l)z('2) _ 1r ,1f

1 (11)T,t = lpim . Z^ j'll zj1 z j2 1{'1='2} 1{j1=j2} jl,j2=0 \

= (tf1' + nr^1' + ^ + ¿fCf» ] , (5.30)

p

1*('l'2'3'4) = l i m V^ C z ('l)z ('2) z ('3)z ('4) 1 (0000)T,t = lpim . Z^ Cj4j3j2 jl zjl zj2 zj3 zj4 '

j1,j2,j3,j4=0

p

1(' 1 '2'3'4) = l i m V^ C I z('l)z('2)z('3)z('4)_

1 (0000)T,t = lp1m . Z^ Cj4j3j2j1 I zjl zj2 z j3 z j4 j1,j2 ,j3 ,j4=0 V

_1 1 z('3)z('4) _ 1 1 z('2)z('4) —

1{'1='2}1{j1=j2}zj3 z j4 1{'1 = '3}1{j1=j3}zj2 zj4

-1 r ■ -,1r- ,,z ('2)z ('3^ 1 r . ,ir. . ,z ('l)z ('4)_

1{'l='4} 1 {jl=j4} zj2 z j3 1{'2 = '3} 1 {j2=j3} zjl z j4

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('l)z('3) _ 1 1 z('l)z('2)

jl zj3 1 {'3='4} 1 {j3 = j4 } z j 1 zj2

—1 -,1 ,z ('l )z ('3) _ 1 .z ('l )z ('2)+

1{'2 = '4} 1{j2 =j4 }zjl zj3 1{'3='4} 1{j3=j4}zjl zj2 +

+ 1{'1='2} 1{jl=j2} 1{'3 = '4} 1{j3=j4} + 1{'1='3} 1{jl=j3} 1{'2='4} 1{j2=j4} +

+ 1{'1 = '4} 1{j1=j4}1{'2='3} 1 {j2 =j3} I , (5.31)

= ¿<T " ((Co!l))4"6 (Co'T +3) w- p- i.

w.p.l, (5.32)

where

+ 1)(2j2 + l)(2j3 + 1)(2J4 + 1)^

— 0 Cj^fcji, (5.33)

1 u z y

Cj4j3j2j1 = j Pj4 (U) y Pj3 (Z) y Pj2 (y) y Pj1 (x)dxdydzdu; (5 . 34)

-1 -1 -1 -1

p

1* ('l '2 '3) = l i m V^ C001 z ('l)z ('2)z ('3) 1 (001)T,t = lpim . Z^ Cj3j2j1 zjl z j2 zj3 ,

j1,j2,j3=0 p

1*('l'2'3) = l i m V^ C010 z('l)z('2)z('3) 1 (010)T,t = lpim . Z^ Cj3j2jl zjl zj2 zj3 ,

j1,j2,j3=0 p

1*('l'2'3) = l i m V^ C100 z('l)z('2)z('3) 1 (100)T,t = lpim . Z^ Cj3j2j1 zjl z j2 zj3 ,

j1j2,j3=°

p

r('l'2'3) = l i m Y^ C001 I z('l)z('2)z^ _ 1 1 z('3)

1 (001)T,t = lpim . Z^ Cj3j2j^l zj1 zj2 zj3 1{'1='2}1{j1=j2}zj3

j1,j2,j3=0

1{'2 = '3}1{j2=j3}zj'1) 1 {' 1='3} 1 {jl = j3} Cjs!2 ) J , (5 . 35)

p

r('l'2'3) = l i m V^ C010 I z('l)z('2)z^ _ 1 1 z('3)_

1 (010)T,t = lpim. Z^ Cj3j2j^ zjl zj2 zj3 1{'1='2}1{j1=j2}zj3

j1,j2,j3=0

1{'2 = '3}1{j2=j3}zj'1) 1 {' 1='3} 1 {jl = j3} Cjs!2 ) ) , (5 . 36)

p

1 (iii2i3) 1 • V^ C100 | z(ii)Z(i2)z(i3) _ i i z(i3)

J(100)T,t _ „jm. Cj3j2jM j zj2 j 1{ii = i2} 1 {ji=j2} zj3

where

ji.j2.j3=0

^{i2=i3} 1 {j2 =j3} zjii) 1{ii = i3} 1{ji=j3}zj22) )' (5.37)

rooi = + 1)(2J2 + l)(2j3 + 1) , ,5/2^001

j3j2.il ^g V / 333231010 ,\5/2010 333231 — ^g K1 L) Ljj3323i'

rioo _ \/(2ji + 1)(2J2 + l)(2j3 + l)r ,5/2^100

333231 J^j V / 333231'

1 z y

j ji _ -J Pj3 (*)/ Pj2 (V) J Pji (x)(x + 1)dxdydz, -1 -1 -1

1 z y

j0ji _ -/ Pj3 (z)/ Pj2 (v)(v + 1)/ Pji (x)dxdydz, -1 -1 -1

1 z y

Cjjji _ -J Pj3 (z)(z + 1)| Pj2 (v) J Pji (x)dxdvdz; -1 -1 -1

((/H)T,i - g ( (Jmt) ~ 3I(l)T,tAt(T,t) ) w. p. 1.

^(nnn) _ 1 / i-(ii) \3 1

— 7: Jmr/ W. p. 1,

[(///)T,t ~~ g \ V)T,t

'mm = ((fe)4 - 6 (fe)2 A<№ + 3 (A,or,)2) w. p.

^(Hiliin) _ J_ f i-(ii) \4 1

Uiui^t - 24 yhm) w- P- -1'

1

where

i

= £ C cf w.p.1, (5.38)

j=0

T T

Ai(T,t) _ /(* - s)2ids, Cj = /(t - s)10 (s)ds;

7-*(«1i2«3i4i5) _l : m V^ C Z(il)Z(i2V(i3)Z(i4)Z(i5)

1 (00000)T,t _ • Z^ Cj5j4j3j2j1 Zji Zj2 Zj3 Zj4 Zj5 '

j1,j2,j3,j4,j5=0

P

P /5

I(i1i2i3i4i5) _l i m V^ C iTTZ(ii) —

1 (00000)T,t _ . Z^ Cj5j4j3j2ji M

j1,j2,j3,j4,j5=0 \l=1

-1 {11 = 12} j1 = Z (13)Z (14) Z (15) - 1{ 11 = 13} =j3}C (12)Z(14) ¿2 Zj4 Z (15 ) -

-1 {11 = 14} j1 = Z (12 ) Z (13)Z (15) - 1{11 = 15} j =j5}C (12)Z(13) ¿2 Zj3 Z (14) -

-1 {12= 13} j2 = Z (11)Z (14) Z (15) - 1{12 = 14} {j2 = =j4}C (11)Z (13) ¿1 Zj3 Z (15 ) -

-1 {12= 15} j2 = Z (11)Z (13)Z (14) - 1{13= 14} {j3 = =j4}C (11)Z (12) ¿1 Zj2 Z (15 ) -

-1 {13= 15} j3 = Z (11)Z (12 )Z (14) =j5} Zj1 Zj2 Zj4 - 1{14 = 15} { j4 = =j5}C (11 )Z (12 ) Z (13) +

+1{11= 12} {j1 =j2 13 1 Z (15 ) =14} 1 { j3=j4} Zj5 + 1{11 = 12} {j1 = =j2} {13=15} {j3 = Z (14) 1 j5} Zj4 +

+1{11= 12} {j1 =j2 14 Z(13) =15} 1 { j4=j5} Zj3 + 1{11 = 13} j =j3} {12=14} {j2 = Z (15 ) 1 j4} Zj5 +

+1{11= 13} {j1 =j3} 12 Z(14) =15} 1 { j2 =j5 } Zj4 + 1{11 = 13} j =j3} {14=15} { j4 = ■ T,Z (12) + j5} Zj2 +

+1{11= 14} {j1 =4} 12 1 Z (15 ) =13} 1 { j2=j3} Zj5 + 1{11 = 14} j =j4} {12=15} {j2 = ■ T,Z (13) + j5} Zj3 +

+1{11= 14} {j1 =4} 13 1 Z (12) = 15 } 1 { j3=j5} Zj2 + 1{11 = 15} {j1 = =j5} {12 = 13} {j2 = ■ T,Z (14) + j3} Zj4 +

+1{11= 15} {j1 =j5} 12 Z(13) = 14} 1 {j2=j4} Zj3 + 1{11 = 15} {j1 = =j5} {13 = 14} {j3 = ■ T,Z (12) + j4} Zj2 +

+ 1{12 = 13} {j2 =j3} 14 1 Z (11) = 15 } 1 { j4=j5} Zj1 + 1{12 = 14} {j2 = =j4} {13 = 15} {j3 = ■ TZ (11) + j5} Zj1 +

+ 1{»2 = »5|1 {j2 =j5} 1{»3=»4}1{j3=j4}Cj(1i1) , (5"39)

= ¿(^ - i)5/2 (((¿,1))5 - 10 (Ci"»)3 + 15^°) w. p. 1,

1 / \ 5

Coir,;1' = ¿>(T-№) w' p-

where

r _ \J(2ji + 1)(2J2 + l)(2j3 + 1)(2J4 + 1)(2J5 + 1) 5/2

^khhhh — 22 ^ ' ^hhhhdi-

1 v u z y

Cj5 j4j3j2 ji ^y Pj5 Pj4 (U) j Pj3 (Z) j Pj2 fe^ Pji (x)dxdvdzdudv;

-1 -1 -1 -1 -1

p

1 »(iii2i3) _l jm V^ C0001 z (ii)z (i2) z (i3)z (i4)

J(0001)T,t lpi;m. Cj4j3j2ji j zj2 j ^'4 '

ji.j2 ,j3,j4=0 p

1 »(iii2i3) _l j m V^ C0010 z (ii)z (i2) z (i3)z (i4)

J(0010)T,t lpJ^m. Z^ Cj4j3j2ji j zj2 j zj4 '

ji,j2 ,j3,j4=0 p

1 »(iii2i3) _ l i m V^ C0100 z(ii)z(i2) z(i3)z(i4)

J(0100)T,t lpJ^m . Z^ Cj4j3j2ji j z j2 j ^'4 '

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ji.j2 ,j3,j4=0 p

1 »(iii2i3) _ l i m V^ C1000 z(ii)z(i2) z(i3)z(i4)

J(1000)T,t lpi;m . Z^ Cj4j3j2ji j zj2 j j '

ji.j2 ,j3,j4=0

p

1 (iii2i3i4) _ l i m V^ C0001 [ z (ii )z (i2 ) z (i3)z (i4)_

J(0001)T,t lpJ^m . Z^ Cj4j3 j2ji I j j z j3 z j4

jij2 ,j3,j4=0 V

1 z (i3)z (i4) _ 1 1 z (i2 ) z (i4)_

1{ii=i2} 1 {ji=j2} ^j^ zj4 1{ii = i3} 1 {ji=j3} ^j^ zj4

1 1 z(i2)z(i3) 1 1 z(ii)z(i4)

1{ii=i4} 1 {ji=j4} ^j^ zj3 1{i2 = i3} 1 {j2=j3} Jj'i j

— 1r- . l1r. (ii )z ^ "1 r . l1r. . ,z (ii )z (i2) +

1{i2 = i4}1{j2=j4}zji j 1 {i3=i4} 1{j3=j4}zji j +

+ I{ii=i2} 1{ji=j2} 1{i3 = i4} 1{j3=j4} + 1{ii=i3} 1{ji=j3} 1{i2=i4} 1{j2=j4} +

+ 1{ii = i4} 1{ji=j4} 1{i2 = i3} 1 {j2 =j3 }

P

I(11121314) _ l i m C 0010 i Z (11 )Z (12 ) Z (13)Z (14 )_

1 (0010)T,t lpi;i( ' Z^ Cj4j3 ¿2:71 I Zj1 ¿2 Zj3 Zj4

¿1^2 J3J4=° V

— 1r- • l1r- . , Z (13)Z (14) _ 1r . l1r. (12 ) Z (14)_

1{11=12}1{j1=j2}Sj3 Zj4 1{11 = 13}1{j1=j3}Sj2 Zj4

1 Z (12 ) Z ^ _ 1 1 Z (11)Z (14) —

1{11=14}1{j1=j4}Zj2 Zj3 1{12 = 13}1{j2=j3}Zj1 Zj4

— 1r- . l1r- . ,Z(11 )Z-1 r . l1r. . ,Z(11 )Z(12) _L

1{12 = 14}1{j2=j4}Sj1 Zj3 ^{13=14} 1{j3=j4}Sj1 Zj2 +

+ 1{11=12} 1{j1=j2} 1{13 = 14} 1{j3=j4} + 1{11=13} 1{j1=j3} 1{12=14} 1{j2=j4} +

+ 1{11 = 14} 1{j1=j4} 1{12 = 13} 1 {¿2 =¿3 }

P i

I(11121314) _ l i m V^ C0100 | Z (11 )Z (12 ) Z (13)Z (14)_ 1 (0100)T,t Jpi;i((' Z^ ¿¿3¿ml Zj1 ¿2 ¿j Zj4

jU2 ,j3,j4=0 V

— 1r- • l1r- . , Z (13)Z (14) _ 1r . l1r. (12 ) Z (14)_

1{11=12}1{j1=j2}Sj3 Zj4 1{11 = 13}1{j1=j3}Sj2 Zj4

_1 1 Z (12 ) Z ^ _ 1 1 Z (11)Z (14) —

1{11=14}1{j1=j4}Zj2 Zj3 1{12 = 13}1{j2=j3}Zj1 Zj4

— 1r- • l1r- . ,Z(11 )Z1r . l1r. . ,Z(11 )Z(12) _L

1{12 = 14}1{j2=j4}Sj1 Zj3 A{13=14} 1{j3=j4}Sj1 Zj2 +

+ 1{11=12} 1{j1=j2} 1{13 = 14} 1{j3=j4} + 1{11=13} 1{j1=j3} 1{12=14} 1{j2=j4} +

+ 1{11=14}1{j1=j4}1{12 = 13} 1 {¿2 =¿3 }

P /

I(11121314) _ l • V^ C1000 I Z (11 )Z (12) Z (13)Z (14)_

1 (1000)T,t lpi;i(( • Z^ C ¿4 ¿3 ¿2:71 I Zj1 Zj2 Zj3 Zj4 ¿1 ¿2 J3J4=0 \

— 1r- • l1r- (13)Z (14) _ 1r . l1r. ..Z (12 ) Z (14)_

1{11 = 12}1{j1=j2}Zj3 Zj4 1{11=13}1{j1=j3}Zj2 Zj4

_1 1 z(12)Z^ _ 1 1 Z(11)Z(14) —

1{11 = 14}1{j1=j4}Zj2 Z ¿3 1{12 = 13}1{j2=j3}Zj1 Z ¿4

_1 1 z (11 )Z - 1 1 Z (11 )Z (12) +

1{12 = 14}1{j2 =¿4^1 ^¿3 1{13=14} 1 {¿3 =¿4} Z 1 ^¿2 +

+ 1{1l = l2}1{j1=j2}1{1з = l4}1{j3=j4} + 1{11=13} "^{¿^=¿3} 1{12=14}1{J2=J4} +

+ 1 {11 ^=14 } 1 {¿1 =¿4 } 1{12=13} 1 {¿2 =¿3 }

where

^0001 _ \J(2ji + 1)(2J2 + l)(2j3 + 1)(2J4 + 1)(rr ,\3n0001 ( ; ; - -~~-( 1 l] ( ./:./,./:./ •

j4j3j2ji

32

0010 + 1)(2J2 + l)(2j3 + 1)(2j4 + 1) _ ,\3/n0010

j3j2ji

32

C

0100

j4j3j2ji

V/(2j1 + l)(2j2 + l)(2j3 + l)(2j4 + l)

32

/ATI ,\3 /n 0100 (T - l) Cj3j2ji ,

1000 _ V(2;/i + 1)(2j2 + l)(2j3 + 1)(2j4 + 1)frr 3 n1000 ( ; ; - -~~-( ' ' ./:/../:./ •

j4j3j2ji

32

1 u z y

Cj ji _ -J Pj4 (u)J Pj3 (z) J Pj2 (v)f Pji (x)(x +1)dxdvdzdu,

111

C 0100

Cj4j3j2ji

C 0010

Cj4j3j2ji

c

0001

j4j3j2ji

u z y

Pj4 (U) j Pj3 (Z) j Pj2 (v)(v +1)/ Pji (x)dxdvdzdu, -1 -1 -1

u z y

j (u)/ Pj3 (z)(z +1) / Pj2 <v) J Pj. «vdzdu,

-1 -1 -1

u z y

- f j(u)(u + «/ pj3 (z) / pj2 (tf/ pj.

-1 -1 -1 -1

1

*(iii2i3i4i5i6)

(000000)T,t

_ l.i.m.

p—

EC z (ii)z (i2) z (i3)z (i4) z (i5) z (i6)

Cj6j5j4j3j2j'i Sji z j2 j j j j '

ji,j2,j3,j4,j5,j6=0

1(iii2i3i4i5i6) _ l i m

1 (000000)T,t lp—-m .

p /6

Cj6j5j4j3j2j'i I H j j'i,j2,j3,j4,j5,j6=0 \/ = 1

_1 1 z^'l2> z^) mm; M'i5) _ 1 1 z (ii)z (i3)z (i4) z (i5)_

1 {j'i=j'6} 1{ii=i6} zj2 zj3 zj4 zj5 zj3 j zj5

1 j3 = j6} {i3= Z (ii)Z (i2)Z (i4)Z i6}Zji Zj2 Zj4 Z j5 — {j4 j6j {i4 =i6}Z (ii)Z(i2) ji Z j2 Z(i3) Zj3 Z (i5)_ Z j5

1 j5 = j6} {i5= Z (ii)Z (i2 ) Z i6}Zji Zj2 Zj3 Zj4 — 1 {ji = =j2 } {ii =i2}Z (i3)Z(i4) j3 Z j4 Z(i5) Zj5 Z(i6) Zj6

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1 ji = j3} {ii= Z(i2^ Z(i4^ Z(i5 ) Z(i6) i3} Z j2 Zj4 Zj5 Zj6 — 1 {ji = =j4} {ii =i4}Z (i2)Z(i3) j2 Zj3 Z(i5) Zj5 Z(i6) Zj6

1 ji = j5} {ii= Z (i2 ) Z (i3)Z (i4) Z (i6) i5} Z j2 Zj3 Zj4 Zj6 — {j2 =j3} {i2 =i3}Z (ii)Z(i4) ji Z j4 Z(i5) Zj5 Z(i6) Zj6

1 j2 = j4} {i2= Z (ii)Z (i3)Z (i5) z (i6) i4} Z j i Zj3 Zj5 Zj6 — {j2 =j5} {i2 =i5}Z (ii)Z(i3) ji Zj3 Z(i4) Zj4 Z(i6) Zj6

1 j3 = j4} {i3= Z (ii)Z (i2 ) Z (i5 ) Z (i6) i4} Zji Z j2 Zj5 Zj6 — {j3 =j5 } {i3 =i5}Z (ii)Z(i2) ji Z j2 Z(i4) Zj4 Z(i6) Zj6

1 {j4=j5}1 {i4 = i5 Z (ii)z (i2 )Z (i3)z (i6) + l"Zji Z j2 Zj3 Zj6 +

+1 j =j2} {ii = i2j {j3 = =j4} Z (i5 )z (i6) {i3=i4} Zj5 Zj6 + 1 {ji = =j2} {ii i2j { j3 =j5} {i3 = Z (i4) Z(i6) I =i5}Zj4 Zj6 +

+1 j =j2} {ii =i2j { j4 = =j5} Z (i3)z (i6) {i4=i5} Zj3 Zj6 + 1 {ji = j3j {ii i3j {j2=j4} {i2 = Z(i5) Z(i6) 1 = i4} Z j5 Zj6 +

+1 j =j3} {ii =i3j {j2 = =j5 } Z(i4)Z(i6) {i2=i5} Zj4 Zj6 + 1 {ji = j3j {ii i3j {j4=j5} {i4 = Z(i2)Z(i6) 1 = i5} Zj2 Zj6 +

+1 j =j4} {ii =i4j {j2 = = j3j Z(i5)Z(i6) {i2=i3} Zj5 Zj6 + 1 {ji = =j4} {ii i4j {j2=j5} {i2 = Z(i3)z(i6) i = i5} Zj3 Zj6 +

+1 j =j4} {ii =i4j {j3 = =j5 } Z(i2)Z(i6) {i3=i5} Zj2 Zj6 + 1 {ji = =j5 } {ii =i5} { j2 =j3 } {i2 = Z(i4)Z(i6) I = i3} Z j4 Zj6 +

+1 j =j5} {ii =i5j {j2 = =j4} Z(i3)Z(i6) {i2=i4} Zj3 Zj6 + 1 {ji = =j5} {ii =i5} {j3=j4} {i3 = Z(i2) Z(i6) + = i4} Z j2 Zj6 +

+ 1 {j2 = =j3} {i2 =i3j { j4 = =j5 } Z(ii)Z(i6) {i4=i5} Zji Zj6 + {j2 =j4} {i2 i4j {j3=j5} {i3 = Z (ii)z (i6) + = i5} Zji Zj6 +

+ 1 { j2 = =j5} {i2 =i5j {j3 = =j4} Z(ii)Z(i6) {i3=i4} Zji Zj6 + {j6 jij {i6 iij {j3=j4} {i3 = Z(i2)Z(i5) I = i4} Z j2 Zj5 +

+ 1 j =ji} {i6 =ii} {j3 = =j5 } Z(i2)Z(i4) {i3=i5} Zj2 Z j4 + {j6 jij {i6 iij {j2 =j5} {i2 = Z (i3)Z (i4) I = i5} Zj3 Zj4 +

+ 1 j =ji} {i6 =ii} {j2 = =j4} Z (i3)Z (i5) {i2=i4} Zj3 Z j5 + {j6 jij {i6 iij {j4=j5} {i4 = Z(i2)Z(i3) 1 = i5} Zj2 Zj3 +

+ 1 j =ji} {i6 =ii} {j2 = =j3} Z(i4) Z(i5) {i2=i3} Zj4 Z j5 + {j6 =j2 } {i6 i2j {j3=j5} {i3 = Z (ii)Z (i4) 1 = i5} Zji Zj4 +

+ 1{j6 = =j2} {i6 =i2j { j4 = =j5} Z(ii)Z(i3) {i4=i5} Zji Z j3 + {j6 =j2} {i6 i2j {j3=j4} {i3 = Z (ii)Z (i5 ) 1 = i4} Z ji Zj5 +

+ 1{j6 = =j2} {i6 =i2j j =j5 } Z(i3)Z(i4) {ii=i5} Zj3 Z j4 + {j6 =j2 } {i6 i2j {ji=j4} {ii= Z (i3)Z (i5 ) I = i4} Z j3 Zj5 +

+ 1{j6 = =j2} {i6 =i2j j =j3} Z(i4) Z(i5) {ii=i3} Zj4 Z j5 + {j6 =j3} {i6 i3j {j2=j5} {i2= Z (ii)Z (i4) 1 = i5} Zji Zj4 +

+ 1{j6 = =j3} {i6 =i3j { j4 = =j5 } Z(ii)Z(i2) {i4=i5} Zji Z j2 + {j6 =j3} {i6 i3j { j2 = j4 } {i2= Z (ii)Z (i5 ) 1 = i4} Z ji Zj5 +

+ 1{j6 = =j3} {i6 =i3j {ji =j5} Z(i2)Z(i4) {ii=i5} Zj2 Zj4 + {j6 =j3} {i6 i3j {ji=j4} {ii= Z(i2)Z(i5) I = i4} Zj2 Zj5 +

+ 1{j6 = =j3} {i6 =i3j {ji = j 2 j Z(i4) Z(i5) {ii=i2} Zj4 Z j5 + {j6 =j4} {i6 i4j {j3=j5} {i3= Z (ii)Z (i2 ) 1 = i5} Zji Zj2 +

+ 1{j6 = =j4} {i6 =i4j {j2 = =j5 } Z(ii)Z(i3) {i2=i5} Zji Z j3 + {j6 =j4} {i6 i4j { j2 = j3 } {i2= Z (ii)Z (i5 ) 1 = i3}Zji Zj5 +

http://doi.org/10.21638/11701 spbu35.2023.110 Electronic Journal. http://diffjournal.spbu.ru/ A.738

+ 1111 z(i2 ) z(i3) + 1 1 1 1 z(i2 ) z(i5) +

+ 1{j6=j4} -M^MHO'^MM^^j j + 1{j6=j4}1{i6 = i4} 1{j'i=j3} 1 {:1=:3} j j +

+ 1111 z (i3)z (i5) + 1 1 1 1 z (ii)z (i2) +

+ 1{j6=j4} "^{¿6 = ¿4} 1{ j'l=j'2} 1{il = i2} zj5 + 1{j'6=j5}1{i6 = i5} 1{j'3=j'4} 1{i3=i4} ^ji j +

+1111 z (ii)z (i3) + 1 1 1 1 z (ii)z (i4)+

+ 1{j6=j5} 1{i6=i5}1{j2=j4}1{i2=i4}Sj,i Sj3 + 1{j6=j5}1{i6=i5}1{j2=j3}1{i2 = i3}Sj,i j +

+ 1111 z(i2 ) z(i3) + 1 1 1 1 z(i2) z(i4) +

+ 1 {j'6=j'5} 1{i6=i5} 1{ j'l=j'4} 1{il=i4} zj2 j + 1{j'6=j'5} 1{i6=i5} 1 {j'l=j'3} 1{il = i3} zj'2 j +

+ 1{j6 = {j6=ji} {j6=ji} {j6=ji} {j6=j2} {j6=j2} {j6=j2} {j6=j3} {j6=j3} {j3=j6} {j6=j4} {j6=j4} {j6=j4} {j6=j5} {j6=j5}

j5 } 1 {¿6: {i6=ii} {i6=ii} {i6=ii} {i6=i2} {i6=i2} {i6=i2} {i6=i3} {i6=i3} {i3=i6} {i6=i4} {i6=i4} {i6=i4} {¿6=i5} {¿6=i5}

¿5} 1{j'l =

{j2=j5} {j2=j4} { j2 =j3 } {j'l=j5} {j'l=j4} {j'l=j3} {j'l=j5} {j'l=j4} {j'l=j2} {j'l=j5} {j'l=j3} {j'l=j2} {j'l=j4} {j'l=j2}

j2}1{ii { ¿2 =¿5 } {¿2=i4} {¿2=i3} {ii=i5} {il=i4} {il=i3} {¿l=i5} {il=i4} {il=i2} {il=i5} {il=i3} {¿l=i2} {il=i4} {il=i2}

z (i3)z (i4) — = i2}Sj3 j

{j6=j5} 1{i6=i5} 1{j'l=j3} 1{il = i3} 1{j2=j4} 1{i2=i4}

{j3=j4} {j3=j5} {j4 =j5} {j3=j4} {j3=j5} {j4 =j5} { j2 = j4 } {j2=j5} {j4 =j5} {j2=j3} {j2=j5} {j3=j5} { j2 =j3 } {j3=j4}

{¿3 = i4} {¿3 = i5} {¿4 = i5} {¿3 = i4} {¿3 = i5} {¿4 = i5} {¿2 = i4} { ¿2 = i5 } {¿4 = i5} {¿2 = i3} { ¿2 = i5 } {¿3 = i5} {¿2 = i3} {¿3 = i4}"

Ahhhhhh) _ 1 (rji 3 ( (Ah) \ _ 1 r

J(oooooo)T,t — 720 VV0

(ii)

+ 45

(¿1)

- 15 w. p. 1,

I±™'> = ±(T - tf (d*1»)" w. p. 1,

(000000)T,t

where

4

2

0

C

Cj6 j5 j4 j3 j2 jl

\J(2Ji + 1)(2j2 + l)(2j3 + 1)(2J4 + 1)(2J5 + l)(2j6 + 1) ^ _

| l) ( ./'• A"./ ./•../ ./ '

C......

Cj6j5j4j3j2ji

1 w v u z y

_ / Pj6 (») / Pj5 (V) / Pj4 M / Pj3 (z) J Pj2 (v) j Pj.

-1 -1 -1 -1 -1 -1

It should be noted that instead of the expansion (5.17) we can consider the following expansion, which is derived by direct calculation

r*(ili2i3)

T*(nWil _ f f r(*3) t-*(«2H) i r(n) j-*(«2«3)\ | 1 r(«3) / 7-*(H«2) _ J-*(«2«l)\ _

J(000)T,i ~~ T — i V i0)^ i10)^ (0)T,iJ(10)T,iy "T" 2 (0)T,i \/(00)T,i J(00)T,iy

- tf2 (k,(,!l)c,(,!s) f4,2) + v/3c!!2) - + -MfA, (5.40)

— i I —

So ^ v S1 ^/5S2 J 1 4

where

to

'Aj ~ A A.- +

i=l, j=0, k=i 2i>k+i-j>-2; k+i-j -even

TO

i=l, j=0, l<k<i-l 2k>k+i-j>-2; k+i-j -even

TO

i=l, j=0, k=i+2 2i+2>k+i-j>0; k+i-j -even

TO

E •V;./7,A'/.. |.;. |cf1}Cjt2)Ck3)~

i=l, j=0, l<k<i+l 2 k-2>k+i-j>0; k+i-j -even

i=l, j=0, k=i-2,k>l 2i-2>k+i-j>0; k+i-j -even

TO

E % A^H+I^-I,^ C(n) Cj*2} ci'3) +

i=l, j=0, l<k<i-3 2k+2>k+i-j>0; k+i-j -even

i=1, j=0, k=i 2i>k+i-j>2; k+i-j -even

OO

i=1, j=0 1<k<i-1 2k>k+i-j>2; k+i-j -even

where

=

(2k + 1)(2j + 1)(2i + 1) '

flm-kak«n-k 2n + 2m - 4k + 1 (2k - 1)!!

j^mnk = - ' -j-5 0>k = -Ti-5 m s n.

' ' am+n-k 2n + 2m — 2k + 1 k!

However, as we will see further the expansion (5.18) is more convenient for the practical implementation then (5.40).

Also note the following relation between iterated Ito and Stratonovich stochastic integrals

(i i i2 i3i4) _ 7-*(iii2i3i4) , 1 n T-*(i3i4) 1 ^ ( /-*(iii4) /-*(iii4)

n*l*2*3*4; _ m*l*2*3*4; _ | m*3*4; _ _ « I 7^*1*4; _ pl'lMJ

j(oooo)t,î — J(oooo)t,i 2 (io)t,i 2 i10)^ (oi)t,i ,

"21{»3=m} ( (T - + 7(0i)T,i) + g(T - î)2i{»i=»2}1{»3=M} w- P-

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Let us denote as

the approximations of iterated Ito and Stratonovich stochastic integrals

I (i 1 ...i k ) 7-*(Î1-"ik )

1 (/1.../k)T,i ' 1 (l1...lk)T,t

defined by (5.3), (5.4), i.e. we replace œ on q in the expansions of these sto-

f*(ï1Î2)q (00)T,t (i1i2) [(00)T,t

chastic integrals. For example, Ioq)^/ is the approximation of the iterated Stratonovich stochastic integral I(oo)y! obtained from (15.10) by replacing to on

q, etc.

It is easy to prove that

1

Moreover, using Theorem 1.3, we obtain for i1 _ i2

M I T*(ili2) _ 1*(ili2)^ 2\ _ mJ ( 1*(ili2) _ 1*(ili2)^ 2

M 1 1(10)T,t 1(10)T,^ f _ M ^(01)7,4 1(01)T,t

(T -1)4 15 1 1

--2V_-__V_

9 4i2 _ 1 (2i

16 y 9 ^ 4i2 - 1 ¿Pi (2i - 1)2(2i + 3)2

y (* + a)2 + (j +1)2 \ (5 42)

For the case i1 _ i2 using Theorem 1.3, we have

m I 1 (iiii) — 1 (iiii)^ 2 i _ mj f ^ _ 1 (iiii)^ 2

IVIS 1 1(10)T,t 1(10)T,^ I I V 1(01)T,t 1(01)T,t

(T~t)4 1 A 1__1

16 I 9 Z^(2i + l)(2i + 5)(2i + 3)2 ^ (2i - l)2(2i + 3)2 I'

v_-__2 V

^(2i + l)(2i + 5)(2i + 3)2 f-'

¿=0 ¿=1

(5.43)

In Tables 5.1-5.3 we have calculations according to the formulas (15.411)-5.43) for various values of q. In the given tables £ means the right-hand sides of these formulas. Obviously, these results are consistent with the estimate (1.223).

Let us consider (5.12), (5.13) for i1 _ i2

Am) _ (T-t)2 ( (AA\2 , 1 Ah) An)

V3

pi'in; _ v^- / i / /in; j____/1 IJ IJ_i_

J(01)T,i- A n> J + t.1 +

, vi 1 ^(h)^(H)__1 /V(ii)Y

tiVV(2i + l)(2i + 5)(2i + 3)^ W2 (2? — l)(2i + 3) V J

(5.44)

,*(hh) _ (T-t)2f (Ah)\2 ,J_Ah)Ah),

\io)T,t - 4 / ^

TO 1 Ji,) Ji,) 1 / )\2

4- v i 1 a'aa'n) 1 /mn)\

¿¿1 \J(2i + l)(2i + 5)(2i + 3) W2 (2i — l)(2i + 3) V J

(5.45)

Table 5.1: Confirmation of the formula (5.41) 2e/(T - t)2 0.1667 0.0238 0.0025 2.4988 ■ 10-4 2.4999 ■ 10-5

q 1 10 100 1000 10000

Table 5.2: Confirmation of the formula (5.42) 16e/(T - t)4 0.3797 0.0581 0.0062 6.2450 ■ 10-4 6.2495 ■ 10-5

q 1 10 100 1000 10000

From (5.44), (5.45), considering (5.7) and (5.8), we obtain

(T - t)2 ft, (,i)\2 1 Ah) Ah) \ _ Ah) Ah) TT7 ^ -,

(10)t,i 1(0l)T,t — 2 V V / /v° I ~ 1(Q)T,t1(l)T,t P'

V3

(5.46)

Obtaining (5.46) we supposed that the formulas (5.12), (5.13) are valid w. p. 1. The complete proof of this fact is given in Sect. 1.7.2 (Theorem 1.10).

Note that it is easy to obtain the equality (5.46) using the Ito formula and standard relations between iterated Itô and Stratonovich stochastic integrals.

Using the Itô formula, we obtain

(id)) V

I(u)T,t = 2 w' p' ( }

In addition, using the Ito formula, we have

(T - t)3

(20)T,i (02)T,i ~~ 1{0)T,t1{2)T,t 3 W" P- -1- W-^Oj

From (5.48), considering the formulas (5.25), (5.26), we obtain

r*(il«l) | r*(ilil) _ r(il) r(il) w p 1 (5 49)

1 (20)T,t + 1 (02)T,t _ 1 (0)T,t 1 (2)T,t w. p. 1. (5.49)

Let us check whether the formulas (5.47), (5.49) follow from (5.22)-(5.24),

Table 5.3: Confirmation of the formula (5.43)

16e/(T-t)4 0.0070 4.3551-10-5 6.0076-10-8 6.2251 ■ 10-11 6.3178 ■ 10-14

q 1 10 100 1000 10000

if we suppose ¿1 = i2 in the last ones. From (I5.22I)-(I524) for ¿1 = i2 we get

(T -t)2

(20)T,i (02)T,i 2 (00)T,i ^ ' J ^J(10)T,i J(01)T,iy "T"

(T -t)2

J-*(h«i) T t / r*(1i1i) 1 7-*(ii«i a , (T -1) Mil a2 J(ii)T,i- 4 J(00)T,i 2 v aw + 24 V1 J '

(5.51)

It is easy to see that considering (5.46) and (5.7)) (HTTP)), we actually obtain the equalities (5.47) and (5.49) from (5.50) and (5.51). This fact indirectly confirms the correctness of the formulas (5.22)-(5.24).

Obtaining (5.47), (5.49) we supposed that the formulas (5.22)-(5.24) are valid w. p. 1. The complete proof of this fact is given in Sect. 1.7.2 (Theorem 1.10).

On the basis of the presented expansions of iterated stochastic integrals we can see that increasing of multiplicities of these integrals or degree indices of their weight functions leads to noticeable complication of formulas for the mentioned expansions.

However, increasing of the mentioned parameters leads to increasing of orders of smallness with respect to T — t in the mean-square sense for iterated stochastic integrals. This leads to a sharp decrease of member quantities in expansions of iterated stochastic integrals, which are required for achieving the acceptable accuracy of approximation. In this context, let us consider the approach to the approximation of iterated stochastic integrals, which provides a possibility to obtain the mean-square approximations of the required accuracy without using the complex expansions like (5.40).

Let us analyze the following approximation of iterated Ito stochastic integral of multiplicity 3 using (5.18)

qi (

I(iii2i3)qi = V"^ c I Z(n)Z(i2V(i3L 1 1 Z(i3) —

1 (000)T,t = Cj3j2ji I j Zj2 Zj3 1 {li=^} 1 { ji=j2} Zj3

ji,j2,j3=0 V

— 1 {¿2=i3}1 { j2 =j3} Zj1i) — 1 {ii = i3}1 {ji=j3}Cj22^ , (5.52)

where Cj is defined by (5.20), (5.21).

In particular, from (5.52) for i1 _ i2, i2 _ i3, i1 _ i3 we obtain

qi

^¿l^M _ V"^ C z(ii)z(i2)z(i3) (5 53)

1(000)T,t _ Z^ Cj3j2 jl Sj'i zj2 zj3 • (5.53)

ji,j2,j3=0

Furthermore, using Theorem 1.3 for k _ 3, we get

|\/1 J / 1 (ili2i3) _ 1 (¿l«2i3)qA 2\ _

IVI S 1 1(000)T,t 1(000)T,^ f _

(T - t)3 qi

E clnn (5.54)

6

jl,j2,j3=0

M J I 1 (ili2i3) _ 1 (ili2i3)?A M _

IVI S 1 1(000)T,t 1(000)T,^ t _

?l ?l 2

6

E Cj3j2ji - E Cj2j3jiCj3j2ji (i1 _ «2 _ i3), (5.55)

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jl,j2,j3=0 jl,j2,j3=0

M J I 1 (ili2i3) _ 1 (ili2i3)?A 2\ _

IVI S 1 1(000)T,t 1(000)T,^ f _

?l ?l 2

6

E Cj3j2jl - E Cj3j2jl Cjij2j3 (i1 _ «3 _ i2), (5.56)

jl,j2,j3=0 jl,j2,j3=0

M J f 1 (ili2i3) _ 1 (ili2i3)?A 2\ _

IVI S 1 1(000)T,t 1(000)T,^ f _

?l ?l 2

E Cj3j2ji - E Cj3jij2Cj3j2ji (i1 _ «2 _ i3). (5.57)

jl,j2,j3=0 jl,j2,j3=0

From the other hand, from Theorem 1.4 for k _ 3 we obtain

jl,j2,j3=0

where i1, i2, i3 _ 1,..., m.

M { ('S " Cm )2} ^ 6 - £ C?A (5.58)

We can act similarly with more complicated iterated stochastic integrals. exampl (see (5.31))

For example, for the approximation of stochastic integral 1(((1012(13:z4t we can write

q2 /

i(lil2l3l4)q2 = V^1 c I Z(1i)z(I2)Z(13)Z(I4)

1 (0000)T,t = Z^ Cj4j3j2 j J j Z j2 Z j3 Zj4 jij2,j3,j4 =0 V

-1 r • ilr- (13)Z (14) _ 1r. . -,1r. .,Z (12 ) Z (14)_

1{1i=12}1{ji=j2}Sj3 Z j4 1{1i = 13}1{ji=j3}Sj2 Zj4

_i 1 Z (12) Z (13L 1 1 Z (1i)z (14) —

1{1i=14}1{ji=j4}Zj2 Z j3 1{12 = 13}1{j2=j3}Zji Zj4

_1 1 Z (1i )Z 1 1 Z (1i )Z (12) +

1{12 = 14}1{j2 =j4}Zji Zj3 1{13=14} 1{j3=j4}Zji Zj2 +

j4j3j2ji

j2 =j4}^ji j {13=14} {j3=j4} ji j

+ 1{1i=12} 1{ji=j2} 1{13 = 14} 1{j3=j4} + 1{1i=13} 1{ji=j3} 1{12=14} 1{j2=j4} +

+ 1{1i = 14} 1{ji=j4}1{12=13} X{j2=j3^ , (5.59)

where Cj j is defined by (5.33), (5.34).

Moreover, according to Theorem 1.4 for k = 4, we get

l\/l J ( r(1i121314) 7-(1i121314)q2 \ / (T —t)4 V- s~i2

IVI | ^(0000)T,t. J(oooo)T,t. / f I 24 Z^

V jij2,j3,j4=0

where i2, i3, i4 = 1,..., m.

For pairwise different ¿1, ¿2, ¿3, ¿4 = 1,..., m from Theorem 1.3 we obtain

l\/l J ( r(1i121314) 7-(1i121314)q2 \ 2\ (T —t)4 ^2

IVI I 1/(0000)T,t J(oooo)T,t ) f — 24 Z^ ( ./:./:■,/■/./ • w-ouj

jij2,j3,j4 =0

Using Theorem 1.3, we can calculate exactly the left-hand side of (5.60) for any possible combinations of i1,i2,i3,i4. These relations were obtained in Sect. 1.2. For example

M J f l(1i 121314) _ i(1H21314)qA =

M ^ 1 1 (0000)T,t 1 (0000)T,t y f =

(T-t)A A

24 / V * ./:./:■,/:•./• | / v | / v °3Aj-ij2jl

Cj4j3j2ji ( ( Cj

ji,j2,j3,j4 =0 V(j'i,j2A(j3,j4)

where i1 = i2 = i3 = i4 and

E

(ji,j2)

means the sum with respect to permutations (j1; j2).

Table 5.4: Coefficients Co^ii

3i = 0 il = =1 ii = 2 ¿1 = 3 ¿1 = 4 il = 5 il = 6

j2 = 0 4 3 -2 3 2 15 0 0 0 0

j2 = 1 0 2 15 -2 15 4 105 0 0 0

j2 = 2 -4 15 2 15 2 105 -2 35 2 105 0 0

j2 = 3 0 -2 35 2 35 2 315 -2 63 8 693 0

j2 =4 0 0 -8 315 2 63 2 693 -2 99 10 1287

j2 = 5 0 0 0 -10 693 2 99 2 1287 -2 143

¿2 = 6 0 0 0 0 -4 429 2 143 2 2145

Table 5.5: Coefficients Cü

Jl = 0 il =1 il = 2 ¿1 = 3 il = 4 il = 5 il = 6

j2 =0 2 3 ! 15 0 2 105 0 0 0

j2 =1 2 15 0 -4 105 0 2 315 0 0

j2 =2 -2 15 8 105 0 -2 105 0 4 1155 0

j2 =3 -2 35 0 8 315 0 -38 3465 0 20 9009

j2 =4 0 -4 315 0 46 3465 0 -64 9009 0

j2 =5 0 0 -4 693 0 74 9009 0 -32 6435

¿2 =6 0 0 0 -10 3003 0 4 715 0

Table 5.6: Coefficients <C2j2j1

il = 0 ii = =1 ii = 2 ¿1 = 3 ii = 4 il = 5 ii = 6

j2 =0 2 15 0 -4 105 0 2 315 0 0

j2 =1 2 15 -4 105 0 -2 315 0 8 3465 0

j2 =2 2 105 0 0 0 -2 495 0 4 3003

j2 =3 -2 35 8 315 0 -2 3465 0 -116 45045 0

j2 =4 -8 315 0 4 495 0 -2 6435 0 -16 9009

j2 =5 0 -4 693 0 38 9009 0 -8 45045 0

¿2 =6 0 0 -8 0 118 0 -4

3003 45045 36465

Table 5.7: Coefficients C3j2j1

3i = 0 3i = l 31 = 2 3i = 3 3i = 4 3i = 5 31 = 6

j2 = 0 0 2 105 0 -4 315 0 2 693 0

j2 = 1 4 105 0 -2 315 0 -8 3465 0 10 9009

j2 = 2 2 35 -2 105 0 4 3465 0 -74 45045 0

j2 = 3 2 315 0 -2 3465 0 16 45045 0 -10 9009

j2 =4 -2 63 46 3465 0 -32 45045 0 2 9009 0

j2 = 5 -10 693 0 38 9009 0 -4 9009 0 122 765765

32 = 6 0 -10 3003 0 20 9009 0 -226 765765 0

Table 5.8: Coefficients Cjj

31 = 0 3i = l 3l = 2 31 = 3 3i = 4 3i = 5 31 = 6

j2 =0 0 0 2 315 0 -4 693 0 2 1287

j2 =1 0 2 315 0 -8 3465 0 -10 9009 0

j2 =2 2 105 0 -2 495 0 4 6435 0 -38 45045

j2 =3 2 63 -38 3465 0 16 45045 0 2 9009 0

j2 =4 2 693 0 -2 6435 0 0 0 2 13923

j2 =5 -2 99 74 9009 0 -4 9009 0 -2 153153 0

32 =6 -4 0 118 0 -4 0 -2

429 45045 13923 188955

Table 5.9: Coefficients (J5j2j1

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31 = 0 3i = l 31 = 2 31 = 3 3i = 4 3i = 5 31 = 6

32 =0 0 0 0 2 693 0 -4 1287 0

32 =1 0 0 8 3465 0 -10 9009 0 -4 6435

32 =2 0 4 0 -74 0 16 0

1155 45045 45045

32 =3 8 0 -116 0 2 0 8

693 45045 9009 58905

32 =4 2 99 -64 9009 0 2 9009 0 4 153153 0

32 =5 2 0 -8 0 -2 0 4

1287 45045 153153 415701

32 =6 -2 4 0 -226 0 -8 0

143 715 765765 415701

Table 5.10: Coefficients C6j2ji

¿1 = 0 ¿1 = 1 ji = 2 ji = 3 Ji = 4 ¿1 = 5 ¿1 = 6

j2 = 0 0 0 0 0 2 1287 0 -4 2145

¿2 = 1 0 0 0 10 9009 0 -4 6435 0

¿2 = 2 0 0 4 0 -38 0 8

3003 45045 36465

j2 = 3 0 20 9009 0 -10 9009 0 8 58905 0

¿2 = 4 10 0 -16 0 2 0 4

1287 9009 13923 188955

¿2 = 5 2 -32 0 122 0 4 0

143 6435 765765 415701

¿2 = 6 2 0 -4 0 -2 0 0

2145 36465 188955

Table 5.11: Coefficients Cooj2j1

¿1 = 0 jl = 1 3i = 2

32 = 0 2 3 -2 5 2 15

32 = 1 -2 15 2 15 -2 21

32 = 2 -2 15 2 35 2 105

Table 5.12: Coefficients Ci0j2j1

jl = 0 jl = 1 ¿1 = 2

32 = 0 2 5 -2 9 2 35

32 = 1 -2 45 2 35 -2 45

32 = 2 -2 21 2 45 2 315

Table 5.13: Coefficients C02j2j1

jl = 0 jl = 1 ¿1 = 2

32 = 0 -2 15 2 21 -4 105

32 = 1 2 35 -4 105 2 105

32 = 2 4 105 -2 105 0

Table 5.14: Coefficients C0ij2j1

j2 = 0 j2 = 1 j2 = 2

ji = 0 ji = 1 ji = 2

_2_

15 _2_

45 -2

35

-2 45 -2 105 _2_ 63

-2 105 2 315 -2 315

Table 5.15: Coefficients <C11j2j1

./i =0 ./, 1 ./, 2

j2 = 0 j2 = 1 j2 = 2

_2_

15

2

105 -4 105

-2

35

0

2

105

0

-2

315

0

Assume that q1 = 6. In Tables 5.4-5.10 we have the exact values of coefficients ôj3j2j1 (j1,j2,j3 = 0,1,...,6). Here and further in this section the Fourier-Legendre coefficients have been calculated exactly using computer algebra system Derive. Note that in [52], [53] the database with 270,000 exactly

calculated Fourier-Legendre coefficients was described. This database was used in the software package, which is written in the Python programming language for the implementation of the numerical schemes (4.65)-(4.69), (4.74)-(4.78).

Calculating the value on the right-hand side of (5.54) for q1 = 6 (i1 = i2, ii = i3, i3 = i2), we obtain the following approximate equality

M { - ^ « 0.01956(T - i)3.

Let us choose, for example, q2 = 2. In Tables 5.11-5.19 we have the exact values of coefficients Cj4j3j2jl (j1,j2,j3,j4 = 0,1, 2). In the case of pairwise

Table 5.16: Coefficients C2oj2j1

ji = 0 ji = 1 ji = 2

j2 = 0 j2 = 1 j2 = 2

A o

15 35 u

-2- 0 ^

105 315

105 105 u

Table 5.17: Coefficients <C21j2j1

32 = 0 32 = 1 32 = 2

31 = 0 31 = 1 31 = 2

_2_

21

2 315 -2 105

-2

45 2 315 2 225

315 -2

225 2

1155

2

Table 5.18: Coefficients <C12j2j1

32 = 0 32 = 1 32 = 2

31 = 0 31 = 1 31 = 2

-2 35 _2_

63 2

105

_2_

45 -2 105 -2

225

-2 105 2 225 -2 3465

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different ii, ¿2,23,24 we obtain from (5.60) the following approximate equality M { (fet - 2} - 0.0236084(T - i)4. (5.61)

Let us analyze the following four approximations of the iterated Ito stochastic integrals (see (I5.,35I)-(I5~391))

q3

j-(Hi2i3)q3 = V"^ c001 / z(n)z(i2V(i3) _ 1 1 z(i3)_

J(001)T,t Z^ j zj2 zj3 -L{i1=i2} ±{j1=j2}zj3

j1 J2,j3 =0

(;0 _ 1 1 z (i2 )

j1 1{;1 = ;3} ±{j1=j3}Zj2

-1{i2=i3} 1{j2=j3} Cj,1 - !{i1 = i3} 1 {j 1 =j3}Zj22 ' (5.62)

q3

I(i1 i2,3)q3 = V"^ c0!0 ( z(i1)C(i2)z^ _ 1 1 z(i3)

J(010)T,t Z^ Cj3j2j^ zj1 z j2 zj3 1{;1=;2} 1{j1=j2} zj3

j1 J2,j3 =0

-1{i2=i3}1 {j2=j3} cj;1) - 1{i1=i3} 1{j1=j3}Cj(22 ) I > (5.63)

j1

q3 /

Ec 100 I z(;1)z(;2)z(;

Cj3j2j1 I zj1 z j2 zj3

,j2,j3 =0 V

1(;1 ;2;3)q3 = V"^ c!00 I z(;1)z(;2)z_ 1 1 z(;3)_

J(100)T,t Z^ Cj3j2j1 I zj1 z j2 zj3 1{;1=;2} 1{j1=j2} zj3

-1{i2=i3} 1 {j2 =j3} Cj;1) - 1{i1 = i3} 1{j1=j3}Cj2;2) I ' (5.64)

Table 5.19: Coefficients C22j2j1

j2 = 0 j2 = 1 j2 = 2

ji = 0 ji = 1 ji = 2

_2

105 2

315

JL. q

1 n C O 1 c KJ

3i5 0

2

3465

-2 1155

0

I

(¿1i2i3i4«5)q4

(00000)T,t

q4

4

C.....

Cj5j4j3 j2j1

j1 ,j2,j3,j4,j5 ='

+ I{«1=i2j + 1{«1=i2j + 1{il=«3j + 1{i1=i4| + !{i1 = i4| + 1{i1 = i5} + 1{i2 = i3|

1{«1=«2} 1{«1=«4} 1{«2=«3} 1{«2=«5} 1{«3 = «5} 1{j1=j2} {j1=j2} {j1=j3} {j1=j4} {j1=j4} {j1=j5} {j2=j3}

Z(i3)Z(i4) Z(i5) _

j1=j2} Zj3 Zj4 Z j5

Z (i2 ) Z (i3)Z (i5) _ j1=j4} Zj2 Zj3 Zj5

Z (i1)Z (i4) Z (i5) _

j2=j3} Zj1 Zj4 Z j5

Z

J=1

(il ) jl

{¿1 = «3}

{¿1 = «5}

{¿2 = «4}

. Z(i1)Z(i3)Z('4L I r

j2=j5} Zj1 Zj3 Z j4 1 {i3 = M}

Z(i1)Z(i2)Z(i4) _ 1

j3=j5} Zj1 Z j2 Zj4 1 {i4=«5}

{¿3=«4} {¿4=«5} { '2=^5 } {¿2=«3} {'3=^5 } {¿2 = «4} {¿4 = 25 }

{j3=j4} Cjj; ) +

Z ('3) + {j4=j5} Zj3 +

Z (i4) + {j2=j5} Zj4 +

■ \C(i5) +

{j2=j3} Zj5 +

1

{¿1=«2}

{¿1=«3}

{¿1=«4}

{j3=j5} j ) +

1

{¿1=«5}

{j2=j4} j ) +

1

{¿1 = «5}

Z(i1) I {j4=j5} Zj1 +

1

{¿2 = «4}

1 {j1 1 j

1 {j2: 1{j3: 1 { j4: 1{j1 1{j1 1{j1 1{j1 1{j1 1{j1 1{j2:

Z (i2 ) Z (i4) Z (i5)_ =j3}Zj2 Zj4 Zj5

Z (i2 ) Z (i3) Z(i4)_ =j5}Zj2 Zj3 Zj4

Z (i1)Z ('3) Z (i5)_

=j4} Zj1 Zj3 Zj5

Z (i1)Z (i2) Z (i5)_

=j4} Zj1 Z j2 Zj5

. Z (i1 )Z (i2 ) Z ('3) I

=j5}Zj1 Zj2 Zj3 +

j2} j3} j3} j4} j5} j5} j4}

1{«3=«5} 1{«2=«4}

1{i4=i5}

1{«2=«5} 1{«2 = «3} 1{«3 = «4} 1{«3 = «5}

{ j3 = j5 } Cj4 +

Z (i5) I {j2=j4} Zj5 +

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Z (i2) I {j4=j5} Zj2 +

■ iZ (i3)+

{j2=j5} Zj3 + {j2=j3} j ) +

{j3=j4} Cjij ) +

Z(i1) I {j3=j5} Zj1 +

+ 1{«2 = «5} 1 {j2 = j5 } 1{«3=«4} 1{j3=j4}C

('1)

j1

(5.65)

Assume that = 2, = 1. In Tables 5.20-5.36 we have the exact values of

Fourier-Legendre coefficients Cj, CT^j^, (7]j2 h (j1, j2, j3 = 0,1, 2), C

j5j4j3j2j1

(jl,...,j5 = 0, 1).

In the case of pairwise different il5..., i5 from Tables 5.20-5.36 we obtain

Table 5.20: Coefficients C°0f2n

3i = 0 3i = 1 3l = 2

j2 = 0 -2 14 15 -2 15

j2 = 1 -2 15 -2 15 6 35

h = 2 2 5 -22 105 -2 105

Table 5.21: Coefficients

31 = 0 3i = 1 3l = 2

j2 =0 -6 5 22 45 -2 105

j2 =1 -2 9 -2 105 26 315

32 =2 22 105 -38 315 -2 315

Table 5.22: Coefficients C°2f2n

31 = 0 3i = 1 31 = 2

32 =0 -2 5 2 21 4 105

32 =1 -22 105 4 105 2 105

32 =2 0 -2 105 0

Table 5.23: Coefficients

3l = 0 31 = 1 3l = 2

32 =0 -2 3 2 15 2 15

32 =1 -2 15 -2 45 2 35

32 =2 2 15 -2 35 -4 105

Table 5.24: Coefficients Cl™n

31 = 0 3i = 1 31 = 2

32 =0 -2 5 2 45 2 21

32 =1 -2 15 -2 105 4 105

32 =2 2 35 -2 63 -2 105

Table 5.25: Coefficients C™h

ji

j i

j2 = 0 j2 = 1 j2 = 2

-2 15 -2 21 -2 105

-2 105 -2

315 -2 315

4

105 2

105

0

1

2

0

Table 5.26: Coefficients

3i = 0 3i = 1 ji = 2

32 = 0 ^ I Ö

32 = 1 ^ 0 ^

32 = 2 ± -f 0

Table 5.27: Coefficients Cjj

j2 = 0 j2 = 1 j2 = 2

31 = 0 31 = 1 3l

-4 5 4 15 ! 105

-4 15 4 105 4 105

4 35 -8 105 0

Table 5.28: Coefficients C^

3i = 0 3i = 1 3i = 2

79 = 0 —

^2 15 105 105

79 = 1 — -A- -A-

^2 ^ 21 105 315

32 = 2 ^ 0 0

Table 5.29: Coefficients Cooo^j

_ 31 = 0 31 = 1

j2 = 0 j2 = 1

J_ -8

15 45

-4 8

45 105

Table 5.30: Coefficients C0i0j2j _ Ji = 0 Ji = 1

j2 = 0 j2 = 1

4_ ^16

45 315 -4 4

315 315

Table 5.31: Coefficients Cii0j2j

_ Ji = 0 Ji = 1

j2 = 0 j2 = 1

8 -2

105 45

-4 4

315 315

Table 5.32: Coefficients C0iij2j1 _Ji = 0 Ji = 1

79 = 0 ^

J2 u 315 315

¿2 = 10

Table 5.33: Coefficients Cooy2ji Ji = 0 ji = l

¿2 = 00 ^

72 = 1 — —

J2 315 105

Table 5.34: Coefficients Ci00j2j1

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ji = 0 ji = 1

¿2 = 0 £

j2

1

315

-4 35 _2_

45

Table 5.35: Coefficients Ci0ij2j

__Ji = 0 ¿i = l

j2 = 0 j2 = 1

fa- 0

315 w

4 -8

315 945

Table 5.36: Coefficients Cmj2j1 _31 = 0 31 = 1

72 = 0 -2- ^

J2 u 105 945

¿2 1 945

105 945

-2- 0

945 w

MA J f T(«1*2^3) _ T2\ =

IVI I ^T(100)T,t T(100)T,t ^ | =

E (^™,)2 - 0.00815429(T-i)!

j1 j2 j3=0

M J ( T(i1i2«3) _ T(i1i2i3)qA 2\ = IVI | \vT(010)T,t T(010)T,t ^ I =

E №)S« 0.01739030(T-t)!

j1 j2 j3=0

M J ( T(i1i2«3) _ T(i1i2i3)qA 2\ = IVI | \vT(001)T,t T(001)T,t ^ J =

E (C^,)2- 0.02528010(7-^

j1,j2,j3=0

m ^ I T(«1 ¿2¿3«4¿5) _ T(¿122232425)^^ 2

11 I (00000)T,t T(00000)T,t

(T"i)5 ¿ ^^,« 0.00759105(7-^

120

j1,j2,j3,j4,j5=0

Note that from Theorem 1.4 for k = 5 we have

^2122232425) /-(2122232425)9^ I ^ 1 on

/ (T - t)5 «

l\/l J i 7-(2122232425) 7(2122 232425 )q4 V /ion / (T -t) V^ ^2

IVI I 1/(00000)1^ J(ooooo)T,t y j - \ 120 ^is^i

where i1,..., i5 = 1,..., m.

j5j4j3j2j1

j1j2 j3j4j5=0

Differential Equations and Control Processes, N 1, 2023 Moreover, from Theorem 1.4 we obtain the following useful estimates

k } \ jU2=0 J

1 } \ jl,j2=0 J

21 (T -t)5

M i ^(100)T,t - 7(100)T,t y J - I —60--^ \yhhh) J'

(T - t)5

V^ //-/010 A2 1

M i y (oio)T,t - (oio)T,t) I - b\ —20--^ \yhhh) J'

V jl,j2 ,j3=0 /

(T -1)5

V^ //-/001 \2 \

M i y (001)T,t - (001)T,t y j - I —10--I '

V jl,j2 ,j3=0 /

m{(S-^)2}<2(^- £ (C&A

^ ^ \ j2,jl=0 J

k } \ j2,jl=0 J

Mi^-«)2}^^^- t (c&A

k j V j2,jl=0 /

^^ (T -1)6

V^ Z/-/1000 \2 1

IVI | VJ(1000)T,i J(1000)T,t J J - I 360 ^ V^jdsjiji) I'

V jlj2 J3J4=0 '

^O, (T - t)6

V^ //-/0100 \2 \

IVI | ^(oioo)T,i J(0100 )T,t ) r- I 120 ^ y^jinnji) I'

V jl,j2 ,j3,j4=0 /

(T - t)6 V^ //-/0010 A2 \

IVI HJ(00i0)T,t J(00i0)T,t / J - I 60 ^ y^hhhh) h

V jlj2j3j4=0 /

^O, (T - t)6

V^ //-/0001 A2 1

IVI | ^(0001)T,i J(0001)T,t y J \ 36 ^ \hhhh) I'

V jlj2j3j4=0 / http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal. http://diffjournal.spbu.ru/ A.757

M ^ i T(212223242526) _ t(212223242526)q\ 21 ^ M S 1 T(000000)T,t T (000000)T,t I ( —

(T -1)6

< 720 -tL - Y C2

— I 720 ''

j1 j2 j3j4j5 j6=0

In addition, from Theorem 1.3 for k = 2 we get

2

im J 1 t(2122) _T(2122)q

M 1 T(10)T,t T(10)T,t

q q

(T - t)4 V^ ^10 V2

E j2 - £ jCj (i1 = i2),

12

j1,j2=0 j1,j2=0

j1,j2=0

i / T(2122) _ T(2122)^ 2

m j vT(01)T,t T(01)T,t

qq

(T - t)4 ^ V2

E (j - E jCj (i1 = i2),

j1,j2=0 j1,j2=0

'l V

j1j2=0

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I I T(2122) _ T(2122)^ 2

M 1 VT(20)T,t T (20)T,t

qq

(T - t)6 ^ ^0 V2

E (O2 - E jj (i1 = i2),

30

j1,j2=0 j1,j2=0

im 1 i t(2122) _T(2122)q

1 1 T(11)T,t T(11)T,t

j1 j2=0 2

q

(T - t)6 A ,„11 \ 2

E j2 - £ Cj (i1 = i2),

18

j1,j2=0 j1ij'2=0

m{ (/,<;$., - = ^ - E (faf «. * y

I I t('ii2L T(2122)q

1 1 T (02) T (02)T,t

j1j2=0 2

(T - t)6 A „<02 V2

q

6

E (j - E j cj (i1 = i2),

j1,j2=0 j1,j2=0

m{ - = fafa - E W* (h * h).

1 } j1,j2=0

Clearly, expansions for iterated Stratonovich stochastic integrals (see Theorems 1.1, 2.1-2.9, 2.30, 2.32-2.35, 2.40 and Hypotheses 2.1-2.3) are simpler than expansions for iterated Ito stochastic integrals (see Theorems 1.1, 1.2 and (11.45l)-(fT75T1)). However, the calculation of the mean-square approximation error for iterated Stratonovich stochastic integrals turns out to be much more difficult than for iterated Ito stochastic integrals. Below we consider how we can estimate or calculate exactly (for some particular cases) the mean-square approximation error for iterated Stratonovich stochastic integrals.

Consider the iterated Stratonovich stochastic integral of multiplicity 2

* T * t2

J*[^(2)kt = J Mt2)J ^1(t1)dft(121)dft(221) (i1 = 1,... ,m), t t

where ^1(r), ^2(t) are continuously differentiable functions on [t,T].

By Theorem 2.2 we have

p

J>(2,]r.t = 1;^. E jcjr'cj:1».

j1j2 =0

Consider the following approximation of the stochastic integral J *

q

J*№(2)]T,t = E CjZj21)cj.21'.

j1 j2=0

T,t

q

According to the standard relation between Stratonovich and Ito stochastic integrals (see (2.452)) and (1.94), we obtain

2

JT,t

m J*[^(2)]T,t - J*№(2)]?

T ^ 2

\j[f1)]T, + \ lMs)i'2(s)ds - £ C^c^ t jl?j2=0

T

1 t q

M { ./[V'(2)]r,f - -mAh + X / Ms}Us)ds

2 I / v ~jljl

t jl=0

T

1 T q

M j (j[V'(2)]t,< - .#<=%)' [+ 2/Ms)Us)ds

t

jl=0

where

/q q

K2 (t1, t2 )dt1dt2 - E Cj2jl - E Cj2jl Cjl j2 +

[tT]2 jl?j2=0 j'l,j2 =0

/1 T q 2

+ o / -

V t jl=0

qq

J= £ Cj2jlcj;'»c,(2:l) - £ j

jl,j2=0 jl=0

is the approximation (see (1.46)) of the iterated Ito stochastic integral

T t2

J i^(2)]T,t = / Ut2)f ^1(t1)dft(l;l)dft(2;l) (i1 = 1,... ,m).

It is not difficult to see that the value

m j (J*i^(2)]T,t - J rr 'JT,t is greater than the value

m Ji^(2)]T,t - Ji^(2)]q

2

T,t

2

by

\ t ji=°

For some particular cases E(il) = 0. For example, for the case ^i(t), ^2(t) = 1 ({0j (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T])) we have

9 ,2 ,2 1,_ - 1T

Î2Cnn = lÎ2fan)2 = l(C0f=1-(T-t) = 1- [ d.s. ¿1=0 ¿1=0 J

However, = 0 in a general case.

Consider the following iterated Stratonovich stochastic integral of multiplicity 3

* T * is * ¿2

S = // / fiXf (M'2,i3 = 1,...,m).

t t t

Taking into account the standard relations between Itô and Stratonovich stochastic integrals (see (2.382)) and Theorem 1.1 (the case k = 3) together with Theorem 2.8, we obtain

M J ( I*(i1i2is) _ I*(i1i2is)A K =

IVI 1 \vJ(000)T,i J(000)T,t ^ | =

T T T

^^ |^(ooo)t!î + J J dsdf!?^ + -l{i2=i3} J j dï^dr — /(0oo)t?

t t t t

T T

M J r(»l»2»3) _ r{i1i2i3)q , T^l^)? , 1 1 f f J.Jffe) ,

IVI S J(000)T,i J(000)T,i ^ J(000)T,i ^ J-{n=i2}2 / / ^

tt

T T x 2

1

/ /¿ew-^ , (5.66)

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tt

where the approximations I^Oo^Tt^, looo)!1^ are defined by the relations (see (5.17), (5.18)) ' '

I,

(ili2i3)q (000)T,t

EC I z (;l)Z (;2)Z ^ _ i i z (;3)

Cj3j2jl | j Zj2 zj3 ±{il = i2}±{jl=j2}zj3

jl ,j2,j3=0

_1 1 z_ 1 1 z(;2)

1{«2=«3} 1 {j2 =j3} zjl 1{«l = «3} 1 {jl =j3} z j2

I

*(ili2i3)q (000)T,t

E

C z (;l)z (;2)z (;3)

Cj3j2jl zjl z j2 z j3 •

jl,j2,j3 =0

(5.67)

(5.68)

Substituting (5.67) and (5.68) into (5.66) yields

q2

(000)T,t 1 (000)T,t T T

= M ^ | I(;l;2;3) _ I(ili2;3)q + 1

1

f(000)T,i J(000)T,i "T" 1{n=i2} I 2

i J df - £ c,3jljlzj33M +

T T

+ I{i2 = «3}

IJ f(;lW - E Cj3j3jlC

tt jl

jl ,j3=0 q

1

tt

jl,j3=0

{»l=»3} ^^ Cjlj2jl C jl,j2=0

(i2) j2

<

(5.69)

4 m I

-(ili2i3^ I(ili2i3)A 1 + 1 F(;3) + lr •1G(:l) + 1r iH(:

(000)T,t J(000)T,ty ( + 1 {;l=;2}F q + 1{i2=;3}Gq + 1{il=;3}Hq

(i2) q

where

Fii3) = M{\\ [ Idsdfrl3)~ E o,.^,

t t jl j3=0

T

q

1

T

= M ^ U/ / E Q^c

t t jl,j3=0

(il) jl

q

2

Hq;2) = m ^ ( e Cjlj2jlzj jl,j2=0

(i2) j2

(5.70)

(5.71)

(5.72)

(5.73)

In the cases of Legendre polynomials or trigonometric functions, we have (see Theorem 2.8) the equalities

lim Fq(i3) = 0, lim G;l) = 0, lim H(2) = 0.

q—

q—TO

However, in accordance with (5.70) the value

M J f I*(ili2«3) _ I*(ni2«3)qX 2

M 1 \vJ(000)T,t J(000)T,t

with a finite q can be estimated by terms of a rather complex structure (see (5.71)-(5.73)). As is easily observed, this peculiarity will also apply to the iterated Stratonovich stochastic integrals of multiplicities k > 4 with the only difference that the number of additional terms like (5.71)-(5.73) will be considerably higher and their structure will be more complicated (the exact calculation of the mean-square error of approximation for iterated Stratonovich stochastic integrals of multiplicities 1 to 4 is presented in Sect. 5.5, 5.6).

Therefore, the payment for a relatively simple approximation of the iterated Stratonovich stochastic integrals (Theorems 1.1, 2.1-2.9, 2.30, 2.32-2.35, 2.40) in comparison with the iterated Ito stochastic integrals (Theorems 1.1, 1.2) is a much more difficult calculation or estimation procedure of their mean-square approximation errors.

As we mentioned above, on the basis of the presented approximations of iterated Stratonovich stochastic integrals we can see that increasing of multiplicities of these integrals leads to increasing of orders of smallness with respect to T — t in the mean-square sense for iterated Stratonovich stochastic integrals (T — t ^ 1 because the length T — t of integration interval [t, T] of the iterated Stratonovich stochastic integrals plays the role of integration step for the numerical methods for Ito SDEs, i.e. T — t is already fairly small). This leads to a sharp decrease of member quantities in the approximations of iterated Stratonovich stochastic integrals which are required for achieving the acceptable accuracy of approximation.

From (5.41) (ii = i2) we obtain

IVI 1 \ (oo)T,t \oo)T,t ) ( ~ 9 4/2 _ x —

2 / 4a;2-1 8

1

¿=9+1

(T — t)

2q + 1

<Ci--S (5.74)

q

where constant C1 does not depend on q.

It is easy to notice that for a sufficiently small T — t (recall that T — t ^ 1 since it is a step of integration for the numerical schemes for Ito SDEs) there

2

exists a constant C2 such that

m { - Ifiw^ C2m { (l*'« - I^)2} , (5.75)

where I*/^1^^ is an approximation of the iterated Stratonovich stochastic integral I(/i. 1 i i

From (5.74) and (5.75) we finally obtain

M { fatm ~ C1«?,)2} < C^t, (5.76)

where constant C is independent of T — t.

The same idea can be found in [79] in the framework of the method of approximation of iterated Stratonovich stochastic integrals based on the trigonometric expansion of the Brownian bridge process. Note that, in contrast to the estimate (15.76), the constant C in Theorems 2.37-2.39 does not depend on p.

We can get more information about the numbers q (these numbers are different for different iterated Stratonovich stochastic integrals) using the another approach. Since for pairwise different ii,..., = 1,..., m

J*[^(k)]T,t = J[^(k)]T,t w. p. 1,

where J[^(k)]T,t, J*[^(k)]T,t are defined by (5.1) and (5.2) correspondingly, then for pairwise different ii,..., = 1,..., m from Theorem 1.3 we obtain

2\ (T — t)4

V^ /^01 \2

M ) 1/(01 )T,t - Aomt ) ( - 4 z^ \y-hh) ' IV/1 I / J-*(«1«2) J-*(«1«2)q\

n (T — t)4

M ) ^(io)T,i ~ 7(io)T,i J f - ^ ^ '

k J j1,j2=0

IV/1 I / J-*(«1«2«3) J-*(«1«2«3)q \

2| (T — t)3 V- ^2 IVI | \/(000)T,i J(000)T,i J f— g Z^ ( MO-.J ■ \D-<1 J

j3j2j1=0

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■V /1 J / 7-*(«1«2«3«4) WH^^M

IVI HJ(0000)T,i J(0000)T,i 7 j ~ 24 Z^ ( ./\K/V./- ' W-'OJ

j1j2j3j4=0

n (T — t)5

lvl | ^(100)T,i J(100)T,i J f - gQ Z^ " ./:',/:./ / '

j1,j2 J3=°

m

m ) i i*(ili2i3) _ i*(iii2«3)q

m ^ 1 J(010))t,t J(010)t,t

m ^ i 1*(i1i2i3) _ i*(i1i2i3)qy

m S 1 J(001)t,t J(001)t,t y

1 (00000)T,t 1 (00000)T,t

m ^ i 1 *(iii2) _ r*(iii2)q

m S 1 1 (20)t,t 1 (20)t,t

m ^ i 1 *(iii2) _ r*(iii2)q

m s 1 J(11)t,t J(11)t,t

M ^ I 1 *(iii2) _ r*(iii2)q

M ^ 1 1 (02)T,t 1 (02)T,t

m ^ i 1 *(iii2«3«4) _ 1 *(iii2i3i4)qv m ^ 1 J(1000)t,t J(1000)t,t y

m ^ i 1 *(iii2«3«4^ _ 1 *(ii«2«3i4)q m ^ 1 J(0100)t,t J(0100)t,t

m ^ i 1 *(iii2«3«4^ _ 1*(iii2i3«4)q m ^ 1 J(0010)t,t J(0010)t,t

m ^ i 1 *(ii«2«3i4) _ 1 *(ii»2«3i4)g m S 1 J(0001)t,t J(0001)t,t

m ^ i 1 *(iii2«3«4i5i6) _ 1 *(iii2«3«4i5»6)q>\ ' m ^ 1 J(000000)t,t J(000000)t,t y

_ V" /W)10 \2 2Q V ./:■,/:./ / '

ji ,j2,j3=0

(T-t)

jU2 j3=0

5 q

£ j

120 / V j5M«3«2ji<

jU2 ,j3j4j5=0

(T~i)6 £ (CS,)2,

30

(r-t) 18

(T t)

j2,ji=0

6 q £

j2,ji=0

6 q

E №)2,

6

360 120

j2,ji=0 6q

E/C1000 \2

VCj4j3j2ji/ '

ji ,j2 ,j3 ,j4 =0 6q

E/C 0100 \2

VCj4j3j2j'J '

60 (T t)6

36 (T t)6

ji j2 J3,j4 =0

6q

E/C 0010 \2

VCj4j3j2j'J '

ji,j2,j3,j4=0

E/C0001 \2

VCj4j3j2j'J '

ji j2 J3,j4=0 q

- E C2

720 / V j6j5j4j3j2ji '

ji j2 ,j3j4j5 j6=0

Recall that the systems of iterated stochastic integrals (I5.1I)-(I54) are part of the Taylor-Ito and Taylor-Stratonovich expansions (see Chapter 4).

The function K(t1;... ,tk) from Theorem 1.1 for the set (5.3) is defined by

K(t1,..., tA) = (t - tA)1k... (t - t1)1i 1{ti<...<tfc}, t1,..., tA G [t, T], (5.79)

2

q

2

where 1A is the indicator of the set A.

In particular, for the stochastic integrals 1(2)Tt ' 1(00)2r t' 1(сэ oc))^?1^ '

t(i l^)1 t(i l^)1 t(i l-i4) t(i l^ t(i lM t(i l^ (• • _ -1 m the func

J(01)T,t ' J(10)T,t ' J(0000)T,i ' 1 (20)T,t' J(ll)T,t ' 1 (02)T,t (ii ' ' ' ' ' i4 _ 1 ' ' ' ' ' m) the func

tions K(t1 ,..., ) defined by (5.79) look as follows

Ki(ti) _ t — ti ' K2(ti) _ (t - ti)2 ' Koo(ti ' t2) _ 1{t l<t2} ' (5.80)

Kooo(tl't2't3) _ 1{t l<t2<t3}' Koi(ti't2) _ (t — t2)1{t^t2}' (5.81)

Ki0(ti't2) _ (t — ti)1{t l<t)}' K0000(ti,t2) _ 1{t l<t2<t?<t4}' (5.82)

K2o(ti't2) _ (t — ti)21{t l<t2}' Kii(ti't2) _ (t — ti)(t — t2)1{tl<t2}' (5.83)

Ko2(ti't2) _ (t — t2)21{t l<t2}' (5.84)

where ti,... ,t4 G [t,T].

It is obviously that the most simple expansion for the polynomial of a finite degree into the Fourier series using the complete orthonormal system of functions in the space L2([t,T]) will be its Fourier-Legendre expansion (finite sum). The polynomial functions are included in the functions (I5.80l)-(I5T841) as their components if /2 +... + /2 > 0. So, it is logical to expect that the most simple expansions for the functions (5.80)—(5784) into generalized multiple Fourier series will be Fourier-Legendre expansions of these functions when /2 + ... + > 0. Note that the given assumption is confirmed completely (compare the formulas (5.8), (5.9) with the formulas (5.85), (5.90) (see below) correspondently). So, usage of Legendre polynomials for the approximation of iterated Ito and Stratonovich stochastic integrals is a step forward.

5.2 Mean-Square Approximation of Specific Iterated Stratonovich Stochastic Integrals of multiplicities 1 to 3 Based on Trigonometric System of Functions

In [1]-[16], [49] on the base of Theorems 1.1, 2.2, 2.6, and 2.8 the author obtained (also see early publications [71] (1997), [72] (1998), [75] (1994), [76] (1996)) the following expansions of the iterated Stratonovich stochastic integrals (5.4)

77]

"(0

[80], [87] excepting the method, in which

(independently from the papers the additional random variables ^ and ^ are introduced)

T*(h) _ /T _ .Ah)

J(0)T,t = v T tz0 ,

C; = ( (¡a - v (i № + )' <5-85'

. r=1

j-*(hh)q _ 1 (T _ ( Ah) Ah) , 1 ^ (Ah) Ah) _ Ah) sA),

J(00)T,t — ' 1 7J- Z^ r V^2r ^2r_1 Z2r-lS2r

r=1

+ v^ (c2;ic2'2) - c^c^)) + ^v^ (i«"»^2' - cAi^A |, (5.86)

1 r^/

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(^>c2!2)c2!3) - + ^AAA) +

1 9 / 1 /

1 ' 1 ' ^(«l) a(«2^(«3) a(«3) a(«2)a(H)

,___ _ M*i) A'AA'A _ A'A A'12'Ah)\ ,

2^/2 ^\7rr v 2r_1'° 0 Z2r-lS0 so J +

_ AAA'AA'A _ oAjA AjA AjA _i_ z(«3M*2M«i)\ i ,

V z2r z0 z0 2S2r z0 z0 + z2r z0 S0 I I +

n2 r2

i Y^ I ^ (Ah)Ah) Ah) _ AA Ai<2AA) _ Ai2) Ai?>AA) , A1^ Ai2AA)\ ,

\ V r Z2r-lS0 S2r-lS2r So S2r-lS2r So S2r-lS2r So J ^

r=1

i ^ ioAh) A'A A'A * AiiAA)A'A _ aA'A Ai3) z^2),

gyj-2j>2 I °S2r-lS2r-lS0 S2r S2r SO oS2r-lS2r-lS0

+3c2r2)1c2;3.)1c0!l) — 2c2;.l)c2r3)c0'2) + c2;3)c2t2)c0'l)) +D<if3)'' , (5.87)

where

9

(hhh)q _ 1 V^ / ^ / AhV^A^ — Ai2VA)AjA T,t - 2n2 Z^ 1 r2 _ /2 I Z2r S»2/ So S2r SO S»2/

r,l = 1

r=l

^V(n) Ah) Ah) _ j_An)Ah) Ah) \__^Ah) Ah) Ah)

' i S2r-lS2/-lS0 fS0 S2r-lS2/-l I ¿S2r-lS0 S2Z-1

E(_^_f_Ah) Ah) Ah) , Ah) Ah) Ah) ,

I rm V Z2r—1>2to—1>2to S2r-lS2r S2to-1~t~

\r, m=1 \

+Z(i1) Z(i2)Z(i3) _ z(i1 )Z(i2) z(i3) ^ I + S2r—1S2m z2m — 1 S2r S2r—1S2m — 1 J +

i__1_ _Ah) Ah) _ Ah) Ah) Ah)_

rn(r + m) v ^2(m+rP2r ^2m t'2(m+r)-lt'2r-lS2m

_Z(i1) Z(i2) z(i3) I z(i1) Z(i2) z(J3M I +

Z2(m+r) — 1Z2r Z2m — 1 + Z2(m+r) Z2r—1Z2m—1) I +

q q / 1

, V^ V^ I / Z^1^ -L ^^^ Ah) Ah) _

m=1 l=m+1 \ ( )

_Z(i1) Z(i2)z(i3) I z(i1) Z(i2) z(i3) ^ I

Z2(l—m) — 1Z2l Z2m—1 + Z2(l—m)Z2l—1Z2m — 1) +

I 1 ( Ah) Ah)Ah),Ah) Ah) Ah) _

/(/ —^(i-mj^m S2i 1" z2(/-to)-1Z2to-1S2/

_Z(i1) z(i2)z(i3) _ z(i1) Z(i2) z(i3)

Z2(l—m) — 1Z2m z2l — 1 z2(l—m)Z2m — 1z2l — 1

T*(hh)q _ _(rji _ ,\2 I 1 Ah) Ah) _ r^rAh) Ah) ,

J(10)T,t - ^ M 6 2a/27T

_I___ 1 Z^ -I__^_(Ah)Ah)_c)Ah)Ahy

+ Trr 2r_1 0 TrV V 2r 0 ^

1 1 I Ji, ) Jio) l

E1 I A*!) Aw i 6 mh) /-(»2) ,

r2 _ ¿2 I z2r S2/ "+" fS2r-lS2/-l J ~r

r,l=1 r=l

V^ f ^ (Ah) Ah) _ Ah) Ah)\ 1 1 qa(»i) z(*2) 1 ^(¿2)^(^1) /coo\

1 4tT?" \ Z2r—1 S2r-lS2r y/"^g7r2f2 \v°C»2r-lS2r-l S2r S2r ' ^O.OOJ

r=1 \

AA = <T - f >2 (-¡<o",<t> (AA"'' - ^'tf1') +

1 s^l 1 ( AA A''A _ nA''A A''A\__(AAA12) _c)A''AA,'A\

"2v/2 ^Wr V52'"-1^0 ZZ2r-lSo J ^2 So Z(>2r So +

1 q 1 /

1 1 / r

E1 I r Ah) Ah) , Ail) Ah) r2 _ ¿2 I I Z2r-lS2/-l ■+■ S2r S2/

r,l=l r=l

El ^ (Ah) Ah) _ Ah) Ah)\ ^ (oAh) Ai 2) _L

I 47r,r ^2r S2r-1 S2r-lS2r J 87T2r2 VZ2r-lS2r-l + S2r S2r

r=1 \

(5.89)

= (r - a'2 (+ IE + akaA -

. r=1

q

11

I • v^V ) )• (5.90)

V2n X^r \ ;=1

where

q ^ = ,- y v — CL-l' aq = ^ ^ l'LV = y ^ ~9 C^r '

^ r=q+1 r=1 V ' q r=q+1

1 v 1 .(,:) n 1 /„^ 1 1

4 q 1 !>

1 r

r=1

T

0 =

4'

where (s) is defined by (11.681) and (g^, (2r, C2;)-1, , Mq^ (r = 1,---,q, i = 1,..., m) are independent standard Gaussian random variables (i1, i2, i3 = 1,..., m).

Note that (5.88), (5.89) imply the following

(rp _ ,\2

M = = (5-91)

j=0 j=0

where

T x

t t T x

eg = / *(x)(t-x) / (y)dydx.

tt

Note that the formulas (5.91) are particular cases of the more general relation (2.10), which has been applied for the proof of Theorem 2.1.

Let us consider the mean-square errors of approximations (I5.86I)-(I5~K91). From the relations (5.86)—(5789) when i1 = = ii = i3 by direct calculation we obtain

m { («> - «)2} = (5.92)

m {(CS - CS")2} = <r - <>'

55 /n4 q l\ 1 q \ 5/4 + 4r4 -3/2f2\

+ 32^ ^90" S J r2/2(r2_/2)2 J' (5"93)

r=l r=l

m

(ii«2)

-I,

*(«i«2)q

(01)T, t (01)T, t

)2} = (T - i)<

1

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8nM 6

^ y ,^>2 r=1

+

4

32n4

90 E r4 ) + zLtt4 ( E

,k,l = 1 k=l

q

£

k,l=i k=l

l2 + k2

k2(l2 - k2)2

(5.94)

5

/ r*(ili2) _ _ (rp _ +\i( _ V^ J_ 1 ,

^(10 )T,t \io )T,t J { M 8TT2 \ 6 r2 I

m <! ( IJfafa - ]21 = (T - t)4 ( ^ ^ - J2 p

JL/^-V^ 1 X] ¿2 + k2 .

+ 32^90 ffa r4)^ 47T4^ ffajl2{l2-k2)2 r [ '

k=l k=l

It is easy to demonstrate that the relations (5.93), (5.94), and (5.95) can be represented using Theorem 1.3 in the following form

m

I*(iii2i3) _ J-*(«i«2Î3)q

1 (000)T,t 1 (000)T,t

q

(T-f)M-- — T-

{ ' \ AK 4^2 Z^ r2

r=1

55

q

E

1

32n4 ^ r

r=1

4

E

5/4 + 4r4 — 3r2/2

4n4 f= r2/2 (r2 - /2)

r,l=1 r=l

(5.96)

m

I*(ii«2) _ r*(ii«2)q

J(10)T,t J(10)T,t

(T -1)4

9

-È1

2n2 r2

r=1

k2 + /2

8n4

1 "

vi-lv

k=l

r=1

(5.97)

m

I*(ii«2) _ I*(n«2)q

J(01)T,t J(01)T,t

(T -1)

9 2tt2 ^ r2

r=1

5

q

vi-iy

^^ r4 n4

/2 + k2

8n4 ^ r

r=1

n4 ^ k2 (P - k2)2

k,l=i k=l

(5.98)

Comparing and (593) (595). we note that

E

k,l=i k=l

/2 + k2

E

/2 + k2

k2 (/2 - k2)2 /2 (/2 - k2)2

k=l

5/4 + 4r4 — 3r2/2 9n4

7T4

48'

oo

E

r,l=i r=l

r2/2 (r2 - /2)2

80

(5.99)

(5.100)

Let us consider approximations of stochastic integrals I^T I(oi)T )t and conditions for selecting number q using the trigonometric system of functions

T*(hii)q _ _(rp _f\2Î^f AhA 2 J(10)T,t - y1 l) \ Q fao

1

2^/27r

(1 i )z(l i )_ qSq z0

2\/27r2

1

Yl—Cin)Cin)+ 1

/ ^ I 7TT 2'"_1 0

2\/2 ^ \ ?rr

v r=1

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^2 r 2

/H1 i v( z2r S0

2

1

2

5

2

1

E

2n2 ^ r2 — l2

r,l=l r=l

8n2

q1

r=1

Ah)Ah) , !_AA AA

S2r S2/ S2r-lS2/-l

+

z(¿1) V

S2r—1J

+

1

j*{hh)q _ trri .n2[ * /7(*iA2 , /^¿(»1)^1)

7(0i)T,f -U (-3^0 ) Co -

1 r^r- 1 V"^ / 1 ,(iA ,(iA 1

V Pqt'A So "t" ^^r-lSO ^2,r2Z2r So ] +

r=1

2^tt2 v 2 V2 \ irr S2'-1S0 ttV So

q

1 V^ 1 I Ah) Ah) , I>(H) z^I) 2-7J-2 Z^ f2 _ /2 I Z2r S2/ + j S2r-lS2/-l

r,l=l r=l

+ 8^2 E f2 (3 (ci-l) + (4°

r=1 ^

Furthermore, we have

M J f TK^O _ T*(:1:1)^ 2 l = M J f TK^O _ T*(:1:1)^ 2

M 1 VT(01)T,t T(01)T,t ^ f = M | \vT(10)T,t T(10)T,t

_(T-t)A i 2 in4 J_(tA_ Y-lV

4 \ 7T4 \ 90 r4 / + 7T4 \ 6 r2 / +

\ \ r=1 / \ r=1 /

^(e-E)^^)- ("0D

\ k,l—1 k,l— 1 / /

k—l k—l

Considering (5.99), we can rewrite the relation (5.101) in the following form

M J (Tk^o _ T*(:1:1)^ 2\ = M J (Tkh^o _ T*(:1:1)^2

M 1 \vT(01)T,t T(01)T,t ^ f = M | \vT(10)T,t T(10)T,t

(:T-t)A( 17

^2 ~~14

240 3n2 ^ r2 n4 ^ r4

r=1 r=1

Table 5.37: Confirmation of the formula (5.96) e/(T -t)3 0.0459 0.0072 7.5722-10-4 7.5973 ■ 10-5 7.5990 ■ 10-6

q 1 10 100 1000 10000

Table 5.38: Confirmation of the formulas (5.97), (5.98)

4e/(T - t)4 0.0540 0.0082 8.4261-10-4 8.4429 ■ 10-5 8.4435 ■ 10-6

q 1 10 100 1000 10000

l/AlV l2 + k2

+ 7r4\^ r2 J 7T4 k2(l2 — k2)2

\T—1 / k,l— 1

k—l

In Tables 5.37-5.39 we confirm numerically the formulas (5.96)-(5.98), (5.102) for various values of q. In Tables 5.37-5.39 the number £ means right-hand sides of the mentioned formulas. Obviously, these results are consistent with the estimate (1.223).

The formulas (5.99), (5.100) appear to be interesting. Let us confirm numerically their correctness in Tables 5.40 and 5.41 (the number eq is an absolute deviation of multiple partial sums with the upper limit of summation q for the series (5.99), (5.100) from the right-hand sides of the formulas (5.99), (5.100); convergence of multiple series is regarded here when p1 = p2 = q ^ o, which is acceptable according to Theorems 1.1, 2.2, 2.6, and 2.8).

Using the trigonometric system of functions, let us consider approximations of iterated stochastic integrals of the following form

* T * ¿2

Jt'Xlt = / -../ dw«; 1 >...dw<:k>,

t t

where A/ = 1 if i = 1,..., m and A/ = 0 if i = 0, l = 1,..., k (w[;) = fT;) for i = 1,..., m and wT0 = t).

Table 5.39: Confirmation of the formula (5.102)

4e/(T - t)4 0.0268 0.0034 3.3955-10-4 3.3804 ■ 10-5 3.3778 ■ 10-6

q 1 10 100 1000 10000

(5.102)

Table 5.40: Confirmation of the formula (5.99)

£q 2.0294 0.3241 0.0330 0.0033 3.2902 ■ 10-4

10 100 1000 10000

1

q

Table 5.41: Confirmation of the formula (5.100) £q 10.9585 1.8836 0.1968 0.0197 0.0020

q 1 10 100 1000 10000

It is easy to see that the approximations

T*(«i«2)q T*(n«2«3)q

J(A1A2)T,t' J(AiA2A3)T,t

of the stochastic integrals

J

;(«1«2)

„ J

= («1«2«3)

(AiA2)T,^ ^(AiA2A3)T,t

are defined by the right-hand sides of the formulas (5 necessary to take

T

j = / j(s)dw

t

(i)

and ¿i, ¿3 = 0,1,..., m. Since

T

j (s)dw(0) =

y/T^t. if j = 0

0 if j = 0

I, (5.87), where it is

(5.103)

then it is easy to get from (5.86) and (5.87), considering that in these equalities Zj:) is defined by (15.103) and i1 ,i2,i3 = 0,1,...,m, the following family of formulas

t *(«i0)q J(10)T,t

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. r=1

r

£'2i + J*

, (5.104)

n \ r

, r=1

>2r- 1

qSq

, (5.105)

s

= (T-i>5/2(+ № + VW ■ -

•S=<r - *>B/S J) - ¿h (t +

= (T - (№ + do (f + V^r)+

r—1

2fa2fa\zfar

1 Up 1 c

+ 2>/27T {¿fa r-

A)_ I /T^rfa i)

2r-1 + V Sq

T*(0i2i3)q _ (rp _ ,\2 ( \fa2)fa3) _ 1 /TTCih) Ah) j_

J(on)T,t ~ y1 l) IgSo SO 2\/27T

_i__L_ Sfa (__L fa3') faA j__— (A&Av) _ yA^A^y

2fa2 ^ I Trr 2r_1 0 far2 V 2r 0 2r 0

__L v^ 1 I Ai2)Ai3) , l_A^ A^ 1 -L.

2^2 Z^ r2 _ /2 I ^2r S.2Z "+" rS2r-lS2Z-l I

r,l— 1 r—l

q ' 1

r—1 1

8n2 r2

+ W ■ (5.106)

T*(hi20)q _ (rji _ ,\2 / V(n)z(*2) , 1 ,

J(H0)T,t - K1 l) I gSo so ■+■ 2y/2ir q

1 V^ / An) , 1 (Ah) An) _ 9A'AA'A\ \ ,

lSo ■+■ n2r2 y>2r So ZS2r So J j +

r=1

1 V^ 1 / rz(n) ^ +AAAA I .

2tT2 r2 _ ¡2 [ I Z2r-lS2/-l ■+■ S2r S2/ J ~r

r,l—1 r—l

, VI 1 (An) Ah) _ Ah) An)'. ,

4Trr lS2r I +

r=1 \

1

8n2 r2

qZ(:1) Z(:2) i Z(:1)Z 3S2r—1S2r—1 + z2r z2r y

T*(»i0i3)9 _ (T _ i\2 I 1Ah)Ai3) , ^ /tt- /t(n)z(*3) _ ¿(^Z^A _i_

J(ioi)T,t ~ ^ ( g^o So ■+■ 2y27T vq q /

2^/2TT

2\/2 1

r=1

1 V I — fA'A A'A _ A'A AA 1 ,

,__L_ />(¿1)^3) I \ _ 1 V 1 AA A'A

+ o o So +S2r So J I 27T2 ^ rl 2r~V ■

n2r2 \^2r SO ^S2r SO ) j Z^ W S2r-lS2/-l

r,l—1 r—l

El /qz(*i) zfe) , z^V^A \

4^2^,2 ^S2r-lS2r-l -r S2r S2r J I •

;=1

5.3 A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Numerical Solution of Ito SDEs

The section is devoted to comparative analysis of efficiency of application the Legendre polynomials and trigonometric functions for the numerical integration

1

of Ito SDEs in the framework of the method of approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series (Theorems 1.1, 2.1-2.9, 2.30, 2.32-2.35, 2.40). This section is written on the base of the papers [20], [39], and [31] (Sect. 4).

Using the iterated Ito stochastic integrals of multiplicities 1 to 3 appearing in the Taylor-Ito expansion as an example, it is shown that their expansions obtained using multiple Fourier-Legendre series are significantly simpler and less computationally costly than their analogues obtained on the basis of multiple trigonometric Fourier series.

Let us consider the following set of iterated Ito and Stratonovich stochastic integrals from the classical Taylor-Ito and Taylor-Stratonovich expansions [79

T

t2

J

(¿1-ifc )

r(ik)

(Ai...Afc )T,t

(5.107)

.T

J

= (H---ifc)

(Ai...Afc )T,t

st2

(5.108)

where w[;) = f^ for i = 1,..., m and wT0) = t, ¿1,..., ik = 0,1,..., m, A/ = 0 for i/ = 0 and A/ = 1 for i/ = 1,..., m (l = 1,..., k).

In [77] Milstein G.N. obtained the following expansion of JH^ on the base of the Karhunen-Loeve expansion of the Brownian bridge process (we will discuss the method [77] in detail in Sect. 6.2)

J

(; ;2) (11)T, t

1

2

- / , S2r-1 S2r-lS:

n ^ r

r=1

4S2r

+

(5.109)

where the series converges in the mean-square sense, i1 = i2, i1, i2 = 1,..., m,

T

j = / to

are independent standard Gaussian random variables for various i or j (i =

1,..., m, j = 0,1,...),

6.j(s) = . { J y/T^t.

1 for j = 0

\/2sin(27rr(s - t)/{T - t)) for j = 2r - 1, (5.110) \/2cos(2ttr{s - t)/{T - t)) for j = 2r

Ah) _ jrp _ ,Ah)

J(1)Ti = V T tz0 ,

where r = 1, 2, . . .

Moreover,

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'(1)T,t _ w ± ~ ^0 where i1 = 1,..., m.

In principle, for implementing the strong numerical method with the convergence order 1.0 (Milstein method [77], see Sect. 4.10) for Ito SDEs we can take the following approximations

, = -yr^cA, (s.iii)

jCmojq _ ^/rp _ ,) ( Ah) Ah) , 2. "V 1 (Ah) Ah) _ Ah) AA i

;=1

+ , (5.H2)

where i1 = i2, i1, i2 = 1, . . . , m. It is not difficult to show that

m { (JAA - «)2| - ^ (i-±h)- <5'113>

r=1

However, this approach has an obvious drawback. Indeed, we have too complex formulas for the stochastic integrals with Gaussian distribution

r(0n) _(T~ f f/2 (Ah) V^ y^ 1 Ah)

7r ¿—f V

r=1

oo oo

r=1

r(00i!) _ (rp ^5/2 I 1Ah) | 1 V 1 Ah) 1 S^^-An)

6^° 2^27Afpr T7~lr

A0h)q _ (T - tff2 fAn) V2 ^ 1 An) J(01 )T,t~ 9 ^o ^ TMr-1

r=1

.(00n)q _ jT _ . v5/2 I K(n) , 1 V^ _ 1 V^ I An)

J(001 )T,t - v v [ g + 2V27T2 ¿f ^ r 2V27T ¿f r^"1

r=1 r=1

where the meaning of notations used in (5.112) is retained.

In [77] Milstein G.N. proposed the following mean-square approximations

on the base of (5.109), (5.114)

« - - v (t № + ) • (5-115)

' r=1

J

^ it _ +\ ( Ah) An) i 1 v^ ^ (A''AA''A _ A^) Ah)< l) I So So "T" \y2r S2r-1 S2r-lS2r

n r

r=1

+ V5 (<£>,<$» - <№,)) + (^"''Co*2' - W) ). (5-116)

where i1 — i2 in (5.116), and

y fÔTa , T 6 T^

^ r=q+1 r=1

where c0!), c2r', c2TL

\ r = ^..., q i = 1,...,m are independent standard Gaussian random variables.

Obviously, for the approximations (5.115) and (5.116) we obtain [77

2

'(01)T,t — J(01)T,t

m ^ I J(0n) - J(0il)M ^ — 0

IVI \ \ J(01)Tt J(01)Tt i ( — 0,

6 r2

r=1

This idea has been developed in [78]-[80]. For example, the approximation JSi, which corresponds to (5.115), (5.116) is defined by [78]-[80]

« = (T-tr( + dp(±+ vW ■ -

+ (5-119)

where have the form (15.1171).

4 q 1

(«-— V -c{i) ß - — -V-

(s) is defined by (5.110), and (0°, (2?, Cir-i, (r — 1,...,q, i =

1,..., m) are independent standard Gaussian random variables.

Moreover,

mJ J(00ii) (00ii)q V! -0

M 1 V J(001)T,t J(001)T,tJ I — 0'

Nevetheless, the expansions (5.115), (5.119) are too complex for the ap-

(0i i ) T (00ii)

(01)T,t' J(001)T,f

proximation of two Gaussian random variables J(00il)

Further, we will see that the introducing of random variables ^ and Mqi) will sharply complicate the approximation of stochastic integral Jlll2)^ (i1, i2, ¿3 — 1,... ,m). This is due to the fact that the number q is fixed for stochastic integrals included into the considered collection. However, it is clear that due to the smallness of T — t, the number q for /(ll^t could be taken significantly less than in the formula (5.116). This feature is also valid for the formulas (5.115), (5.119).

On the other hand, the following very simple formulas are well known (see

(OM52I))

= (5.120)

Aoon) _ (T ~ f )5/2 (Ah) , ^A'n) 1 Ah) 'mm - Q ^o

where Z((i), Z(i), (i — 1,... ,m) are indepentent standard Gaussian random variables. Obviously, that the formulas (5.120)-(5.122) are part of the method based on Theorem 1.1 (also see Sect. 5.1).

To obtain the Milstein expansion for the stochastic integral (5.2) the truncated expansions of components of the Wiener process fs must be iteratively substituted in the single integrals in (5.2), and the integrals must be calculated starting from the innermost integral. This is a complicated procedure that obviously does not lead to a general expansion of (5.2) valid for an arbitrary

multiplicity k. For this reason, only expansions of simplest single, double, and

triple integrals (5.2) were obtained [77]-[80], [87], [88] by the Milstein approach

77] based on the Karhunen-Loeve expansion of the Brownian bridge process.

At that, in [77], [87] the case ^i(s), ^2(s) = 1 and i^i2 = 0,1,...,m

(il = i2) is considered. In [78]-[80], [88] the attempt to consider the case ^1(s),

^2(s), ^3(s) = 1 and i1,i2,i3 = 0,1,...,m is realized. Note that, generally speaking, the mean-square convergence of J*1(11)2T3i)q to J*/^2^ if q ^ 00 was not proved rigorously in [78]-[80], [88] within the frames of the Milstein approach

77] together with the Wong-Zakai approximation [68]-[70] (see discussions in

Sect. 2.20, 2.21, 6.2).

5.3.1 A Comparative Analysis of Efficiency of Using the Legen-dre Polynomials and Trigonometric Functions for the Integral

T (h«2) J(11)T,t

Using Theorem 1.1 and complete orthonormal system of Legendre polynomials in the space L2([t,T]), we have (see (5.11))

T{hi2) _ T ~ t\ An) A^ I V^ 1 (An) Ah) Ah)Ah)\ i i

J(n)T,t~ 2 ' ' _ i y*"1^ ^ ^-v *-{h=n} y

(5.123)

where series converges in the mean-square sense, i1, i2 = 1,..., m,

T

j = J to Wf

t

are independent standard Gaussian random variables for various i or j,

= ,-=0,1,2..... (5124)

where Pj(x) is the Legendre polynomial.

The formula (5.123) has been derived for the first time in [71] (1997) with using Theorem 2.10.

Remind the formula (5.41) [71] (1997)

M { (jg& - «)*} (5.125)

Table 5.42: Numbers qtrig, 9trig, ipoi

T - t 2-5 2-6 2-7 2-8 2-9 2-io 2-11 2-12

qtrig 3 4 7 14 27 53 105 209

qtrig 6 11 20 40 79 157 312 624

qpol 5 9 17 33 65 129 257 513

where

J

(¿i«2)q (11)T,t

T -t ( An) Aii) , V 1 (Ah)Ai2) _ AAAlA So So rrô-S i Si—l

V4i2 - 1

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Let us compare (5

Z(i1)Z(i2) _ Z(i1)Z_ -1 1

Si — 1 Si Si Si — A{il=i2} I •

(5.126)

with (5.116) and (5.125) with (5.118). Consider

minimal natural numbers qtrig and qpoi, which satisfy to (see Table 5.42)

(T -1)2 /1

qpoi

9 S Aï.

1

< (T -t)3,

2 \ 2 ^ 4i2 - 1

¿=1

(T - t)22 (n2 1 1 ^ ^

(5.127)

2n2

r=1

Thus, we have

qpoi

qtrig

« 1.67, 2.22, 2.43, 2.36, 2.41, 2.43, 2.45, 2.45.

From the other hand, the formula (5.116) includes (4q + 4)m independent standard Gaussian random variables. At the same time the folmula (5.126) includes only (2q + 2)m independent standard Gaussian random variables. Moreover, the formula (5.126) is simpler than the formula (5.116). Thus, in this case we can talk about approximately equal computational costs for the formulas (5.116) and (5.126).

There is one important feature. As we mentioned above, further we will see that the introducing of random variables ^ and Mqi) will sharply complicate the approximation of stochastic integral /(H^i (¿1, ¿2, ¿3 — 1,..., m). This is due to the fact that the number q is fixed for all stochastic integrals, which included into the considered collection. However, it is clear that due to the

smallness of T — t, the number q for J(

(¿1«2«3)

(111)T,t

could be chosen significantly less

than in the formula (5.116). This feature is also valid for the formulas (5.115), (5.119). However, for the case of Legendre polynomials we can choose different numbers q for different stochastic integrals.

From the other hand, if we will not introduce the random variables and ^ , then the mean-square error of approximation of the stochastic integral Jl^y t will be three times larger (see (15.1131)). Moreover, in this case the

stochastic integrals J((01)T t, J(ooi)Tt (with Gaussian distribution) will be approximated worse.

Consider minimal natural numbers q*rig, which satisfy to (see Table 5.42)

3(T - i)2 I n2 ^g 1

2n2 \ 6 r2

\ r=1

In this situation we can talk about the advantage of Legendre polynomials (q*rig > and (5.116) is more complex than (5.126)).

5.3.2 A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Integrals

T (i 1 ) T ( i 1 i 2 ) T(0h) T (il0) T(ili2«3)

J(1)T,t , J(11)T,t , J(01)T,t , J(10)T,t , J(111)T,t

It is well known [77]-[80], [87] (also see [14]-[16]) that for the numerical real-

ization of strong Taylor-Ito numerical methods with the convergence order 1.5 for Ito SDEs we need to approximate the following collection of iterated Ito stochastic integrals (see Sect. 4.10)

T (i 1) T (i 1 i2) T (Oil) T (ilO) t (¿1«2i3) J(1)T,t ' J(11)T,t ' J(01)T,t ' J(10)T,t ' J(111)T,t-

Using Theorem 1.1 for the system of trigonometric functions, we have (see Sect. 5.2)

J^t = VT—t,(t\ (5.128)

T(hi2)q _ ^(t — f\( Ah)An) i 2. V^ 1 (A^A^ — A1^ Ai2Al (ll)T,t 2 I S2r-lS2r

\ r=1

-(i1) Z(i2) _

r— ISO zo S2r-1

+—^ (iAdr-' - W) - !{«=«} ) • (5-129)

n

(r - A'2 (Itf'1 (e™ W - +

r(iii2i3)q

^(iii)T,i ~~ ^ igSo so so Vs<? so so s<? so so

1

4

i 1 V^ ( 1 MA An) _ An) AiAAn)\,

+ 7rr V 2r_1 0 0 ^r-iso so j +

i 1 / Ah) An) Ah) _ 0 Ai2) Ah) Ah) , | ,

~^7r2,r2 ^0 So zS2r So So "T" S2r SO SO ] | +

1 f Ah)Ah) Ah) Ah) Ah)Ah) Ah) Ah)Ah) , Ah) Ah)Ah)

' I 4trr V

r=1

, "VI i AH>Ai2> An> — An> At2>At3> — A^) At3>An> _i_ Ai2>AH> \ _i_

t 4 IS 2/' S2r-lSo S2r-lS2r So S2r-lS2r So S2r-lS2r SO I

I 1 /qZ(n) Z(«2) Z(«3) , /-(HM^M^) _ cAA AA AA _|_ ^g7r2,r2 I °S2r-lS2r-lSo ~r S2r S2r So °S2r-lS2r-lSo

+3zir2)1zir3)1 Z0"1'—2Z2:1»z2;3)z0*2)+zir3)z2r2)z0*i)) +Dir3)M, (5.132)

where in (5.132) we suppose that i1 — i2, i1 — i3, i2 — i3,

q

\ihhh)q _ 1 "V I 1 | /-(«lM^M^) _ A''A A'AA'A T,t ~ 2n2 Z^ 1 f2 _ /2 I S2r S2/ SO S2r So S2I

r,l = 1 r=l

r_AA AA AA — LAAAA AA 1__'LAA AAAA

S2r-lS2/-lS0 SO S2r-lS2/-l J f^S2r-lSo S2/-1

El (_A;A A;A A;A _l An"> AAAA _L

1 -r-m V ^2/•—1S2;?? — 1S2;?? S2r-\S2r S2m-l~r \ / m \

4\/27r2 I ^ I rm

1

where

+ m(

I Z(i1) Z(i2)Z(i3) _ Z(i1 )Z(i2) Z(i3) ^ +

+ S2r-lS2m S2m-1 S2r S2r-1S2m-1 J +

^ / z(n) An)An) - Z(n) Ah) Ah)_

(r + m) V Z2(m+r)Z2r Z2m Z2(m+r)-1 Z2r-1Z2m

_Z(i1) Z(i2)Z(i3) I Z(i1) Z(i2) Z(i3) \ 1 +

Z2(m+r)-1Z2r Z2m-1 + Z2(m+r)Z2r-1S2m-1j I +

, v^ v^ I 1 fz(n) An)An) -L z(n) z(i2) z(i3)_

m=1 /=m+1 \ ( )

_Z(i1) Z(i2)Z(i3) + Z(i1) Z(i2) Z(i3) ^ +

Z2(/-m)-1Z2/ Z2m-1 + Z2(1-m)Z21-1Z2m-y +

-m)-1^2/ S2m-1 1 S2(/-m)^2/-r

i ^ ( An) An)An) -L An) A'A A'^-

1(1 — m)\ ^(l-nifam A>1 "+" S>2(/-m)-l<!>2m-l<!)2/

Z(i1) Z(i2)Z(i3) _ Z(i1) Z(i2) Z(i3)

"Z2(/-m)-1Z2m Z2/-1 Z2(/-m)Z2m-1Z2/-1

1 ^ 1 ^2 q 1

.- / v -3 2?'—1' VJL(l ~ r / v 01

«a 6 f

^ r=q+1 r=1

Z*> - — V - Z*> 8 - — - V

4

and Zoi), Z2r), Z211, 6(i), (r = 1,..., q, i = 1,... ,m) are independent standard Gaussian random variables.

The mean-square errors of approximations ((5.129l)-((5.132l) are represented by the formulas

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m J I , _ ,

(01)T,t J(01)T,t

M J I T(0i1) - T(0i1)M \ = 0

IVI 1 J(01)Tt J(01)Tt ( = 0,

M J I T(i10) - T(i10)M \ = 0 M i I J(10)T,t J(10)T,t / f = 0,

M {- = ■^ [i - £ i) ■ <5-133'

r=1

q

1

Table 5.43: Confirmation of the formula (5.134) e/(T -1)3 0.0459 0.0072 7.5722-10-4 7.5973 ■ 10-5 7.5990 ■ 10-6

q 1 10 100 1000 10000

m

' T(¿1«2«3) _ T(¿1«2«3)q

J(111)T,t J(111)T,t

)2} = (T - t):

45 47T2 ^ r2

55

32n4

E

1

E

r,l=1 r=l

5/4 + 4r4 - Sr2l2\ r2l2{r2-l2)2 )'

(5.134)

where ¿1 = ¿2, ¿1 = ¿3, ¿2 = ¿3.

In Table 5.43 we can see the numerical confirmation of the formula (5.134) (e means the right-hand side of (5.134)).

Note that the formulas (5.128), (5.129) have been obtained for the first time in [77]. Using (5.128), (5.129), we can realize numerically an explicit one-step strong numerical method with the convergence order 1.0 for Ito SDEs (Milstein method [77]; also see Sect. 4.10).

An analogue of the formula (5.132) has been obtained for the first time in [78], [79].

As we mentioned above, the Milstein expansion (i.e. expansion based on the Karhunen-Loeve expansion of the Brownian bridge process) for iterated stochastic integrals leads to iterated application of the operation of limit transition. An analogue of (5.132) for iterated Stratonovich stochastic integrals has

been derived in [78], [79] on the base of the Milstein expansion together with the Wong-Zakai approximation [68]-[70] (without rigorous proof). It means that the authors in [78], [79] formally could not use the double sum with the upper limit q in the analogue of (5.132). From the other hand, the correctness of (5.132) follows directly from Theorem 1.1. Note that (5.132) has been obtained reasonably for the first time in [1]. The version of (5.132) but without the introducing of random variables and ^ can be found in [71] (1997).

Note that the formula (5.133) appears for the first time in [77]. The mean-square error (5.134) has been obtained for the first time in [76] (1996) on the base of the simplified variant of Theorem 1.1 (the case of pairwise different

¿1,..., ).

1

The number q as we noted above must be the same in (5.129) (5.132). This is the main drawback of this approach, because really the number q in (5.132) can be chosen essentially smaller than in (5.129).

Note that in (5.132) we can replace with J(\(11i)2T3t)q and (15.132) then

will be valid for any i1, i2, i3 — 0,1,..., m (see Theorems 2.6-2.8).

Consider now approximations of iterated stochastic integrals

T (i1) 7 (i1i2) j (0i1) T(i10) T (i1i2i3) (• • • — 1 m)

J(1)T,t , J(11)T,t , J(01)T,t , J(10)T,t , J(111)T,t (i1 , ¿2 , ¿3 — 1 , . . . , m)

on the base of Theorem 1.1 (the case of Legendre polynomials) [1]-[16], [31

J,(;i» = VT~t C«"», (5.135)

(1)T,t _ v ^ vS0

j{hh)q _ T — t I Ji^Jio) V-^ 1 (Ah) An) _ A'AA'A

J(ii)T,t. ~ 2 I 0 0 J^-2 _ i A ^-i \ i=1

Ah)A'12) _i_ "V 1 (AiiAA) _ AAAA\ _ i i

So So r^.2-1 ^Si-lSi Si Sh-1 J *-{h=h} I '

i=1 /

(5.136)

Jmr,t = (c(-.) + -^c!"'), (5.137)

= ^ rcA"' - ^Ci'^O, (5.138)

r(i1i2i3)q1 _ n I X^M^M^) i i z(i3)

¿1 = i2}1{j1=j2}Sj3

j1 ,j2 ,j3=0

■1{i2 = i3}1{j2=j3}Zj1i1) — !{i1=i3} !{j1=j3} C^ ) , q1 ^ q, (5.139)

= -3C

where

T z y

Cj3j2j1 — j (z ) / j M / j (x)dxdydz

v/(2j1 + 1)(2j2 + 1)(2j3 + 1)(t _ i)3/2^

8

j3j2j1;

jj = p3(z) / pj2 M / pji (x)dxdydZ

(5.140)

1

(x) is defined by (5.124) and P*(x) is the Legendre polynomial (f = 0,1, 2,...).

The mean-square errors of approximations (5.136), (5.139) are represented by the formulas (see Theorems 1.3 and 1.4; also see Sect. 5.1)

M J I T(iii2) _ j(iii2)q M S I J(n)T,t J(11)T,t

[T-tf (1

E

¿=1

1

4z2 — 1

(¿1 = ¿2), (5.141)

m

6

qi

T (¿1*2*3) _ 7-(iii2«3)qi

J(111)T,t J(111)T,t

(¿1 = ¿2,¿1 = ¿3,¿2 = ¿3),

E C

2

j3 j2ji

j3,j2 ,j'i=0

(5.142)

m

7-(¿1*2*3) _ T-(*i*2*3)qi

J(111)T,t J(111)T,t

(T-ty 6

qi

E C

j3,j2,j'i=0

2

j3j2ji

qi

y ^ Cj2j3j1 Cj3j2j1 (¿1 = ¿2 = ¿3), j3,j2,j'i=0

(5.143)

m

t (*1*2*3) _ 7-(*1*2*3)qi

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J(111)T,t J(111)T,t

6

qi

E C

2

j3j2ji

qi

E n

j3j2,j'i=0

. n. . .

j3j2ji nj1j2j3

j3,j2,j'i=0

(¿1 = ¿3 = ¿2),

(5.144)

m

t (*1*2*3) _ T-(*1*2*3)qi

J(111)T,t J(111)T,t qi

6

qi

E C

2

j3j2ji

E c

j3,j2,ji=0

. n. . .

j3jij2 nj3j2j1

j3,j2,j'i=0

(¿1 = ¿2 = ¿3),

(5.145)

m

J,

(*1*2*3)

J

(*1*2*3)qi

(111)T,t ^(111)T,i

6

qi

E C

2

j3j2ji

j3,j2,j'i =0

, (5.146)

1

y

z

2

where ¿1, i2, ¿3 = 1,..., m in (5.146).

Let us compare the efficiency of application of Legendre polynomials and trigonometric functions for the approximation of iterated stochastic integrals

T (¿1*2) 7- (¿1*2*3)

J(11)T,t' J(111)T,r

Consider the following conditions (i1 = i2, i1 = i3, i2 = i3)

j1,j2,j3=0

2n2 \ 6 ¿—f r2

r=1

(T -t)

where

P1

45 4tt2 ^

r2

55

pi

32n4

pi

^ J_ _ J_ ^ 5/4 + 4r4 - 3r2/2

4n4 r2/2 (r2 - /2)

r,i=1 r=i

C- ■ ■ —

Cj3j2j1

v/(2j1 + 1)(2j2 + 1)(2j3 + 1) _ 3/2^ 8 :

j3j2ji:

< (T-1)4, (5.150)

GW?1 — J Pj3 (z)y 11

Pj2 (y) / Pj (x)dxdydz,

1

where P^x) is the Legendre polynomial.

In Tables 5.44 and 5.45 we can see the minimal numbers q, qi, p, pi, which satisfy the conditions (I5.147l)-(l5.!50l). As we mentioned above, the numbers q, q1 are different. At that q1 ^ q (the case of Legendre polynomials). As we saw in the previous sections, we cannot take different numbers p, p1 for the case of trigonometric functions. Thus, we should choose q = p in (I5.1'29)-(5.13'2). This leads to huge computational costs (see the fairly complicated formula (5.132)).

From the other hand, we can take different numbers q in (5. 129)—(5. 132).

At that we should exclude random variables

(i)

(i) Mq

from

.132). At

this situation for the case i1 = i2, i2 = i3, i1 = i3 we have

1

y

z

Table 5.44: Numbers q, q\

T - t 0.08222 0.05020 0.02310 0.01956

Q 19 51 235 328

qi 1 2 5 6

Table 5.45: Numbers p, pi, p*, p\

T - t 0.08222 0.05020 0.02310 0.01956

p 8 21 96 133

Pi 1 1 3 4

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p* 23 61 286 398

Pi 1 2 4 5

^-^Î-Èè I (5,51,

2n2 \ 6 r2

r=1

(T-oM-- —f-- —f--

v M 36 2tt2 ^ r2 32TT4 r4

r=1 r=1

4n4 ^ r2/2 (r2 - l2)2

r,i = l r=i

where the left-hand sides of (5.151), (5.152) correspond to (5.129), (5.132) but without In Table 5.45 we can see minimal numbers p*, p*, which satisfy

the conditions (5.151), (5.152).

Moreover,

m j i j(0ii) - r(0ii)q \2\ — m J f r(ii0) _ j(ii0)qx 2

ivi S i J(0i)T,t J(01)T,t J f — m 1 lJ(10)T,t J(10)T,t

Table 5.46: Confirmation of the formula (5.152)

e/(T - t)3 0.0629 0.0097 0.0010 1.0129 ■ 10-4 1.0132 ■ 10-5

Q 1 10 100 1000 10000

t-E^ ¿o, <5-153>

(T-tf Ui 2tt2 I 6 r'-

\ r=1

where , are defined by (51m (5131 but without

It is not difficult to see that the numbers qtrig in Table 5.42 correspond to minimal numbers qtrig, which satisfy the condition (compare with (5.153))

From the other hand, the right-hand sides of (5.137), (5.138) include only 2 random variables. In this situation we again can talk about the advantage of Ledendre polynomials.

In Table 5.46 we can see the numerical confirmation of the formula (5.152) (e means the left-hand side of (5.152)).

5.3.3 A Comparative Analysis of Efficiency of Using the Legen-dre Polynomials and Trigonometric Functions for the Integral

T*(°«1«2)

J(011)T,t

In this section, we compare computational costs for approximation of the iterated Stratonovich stochastic integral J*^)^! (i1, = 1,...,m) within the framework of the method of generalized multiple Fourier series for the Legen-dre polynomial system and the system of trigomomenric functions.

Using Theorem 2.1 for the case of trigonometric system of functions, we obtain [6]-[16], [39]

J*{0hi2)q _ (rji _ ,\2 I \Aii)An) _ 1 /TTAn) Ah) I

■J(011 )T,t — y1 l) I gH) So 2\/27T

1 v^ / _ 1 A''A A'*a _i__— f A''AA''A _ oA''AA''A\ _

Z^l Kr 2'"10 7T2T2 \ ' So )•

2 a/2 1 \ 7TT * 1 u 7r2r

v r_i \

1 v^ 1 / -l LA'11') A'A

'27r2 r2 - 12 \ 2r 21 V

r,l=1 r=l

Table 5.47: Confirmation of the formula (5.155)

4e/(T - t)4 0.0540 0.0082 8.4261-10-4 8.4429 ■ 10-5 8.4435 ■ 10-6

Q 1 10 100 1000 10000

Table 5.48: Confirmation of the formula (¡à. 157)

16e/(T - -i)4 0.3797 0.0581 0.0062 6.2450 ■ 10-4 6.2495 ■ 10-5

Q 1 10 100 1000 10000

I VI 1 f Ah) Ai 2) _ Ai 1) z

-(«1) Z_l_ -lS2r I +

+

r=1 1

8n2 r2

qZ(«1) Z(«2) _L Z(«2)Z 3S2r-1S2r-1 + Z2r Z2r J

(5.154)

For the case i1 = i2 from Theorem 1.3 we get [6]-[17], [30], [39

2 ï (t - t)4

m

t*(0h«2) _ r*(0ii«2)q J(011)T,t J(011)T,t

q

5

q 1 1 q

,r4 tj-4

49

k2 + l2

I__L V —

q 27t2 r2

r=1

8n4 ^ r

r=1

^ k,7r112 (12 - k2)'

k=l

(5.155)

Analogues of the formulas (5.154), (5.155) for the case of Legendre polynomials will look as follows [6]-[17], [30], [39]

T*(Oiii2)<7 _ T ~ t 7-*(j1j2)qr {T ~ I 1 Jj2) (oii)T,i - 2 4 I ^

¿¿V y/(2i + l)(2i + 5)(2i + 3) (2i-l)(2i + 3)

Z («i)Z («2 )

, (5.156)

where

J,

*(«i«2)q (11)T,t

T-t

/ q

(c0,i)z0*2)+£ ^ «=1

1

V4i2 - 1

Z(«1)ZW _ Z(«1) Z(«2)

M f ( T*(0iii2) _ ^(On^)^2! _ f 5 _ _1

11 ] \J(011)T,t (011)T,i y f- 1fi Q

16 \ 9 ^ 4i2 - 1

\ ¿=2

V 1 y-v (? + 2)2 + (j. + l)2 \

¿i(2i-l)2(2i + 3)2 ¿-J (2i + l)(2i + 5)(2f. + 3)2J >

where i1 = i2.

In Tables 5.47 and 5.48 we can see the numerical confirmation of the formulas (5.155) and (5.157) (e means the right-hand side of (5.155) or (5.157)).

Let us compare the complexity of the formulas (5.154) and (5.156). The formula (5.154) includes the double sum

1 V^ 1 (AA An) , l_A'h) Ah) 2tj-2 Z^ r2 _ /2 I ^2r s,2/ -r fS2r-lS2/-l

r,l = 1 r=l

Thus, the formula (5.154) is more complex, than the formula (5.156) even if we take identical numbers q in these formulas. As we noted above, the number q in (5.154) must be equal to the number q from the formula (5.129), so it is much larger than the number q from the formula (5.156). As a result, we have obvious advantage of the formula (5.156) in computational costs.

As we mentioned above, if we will not introduce the random variables and then the number q in (15.154) can be chosen smaller, but the mean-

thr

r(0»i)

square error of approximation of the stochastic integral J^y t will be three

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times larger (see (15.1131)). Moreover, in this case the stochastic integrals J^jT t,

J(io)T t' J(ooi)T t (with Gaussian distribution) will be approximated worse. In this situation, we can again talk about the advantage of Ledendre polynomials.

5.3.4 Conclusions

Summing up the results of previous sections we can come to the following conclusions.

1. We can talk about approximately equal computational costs for the formulas (5.129) and (5.136). This means that computational costs for realizing the Milstein scheme (explicit one-step strong numerical method with the convergence order y = 1.0 for Ito SDEs; see Sect. 4.10) for the case of Legendre

polynomials and for the case of trigonometric functions are approximately the same.

2. If we will not introduce the random variables (see (5.129)), then the mean-square error of approximation of the stochastic integral J((11)2T) t will be three times larger (see (5.113)). In this situation, we can talk about the advantage of Ledendre polynomials in the Milstein method. Moreover, in this case the stochastic integrals Jq^T t, J(1o)T t, J(oo1)t t (with Gaussian distribution) will be approximated worse.

3. If we talk about the explicit one-step strong numerical scheme with the convergence order 7 = 1.5 for Ito SDEs (see Sect. 4.10), then the numbers q, q1 (see (5.136), (5.139)) are different. At that q1 ^ q (the case of Legendre polynomials). The number q must be the same in (5.129)) (5.132) (the case of trigonometric functions). This leads to huge computational costs (see the fairly complicated formula (5.132)). From the other hand, we can take different numbers q in (5. 129)—(5. 132). At that we should exclude the random variables £q4), Hq^ from (5.129)) (5.132). This leads to another problems which we discussed above (see Conclusion 2).

4. In addition, the author supposes that effect described in Conclusion 3 will be more impressive when analyzing more complex sets of iterated Ito and Stratonovich stochastic integrals (when 7 = 2.0, 2.5, 3.0, ...). This supposition is based on the fact that the polynomial system of functions has the significant advantage (in comparison with the trigonometric system) when approximating the iterated stochastic integrals for which not all weight functions are equal to 1 (see Sect 5.4.3 and conclusion at the end of Sect. 5.1).

5.4 Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals Based on Theorem 1.1 and Multiple Fourier—Le-gendre Series

This section is devoted to optimization of the mean-square approximation procedures for iterated Ito stochastic integrals (5.3) of multiplicities 1 to 4 based on Theorem 1.1 and multiple Fourier-Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence orders 1.0, 1.5, and 2.0 for Ito SDEs with multidimensional non-commutative noise (see (4.65)-(4.67)). We show that the lengths of sequences of independent standard

Gaussian random variables required for the mean-square approximation of iterated Ito stochastic integrals (5.3) can be significantly reduced without the loss of the mean-square accuracy of approximation for these stochastic integrals. This section is written on the base of paper [55]. An extension of the mentioned results to iterated Ito stochastic integrals of multiplicity 5 can be found 54].

in

Using Theorem 1.1 and the system of Legendre polynomials, we obtain the following approximations of iterated Ito stochastic integrals (5.3)

J(0)T,t — v T tz0 ,

I

(il) (1)T,t

I

(¿i«2)q (00)T,t

T-t

z0:i)z0:2) + E

¿=1

V4i2 - 1

Z(i1)Z(i2) _ Z(i1)Z_ -1 \

Zi-1 Zi Zi Zi-1J 1{ti=t2H ,

(5.158)

I

(iii2i3)qi (000)T,t

qi

Er\Z(ii)Z(i2)Z_ "1 r . ,1r. ,lZ(i3)_

Cj3j2ji | Zji Zj2 Zj3 1{ii=i2} 1{ji=j2} Zj3

ji J2,j3 =0

1 1 Z — 1 1 Z (i2)

1{i2=i3} 1 { j2 =j3} Zji 1{ii = i3} 1 { j i =j3} Z j2

(5.159)

I

(iii2)q2 (10)T,t

q2

EC10 / Z(ii)Z(i2) 1 Cj2ji Zji Zj2 - 1

I

(ii«2)qr2 (01)T,t

ji ,j2=0 </2

E ji zfzji2' -1

jij2=0

L{ii = i2> 1{ji=j2>

(5.160)

(5.161)

I

(iii2i3i4)q3

(0000)T,t

q3

EC i Z (ii)Z (i2) Z (i3 )Z (i4)_

Cj4j3j2 ji Zji Z j2 Zj3 Zj4

ji J2,j3,j4 =0

1 1 Z (i3)Z (i4) _ 1 1 Z (i2 ) Z (i4)

1{ii=i2}1{ji=j2}Zj3 Z j4 1{ii = i3}1{ji=j3}Zj2 Zj4

— 1r- • l1r- • Z (i2 ) Z (i3^ "1 r . l1r. .,Z (ii)Z (i4)_

1{ii=i4}1{ji=j4}Sj2 Sj'3 1{i2 = i3} 1 {j2=j3} Sji Sj4

_1 1 z(ii)Z- 1 1 Z(iiV(i2) +

1{i2 = i4} 1{j2 =j4}Zji Zj3 1 {i3=i4} 1{j3=j4}Zji Zj2 +

1

+ l{n=«2} 1{j'l=j2} 1{«3 = i4} 1{j3=j4} + 1{il=i3} 1{j'l=j3} 1{i2=i4} 1{j2=j4} +

+ !{il = i4} 1{j1=j4}1{i2=i3}1{j2=j3^ , (5.162)

where 1A is the indicator of the set A,

T

j = J h (sf (i = 1,..., m, j = 0,1,...)

t

are independent standard Gaussian random variables for various i or j, {hj (x)}°=0 is a complete orthonormal system of Legendre polynomials in the space L2([t,T]) (see (5.5)),

^ './:■,/: ./ = g^jl^jsi^1 ~~ Cj2jl = g^2 (Î1 ~~ ^^jtjl' (5-163)

^k/'l ~~ gLj^oiT ( '/:/;./. / — I'J J:JaJ:( I' I )~( • (5.164)

^,2 = V/(2ji + l)(2j2 + l), ^2,3 = V/(2ji + l)(2j2 + l)(2j3 + l),

v/(2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1),

1 z y

Cj = j Pj3 (Z) j Pj2 (y) j Pj1 -1 -1 -1

1 u z y

Cj4j3j2j1 = j Pj4 (U) y Pj3 (Z) y Pj2 (y) y Pj1 (x)dxdydzdu, -1 -1 -1 -1 1 y

Cjj = -J (1+ y)Pj2 (y) y Pj1 (x)dxdy, -1 -1 1 y

1o

j = -J Pj2(y)J (1 + x)P,1 (x)dxdy, -1 -1

Pj (x) is the Legendre polynomial (see (5.6)).

Combining the estimates (4.70) and (1.127) for p1 = ... = pk = p, we obtain

I . p \

k!

J K 2(ti,...,tk )dti ...dtk - E Cjk .j

V[t,T ]k j'i>->j'fc=o y

< C(T - t)r+1, (5.165)

where K(tl5...,tk) is defined by (5.79) (see (5.80)-(5.82)), r/2 is the strong convergence orders for the numerical schemes (I4.65l)-(I4.67I). i.e. r/2 = 1.0,1.5, and 2.0; constant C is independent of T — t.

It is not difficult to see that the multiplier factor k! on the left-hand side of the inequality (5.165) leads to a significant increase of computational costs for approximation of iterated Ito stochastic integrals. The mentioned problem can be overcome if we calculate the mean-square approximation error for iterated Ito stochastic integrals exactly (see Theorem 1.3 and Sect. 1.2.3). In this section, we discuss how to essentially minimize the numbers q, q1, q2, q2, q3 from (5.158)-(5.162). At that we will use the results from Sect. 1.2.3.

Denote

E^..« = M j (/i,^ — J , (5.166)

where /i81'"^, is the iterated Ito stochastic integral (5.3) and is the

mean-square approximation of this stochastic integral. More precisely, the approximations /g^gf, /f^, , /(0^ are defined by (5H5E)-(5.162).

The results of Sect. 1.2.3 give the following formulas for the case of Legendre polynomials

Em» = iT_tf(l_l_ g LjlM3{CKKJA, (5.168)

V jl j2 j3=0 /

where ii = ¿2, ¿1 = ¿3, ¿2 = ¿3, £,ooo, = (T_t)s/l_£ £ ¿2^ ((CJ№31)2 + ) , (5.169)

V ilj2j3=0

where i1 = i2 = i3,

B,000) = (T _ t)3 | 1 _ i_ g L2 ^ + CnnllCnl2J) ) , (5.170)

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j1 j2 j3=0

where i1 = i2 = ¿3,

<40) = (T-tf ( g - gl ^ ./:./:■■ (f^ '/•../ ./ ) + './:■../:./ './ ./:./:■.) j > (5-171)

V jl ,j2, j3=0 /

where i1 = i3 = i2,

V jl ,j2=0 /

= 1,1*0%, I£ Ö&) ). n = <* <M»)

\ j1 j2=0 \(ji,j2) ) )

jl,j2=0

= (T"i)4 E ¿M, (E )> (5-175)

\ j1,j2=0 \(jU2) ) )

40000» = (T _ t)4 fl_ _ JL £ (c,..,)2) , (5.176)

V jl,-,j4 =0 /

where il5..., are pairwise different,

= E (v<\ ))- (5-177)

\ j1,--,j4=0 \(jl,j2) ) )

where i1 = i2 = i3, i4; i3 = i4, ^r1 = (T - if (1 - g ^..A-,, ( £ <?, ,,) ) , (5.178)

\ jl,-,j4=0 \(jl,js) / /

where i1 = i3 = i2, i4; i2 = i4,

^' = ^-^(¿-¿6 E (v<-. ))- (5-179)

\ jl,--,j4=0 \(jlj4) ) )

http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal, http://diffjournal.spbu.ru/ A.798

where ¿1 = ¿4 = ¿2, ¿3; ¿2 = ¿3,

^' = ^-^(¿-¿6 £ '•' (v<\ ))• (5-18°)

\ j'l,-j4=0 \(j2,j3) / )

where ¿2 = ¿3 = ¿1, ¿4; ¿1 = ¿4,

C001 = "^ (Ä"i E ( E)) • <5-181)

\ jl,-j4=0 \O2j4) ) )

where ¿2 = ¿4 = ¿1, ¿3; ¿1 = ¿3,

^r01 = -o4fe-¿e E (v<-. ))• <5-182)

\ j'l,-j4=0 \(j3,j4) ) )

where ¿3 = ¿4 = ¿1, ¿2; ¿1 = ¿2,

<r = (r ■-04 (^ - ¿6 E ( E ■ . )) • (5-183)

\ J'l,-j4=° \(jl ,j2,j3) / /

where ¿1 = ¿2 = ¿3 = ¿4,

C = E A.. A A E Aw, )), (5-184)

\ J'l,-j4=° \(j2 ,j3,j4) / /

where ¿2 = ¿3 = ¿4 = ¿1,

<T = (r ■"i)4 ( Ya - ¿6 E Lh-.h°i>-h ( E ' • ) ) . (5-185)

where ¿1 = ¿2 = ¿4 = ¿3, C?01 = (T - if (1 - g L|„„A,, ( £ ■ . ) ) , (5.186)

\ jl,-j4=0 \(jl J3J4) / /

where ¿1 = ¿3 = ¿4 = ¿2,

<T> = (r■-i)4 (i-¿6 E L!-jA>-h (E ( E ))).

\ j'l,-,j'4=0 \(jl,j2) \(j3 ,j4) I))

(5.187)

where i1 = i2 = i3 = i4,

1

93,13

24 256

jl,...,j4=0

E E ^

,(j1,j3) \(j2 ,j4)

jl

where i1 = i3 = i2 = i4,

(5.188)

E(oooo) = (T -1)

1 1 93,14

24 ~ 256 ^

L2 C

Lj1-j4 Cj4--j1

j1,...,j4=0

(5.189)

where i1 = i4 = i2 = i3.

Obviously, the conditions (5.167) (5.189) do not contain the multiplier factors 2!, 3!, and 4! in contrast to the estimate (1.127) (see Theorem 1.4). However, the number of the mentioned conditions is quite large, which is inconvenient for practice. In this section, we propose the hypothesis [52]-[55] that all

the formulas (I5.167l)-(l5.189l) can be replaced by the formulas (15.167), (15.168), 172), (5.174), (5.176) in which we can suppose that i1,i2,i3,i4 = 1,...,m.

At that we will not have a noticeable loss of the mean-square approximation accuracy of iterated Ito stochastic integrals.

It should be noted that unlike the method based on Theorem 1.1, existing approaches to the mean-square approximation of iterated stochastic integrals based on Fourier series (see, for example, [77]-[80], [87], [91]) do not allow to choose different numbers p (see (5.166)) for approximations of different iterated stochastic integrals with multiplicities k = 2,3,4,... Moreover, the noted approaches exclude the possibility for obtaining of approximate and exact expressions similar to (1.75), (1.127) (see Theorems 1.3, 1.4). The detailed comparison of Theorem 1.1 with methods from [77]-[80], [87]-[89], [91], given in Chapter 6 of this monograph.

Consider the following conditions

is

£(°0) < (T - t)4, < (T - t)4, i = 1,..., 4,

(5.190)

and

£(°0) < (T - t)5, < (T - t)5, E^j < (T - t)5, (5.191)

92 ,j —

E^ < (T - t)5, E^0 < (T - t)5,

(5.192)

where i =1,..., 4; j = 1, 2; k = 1,..., 14.

Let us show by numerical experiments that in most situations the following inequalities are fulfilled (under conditions (5.190) and (5.191), (5.192))

qi,i > i = 2,3,4, (5.193)

92,1 > 92,2, 92,1 > 92,2, (5.194)

93,1 > 93,k, k = 2,..., 14, (5.195)

where q1,i, q2j, q"2j, q3,k (i = 1,..., 4; j = 1, 2; k = 1,..., 14) are minimal

natural numbers satisfying the conditions (5.190) and (5.191), (5.192).

In Tables 5.49-5.56 we can see the results of numerical experiments. These results confirm the hypothesis proposed earlier in this section. Note that in Tables 5.54-5.56 we calculate the mean-square approximation errors of iterated Ito stochastic integrals in the case when

91.1 = 91,1, i = 2, 3, 4,

92.2 = 92,1, 92,2 = 92,1, 93,k = 93,1, k = 2,..., 14,

where 91,1, 92,1, 92,1, 93,1 are minimal natural numbers satisfying the conditions (5.190) and (5.191), (5.192). In this case, there is no noticeable loss of the mean-square approximation accuracy of iterated Ito stochastic integrals (see Tables 5.54-5.56). This means that all the formulas (5. 167)—(5. 189) can be replaced by the formulas (5.167), (5.168), (5.172), (5.174), (5.176) in which we can suppose that i1, i2, i3, i4 = 1,..., m.

Let 91,1 and 93,1 be minimal natural numbers satisfying the conditions

<0) < (T — t)4, (5.196)

E(0°°0) < (t — t)5, (5.197)

where the left-hand sides of these inequalities are defined by the formulas (5.168) and (5.176), respectively.

Let p1,1 and p3,1 be minimal natural numbers satisfying the conditions

3! • £P°00) < (T — t)4, (5.198)

4! • EP(00i00) < (T — t)5, (5.199)

Table 5.49: Conditions < (:T-t)\ i = 1,.. .,4.

T - t 0.011 0.008 0.0045 0.0035 0.0027 0.0025

91,1 12 16 28 36 47 50

qi,2 6 8 14 18 23 25

qi,3 6 8 14 18 23 25

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qi,4 12 16 28 36 47 51

where the values and Ep00|oo) on the left-hand sides of these inequalities

are defined by the formulas (5.168) and (5.176), respectively.

In Tables 5.57, 5.58 we can see the numerical comparison of the numbers g1;1 and q3j1 with the numbers p1;1 and p3;1, respectively. Obviously, excluding of the multiplier factors 3! and 4! essentially (in many times) reduces the calculation

costs for the mean-square approximations of iterated Ito stochastic integrals. Note that in this section we use the exactly calculated Fourier-Legendre coefficients using the Python programming language [52], [53].

As we mentioned above, existing approaches to the mean-square approximation of iterated stochastic integrals based on Fourier series (see, for example, 77]-[80], [87], [91]) do not allow to choose different numbers p (see Theorem 1.3)

for approximations of different iterated stochastic integrals with multiplicities k = 2,3,4,... and exclude the possibility for obtaining of approximate and exact expressions similar to the formulas (1.75), (1.127) (see Theorems 1.3, 1.4). This leads to unnecessary terms usage in the expansions of iterated Ito stochastic integrals and, as a consequence, to essential increase of computational costs for the implementation of numerical methods for Ito SDEs.

In this section (also see [54], [55]) we have optimized the method based on

Theorems 1.1 and 1.3, which makes it possible to correctly choose the lengths of sequences of standard Gaussian random variables required for the approximation of iterated Ito stochastic integrals. Thus, the computational costs for the implementation of numerical methods for Ito SDEs are significantly reduced.

On the base of the obtained results we recommend to use in practice the following conditions (for any i1,...,i4 = 1,...,m) for correct choosing the minimal natural numbers q, q1, q2, q2, q3

E(00) < C(T - t)3

Table 5.50: Conditions < (T - t)5, i = 1,... ,U.

T -1 0.011 0.008 0.0045 0.0042 0.0040

93,1 6 8 14 15 16

93,2 4 5 10 11 11

93,3 6 8 14 15 16

93,4 6 8 14 15 16

93,5 3 5 9 9 10

93,6 6 8 14 15 16

93,7 4 5 10 11 11

93,8 2 3 4 5 5

93,9 2 3 4 5 5

93, io 4 6 10 11 11

93, ii 4 6 10 11 11

93,i2 2 3 5 6 6

93,i3 6 8 14 15 16

93,14 3 5 9 9 10

Table 5.51: The conditions (5.191), (5.192).

T - t 0.010 0.005 0.0025

92,1 4 8 16

92,2 1 1 1

92,1 4 8 16

92,2 1 1 1

Table 5.52: The condition (5.190).

T - t 2-1 2-3 2-5 2-8

9 1 8 128 8192

9u 0 1 4 32

qi,2 0 0 2 16

qi,3 0 0 2 16

9M 0 0 4 33

Table 5.53: The conditions (5.191), (5.192).

T-t 2-i 2-3 2-5 2-8

9 1 8 64 512

9i,i 0 2 4 32

qi,2 0 1 4 16

qi,3 0 1 4 16

qi,4 0 2 8 33

92,1 0 0 1 1

92,2, 92,i) 92,2 0 0 0 0

q3,ij • • • ) q3,i4 0 0 0 0

Table 5.54: Values • (T - t)~3 = Eqi ., i = 1,..., 4.

T-t 0.011 0.008 0.0045 0.0035 0.0027 0.0025

9l,l 12 16 28 36 47 50

Eqi,l 0.010154 0.007681 0.004433 0.003456 0.002652 0.002494

9i,2 12 16 28 36 47 50

E Eq 1,2 0.005077 0.003841 0.002216 0.001728 0.001326 0.001247

9i,3 12 16 28 36 47 50

E 0.005077 0.003841 0.002216 0.001728 0.001326 0.001247

9i,4 12 16 28 36 47 50

Eqi,l 0.010308 0.007787 0.004480 0.003488 0.002673 0.002513

Table 5.55: Values Efaj • (T - t)~4 = E-q2 p • (T - t)~4 = I-],. . j = 1,2.

T — t 0.010 0.005 0.0025

92,1 4 8 16

E- 0.008950 0.004660 0.002383

92,2 4 8 16

E- 0.000042 0.000006 0.000001

92,1 4 8 16

E Eq2,i 0.008950 0.004660 0.002383

92,2 4 8 16

Eq2,2 0.000042 0.000006 0.000001

(for the Milstein scheme (4.65)),

E(00) < (t — t)4, E^00 < C(T — t)4 (for the strong scheme (4.66) with order 1.5), and

E(00) < C(T — t)5, Eq000) < C(T — t)5, Eq1 0) < C(T — t)5,

'q — ~ V^ qi,1 — ^ \ > ' q2 ,1 —

Eq10,!) < C(T — t)5, Eq30000) < C(T — t)5

(for the strong scheme (4.67) with order 2.0). Here the left-hand sides of the above inequalities are defined by the relations (5.167), (5.168), (5.172), (5.174), (5.176) and C is a constant from the condition (4.70).

Taking into account the results of this section (also see [54]), we recommend to use in practice the following condition (for any i1,..., ik = 1,..., m) on the mean-square approximation accuracy for iterated Ito stochastic integrals

m J i /(i1-"ifc) _ i(«1---«fc)p x 2 m < 11(h...ik)T,t 1(h...ik)T,t

K2(t1,...,tk)dt1 ...dtk — E C2k...j1 < C(T — t)r

j1 ,...,jk=0

p

Table 5.56: Values • (T -1)~4 = Eq:ik, k=l,...,U.

T-t 0.011 0.008 0.0045 0.0042

¿3,1 6 8 14 15

E 0.009636 0.007425 0.004378 0.004096

¿3,2 6 8 14 15

E 0.006771 0.005191 0.003041 0.002843

¿3,3 6 8 14 15

E 0.009722 0.007502 0.004424 0.004139

¿3,4 6 8 14 15

E 0.009641 0.007427 0.004379 0.004097

¿3,5 6 8 14 15

E 0.005997 0.004614 0.002720 0.002545

¿3,6 6 8 14 15

E 0.009722 0.007502 0.004424 0.004139

¿3,7 6 8 14 15

E 0.006771 0.005191 0.003041 0.002843

¿3,8 6 8 14 15

E 0.003095 0.002364 0.001379 0.001290

¿3,9 6 8 14 15

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E 0.003095 0.002364 0.001379 0.001290

¿3,10 6 8 14 15

E 0.006885 0.005282 0.003090 0.002889

¿3,11 6 8 14 15

E 0.006885 0.005282 0.003090 0.002889

¿3,12 6 8 14 15

E 0.003690 0.002834 0.001663 0.001555

¿3,13 6 8 14 15

E 0.009756 0.007545 0.004457 0.004170

¿3,14 6 8 14 15

E 0.006010 0.004621 0.002722 0.002547

Table 5.57: Comparison of numbers qi,i and pi,i.

T - t 2^ 2-2 2-3 2-4 2-5 2-6

91,1 0 0 1 2 4 8

(9i,i + I)3 1 1 8 27 125 729

Pi,i 1 3 6 12 24 48

(pi,i + 1)3 8 64 343 2197 15625 117649

Table 5.58: Comparison of numbers q3;1 and p3,i.

T- t 2-1 2-2 2-3 2-4 2"5 2-6

93, i 0 0 0 0 0 0

(93,i + 1)4 1 1 1 1 1 1

P3,i 3 4 6 9 12 17

(p3>1 + 1)4 256 625 2401 10000 28561 104976

where I^ ' ' is the iterated Ito stochastic integral (15.31). I^ ' ' is the mean-square approximation of this stochastic integral based on Theorem 1.1 and multiple Fourier-Legendre series, p and k G N,

K (ti,...,tk) = (t - tk )lk... (t - ti)11 1{t1< ' ' ' <tfc}, G [t,T],

1A is the indicator of the set A, li,..., lk = 0,1,..., C and r have the same meaning as in the formula (4.70).

5.5 Exact Calculation of the Mean-Square Approximation Errors for Iterated Stratonovich Stochastic Integrals T-*(n ) T-*(n ) T *(i1i2 ) I*(i1i2i3) tegrais J(0)T,t' J(1)T,t' J(00)T,t' J(000)T,t

Consider the question on the exact calculation of the mean-square approximation errors for the following iterated Stratonovich stochastic integrals

t*(ii) t-*(h) wn^) wH^s) • • • _ 1 m (5 200) J(0)T,t , J(1)T,t , J(00)T,t , J(000)T,t , i1 , i2 , i3 _ 1 , • • • , (5-200)

We assume that the stochastic integrals (5.200) are approximated using

(ii) = (0)T,t =

Theorems 1.1, 2.1, 2.8 and the Legendre polynomial system. Since I(il)

I

=(n) i (ii)

= I,

(0)T t' J(1)T t ~ J(*i)T t w. p. 1, we can use (15.71), (15.81) to approximate the

stochastic integrals I*^ t, I*/^ t. In this case, we will have zero mean-square approximation errors.

To approximate the iterated Stratonovich stochastic integral I use the formula (see (5.10))

= (Ù«2) (00)T,t

we can

I

*(n«2)q (00)T,t

T~tl cAcA + E

1

¿=1

V4i2 - 1

Z (¿1 )Z (i2) _ Z (il) Z (¿2)

— 1 — 1

. (5.201)

The mean-square approximation error for (5.201) will be determined by the formula (5.41) (i1 = ¿2). For the case ¿1 = ¿2 we can use the formula (see (6.75))

I

= (Ù«l) (00)T,t

T-t

d<0)2 w.p. 1.

Consider now the iterated Stratonovich stochastic integral I*0(00i)2T3t) of multiplicity 3 (i1, ¿2,¿3 = 1,..., m). For the case of pairwise different ¿1, ¿2, ¿3 we can use the formula (5.77). In the case ¿1 = ¿2 = ¿3, to approximate the stochastic

integral I

=(«i«i«i) (000)T,t

, we use the formula (5

Thus, it remains to consider the following three cases

¿1 = ¿2 = ¿3, ¿1 = ¿2 = ¿3,

Consider the case

¿1 = ¿3 = ¿2-

From (5.69) we obtain

(5.202)

(5.203)

(5.204)

m

I*(«l«2«3) _ I*(il«2«3)q

1 (000)T, t 1 (000)T,t

m

Mihi-i) _ j{hhh)q , ^ J(000)T,i J(000)T,i "T" 2

T

J J dsdf(i3) - ^ C

j3jljlS j3

Z

t t

jl, j3=0

. (5.205)

According to the formulas (1.76), (1.85), the quantity

I (i 1 i2 i3 ) _ I (ili2«3)g

J(000)T, t J(000)T, t

2

includes only iterated Ito stochastic integrals of multiplicity 3. At the same time, the quantity

1 T t q

J J df - £ cj zj

2ll ------t z._/ "Jwun j3

t t ji,j3=0

contains only iterated Ito stochastic integrals of multiplicity 1. This means that from (5.205) we get

m i (i*(!ii2*3j - iZi^m !> = m ^ (i

■*(ili2«3) _ I *(il«2«3)^ I _ TA I ( I (¿1«2«3) _ I (¿1«2«3)^ I |

(000)T,t J(000)T,t y f _ IVI j \vJ(000)T,t J(000)T,ty f +

• Mi I' /' Odffc' - è C3uuicj:s) j J • (5.206)

The relation (1.101) implies that

1 (T - i)3

L(000)T,i J(000)T,i j ( g

q

y2

Cj3j2jl ^^ Cj3j1j2 Cj3j2j1 , (5.207)

j1j2j3=0 j'l,j2 J3=°

where ii = i9 = i3. We have

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m|| j(r-t)dt™- £ cJSJ1J1ci'j J ' /' ¿)2<ir-

T 9

q ^ q / q \ 9

- Cj3j1j1 J (t - t)^j'3 (t)dT + ^ ( X] Cj3j1j J ,

j1 ,j3=0 t j3=° \j1=0 /

where j(t) is the Legendre polynomial defined by (5.5). According to (2.139), we obtain

1, j3 = 0

T (T -1)3/9

(5.208)

J (t - i)0j3(t)dT _ t

l/>/3, is = 1- (5.209)

0, j3 > 2

Combining (

'5.209), we get

m j i i*(il«2«3) _ i*(«li2«3)q m ^ 1 J(000)T,t J(000)T,t

(T -1)3

q q

2

j3j2jl

jl ,j2 ,j3=0 jl ,j2 ,j3=0

Cj3 j2 jl ^ Cj3jl j2C

j3j2jl-

(T -1)3/2

y^ ^ '"./ ./ + yy^ V/ ./ ^

+

jl=0 qq

+ ^^ I Cj3jUl j3=0 \j'l=0

where ¿1 = ¿2 = ¿3.

Consider the case (5.203). From (5.69) we obtain

m ) l i*(il«2«3) _ i*(«li2«3)q m S 1 J(000)T,t J(000)T,t

(5.210)

m

j(ni2«3) _ j{hhi?M ^

T

(ooo)T,i J(ooo)T,i 2 ' '

tt

J J df^W - £ Cj j'

jl,j3=0

m

Ahhh) _ Ahhh)q _ J(ooo)T,t J(ooo)T,t 2

+ 1-j(T-s)dt:<•■»- ]T c^cj;1')

t jl,j3=0 /

m

I (¿l«2«3) _ I (¿l«2«3)q\ 1 I

J(000)T,t J(000)T,ty f +

+m

(\ J(T-s)dfM- £ c^cji1^

V t jl,j3=0 y

= m ^ I I(ili2i3) _ I(ili2i3)^ 1 +

= m ^ 1 J(000)T,t J(000)T,^ [ +

T

\ I (T s)2ds - £ C^ j(T s)0r(»)<ls.

jl,j3=0

2

9

q q 9

+ E Ej , (5.211)

j1=0 Vj3=0

where j(t) is the Legendre polynomial defined by (5.5). The relation (1.102) implies that

M J ( I(i1«2i3) _ I(i1«2«3 )q\9\ _ (T - t)

"(000)T,i J(000)T,i J j q

qq

— ^^ Cj3j2 j1 — ^^ Cj2j3j1 Cj3j2j1 , (5.212) j1,j2,j3=0 j1,j2 ,j3=0

where ii = i9 = i3. Moreover,

1, ji = 0

f(T- s)Oj. {s)(ls = {T i -1/V3, ji = 1. (5.213)

0, ji > 2

Combining (523HH5233), we get

l\/l J I 1«2«3) 7-*(H«2«3 )qA9] _ (T - t)3 IVI ^ 1 J(000)T,t J(000)T,t

qq

E Cj3 j2 j1 - E Cj2j3 j1 Cj3j2jV j1,j2 ,j3=0 j1,j2,j3=0

2

j3=0

(T - i)3/2 x /,

qq

+ E Ej , (5.214)

j1=0 j3=0

where ii = i9 = i3.

Consider the case (5.204). From (5.69) we obtain

m j i I*(«1«2«3^ _ I*(h«2«3)qN 9 m ^ 1 I (000)T,t I (000)T,t

9

m

I(¿1«2«3) _ I(¿iÎ2Î3)q

T(000)T,t T(000)T,t

C212221 C

(i2) 22

j1,j2=0

m j i t(i1i2i3) t^1^3^ m + m m < 1 T(000)T,t - T(000)T,ty f + m

£

j1,j2=0

C z (i2) C212221 zj2

m

q / q

T(Î1Î2Î3) _ T(i1i2«3)q\ I 1 / C

T (000)T,t T (000)T,t J f + 1 Z^ Cj1j2j1

22=0 \j1=0

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(5.215)

The relation (1.103) implies that

M J 1 T(i1i2i3) _ T(i1i2i3)q

IVI S 1 T(000)T,t T(000)T,t

(T-tf 6

y^ Cj3j2 21 ^ Cj3j2j1 Cj1j2j3 )

21,22,23=0 21,22,23=0

(5.216)

where i1 = i3 = i2.

Combining (5.215) and

I, we obtain

M J I t*(Î1Î2Î3) _ T*(Î1Î2«3)q

M ^ 1 T(000)T,t T(000)T,t

(T-tf 6

qq 2

232221

21,22,23=0 21,22,23=0

y^ C232221 C23222'1 C212223 +

qq

+ ^^ I y^ C212221

j2=0 j1=0

(5.217)

where ¿1 = ¿3 = i2.

Thus, the exact calculaton of the mean-square approximation error for the iterated Stratonovich stochastic integral Ijooo^Tt (¿1, ¿2, ¿3 = 1,... ,m) is given by the formulas (5.77), (5.210), (5.214), and (5.217).

2

q

2

5.6 Exact Calculation of the Mean-Square Approximation Error for Iterated Stratonovich Stochastic Inte-

1 T*(i1 ¿2^3 ¿4) gral 1(0000)T,t

Consider now the iterated Stratonovich stochastic integral 1*0(00o)Tt) of multiplicity 4 (i1,i2,i3,i4 = 1,...,m). For the case of pairwise different i1,i2,i3,i4 we can use the formula (5.78). In the case ¿1 = ¿2 = ¿3 = ¿4, to approximate the iterated stochastic integral 1(o0oo)Ti^ we use the formula (15.32).

Thus, it remains to consider the following 13 cases

¿1 = ¿2 = ¿3, ¿4; ¿3 = ¿4, (5.218)

¿1 = ¿3 = ¿2, ¿4; ¿2 = ¿4, (5.219)

¿1 = ¿4 = ¿2, ¿3; ¿2 = ¿3, (5.220)

¿2 = ¿3 = ¿1, ¿4; ¿1 = ¿4, (5.221)

¿2 = ¿4 = ¿1, ¿3; ¿1 = ¿3, (5.222)

¿3 = ¿4 = ¿1, ¿2; ¿1 = ¿2, (5.223)

¿1 = ¿2 = ¿3 = ¿4, (5.224)

¿2 = ¿3 = ¿4 = ¿1, (5.225)

¿1 = ¿2 = ¿4 = ¿3, (5.226)

¿1 = ¿3 = ¿4 = ¿2, (5.227)

¿1 = ¿2 = ¿3 = ¿4, (5.228)

¿1 = ¿3 = ¿2 = ¿4, (5.229)

¿1 = ¿4 = ¿2 = ¿3. (5.230)

By analogy with (5.69) and using (2.383), (1.48), we obtain

M J ( 1*(*2^4) _ I* (iii2«3«4)<A 2 I = M S 1 1 (0000)T,t 1 (0000)T,t / f =

T t4 is

M <! I + iff dkd+

t t t

T t4 t2 T t3 t2

J J J dw^dtodw^ + J J J dw^dwifdhA

t t t t t t

T t2

+ 41{*i=^o}l{i3=^o} J J dhdto - /(oooo)T]9

tt

q q q q

i V^ V^ c Z(i3)z(i4) _ n V^ V^ C Z(i2)Z(i4)

1{il=i2=0} / „ Cj4j3jljlZ j3 Zj4 1{il=»3=0} / ,CJ4jU2jlZ j2 Zj4

j4,j3=0 jl=0 j4,j2=0 jl=0

q q q q

1 V^ Z (i2) Z (i3) _ ■ V^ Z (il)Z (i4)

1{il=i4=0} / v / v Cjlj3j2jl j Zj ■1{»2=i3=0} / v / v Cj4j2j2j'l j j

j3,j2=0 jl=0 j4,jl=0 j2=0

q q q q

_1 V^ Z(il)Z(*) _ ■ V^ Z(il)Z(i2) I

■1{i2 = i4=0} / v / v Cj2j3j2j'l Zj j ■1{»3 = i4=0} / v / v Cj3j3j2j'l Zj Zj2 +

j3,jl=0 j2=0 j2,jl=0 j3=0

qq

+ 1{il = i2=0}1{i3=i4=0} Cj3j3j'lj'l + ■{¿l = »3=0}1{i2 = i4=0} ^ v Cj2j'lj2j'l +

j3 ,jl=0 j2 ,jl=0

q VI

+ 1{il=i4=0}1{i2 = i3=0} ^ Cjlj2j2jl ) / , (5.231)

j2,j'l=0 / J

where I^o3^ is defined by (15.162).

Consider the case (5.218). From (5.231) we get

m i ^ I*(ili2i3i4) _ I*(il«2i3«4)^ M =

m I(0000)T,t I(0000)T,t =

T t4 t3

M < I (0000)T,f - (0000)T,f +2 J J J dtldwt3 dwt4 -

ttt

q q \ 2>|

E E CjjjjlZjj ■ (5.232)

j4,j3=0 j'l =0 / I

Note that

T t4 T t3

Zj(33)Zj(44) = / j M / j (t3)dwt(33)dwt(44) + / j (,)/ j (t4)dwt(44)dwt(33)

(5.233)

w. p. 1, where ¿3 = ¿4.

According to the formulas (1.76), (1.85), the quantity

I(i1i2^3^4) _ I(¿li2«S«4)q

1 (0000)T,t 1 (0000)T,t

includes only iterated Ito stochastic integrals of multiplicity 4. At the same time (see (5.233)), the quantity

T t4 ts q p

\ / / /'iiirfw?,'iw<(;4)- E

t t t j4,jS=0 j1=0

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contains only iterated Ito stochastic integrals of multiplicity 2. This means that from (5.232) we have

m J ( 1 *(i 1 ^2isi4) _ 1 *(il«2is«4)q\ 2 1 = m J ^ 1 (i 1 i2isi4) _ 1 (¿li2«S«4)^ M ,

m 1 \vJ(0000)T,t J(0000)T,t y f = m 1 \vJ(0000)T,t J(0000)T,t ^ f +

2

q q

+M/(¿3-odw^w^- v ¿Q^.cir^l4')

V t t j4,js=0 j1=0 /

T t4

= M ^ ( ^oooo't] - /(0000)T4j<?) f + 4 / / ~

qq

Cj4jS .1.1

tt 2

qq

+ E Ec

.4.3=0 \j1=0

T t4

E ECj4jsj1jW . (t4M . (t3)(t3 - t)dt3dt4 =

j4,jS =0 j1=0 { {

(i1i2isi4) (i1i2isi4 )q \2\ , (T - t)4 , r ir^ 1 ,

~~ IVI i ^J(oooo)T,i J(oooo)T,i 48 Z^ I Ahjiji I

L J j4,js=0 Vj1=0 /

qq

+ E E C./:./s.j j CjSs, (5.234)

.4 js =0 .1=0

where (see (5.14))

T t4

. = J .(u) J .(t3)(t - t3)dt3dt4. (5.235)

tt

Using (1.108) and (5.234), we finally get

m ) I I*(ili2«3«4) _ I*(ili2i3i4)q m ^ 1 I(0000)T,t I(0000)T,t

(T-ty

16

q

Cj4j3j2jl I Cj4j3j2j'l I + ^^ I Cj4j3j'm

j'l,j2,j3,j4=0 V(j'l,j2) / j4,j3=0 \j'l=0

+

qq

+ ^ y^ Cj4j3j'lj'l Cj40j3 , (5.236)

j4,j3 =0 jl=0

where ¿1 = ¿2 = ¿3, ¿4; ¿3 = ¿4.

Consider the cases (5.219), (5.220) by analogy with the case (5.218) using (1.109), (1.110). We have

m j I I*(ili2^3^4) _ I*(ili2i3i4)q

m I(0000)T,t I(0000)T,t

(T-ty

24

qq

Cj4j3j2j'l I y^ Cj4j3j2j'l I + ^^ I y^ Cj4j'lj2^l j'l,j2,j3,j4=0 V(j'l,j3) / j4,j2=0 \j'l=0

where ¿1 = ¿3 = ¿2, ¿4 and ¿2 = ¿4;

m ^ I i * (i l i2 ^3^4) _ i *(ili2i3i4)q

m S 1 I(0000)T,t I(0000)T,t

24

qq

y^ Cj4j3j2j'l I y^ Cj4j3j2j'l I + ^^ I y^ Cj'lj3j2^l j'l,j2,j3,j4=0 V(j'l,j4) / j3,j2=0 \j'l=0

where ¿1 = ¿4 = ¿2, ¿3 and ¿2 = ¿3.

Consider the case (5.221) by analogy with the case (5.218). We have

m

+m

I*(ili2i3i4) I*(ili2i3i4)q

I (0000)T,t I (0000)T,t

m

I(ili2i3i4) _ I(ili2i3i4)q\ I

I(0000)T,t I(0000)T,t ] +

2

T t4 t2

/,> q q

/ / dwiil>dt2dwi:4> - £ £ c

«/ «/ ' _A ' _A

ttt

m

j4 ,j l=0 j2=0

I(ili2i3i4) _ I(ili2i3i4)q

I (0000)T,t I (0000)T,t

Z(il)Z(i4) j4j2 j2 jl Sj'l Zj4

+

1'

q

1'

2

q

2

q

2

+MMU± icJinnncfcf)

\ t t .4.1=° j2=0 J

4q

— I\/I J I r(i1i2isi4) r(i1i2iSi4)q\ n , (T-i)" , v- /v^ ~ IVI i ^J(oooo)T,t J(oooo)T,t J \ 4g Z^ 1 jij2j2jl

.4 ,.1=0 \.2=0

q q T t4

E ECj4j2j2j\y 0.4 (i4) J 0.1 (i1) (i4 - i1 )di3di4.

.4 ,j1=0.2=0 t t

Then using (1.111), we obtain

1^1 i / j*{hnhh) _ j*{hnhu)q\ _

(0000)T,i J(0000)T,i y [

C.4.3.2.1 I ^3 C.4.3.2.1 I +

. 1. 2 .4=0 \ (.2 ,.3 ) /

2

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q / q \ 2 q q

+ E ( 53^.4.2.2.^1 53 53^.4.2.2.1 (C.4.1 C.4.J ' .4,.1=0 \.2=0 / .4,.1=0.2=0

where ¿2 = ¿3 = ¿1; ¿4 and ¿1 = ¿4; C10 is defined by (5.235) and

j4j1

t t4

c. = / .(i4)(i - i4)/ 0.1 (i1)di1 di4. (5.237)

j4j1

tt

For the case (5.222) by analogy with the case (5.218) and using (1.112), we

get

i / J*(»li2«3«4) _ J*(ili2i-iU)q\ _ (T — t)

(oooo)T,t J(oooo)T,t J f 24

53 C.4.s.2.1 I ^3 C.4.s.2.1 I + I 53 C

.1,.2,.S,.4=0 \(.2,.4) / .S,.1=0 \.2=0

q

C.2.2.1

where ¿2 = ¿4 = ¿1; ¿3 and ¿1 = ¿3.

2

q

Consider the case (5.223) by analogy with the case (5.218). We have (see Example 3.1 in Sect. 3.6)

m j f I * (i l i2 i3i4) _ I *(ili2i3i4)q\ 2 1 = m j ( I (i l i2 i3 i4 ) _ I (ili2i3i4)q\ m ,

m ^ 1 I(0000)T,t I(0000)T,t J f = m 1 \vI(0000)T,t I(0000)T,t y [ +

T t2 q q

+M /' / / [(¿wji'Wif - £ Ec^cfef

M { { j2,jl=0 j3=0

2 ^ (T — t)4 JL

(ili2i3i4) (ili2i3i4)q n . (T-t)4 , v-

- IVI ^ ^(0000)T,i J(oooo)T,t ) \ 48 Z^ 1 j3j3j2jl

j2,jl=0 \j3=0

q q T t2

X] X^Cj3j3j2jW (T-t2)0j2 (t2M j (t1 )dt1dt2.

j2,j l=0 j3 =0 t t

Then using (1.113), we obtain

(0000)T,i J(0000)T,i y [ -^g

Cj4j3j2j'l ( Cj4j3j2jl j +

j'l,j2,j3,j4=0 V(j3,j4) /

q / q \ 2 q q

+ ^^ I y^ Cj3j3j2j'l I - y^ y^ Cj3j3j2j'l ((T - t)Cj2j'l + Cj2j'l) ,

j2,jl=0 \j3=0 / j2,jl=0 j3=0

where ¿3 = ¿4 = ¿1, ¿2 and ¿1 = ¿2; is defined by (5.237) and

T t2

Cj2jl = j j (t2^ j (t1 )dt1dt2. tt

Consider the case (5.224). From (5.231) we have

^ / T t4 t3

m {(c^f - Co,«")2}=m S!+\j 11

ttt

T t4 t2 q q

+2 /K'Ww'«-/'™£ ^c^ifc

t t t .4,.3=0.1=0

2

q q q q

■ E 53^4.1.2.1 C.21 C.44 - 53 53^.4.2.2.1 C C.4M i . (5.238)

.4,.2=0 .1=0 .4,.1=0 .2=0

Furthermore,

T t4 t3 T t4 t2

||| dwi3')dw{:4)| dw«i'> di2dw«:4>=

t t t t t t T t4 T t4

= / Jik - i'dw^dwfi*' + | |(i4 - i1)dw<:1'dw«:4> =

t t t t T t4

= ^4 - i)/ dwt(i1)dwt(44) w. p. 1. (5.239)

tt

From (5.238) and (5.239) we get

M i ^ 1 *(i1*1i1i4) _ 1 *(i1Hi1i4)q\ 2 I = M J ^ 1 (i 1 i 1 i 1 i4) _ 1 (i 1i 1i 1i4)q^ 2 I +

M ^ 1 J(0000)T,t J(0000)T,t J r = M | \vJ(0000)T,t J(0000)T,t y f +

i ( T t4

+M | ii|(i4-i)Jdw™dw™-tt

\ 2

q q 2

53 53 (C.4.1.2.2 + C.4.2.1.2 + C.4.2.2.1 ) C.i C.

(i1^(i4) .4

.4 ,.1=0 .2=0

(i1i1i1i4) (i1i1i1i4)q n , (T - i)4

~~ IVI | y (0000)T,t J(0000)T,i 7 | 16

q ( q \2

+ y ^ l 53 (C.4.1.2.2 + C.4.2.1.2 + C.4.2.2.1 ) I -

.4,. 1 =0 V.2 =0 /

T t4

53 53 (C.4.1.2.2 + C.4.2.1.2 + C.4.2.2.1 ) /(t4 - t)0.4 (t4M 0.1 (i1)di1di4. .4,.1=0.2=0 { {

(5.240)

qq

Using (1.114) and (5.240), we finally obtain

l\/l J I r*(ili2i3i4) r*(ili2i3i4)q\

n 5(T -1)4 IVI ^ 1 J(oooo)T,i J(oooo)T,i ) \ ~ 48

Cj4j3j2j'l ( Cj4j3j2j'l ) +

j'l,j2,j3,j4=0 V(j'l,j2,j3)

qq

+ y ^ l (Cj4j'lj2j2 + Cj4j2j'lj2 + Cj4j2j2j'l ) I +

j4,jl=0 j2=0 qq

+ ^ ^ ^^ (Cj4j'lj2j2 + Cj4j2j'lj2 + Cj4 j2j2j'l ) Cj4j2 ,

j4 jl =0 j2=0

where ¿1 = ¿2 = ¿3 = ¿4.

Consider the case (5.225). From (5.231) we have

{/ T t4 t2

I (0000)T,t +2 J J J dwh dp2dwh + ttt

T t3 t2 q q

+ 2 // / dwh)dwh)dt3 ~ ^oooo)^- ¿ ¿Q^iCj^Cj;2-

t t t j4,jl=0 j2=0

q q q q 2

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- ^ y^Cj2j3j2jl j j - ^ y^Cj3j3j2jl j Zj22 j • (5.241) j3,jl=0 j2=0 j2,jl=0 j3=0 / I

Moreover,

T t4 t2 T t3 t2

III dw«il>di2dw«i2>+jj | dw«il>dw^^=

t t t t t t T t4 T t4

I |(i4 -11 )dw<;l) dw( i2) + | |(T -14 )dw< « dw<42) =

t t t t T t4

= J J(T - t1 )dw( I1'dwt(42) w.p. 1. (5.242)

tt

q

2

From (5.241) and (5.242) we get

m J ( 1 *(i1i2i2i2) _ 1 *(i1i2i2i2)q\ 2\ = m j ( 1 (i 1 i2i2i2) _ 1 (ui2i2i2)q\ M +

m 1 I J(0000)T,t J(0000)T,t i f = m 1 I J(0000)T,t J(0000)T,t ] f +

i ( T t4

1(5/ JP-V^dwM-

tt

+M:: J

2

q q 2

(i1) (i2) .4

.4 ,.1=0 .2=0

53 (C.4.2.2.1 + C.2.4.2.1 + C.2.2.4.1 ) C.i C.

qq

_ M J i (i 1 i2i2i2) r(i1i2i2i2)q\ 21 , (T - i)4,

~~ IVI | y (0000)T,f J(0000)T,i y [ 16

q ( q \2

+ y ^ l 53 (C.4.2.2.1 + C.2.4.2.1 + C.2.2.4.1 ) I -

.4,. 1 =0 V.2 =0 /

T t4

53 53 (C.4.2.2.1 + C.2.4.2.1 + C.2.2.4.1 ) / 0.4 (t4M (T - t1)0.1 (t1)dt1dt4. .4,. 1 =0.2=0 i {

(5.243)

Using (1.115) and (5.243), we finally obtain

l\/l J i 7-*(i 1 i2i3i4) r*(ili2isi4)q^

21 5(T - i)4 IVI ^ 1 J(oooo)T,i J(oooo)T,i ) \ ~ 48

53 C.4.3.2.1 ( 53 ^74.3.2.1 I +

. 1 ,.2 ,.4 =0 V(.2,.3,.4) '

q ( q \2

+ y ^ l 53 (C.4.2.2.1 + C.2.4.2.1 + C.2.2.4.1 ) I -

.4 ,.1=0 \.2=0 /

q q

- ^ ^ 53 (C.4.2.2.1 + C.2.4.2.1 + C.2.2.4.1 ) ((T - t)C.4.1 + C.4.J '

.4 ,.1=0 .2=0 where ¿2 = ¿3 = ¿4 = ¿1.

For the cases (5.226), (5.227) by analogy with the case (5.225) and using (1.116), (1.117), we get

1^1 i / j*{hnhh) _ j*{hnhu)q\ _ —

(0000)T,i J(0000 )T,t j ( 16

Cj4j3j2j'l I Cj4j3j2j'l ] +

j'l,j2,j3,j4 =0 V(j'l,j2,j4)

q

+ ^ ^ I y^ (Cj4j3j'lj'l + Cj'lj3j4j'l + Cjlj3jl j4 ) I +

j4 ,j3 =0 \jl=0 /

qq

+ ^^ y^ (Cj4j3j'lj'l + Cj'lj3j4j'l + Cj'l j3j'lj4 ) Cj4j3 ,

j4,j3 =0 jl=0

where i1 = i2 = i4 = i3;

i i j*(»i«2«3«4) _ j-*{hhhu)q\ _

(0000)T,i J(0000)T,i y [ -^g

Cj4j3j2jl Cj4j3j2jl +

j'l,j2,j3,j4=0 V(j'l,j3,j4) /

q / q \2

+ ^ ^ I y^ (Cj4j'lj2j'l + Cj'lj4j2j'l + Cj'lj'lj2j4 ) I -

j4,j2=0 \j'l=0

q q

- ^ ^ y^ (Cj4j'lj2j'l + Cj'lj4 j2 jl + Cj'lj'l j2j4) ((T - t)Cj2j3 + Cj2 j3 )

j4 ,j2 =0 jl=0

where i1 = i3 = i4 = i2.

Let us consider the case (5.228). Using (5.231), we obtain

q2

L (0000)T,t I (0000)T,t

m { [ + -

T t4

I/(t3 - t)dwt(33)dwt(i3) +

(oooo)T,i 1 2

tt

T t3 t2 2

+\J JI '^d^'dh + ^^ -ttt

q q q q q

Z(i3,Z(i3, __Z(il)Z(il) + V^ C

/ V Cj4j3jljl Zj3 Zj4 / v / v Cj3j3j2j'l j Zj2 + / v Cj3j3jljl

j4,j3 =0 jl=0 j2 ,jl=0 j3=0 j3,j'l =0

2

q

T t4

L(0000)T,i _ J(0000)T,i

= m < | - + -j J(t3 - /V/w/:;:; tlw/r

t t

q q

E E Cj4j3jljl (Cj3 ^Cj^ ^ 1{j3=j4}) +

j4 ,j3=0 jl=0

1 T t3 t2 q q

/ / dw«:'»dw«:i>di3 - £ (cj;i)cj:i) -ij=*}) +

t ï ï J«; =0 j3=0

- E }■. (5.244)

J3,ji=0

Note that

Z (i3)z( ^ _ 1 _

ZJ3 cj4 1 {j3 =j4 }

T t4 T t3

_ i jM / j(t3)dwi:3)dwi43) + i j(*)/ j(t4}dwt(;3)dwt(33), (5.245)

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z( ;i)z( ^ _ 1 _

cji cj: 1{ji=j:}

T t: T ti

_/ jte)/ j(ii)dwt(;i)dwi;i) + / j(h) J j(t2}dwt;i)dw(;i) (5.246) t t t t w. p. 1.

The relations (I5.'244)-(I5.'246) and Example 3.1 in Sect. 3.6 imply the following

m i ^i*(;i;i;3;3) _ i*(;i;i;3;3)q\m _ |m j ^i(;i;i;3;3) _ i(;i;i;3;3)q\m + m ^ 1 J(0000)T,t J(0000)T,t y [ _ m j \vJ(0000)T,t J(0000)T,t ^ [ +

i / T t4

u/ Jte-Q^^u-tt

+m : : j

qq

E CJ4J3JIJI (j )(],; ) 1{j3=j4}) l r +

J4,J'3 =0 ji=0

2

T t3 t2

+M < ( IJ 11 dw^dw^dts-

ttt

q q \

^ ^Cj'3j3j2jl l' Z,j2l) - 1{jl=j2^ ( +

j2,j'l =0 j3=0

2

= m <( ( I(ilili3i3) _ I(ilili3i3)^ I + m

(0000)t,t j(0000)t,t J ( i i 2

j3 ,jl=0

21

if Tt4

u /

tt

qq

Yl ^Cj'4j3jljl (Zj(3i3)Zi43) - 1{j3=j4m f +

j4,j3 =0 jl=0

i / T t2

+M \[lJ(T-h) jdw/ ,/w,'

tt

q q \ - |

^ ^Cj3j3j2jl (Zj(il)Zj(2il) - 1{j'l=j'2}) ) f +

j2 ,j l =0 j3=0

2 q 2

I V c.

"T" [ g / v ^JsJsJUi

j3 ,jl=0

r(ilili3i3) T-(ilili3i3 M2\ , (T-t)

— KA J i Ahhhis) _ T(hhi3i3)q\ I , U ^ , ^ ^10 , rAQ \ ,

- IVI i VJ(0000)T,i J(0000)T,t y I 48 Z^ / , ^jdddi V^hh ^ ^hh)

qq

j4,j3=0 jl=0

2

+M E ECj4j3jljl (Zj33,Zj(43) - %=«) +

j4,j'3=0 jl=0

\4 q q

~ AO S ^hhhh ((T -t) (Cjlj2 + Cj2jx) + C^J2 + C^-J +

+ 48

j2 ,jl=0 j3=0

+M ( E E Cj3j3j2jl (Z«i')Cj,2l) - 1(jl=j2^ +

j2,j'l=0 j3=0

2

(t -1)2 ' 12

+ ( '—fa ~ e ) ■ (5.247)

.3.1=0

Furthermore,

m!( E Ec«^(CC-11.3=.})

qq

.4 ,.3=0 .1=0

q q \2) ( q \2

( E .. -2 E ..J +

.4 ,.3=0.1=0 ' I V.3 ,.1=0 '

+ ( E c

.3,.1=0

q

Cjsjsjljl

2

q q 2 q

Mil £ > f- ^ ) , (5.248)

.4 ,.3=0.1=0 / I \.s,.l=0

qq

m E £.1(cí:l)c)21,-i..

.2 ,.1=0 .3=0

.(ilMil)

2

q q 2 q

= M\ £ £..1 Cf^M - ^ ) . (5.249)

^ \.2,.l=0.3=0 / J V. 1.3=0

Using (2.307), we get

m|( E £.1zf.

2

q q 2

.4

.4,.3=0 .1=0

q \ 2 q .4 1 / q q \ 2

^.3.3.1.1 I + ^3 1 53 ^.3.4.1.1 ^53 C.4.3.1.1 I +

Cl=0 / .4=0.3=0 \.l=0 .1=0 /

q ( q \2

+ 2£ £..1 . (5.250)

.4=0 .1=0

2

From (5.248) and (5.250) we have

qq

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m £ £ jl (Zj33,Zj(43) - 1{j3=y.>)

j4,j'3=0 jl=0

q j4 1 / q q \ 2 q/q \ 2

y^ y^ ( y] Cj3j4j'lj'l + y] Cj4j3j'lj'l ] + 2 ^ i y] Cj4j4j'lj'l ] • (5.251)

j4=0j3=0 \j'l=0 j'l=0 / j"4=0 \j"l=0 /

By analogy with (5.251) we obtain

M) Yl y^Cj3j3j2jl (Zj(il)Zj(2il) - 1{jl=j2})

q j'2-W q q \ 2 q/q \ 2

= Cj3j3jlj2 + Cj3j3j2jl + 2 Cj3j3j2j2 . (5.252)

j2=0 j'l=0 Vj3=0 j3=0 / j2=0 \j'3=0 /

Combining (1.118), (5.247), (5.251), and (5.252), we finally have

^*(ili2i3i4) _ I*(ili2i3i4)q |

(T -1)

IVI ^ 1 J(0000)T,i J(0000)T,i J f ~ 12~

qq

y^ Cj4j3j2j'l l y^ ( y] Cj4j3j2j'l ) ) + ^ y^ Cj4j3 j'lj'l (Cj3j4 + Cj4 j3) + j'l,j2,j3,j4=0 V(j'l,j'2^(j'3,j'4) / / j4,j3=0 j'l=0

q j4 1 / q q \ 2 q/q x 2

+ ^^ y^ I y^ Cj3j4j'lj'l + y^ Cj4j3j'lj'l I + 2 ^^ I y^ Cj4j4j'lj'l

j4=0 j3=0 Vj'l=0 j'l=0 / j"4=0 Vj"l=0

q q

- y] Cj3j3j2j'l ((T - t)Cj'l Cj2 + jj + jj'l) +

j2 ,j l =0 j3=0

q j 2 - 1 q q 2 q q 2

+ y^ I y^ Cj3j3j'lj2 + y^ Cj3j3j2j'l I + 2 ^^ I y^ Cj3j3j2j2 I +

j2=0 j'l=0 Vj3=0 j3=0 J j2 =0 V j3 =0 /

g / V °J-iJ-iJlJl

j3,jl=0

2

q

where ¿1 = ¿2 = ¿3 = ¿4 and

t (y/r=t, j = o

C. = 0. (t)dT = ^ .

t 0, j = 0

Consider the case (5.229) by analogy with the case (5.228). Using (5.231), we obtain

m 1 *(i1i2i1i2) 1 *(i1i2i1i2)q 2 = m 1 (i1i2i1i2) 1 (i1i2i1i2)q m ^ 1 J(0000)T,t J(0000)T,t J f = m ] \ J(0000)T,t J(0000)T,t

\ 2 ■

q q q q q 2

z(i2)z(i2) _ V^ z(il)z(il)+ V^ C

/ j C.4.1.2.1 z.2 z.4 / v C.2.3.2.1 z.l Z.3 + Z^ C.2j1j2j1

.4,.2=0 .1=0 .3,.1=0 .2=0 .2,.1=0

(( q q

= m j i 1 (i1i2i1i2) _ 1 (i1i2i1i2)q __V^ c /V(i2) Z(i2) _ 1 ^ _

= lvl S I J(0000)T,t J(0000)T,t Z^ / c.4.1.2.1 ^S.2 Z.4 1{.2=.4^y

I V .4 ,.2=0 .1=0

q q q

£ £ C.2.3.2.1 (c]i ). ) - 1{.1=.3^ - E C.

.3,.1=0 .2=0 .2,.1=0

= m 1 (i1i2i1i2) 1 (i1i2i1i2)q 2 +

= m ] I'1(0000)T,t J(0000)T,t / f +

+ ^ ( EE (C - .A}) J +

.4 ,.2=0 .1=0 qq

+m{( £ (zii1. -1.1=«)) [ +

.3.1=0 .2=0

q

+ ( E C.2.1.2.1 ) • (5.253)

.2 ,.1=0

Using (1.119) and (5.253), we finally get

l\/l J ( r*(ili2i3i4) r*(ili2i3i4)q \

(T - i)

IVI s 1 J(oooo)T,i J(oooo)T,i J f ~ 24

2

C.4.3.2.1 1 ^^ I 53 C.4.3.2.1 ] ] +

. 1 ,.2 ,.3,.4=0 \(.1,.3) \(.2 ,.4)

q . 4 - 1 q q 2 q q

+ ^3 y3 I 53 C.2.1.4.1 ^53 C.4.1.2.1 I + 2 ^3 I 53 C.4.1.4.1 ] +

.4=0.2=0 \.1=0 .1=0 J .4=0 V. 1=0

q . 3 - 1 q q 2 q q

+ ^3 y3 1 53 C.2.1.2.3 ^53 C.2.3.2.1 I + 2 ^3 1 53 C.2.3.2.3 ) + .3=0.1=0 \.2=0 .2=0 / .3=0 \.2=0

( q \ 2

+ 1 ^3 C.2313231 \.2,.1=0

where ¿1 = ¿3 = ¿2 = ¿4.

Consider the case (5.230) by analogy with the cases (5.228) and (5.229). Using (5.231), we obtain

m 1 *(i1i2i2i1) 1 *(i1i2i2i1)q 2 m ^ 1 J(0000)T,t J(0000)T,t

M J I r(*i*2«2H) I 1 / / / 1 (n)i. t-ufa'i) Ahhi2ii)q

M < I (0000)T,t +2 J J J dwti dp2dwh - \oooo)T,t

ttt

q q q q q

Z(i2) Z(i2) _ V^ Z(il)Z(il) + V^ C

/ „ C.1.3.2.1 Z.2 Z.3 Z^ Z^C.4.2.2.1 Z.l Z.4 + C.1j2j2.1

.3,.2 =0 .1 =0 .4,.1=0 .2=0 .2,.1 =0

T t4 t2

= M 1 (i1i2i2i1) 1 (i1i2i2i1)q+ = M ^ 1 J(0000)T,t J(0000)T,t +

T r r q q

: / / dw^dw™- 5: 53 r,,,,,^; ,/ 1,, ,,;)

i t t .4 ,.1=0.2=0

\ 2 ■

q q q x 2

- 53 53 C.1.3.2.1 (C. )c];j ) - 1 {.2 =.3}) - 53 C.1.2.231

.3,.2=0 .1=0 .2 ,.1=0

_ |W| J I Annnn) _ j{nnnn)q\2 \ , IWI J I 1

— lvl S 1 J(0000)T,i J(0000)T,i y I I I 2

i ( T t4

u/ JiU-tJdwMdwM-tt

q

2

qq

Yl S Cj4j2j2jl (Zj(r)Zj(4il) - 1{jl=j4m } +

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j4 ,jl=0 j2=0 ' J

+ E E Cj (jj - 1(j2=j3}^ [ +

V?3,j2=0 jl=0

+ Cj l j2 j 2 l j2,jl=0

= m ^ I i(ili2i2il) _ i(ili2i2il)^ I . +

= m I(0000)T,t - I(0000)T,t +

^ (T -1)4

48

qq

/ T t4

X] ^Cj4j2j2jl l J j (U) j (t4 - t1)0jl (t1 )dt1dt4 + j4,jl=0 j2=0 t t

T t4 \

+ J j (A) J(t4 - ¿1) j (t1)dt1dtj + tt

+M H E E Cj4j=j2jl (Zji')Zj(4l) - 1{jl=A>)) I +

j4,j'l=0 j2=0 qq

( E ECj,j3j2jl (Z]:2,Zj(32) - 1{j2=j3})) \ +

Vj3 ,j2=0 jl=0

/ q \2

+ Cj l j 2 j 2

j2,j'l=0

Using (1.120) and (5.254), we finally get

m I*(ili2i3i4) I*(ili2i3i4)q

m ^ 1 I(0000)T,t I(0000)T,t

2] (T -1)4

16

(5.254)

- Cj4j3j2jl Cj4j3j2 l

j'l,j2,j3,j4=0 \(j'l,j'4^(j'2,j'3)

q q

Ev c /c 10 + C 10 — C01 - C01 ^ +

/ ,Cj4j2j2j,l \Cj4jl + Cjlj4 Cj4jl Cj'lj'J +

j4 ,j l =0 j2=0

2

q

q j'4-1 / q q

+ ^^ I y^ C31323234 + y^ C34323231

34=0 jl=0 \j2=0 32=0

q j3-W q q

+ ^^ y^ I y^ C31323331 + y^ C31333231 33=0 32=0 \31=0 31=0

2 q f q \2

+ 2 ^^ i y^ C34323234 ] + 34=0 V/2=0 /

2 q ( q x 2

+ 2 ^^ ( ^^ C31333331 I + 33=0 \31=0

where ii = i4 = i2 = i3.

+ ( y^ C31323231 32,31=0

5.7 Optimization of the Mean-Square Approximation Procedures for Iterated Stratonovich Stochastic Integrals Based on Theorems 2.2, 2.8 and Multiple Fourier—Legendre Series

This section is devoted to optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals (5.4) of multiplicities 1 to 3 based on Theorems 2.2, 2.8 and multiple Fourier-Legendre series [65]1.

The mentioned stochastic integrals are part of strong numerical methods with convergence orders 1.0 and 1.5 for Ito SDEs with multidimensional non-commutative noise (see (4.74), (4.75)).

We show that the lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Stratonovich stochastic integrals (5.4) can be significantly reduced without the loss of the mean-square accuracy of approximation for these stochastic integrals.

Using Theorems 2.2, 2.8 and the system of Legendre polynomials, we obtain the following approximations of iterated Stratonovich stochastic integrals (5.4)

T*{h) _ /rri _ .An) I(0)T,t = v T tz0 ,

I

in) (1)T,t

(T -t)3/2

z0i1) +

:C

(n)

1The results of this section were obtained jointly with Kuznetsov M.D., who is also a co-author of the publications [52]-[55], [57], [59], [65].

2

q

1

fahh)q _ £_T_ / Ah) Ah) I Y^ 1 ( Ah) Ah) _ Ah)Ah)\ 1 /r 9rr\

J(00 )T,t. — 9 \ So "1" / , ^.Q _ ^ l^i-iSi S.Î Si-1 J J' zoo;

where

qi

I*(;i;:;3)qi _ Y"^ c c(;i)c(;:)c(;3) ^

J(000)T,t _ Cj3j: ji j zj: j ' (5.256)

ji ,j:,j3=0

T

j _ fa(sf (i _1,...,m, j _ 0,1,...)

j — / j

t

are independent standard Gaussian random variables for various i or j, {faj(x)}°=0 is a complete orthonormal system of Legendre polynomials in the space L2([t,T]) (see (5.5)),

_ v/(2j1 + 1)(2j2 + 1)(2j3 + 1)^

( ./:•../ ./ ~~ g 'V ( ./:•../ ./ •

1 z y

j-j _/ pj3 (z) J Pj: (y) J pji (x)dxdydz, -1 -1 -1

Pj (x) is the Legendre polynomial (see (5.6)). Denote

E*(li-lk) _f M J 7 t*(;i-;fc) _ t*(;i--;fc)A 1

_ M1 ^(/i.../fc)T,t I(h...ik)T,t) fa

where I*/^/^^ is the iterated Stratonovich stochastic integral (5.4) and I*i^'ilyrt is the mean-square approximation of this stochastic integral. More precisely, the approximations I*™)^/, Iioo0vr3t)qi are defined by ((5.2551), (5.256).

(00)T,t (000)T,t

Using (5.41), (5.77), (5.210), (5.214), (5.217), we get

(T-tfa^l ^ 1 (T -1)3 qM

ft*«. (5257)

^(000) = _ £ c,.. fa^fa i^fa (5.258)

qM 6

ji,j: J3=0

3 qi,: qi,:

F*(000) _ ~ ( J _ V^ _ V^ fi fi

qi,2 ~ 4 Z^ ./:■,/:•./• Z^ ( ./:■,/ ( ./:•../ ./

Ji,j:,j3=0 Ji,j:,j3=0

(T -1)3

qi,2 , „ \ qi,2 / qi,2

^ E ( <»./•./• + V/ ./ ) E ( E f ) ~ /

2 / v i 1 pr

ji=0 V V3 7 js =0 \ji=0

(5.259)

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3 qi,s qi,s

A

91,3 4

ji,j2,js=0 ji,j2,jS=0

(T - t)3/2 qi'3 / 1 \ qi's / qi'3 \2

E*(000) _

^1,3 ~~ ¿1 Z^ ./::./: ./ Z^ ( ./V./':«./'-1 ./-./: ./

./:■■ <» V ^ 7 ./ <» \J3=0

(t -t)3 9m qm

(5.260)

T^*(000) _ y-1 ~ l ) _ V^ f /2 _ V^ r* -U

y 1,4 ~~ g Z^ ./"•./../ Z^ ( ./3./2./ ( ./ ./: ./- '

ji,j2,jS=0 ji,j2,jS=0

qi,4 / qi,4 \ 2

+ E ECjij2ji (ii = i3 = i2). (5.261)

j2=0 \ji=0 /

Note that the number of conditions (I5.258l)-(l5.261l) is quite large, which is inconvenient for practice. In this section, we propose the hypothesis that all the formulas (5.258)—(k5.261) can be replaced by the formula (5.258) in which we can suppose that i1, i2, i3 = 1,..., m. At that we will not have a noticeable loss of the mean-square approximation accuracy of iterated Stratonovich stochastic integrals.

Consider the following condition

E*(00) < (T - t)4, £;(000) < (T - t)4, i = 1,..., 4. (5.262)

Let us show by numerical experiments that in most situations the following inequality is fulfilled

qi,i > i = 2,3,4, (5.263)

where q1;i (i = 1,..., 4) are minimal natural numbers satisfying the condition

In Tables 5.59-5.61 we can see the results of numerical experiments. These results confirm the hypothesis proposed earlier in this section. Note that in Table 5.61 we calculate the mean-square approximation errors of iterated Stratonovich stochastic integrals in the case when

qi,i = qi,i, i = 2, 3, 4,

Table 5.59: Conditions E,*-000- < (T-t)4, i = 1,.. .,4.

T- -t 0.011 0.008 0.0045 0.0035 0.0027 0.0025

12 16 28 36 47 50

qi,2 6 8 14 18 23 25

qi,3 6 8 14 18 23 25

qi,4 12 16 28 36 47 51

Table 5.60: The condition (5.262).

T-t 2-l 2-3 2-5 2-8

Q 1 8 128 8192

qi,i 0 1 4 32

qi,2 0 0 2 16

qi,3 0 0 2 16

qi,4 0 0 4 33

Table 5.61: : Values <°00) • (T - t)~3 = El ,i = 1,. .., 4.

T - t 0.011 0.008 0.0045 0.0035 0.0027 0.0025

Qhi 12 16 28 36 47 50

0.010154 0.007681 0.004433 0.003456 0.002652 0.002494

qi,2 12 16 28 36 47 50

z?* 0.005102 0.003855 0.002221 0.001731 0.001328 0.001248

qi,3 12 16 28 36 47 50

* 0.005102 0.003855 0.002221 0.001731 0.001328 0.001248

qi,4 12 16 28 36 47 50

* 0.010407 0.007845 0.004500 0.003501 0.002680 0.002519

Table 5.62: Comparison of numbers qi;i and pi;i.

T - t 2^ 2-2 2-3 2-4 2-5 2-6

91,1 0 0 1 2 4 8

(91.1 +1)3 1 1 8 27 125 729

pi,i 1 3 6 12 24 48

(pi,i + 1)3 8 64 343 2197 15625 117649

where q1;1 is the minimal natural number satisfying the condition (15.262). In this case, there is no noticeable loss of the mean-square approximation accuracy of iterated Stratonovich stochastic integrals (see Table 5.61). This means that all the formulas (I5.258l)-(l5.261l) can be replaced by the formula (5.258) in which we can suppose that i1, i2, ¿3 = 1,..., m.

Let q1;1 be the minimal natural number satisfying the condition

£(000) < (T _ t)4, (5.264)

where the left-hand side of (5.264) is defined by the formula (5.258). Let p1;1 be the minimal natural number satisfying the condition

3! • Ep^ < (T _ t)4, (5.265)

where the value £P°010) on the left-hand side of (5.265) is defined by the formula

(recall that 3! is included in the inequality (5.165) for the case k = 3).

In Table 5.62 we can see the numerical comparison of numbers q1;1 and p1;1. Obviously, excluding of the multiplier factor 3! essentially (in many times) reduces the calculation costs for the mean-square approximations of iterated Stratonovich stochastic integrals. Note that in this section we use the exactly

calculated Fourier-Legendre coefficients using the Python programming language

Chapter 6

Other Methods of Approximation of Specific Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 4

6.1 New Simple Method for Obtainment an Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on the Wiener Process Expansion Using Leg-endre Polynomials and Trigonometric Functions

This section is devoted to the expansion of iterated Itô stochastic integrals of multiplicity 2 based on the Wiener process expansion using complete orthonormal systems of functions in L2([t, T]). The expansions of these stochastic integrals using Legendre polynomials and trigonometric functions are considered. In contrast to the method of expansion of iterated Ito stochastic integrals based on the Karhunen-Loeve expansion of the Brownian bridge process [77 -79], this method allows the use of different systems of basis functions, not only

the trigonometric system of functions. The proposed method makes it possible to obtain expansions of iterated Itô stochastic integrals of multiplicity 2 much easier than the method based on generalized multiple Fourier series (see Chapters 1 and 2). The latter involve the calculation of coefficients of multiple Fourier series, which is a time-consuming task. However, the proposed method can be applied only to iterated Ito stochastic integrals of multiplicity 2.

It is well known that the idea of representing of the Wiener process as a functional series with random coefficients (that are independent standard Gaussian random variables) with using the complete orthonormal system of

trigonometric functions in L2([0, T]) goes back to the works of Wiener [147

(1924) and Levy [148] (1951). The specified series was used in [147] and [148 for construction of the Brownian motion process (Wiener process). A little later, Ito and McKean in [149] (1965) used for this purpose the complete orthonormal system of Haar functions in L2([0,T]).

Let fT, t E [0,T] be an m-dimestional standard Wiener process with inde-dent com We have

pendent components , i = 1,... ,m.

s T

f<-> - f'!) = | f =| 1{T<s}df<",

t t

where

T

J 1{T<f

t

is the Itô stochastic integral, t > 0, and

1, t < s 1{T<s} H , T,s E [t,T].

0, otherwise

Consider the Fourier expansion of 1{T<s} E L2([t,T]) at the interval [t,T] (see, for example, [115])

t s

oo T oo s

E / 1{t <s}fa(t )dT0j (t ) = ^ i fa (t )dT0j (t ), (6.1)

j=° { j=o {

where {fa(t)}j=° is a complete orthonormal system of functions in the space L2([t,T]) and the series (6.1) converges in the mean-square sense, i.e.

T / s \ 2

q s

1{T<s} — £ /fa(t)dTfa(t) dT ^ 0 if q ^ to.

t \ j=0 t J

Let fs(it)q be the mean-square approximation of the process fs(i) — ft(i), which

has the following form

T ( q s \ q s T

fSHq = J £ J fa(t)dTfaj(t) df« = £ J fa(t)dTj fa(tf (6.2)

.j=° t / j=0 t t

Moreover,

m

f (i) _ f№ _

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1

J2 J fa (t)dTfa (t) df«

|T<s} / v I VjW^'Vj' j=0

^ J fa (t)dTfa (t) j dT ^ 0 if q ^ œ. (6.3)

-jr<s} / v I YJK'J^'YJ t \ j=0 t

In [82] it was proposed to use an expansion similar to (6.2) for the expansion of iterated Ito stochastic integrals

T t2

(i1«2) (00)T,t

df^'dft;2'

(ii,i2 = 1,... ,m).

(6.4)

tt

At that, to obtain the mentioned expansion of (6.4), the truncated ex-of components of the Wiener process fs have been iteratively

pansions

substituted in the single integrals [82]. This procedure leads to the calculation of coefficients of the double Fourier series, which is a time-consuming task for not too complex problem of expansion of the iterated Ito stochastic integral (6.4). In [82] the expansions on the base of Haar functions and trigonometric functions have been considered.

In contrast to [82] we subsitute the expansion (6.2) only one time and only into the innermost integral in calculation of the coefficients

This procedure leads to the simple

fa (t )dT (j = 0,1, 2,...)

of the usual (not double) Fourier series.

Moreover, we use the Legendre polynomials [50], [66] for the construction of the expansion of (6.4). For the first time the Legendre polynomials have been applied in the framework of the mentioned problem in the author's papers [71] (1997), [72] (1998), [73] (2000), [74] (2001) (also see [1]-[66]) while in the papers of other author's these polynomials have not been considered as the

s

2

s

basis functions for the construction of expansions of iterated Ito or Stratonovich stochastic integrals.

Theorem 6.1 [14]-[16], [50], [66]. Let fa(t) (j = 0,1,...) be an arbitrary complete orthonormal system of functions in the space L2 ([t,T]). Let

T T T s

J f^f = £ J to (r)df<:i) | J to(r)drdf<'2» (6.5)

t j =0 t t t

be an approximation of the iterated Ito stochastic integral (6.4) for ii = i2. Then

T

= fij"df<-2> (ii = ¿2),

t

where ii, i2 = 1,..., m.

Proof. Using the standard properties of the Ito stochastic integral as well as (6.3) and the property of orthonormality of functions toj(r) (j = 0,1,...) at the interval [t, T], we obtain

T s T

mN i idf(ii)dfi(i2)-Jfi;ti)qdfs(i2) t t t

J m{ f(:i) - f:i) - fi:^)2} ds

T T / q i N 2

J fa (t)dTfa (T m dTds

1{T<s} - ^ J j )u'l j

t t \ j=0 t

T / q (} \2\

I (s _ t) _E /fa (t)dT

t \ j=0 V t J )

ds. (6.6)

Using the continuity of the functions uq(s) (see below), the nondecreasing property of the functional sequence

2

q ' s

uq(s) = £ I J fa (t)dT

t

and the continuity of the limit function u(s) = s—t according to Dini's Theorem, we have the uniform convergence uq(s) to u(s) at the interval [t,T].

Then from this fact as well as from (6.6) we obtain

T

= l^S- /fi^dfi"'. (6.7)

t

Note that we could also use Lebesgue's Dominated Convergence Theorem in (6.6) to obtain (6.7).

Let {fa (t)}s=0 be a complete orthonormal system of Legendre polynomials in the space L2([t,T]), which has the form (5.5). Then

LjiT)dT = Lzl ( _ j^wJ} for . > j (6 8)

J 2 ww + wj + 3) VW^ï

Let us denote

T

j =i fa(t)f (i = 1,..., m).

From (6.5) and (6.8) we obtain

T T ^_

y/T^t. " t t

T~* V r"'1 I 1 ,-fe) _ 1 | =

2 \ \/(2j + l)(2j + 3) VF^ '"1'

2

, T ~f ( 1 1 Ai2) 1 =

2 \vWTT)(2jT3) '

T -t I Jtl)Ji2) 1 (Ah) Ah) _ A'AAA\ \ ,

Z^ _ 1 V ' 1 ' ' 1 / I

2 C<? 1 \J i^q + l)(2g + 3) ' ( j

Then from (6.7) and (6.9) we get

T

T(i1i2) = l i m / f(i1)qdf(i2) = 1 (00)T,t = lqi;:m . / fs,t dIs =

t

T-t ' œ

Ah) Ah) i V^ 1 i Ah) Ah) Ail)Ai2)\ I (a irh So so • z^ I 1^./ ^ ^ I J I • ((K|U)

It is not difficult to see that the relation (6.10) has been obtained in Sect. 5.1 (see (5.11)).

Let {fa-(t)}j=0 be a complete orthonormal system of trigonomertic functions in the space L2([t,T]), which has the form (5.110).

We have

s T_A fa2r_1(s), j = 2r

JUT)dT = — l , (6.11)

t [V2(f)0{s) - far(-s), j = 2r — 1

where j > 1 and r = 1, 2,...

From (6.5) and (6.11) we obtain

T T

tt

i T ~f v^ J_ / fA'h)A'1^ - A11') _i_ ^ -

2 ^ ITT \ \ S2r-lS2r y So S>2?-— 1 y —

r=1

1 ^{T-tf/2 ( 1 Ji2)

y/T=t* 2 P 7T ¿^r^-1

T -1 sfa J_ ((Ah) Ah) _ ^ _l ^

irr v v' S2r-lS2r y So S>2?-— 1J

2 nr

r=1

\ r=1 ^

0 So "T" _ / v I S2/' S2r-1 S2r-lS2r

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+ V5 - <$»<&))) - ^¿tf1' £ ¿<82,. (6.i2)

From (6.12) and (6.7) we get

= 1.1m. J f^df^ = \(T - t) ( W + t ^

+ ^E K^1'^-1" (c^ici!2) - «ii))) ■ (6.13)

r=1 ^ ' /

where i1 = i2.

It is obvious that (6.13) is consistent with (5.86) for i1 = i2 (we consider here (15.86) without the random variables ^^

6.2 Milstein method of Expansion of Iterated Ito and Stratonovich Stochastic Integrals

The method that is considered in this section was proposed by Milstein G.N. [77] (1988) and probably until the mid-2000s remained one of the most famous methods for strong approximation of iterated stochastic integrals (also see [78 -

80], [86]-[88], [91], [92]). However, in light of the results of Chapters 1 and 2 as

well as Sect. 5.1 and 5.3, it can be argued that the method based on Theorem 1.1 is more general and effective.

The mentioned Milstein method [77] is based on the expansion of the Brown-ian bridge process into the trigonometric Fourier series with random coefficients (version of the so-called Karhunen-Loeve expansion).

Let us consider the Brownian bridge process

fi-^fA, ¿e[0,A], A > 0, (6.14)

where ft is a standard Wiener process with independent components ft(i), i = 1,..., m.

The componentwise Karhunen-Loeve expansion of the process (16.14!) has the following form

oo

•(i) t (i) 1 , / 2nrt . 2nrt

'■t - = + Z^ ( <VC0S-^- + 6vrsm-^-

r=1

where the series converges in the mean-square sense and

A

Hr = \ J (fsW - ¿fA}) cos^^ete, 0

A

=! / (f-1 - ™-irds-0

r = 0,1,..., i = 1,..., m.

It is easy to demonstrate [77] that the random variables aijr, bijr are Gaussian ones and they satisfy the following relations

m {«i,r} = m {a^} = 0, m {«i,ra^} = m } = 0,

m {";„";,.,} = m {!>,.,.!>,..,.} =0, m {a?r} = m {&?r} =

where i, i1, i2 = 1,..., m, r = k, ii = i2. According to (6.15), we have

= fA^ + 7^,0 + £ iavrcos^i + bhr sin^p J , (6-16)

r=1 V /

where the series converges in the mean-square sense.

Note that the trigonometric functions are the eigenfunctions of the covari-ance operator of the Brownian bridge process. That is why the basis functions are the trigonometric functions in the considered approach.

Using the relation (6.16), it is easy to get the following expansions [77]-[79

I df^ = -^-f^ + iah0 + £ ( C0S~W~ + ) > (6-1'7)

i r=1 ^ '

t

df^d-

t =

2A

f + ^ +

00

oo

A ^ 1

2tt r

r=1

2nrt 7 avrsm—--bi-

2

' 2nrt

COS—:--1

A

(6.18)

t

T

2

t

t T t t T

J J drfaf^ = t J df« -J J rffWrfr = +

0 0 0 0 0

^ ( 2nrt 7 2nrt \ ( a»,rCos-^- + 5iirsin— 1 -

r=1 ^ '

A ^ 1 ( 2nrt 7 ( 2nrt \\ .

£ r ^fli'rSm~A" " J J ' ( ^

t T t T t

f f df^df^ = if^ f f dndl7 + iail>0 f +

0 0 0 0 0

oo

r=1

1 ^ ^ ^ ^ ( 4nrt \

+ 4 Z^ I - oil)r&i2)r) I 1 - cos-^- I +

r=1

f h h \ • 47rri

Vii,rO'i2,r) Sill — h

2 ,(:2 7 . 2nrt ( 2nrt

-—fA ( ai^sin— + biur ( cos—

« « / /cos cos

+ E £ k[aH,r^2, J (jfe } + 2{k_r) +

k=1 r=1(r=k) \ \

'sin (M^lli) sin +a<i A,* ( 2(A; + r) + ^[k — r) 7 I +

+ ft |C0H A j cos(^)__

A / sm A sm

2(fc + r) - 2(fc - r) ' 1 1 (6'20)

converging in the mean-square sense, where we suppose that i1 = i2 in (6.20).

It is necessary to pay a special attention to the fact that the double series in (6.20) should be understood as the iterated one, and not as a multiple series (as in Theorem 1.1), i.e. as the iterated passage to the limit for the sequence of double partial sums. So, the Milstein method of approximation of iterated stochastic integrals [77] leads to iterated application of the limit transition (in contrast with the method of generalized multiple Fourier series (Theorem 1.1), for which the limit transition is implemented only once) starting at least from the second or third multiplicity of iterated stochastic integrals (we mean at least double or triple integration with respect to components of the Wiener process). Multiple series are more preferential for approximation than the iterated ones, since the partial sums of multiple series converge for any possible case of joint converging to infinity of their upper limits of summation (let us denote them as p1;... ,pk). For example, when p1 = ... = pk = p ^ to. For iterated series, the condition p1 = ... = pk = p ^ to obviously does not guarantee the convergence of this series. However, as we will see further in this section in [78] (pp. 438-439), [79] (Sect. 5.8, pp. 202-204), [80] (pp. 82-84), [88] (pp. 263-264) the authors use (without rigorous proof) the condition p1 = p2 = p3 = p ^ to within the frames of the Milstein method [77] together with the Wong-Zakai approximation [68

(also see discussions in Sect. 2.20, 2.21). Furthermore, in order to obtain the Milstein expansion for iterated stochastic integral, the truncated expansions (6.16) of components of the Wiener process ft must be iteratively substituted in the single integrals, and the integrals must be calculated, starting from the innermost integral. This is a complicated procedure that obviously does not lead to the expansion of iterated stochastic integral of multiplicity k (k G N).

Assume that t = A in the relations (6.17) (16720) (at that double partial sums of iterated series in (6.20) will become zero). As a result, we get

A

dfW = 4°, (6.21)

A t

J J df^dr = ^A (f£> + ai>0) , (6.22)

00 A t

J J dndfW = ^A (if - ai>0) , (6.23)

00

A t

1 „ (^ Win ) 1

Jf(n) Jf(i2) _ 1f(«l)f(«2) 1 7 An) n f^A I

dVTl >dt± - -tA tA - - ^ai2,oiA - aii,oiA ) +

0 0

œ

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(a

r=1

+ n r (a«i,rb«2,r - ai2,r) (6.24)

converging in the mean-square sense, where we suppose that i1 = i2 in (6.24). Deriving (l6.21l)-(l6T2il), we used the relation

œ

a^ = , (6.25)

r=1

which results from (6.15) when t = A.

Let us compare expansions of some iterated stochastic integrals of first and second multiplicity obtained by Milstein method [77] and method based on generalized multiple Fourier series (Theorem 1.1).

Let us denote

_ A _ A

2 f . 2nr.s (i) w /2 f 2ttr.s {l)

Â J ' & = V ÂJ ' (6-26) 00

A

Co' = 4r /"rfff, (6.27)

,0 0

where r = 1, 2,...,i = 1,...,m.

Using the Ito formula, it is not difficult to show that

a,i,r = y ydli, V = ¿y w- P- 1- (6-28)

From (6.25) we get

œ 1

^ — E^-r (6-29)

n z—' r

r=1

After substituting (6.28), (6.29) into (6.21)-(6.24) and taking into account (6.26), (6.27), we have

A

J dfM = VEÇ^K (6.30)

A

dndfTil)

i) —

A3/2

00 A t

df^dr =

A3/2

00

oo

n ^ r

r=1

oo

2r— 1

n ^ r

r=1

2r— 1

(6.31)

(6.32)

A

dfT(il)df(i2)

A I AA An) , 1 V^ 1 _ Ah) Ah) :

2 I H) So S2r-1 S2r-lS2r

1 A 1

00

n r

r=1

(6.33)

Obviously, the formulas (6.30) (167331) are consistent with the formulas (5.7), (5.86), (5.104), (5.105). It testifies that at least for the considered iterated stochastic integrals and trigonometric system of functions, the Milstein method and the method based on generalized multiple Fourier series (Theorem 1.1) give the same result (it is an interesting fact, although it is rather expectable).

Further, we will discuss the usage of Milstein method for the iterated stochastic integrals of third multiplicity.

First, we note that the authors of the monograph [79] based on the results of Wong E. and Zakai M. [68], [69] (also see [70]) concluded (without rigorous proof) that the expansions of iterated stochastic integrals on the basis of (6

(the case i1 ,i2,i3 = 1,... ,m) converge to the iterated Stratonovich stochastic integrals (see discussions in Sect. 2.20, 2.21). It is obvious that this conclusion is consistent with the results given above in this section for the case i1 = i2.

As we mentioned before, the technical peculiarities of the Milstein method 77] may result to the iterated series of products of standard Gaussian random variables (in contradiction to multiple series as in Theorem 1.1). In the case of simplest stochastic integral of second multiplicity this problem was avoided as we saw above. However, the situation is not the same for the simplest stochastic integrals of third multiplicity.

Let us denote

= T

■M

J

= (il...ifc ) (Ai...Afc )T,t

dw(il)... dw

(ik) tk '

where A/ = 1 if i = 1,..., m and A/ = 0 if i = 0, l = 1,..., k, wT^ = f-^ for

i = 1,..., m and w-0) = t.

Let us consider the expansion of iterated Stratonovich stochastic integral of third multiplicity obtained in [78]-[80], [88] by the Milstein method [77

j*(hhi3) _ J_ t*(H) T*(0«2«3) 1 (111)A,0 ~~ ^ (1)A,0 (011)A,0

_l!/7 7*(*2«3) I J_7 J*(h) _ A T*(«2) R I

'i'0 (11)A,0 "T" 2TJ- n (1)A,0'j(1)A,0 z-1'7(1)A,0-DîIÎ3"1"

+ A J

= (is)

1

(1)A,0 I 2

A- • — O- • 1 + A3/2D- ■ •

(6.34)

where

T*(0i2«3) _ 1 T*(«2) J*(h) _ }_ A T*(«3) 7 I

(011)A,0 ~~ g'J(l)A,0'J(l)A,0 7rUiJ

2D 7*(«2) , 1 AT, T*^ , , 1 \ 2

i ajd __An T{2) -L—Ah rwl _i_ A^C1 -l1AzA

00

r=1

O

°«2«3

œœ

¿E E

r

/=1 r=1(r=/)

r2 l2

(ra^2,r a«3,/ + lb«2,r b«3,/)

1 œ œ 1

r=1

Di^iz — 2A3/2 ^ ^ 53 ^ ( ai2,1 (ai3,l+rbii,r +

/=1 r=1

/+r) +

œ /-1

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+ ~ . o,0 53 53 M ^ {ah,rbi3,i-r + aisj-rbi^r)

2A3/2

/=1 r=1

(a«l,r,l —r — r) I +

TO TO

+ ~ » o /n £ £ I I «»2,/ - a>h,rbi3,r-l) +

2A3/2

l=1 r=l+1

+ (aii,ra«3,r—l + bii,rb«3,r—l) I •

From the expansion (6.34) and expansion of the stochastic integral Jol^o we can conclude that they include iterated (double) series. Moreover, for approximation of the stochastic integral jl^Ao in the works [78] (pp. 438-439), 79] (Sect. 5.8, pp. 202-204), [80] (pp. 82-84), [88] (pp. 263-264) it is proposed to put upper limits of summation by equal q (on the base of the Wong-Zakai approximation [68]-[70] but without rigorous proof; also see discussions in Sect. 2.20, 2.21).

For example, the value A1i2i3 is approximated in [78]-[80], [88] by the double sums of the form

D1ll2i3 = ~2A3/2 E E M (ai3,i+rK,r - ah,rbi3,i+r) +

l=1 r=1

l+r) +

q l —1

+ ~ . o /» £ £ 11 a-ioj {ah,rbi3,i-r + ah,i-rbiur)

2A3/2

l=1 r=1

bi2,l (aii,ra«3,l—r bii,rb«3,l—r) I +

7t

2A3/2

q 2q

^ ^ ^ ^ M a«2,l (a«3,r—lbii,r — aii,rb«3,r—l) + l=1 r=l+1 V

+ b«2,l (aii,ra«3,r—l + bii,rb«3,r—l) I •

We can avoid the mentioned problem (iterated application of the operation of limit transition) using the method based on Theorems 1.1, 2.1-2.9, 2.30, 2.32-2.35, 2.40.

From the other hand, if we prove that the members of the expansion (6.34) coincide with the members of its analogue obtained using Theorem 1.1, then we can replace the iterated series in (6.34) by the multiple series (see Theorems 1.1, 2.1-2.9, 2.30, 2.32-2.35, 2.40) as was made formally in [78]-[80], [88]. However, it requires the separate argumentation.

6.3 Usage of Integral Sums for Approximation of Iterated Ito Stochastic Integrals

It should be noted that there is an approach to the mean-square approximation of iterated stochastic integrals based on multiple integral sums (see, for example, [77], [87], [89], [150]). This method implies the partitioning of the in-

tegration interval [t,T] of the iterated stochastic integral under consideration; this interval is the integration step of the numerical methods used to solve Ito SDEs (see Chapter 4); therefore, it is already fairly small and does not need to be partitioned. Computational experiments [1] (also see below in this section) show that the application of the method [77], [87], [89], [150] to stochastic in-

tegrals with multiplicities k > 2 leads to unacceptably high computational cost and accumulation of computation errors.

As we noted in the introduction to this book, considering the modern state of question on the approximation of iterated stochastic integrals, the method analyzed in this section is hardly important for practice. However, we will consider this method in order to get the overall view. In this section, we will analyze one of the simplest modifications of the mentioned method.

Let the functions (t), l = 1,..., k satisfy the Lipschitz condition at the interval [t, T] with constants Cl

(T1) — ^l(T2)| < Cl|T1 — T2| for all T1,T2 G [t,T]. (6.35)

Then, according to Lemma 1.1 (see Sect. 1.1.3), the following equality is correct

N—1 j2 — 1 k

J[^k)]T, = l.i.m. £ ...¿n^l(j^wj w.p. 1, where notations are the same as in (1.12).

Let us consider the following approximation

N — 1 j2-1 k

j №(k)iN, = E ■ • . E n^' (j >AwT:;) (6-36)

jk=0 ji=0 '=1

of the iterated Ito stochastic integral J[7k)]i> The relation (6.36) can be rewritten as

N—1 j2 — 1 k

•w'% = E EII V^Mi-xWi?, (6.37) jk =0 ji=0 '=1

where u^ =f (w^ — /y7Atj, i = 1,... ,m are independent standard

Gaussian random variables for various i or j, u^ = y7Arj. Assume that

Tj = t + j A, j = 0,1,...,N, tn = T, A > 0. (6.38)

Then

N—1 j2 1 k

J[^(k)lNf = Ak/2 E .. ^ n (t + j'A)uj;;), (6.39)

jk=0 ji=0 '=1

where uf S (w«(j+1)i - w« A)/VS. i = l.....m, uf = VS.

Lemma 6.1. Suppose that the functions fa(t), l = 1,...k satisfy the Lipschitz condition (16.35) and {Tj}N=0 is a partition of the interval [t,T], which satisfies the condition (6.38). Then for a sufficiently small value T — t there exists a constant Hk < to such that

N\21 ^HfaT-tfa

N

m 4 ( J[fa(k)]T,t — J[#k)]Nt) > <

Proof. It is easy to see that in the case of a sufficiently small value T — t there exists a constant Lk such that

m { (J№(k)]r,t — J[<A(t»]N()2} < Lkm { (J[V/2)]„ — J[fa(2)]N/ 2

where

3

J [fa(2)]T,t — J [fa(2)]Nt = E j,

j=1

N-1 j+1

sN = E J Mh)J ^i(ti}dwt(;i)dwi;2),

j1=0 T, T,

N-1 j+1 ji-1 j+1

S2N = £ / (^2} - ^2 (j }} dw£2'£ / ^i(ti}dwi i1}

' _A ^ ' _A ^

j1=0 T. j2=0 ,

'31 32

N-1 j1-1 Tj2+1

sN = £ «T*^'E / (wo - }} dwt 11'.

j1=0 j2=0 j

'32

Therefore, according to the Minkowski inequality, we have

(m { (J[«/> (2>]T,( - J^'IN,)^1/2 < j (m {(SN)2})1 /2.

Using standard moment properties of stochastic integrals (see (1.26 (1.27}}, let us estimate the values m { (SN)2} , j = 1, 2,3.

Let us consider four cases. Case 1. i1,i2 = 0 :

M f ioN\2\ ^ ^(rr ^ .J.2(„\.J.2,

{(Sf)2} < "2 (r - t) max

2 sG[i,T ]

{ , N) 2^ /

m

A 2

) 6 s€[t,i J

A 2

J 6 sG[t,T J

Case 2. ¿1 = 0, ¿2 = 0 :

2

m i (Sf )2} < —(T - t)3 (C2)2 max $ (s),

n _______,2,

M i (Sf)2} < —(T - tf (C\f max V'K-s).

J 3 sG[t,i J

Case 3. i2 = 0, i1 = 0 :

m

m m

^ J 3 sG[t,T]

J 3 sG[i,T J

A 0

(Sf)2} < —(T - f)3(C.'!)2 max

J 8 seItj l

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Case 4. i1 = i2 =

m

m m

0:

4 ^ ' sG[i,T J A2

J 4 se [t,T J

J 4 se[t,T J

4 A

According to the obtained estimates, we have

m

J

T,t

- J

N

where Hk < to. Lemma 6.1 is proved.

It is easy to check that the following relation is correct

m

I

(¿1«2)

-I,

(ii»2)N

(00)T,t J(00)T,t

2 N

(6.40)

where i1, i2 = 1,..., m and I(o0)2rN is the approximation of the iterated stochastic integral I(00)^ (see (6.4)) obtained according to the formula (6.39).

Finally, we will demonstrate that the method based on generalized multiple Fourier series (Theorem 1.1) is signficiantly better, than the method based on multiple integral sums in the sense of computational costs on modeling of iterated stochastic integrals.

Let us consider the approximations of iterated Ito stochastic integrals obtained using the method based on multiple integral sums

q—1

IWq -x/AVr

(6.41)

j =0

2

Table 6.1: Values Tsum/Tpoi.

T — t 2-5 2-6 2-7

Tsum/Tpol 8.67 23.25 55.86

q—1 /j—1 \

Ck = A£ £ii " if, (6.42)

j =0 \ «=0 J

where

are independent standard Gaussian random variables, A = (T — t)/q, 1(oo)t t, /(o)t t are approximations of the iterated Ito stochastic integrals /(o0)T t (see

(eD, /(¡^ = 41' — ft11. '

Let us choose the number q (see (6.41), (6.42)) from the condition

Let us implement 200 independent numerical modelings of the collection of iterated Ito stochastic integrals /(o0)T t, /(ojy t using the formulas (16.41), (16.42) for T — t = 2—j, j = 5, 6, 7. We denote by Tsum the computer time which is necessary for performing this task.

Let us repeat the above experiment for the case when the approximations of iterated Ito stochastic integrals /(o0)T t, /(ojy t are defined by (15.135), (15.136) and the number q is chosen from the condition (5.127) (method based on Theorem 1.1, the case of Legendre polynomials). Let Tpoi be the computer time which is necessary for performing this task.

Considering the results from Table 6.1, we come to conclusion that the method based on multiple integral sums even when T— t = 2—7 is more than 50 times worse in terms of computer time for modeling the collection of iterated Ito stochastic integrals /00^, /0^, than the method based on generalized multiple Fourier series.

It is not difficult to see that this effect will be more essential if we consider iterated stochastic integrals of multiplicities 3, 4,... or choose value T—t smaller

than 2—7.

6.4 Iterated Ito Stochastic Integrals as Solutions of Systems of Linear Ito SDEs

Milstein G.N. [77] (also see [94])) proposed an approach to numerical modeling of iterated Ito stochastic integrals based on their representation in the form of systems of linear Ito SDEs. Let us consider this approach using the following set of iterated Ito stochastic integrals

S ¿2

J(¿1) =

df

(¿1)

I

(¿1*2)

df^dff,

1(0)s,t _ J "Hi 1 *(00)s,t-J J "Hi vxt2 ' (6.43)

t t t

where i1,i2 = 1,...,m, 0 < t < s < T, fs(i) (i = 1,...,m) are independent standard Wiener processes.

Obviously, we have the following representation

/j(¿1) \

'0)s,t (1 o\ /f(*1)\

1 0 fs (6.44)

/J(i1) \ I J(0)s,t

d

J (i1i2) I V(00)s,t/

00 10

J(i1i2)

VJ(00)s,V

df(i2) +

00

d

^(¿2)

It is well known [77], [79] that the solution of system (6.44) has the following integral form

(I(n) \

J(0)st

¿1) L(0)s,t

(¿1«2)

1 0,

I(«1«2) / V(00)s,t/

s 0 0^

=/'

t

f(®2)— f(®2)

1 ^ d if»'1 r 0 Of d iff'

(6.45)

where eA is a matrix exponent

eA =f

Ak

S k\ '

k=0

A is a square matrix, and A0 =f I is a unity matrix.

Numerical modeling of the right-hand side of (6.45) is unlikely simpler task than the jointly numerical modeling of the collection of stochastic integrals (6.43). We have to perform numerical modeling of (6.43) within the frames of the considered approach by numerical integration of the system of linear Ito SDEs (16.44). This procedure can be realized using the Euler (Euler-Maruyama)

s

method [77]. Note that the expressions of more accurate numerical methods for the system (6.44) (see Chapter 4) contain the iterated Ito stochastic integrals (6.43) and therefore they useless in our situation.

Let {Tj}N=0 be the partition of [t, s] such that

Tj = t + jA, j = 0,1,..., N, tn = s.

Let us consider the Euler method for the system of linear Ito SDEs (6.44)

/ yp+i ^ (yP"} ( f ^

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+

Vy^'Af^y

VyP+i2)/

U!1!2)/

y0!,) = 0, y0!1,2) = 0, (6.46)

where

y(ii) def y(ii)

j t; J p '

y

(¿1«2) = y(«1«2)

= yp

are approximations of the iterated Ito stochastic integrals T( tained using the numerical scheme (16.461). AfT;' = fTp+1 — fTp', i = 1,... ,m. Iterating the expression (6.46), we have

(ii)

1) T (i1i2) ob_ (0)T;,t' T(00)T;,t ob

yN

N- i

N —1 q—1

£ Af''1», y^!2) = ££Af<'f,

1=0 q=0 1=0

(6.47)

where ^ =f 0.

Obviously, the formulas (6.47) are formulas for approximations of the iterated Ito stochastic integrals (6.43) obtained using the method based on multiple integral sums (see (6.41), (6.42)).

Consequently, the efficiency of methods for the approximation of iterated Ito stochastic integrals based on multiple integral sums and numerical integration of systems of linear Ito SDEs on the base of the Euler method turns out to be equivalent.

;

6.5 Combined Method of the Mean-Square Approximation of Iterated Ito Stochastic Integrals

This section is written of the base of the work [151] (also see [16]) and devoted

to the combined method of approximation of iterated Ito stochastic integrals based on Theorem 1.1 and the method of multiple integral sums (see Sect. 6.3).

The combined method of approximation of iterated Ito stochastic integrals provides a possibility to minimize significantly the total number of the Fourier-Legendre coefficients which are necessary for the approximation of iterated Ito stochastic integrals. However, in this connection the computational costs for approximation of the mentioned stochastic integrals are become bigger.

Using the additive property of the Ito stochastic integral, we have

N-1

4 W = w.p.1, (6.48)

k=0

N —1

1<iîT,( _ £ (^L.Tk — A3/2<.)) w. p. 1, (6.49)

k=0

N —1 k-1 N —1

'<!0& _A££z0;;)c0¥ + £/S^ w.p. 1, (6.50)

N-1 k-1 1-1

T<;l;2;3) _ V^ <;i)Z<;2)Z<;3) I

J<000)T,t _ A Z^ Z^ Z^ Z0,q Z0,1 Z0,k +

k=0 1=0 q=0

N-1 k-1 AT-1

r<;i;2) z<i3) + Z<;i) I<i2i3) 1 r<;i;2;3)

1 <00)r;+i,T; Z0,k + Z0,1 J<00)rfc+i,r^ k=0 1=0 ' k=0

N -1 k-1 N -1

+ vS£ £ (i&t^f + J + E P- <6-51)

where stochastic integrals

I (il) I (il) I (il i2 ) I (i 1 i2 i3 )

J(0)T,t' J(1)T,i' J(00)T,t' J(000)T,t

have the form (5.3), i1,... = 1,... ,m, T - t = N A, Tk = t + kA,

Tk+i

Ji) def J_ f {i)

~ Va J s '

Tk

k = 0,1,..., N — 1, the sum with respect to the empty set is equal to zero. Substituting the relation

I{H) - A3/2 (t{n) + 1 w n 1

into (6.49), where (0V, dV

are independent standard Gaussian random variables, we get

w-1

= -A3/s £ ( {{+k) +w-p-L (6'52) k=0

Consider approximations of the following iterated Ito stochastic integrals using the method based on multiple Fourier-Legendre series (Theorem 1.1)

J('1'2) J(¿2'3) J(¿1'2'3)

(00)Tfc+1,Tk ' (00)Tfc+1,Tk ' (000)Tfc+1,Tk .

As a result, we get

N-1 k-1 N-1

_aEE w +E . (6.53)

k=0 1=0 k=0

N-1 k-1 1-1

T(:i:2:3)N:qi:q2 _ V"^ V^ z(:i)Z(^V(:3) +

J(000)T:t _ ^ Z0,q Z0,1 Z0,k +

k=0 1=0 q=0

N-1 fc-1 AT-1

__l_ ./XV V^ ( T^hh)qi Ah) I Ah) j{hh)qi \ , V^ Ahhh)q2 (n rA\ + \ ^ ^ ^(00)^+1 :Tl Z0,k + Z0,1 T(00)rfc+1:T^ + T(000)rfc+1:Tfc ' (D'04) k=0 1=0 k=0

where we suppose that the approximations

j (:i:2)q j(:i:2)qi ^hhh^

(00)rfc+1 :Tfc ' (00)rfc+i:Tfc ' (000)rfc+i:Tfc

are obtained using Theorem 1.1 (the case of Legendre polynomials).

In particular, when N _ 2, the formulas (6.48), (6.52)-(6.54) will look as follows

/<;;>,, = VÂ (<<« + $>) w. P. i, (6.55)

c, = -AV2 ({<*? + §<fi' + {<&+) w. p. 1, (6.56)

r(:i:2)2:q _ a ( Ah)Ah ) , T(:i :2)q , /-(:i:2)q ^ fn r7\

T(00)T:t _ ^ lS0:0 Z0:1 + T(00)T:^ + T(00)^:T ^ ' (6.57)

(00)T,t - ^ \^0,0 W T" (00)tl,To (00)T2,T1^

Ahhh)2,qi,q2 _ rr (r{hh)qi Ah) , Ah) Ahh)qi \ ,

J(000)T,t A y (00)T1,ToZ0,i + S0,0 J(00)T2,T^ +

+ I((00m3)q2 + T((^00'3)q2, (6.58)

(000)T1,To (000)T2,T 1' V )

where A = (T — t)/2, Tk = t + kA, k = 0,1, 2.

Note that if N =1, then (6.48), (6.52)-(6.54) are the formulas for numerical modeling of the mentioned stochastic integrals using the method based on Theorem 1.1.

Further, we will demonstrate that modeling of the iterated Ito stochastic integrals

I

<;i)

<0)T,t'

using the formulas (I6.55I)-(I6

I

<;i)

I

<;i;2)

I

<;i;2;3)

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(1)T,t' J(00)T,t' J(000)T,t

results in abrupt decrease of the total number of Fourier-Legendre coefficients, which are necessary for approximation of these stochastic integrals using the method based on Theorem 1.1.

From the other hand, the formulas (6.57), (6.58) include two approximations of iterated Ito stochastic integrals of second and third multiplicity, and each one of them should be obtained using the method based on Theorem 1.1. Obviously, this leads to an increase in computational costs for the approximation.

Let us calculate the mean-square approximation errors for the formulas (6.53), (6.54). We have

en _f m

<;i;2)

-I,

<;i ;2)N,q

<00)T,t J<00)T,t

N 1

m

k=0

<;i;2)

I

<;i;2)q

<00)rfc+i,rfc <00)rfc+i,rfc

_ N

A2 / 1

2 \ 2

E

1=1

1

4/2 — 1

(T-ty-

2 N

2

E

1=1

1

4/2- 1

(6.59)

E9i,?2 _f m j ( i^ïO _ i<;i;2;3)N,qi,q2

N

m

N 1 k 1

vk=0 V 1=0 v v

L <000)T,t J <000)T,t

_ I<;i;2)qi

<00)r; + i,T; I<00)ri + i,rj +

i z<;iH i<;2;3) _ ifo^gi \ \ , /<;i«2;3) _ i<«i;2;3)q2

+ Z0,1 I I<00)rfc+i,Tfc I<00)rfc+i,r7 + I<000)rfc+i,Tfc 1 <000)rfc+i,Tfc

JV-1 r / fe-1 , ,

Vm (v aV(. ('

k=0 1=0 v v

<;3W I<;i;2) _ I<;i;2)qi \ ,

0,k I I<00)ri+i,ri I<00)ri+i,rj +

<;i) <;2;3)

+C0ÏM I

i

<;2;3)qi

0,1 I <00)rfc+i,Tfc <00)rfc+i,Tfc

i i<;i;2;3) _ /<ù;2;3)q2

+ I<000)rfc+i,Tfc I<000)rfc+i,Tfc

g (aJ fe g (t,

k=0 V I \ 1=0 v

where

(«1«2) _ j('1'2)q1 +

0,k Z^ I T(00)T|+1,T| J(00)T|+1,T| +

+ AM J ( ( T('2'3) _ T('2'3)q1 \ k-i ZI + H('1'2'3)

+ AM ^ 1 I J(00)Tk+1,Tk J(00)Tk+1,Tk j Z0,1 J f + Hk,q2

1=0

N— i / k—i ( , v 2'

e ae m T{00')t,T, — & }+

k=0 1=0

+ k Am^ i T ('2'3) _ T ('2'3)q1 \ I + H ('1'2'3)

+ kaivi< ( T(00)Tk+1,Tk T(00)Tk+1,Tk ) f + Hk,q2

N— i / r / \ 2'

E 2kam fe^ — I^J K HkW*3) k=0

_ 3N(N — 1) /1 ^ 1 ^ , ^ „(ili2i3) \ 1=i / k=0

N-i

+ E Hk;qi2!3), (6.60)

k=0

H ('1'2'3) = mJ | T (*1*2'3) _ J ('1«2«3)q2

Hk,q2 = m i J(000)Tk+1,Tk J(000)Tk+1,Tk

Moreover, we suppose that ii = i2 in (6.59) and not all indices ii,i2,i3 in (6.60) are equal. Otherwise there are simple relationships for modeling the

mtegrflls T('1'2) T('1'2'3) integrals J(00)T,t, J(000)T,t

2

(000 )T,t Q

where

r(iliiii) = }_^ _ ^3/2 QAh)

himA = -(T-ty<<[(Cr -¿Co" w.p. 1,

T

Ah) _ 1 f jf(n)

Co ~ vr^t, J s t

is a standard Gaussian random variable.

For definiteness, assume that i1,i2,i3 are pairwise different in (6.60) (other cases are represented by (I5.55l)-(I5T581)). Then from Theorem 1.3 we have

(1 ® c 2 \

j1, j2, j3 =0

where

_ v/(2j1 + 1)(2j2 + 1)(2j3 + 1)a3/2^

( AJA- — 8 ( AJA- ■

+

Cj = J Pj3 (z) J Pj2 y Pj1 1 1 1

and Pi(x) (i = 0,1, 2,...) is the Legendre polynomial. Substituting (6.61) into (6.60), we obtain

^ = 2(T " i)3 (]V " A^) (2 " fj 4/^1

1 6 _ -64-i (6"62)

\ j1 ,j2 ,j3=0 /

Note that for N =1 the formulas (6.59), (6.62) pass into the corresponding formulas for the mean-square approximation errors of the iterated Ito stochastic integrals /((00i)2T t, 1(000)1^ (see Theorem 1.3).

Let us consider modeling the integrals /(0)T t, /((00i)2T t. To do it we can use the relations (6.48), (6.53). At that, the mean-square approximation error for the integral /((00)r t is defined by the formula (6.59) for the case of Legendre polynomials. Let us calculate the value E^ for various N and q

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E32 « 0.0167(T — t)2, E3 « 0.0179(T — t)2, (6.63)

1

y

z

Table 6.2: T-t = 0.1.

N q qi q2 M

1 13 - 1 21

2 6 0 0 7

3 4 0 0 5

Table 6.3: T — t. = 0.05.

N Q Qi q-2 M

1 50 - 2 77

2 25 2 0 26

3 17 1 0 18

E6 « 0.0192(T - t)2. (6.64)

Note that the combined method (see (6.63)) requires calculation of a significantly smaller number of the Fourier-Legendre coefficients than the method based on Theorem 1.1 (see (6.64)).

Assume that the mean-square approximation error of the iterated Ito stochastic integrals I^yl t, I(ooo)t\ equals to (T — t)4.

In Tables 6.2-6.4 we can see the values N, q, qi, q2, which satisfy the system of inequalities

EN < (T — t)4

(6.65)

En'q2 < (T — t)4

as well as the total number M of the Fourier-Legendre coefficients, which are ne-sessary for approximation of the iterated Ito stochastic integrals /(oO^T I((i0(0)ir)t when T — t = 0.1, 0.05,0.02 (the numbers q, qi, q2 were taken in such a manner that the number M was the smallest one).

From Tables 6.2-6.4 it is clear that the combined method with the small N

Table 6.4: T t = 0.02.

N Q Qi Q2 M

1 312 - 6 655

2 156 4 2 183

3 104 6 0 105

(N = 2) provides a possibility to decrease significantly the total number of the Fourier-Legendre coefficients, which are necessary for the approximation of the iterated Ito stochastic integrals /(¿O^T t, 1(ooo)t\ in comparison with the method based on Theorem 1.1 (N = 1). However, as we noted before, as a result the computational costs for the approximation are increased. The approximation accuracy of iterated Ito stochastic integrals for the combined method and the method based on Theorem 1.1 was taken (T — t)4.

6.6 Representation of Iterated Ito Stochastic Integrals of Multiplicity k with Respect to the Scalar Standard Wiener Process Based on Hermite Polynomials

In Chapters 1,2, and 5 we analyzed the general theory of the approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. However, in some narrow special cases we can get exact expressions for iterated Ito and Stratonovich stochastic integrals in the form of polynomials of finite degrees from one standard Gaussian random variable. This and next sections will be devoted to this question. The results described in them can be found, for example, in [103] (also see [77], [79]).

Let us consider the set of polynomials Hn(x,y), n = 0,1,... defined by

Hn(x,y) = (J^eax-0?y/2

' a=0

It is well known that polynomials Hn(x,y) are connected with the Hermite polynomials

dn . . K[x)=(-1)ne*2d^ (e"s)

by the formula

'Vr/2L t x \ ..n/2rr ( x\

.2/ ""Vv/%; "

where Hn(x) is the Hermite polynomial (1.263). Using the recurrent formulas

dhn

dz

■(z) = 2nhn—i(z), n = 1, 2,...,

hn(z) _ 2zhn - 1(z) - 2(n - 1)hn-2(z), n _ 2, 3,..., it is easy to get the following recurrent relations for polynomials Hn(x,y)

dHn

= nHn-i(x,y), n = 1,2,..., (6.66)

dH" n nx

=—Hn{x,y) -—Hn_i{x,y), n = 1,2,..., (6.67)

= n = 2,3,... (6.68)

From (666) - (668) it follows that

dHn, , 1 d2Hn , ,

+ 2-d^{x'v) = n = 2'3"" <6'69»

Using the Ito formula, we have

t t

Hn(f„t)-Hn(0,0) = f^{f„sW. + j ffiv.+

00

(6.70)

w. p. 1, where t £ [0, T] and /t is a scalar standard Wiener process.

Note that Hn(0, 0) = 0, n = 2,3,... Then from (6.69) and (6.70) we get

Hn(/t ,t) = y nHn—1(/s,s)dfs, w.p.1 (n = 2, 3,...). (6.71) 0

Furthermore, by induction it is easy to get the following relation (see (6.71)) t t2

j ...j<IJ)....<IJ), w.p. 1 (n = 1,2,...). (6.72)

00

Let us consider one generalization of the formula (6.72) [103

t t2

j«-» s I ■Wr.)...I mm, ■ ■ ■ m = g"(y(t)) w. P. i, (6.73)

n!

00

t

where t e [0,T], n = 1, 2,..., and

t t

xt d= J ^(s)d/s, y(t) = J ^2(s)ds,

0 0

where ^(s) is a continuous nonrandom function at the interval [0,T].

To prove the equality (6.73), we apply the Ito formula. Using the Ito formula and (6.66), (6.69), we obtain w. p. 1 (Hn(0, 0) = 0, n = 2,3,...)

t

Hn(xuy(t))-Hn(0,0) = J l/j(s)^(xs,y(s))dfs+

0

t 2 + J + d.s =

0

t

= J ■i/>{s)-S^{xs,y{s))dfs+ 0

t 2

0

t

= J ■i/>{s)-S^{xs,y{s))dfs+ 0

t 2

0

t t

/d J! i*

■&(s)-7^{xs,y{s))dfs = / i/j{s)nHn_i{xs,y{s))dfs =

0 0

t s

= J ^(s)y ^(t )n(n — 1)Hn—2(xT ,v(t ))d/T df = ... 0 0

t t2

... = n! J ^(tn) ... | ^(t1)d/t1... dftn. (6.74)

0 0

From (6.74) we get (6.73).

It is easy to check that first eight formulas from the set (6.73) have the following form

^ = Y\Xu

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= ^ (to)3 - ^ty(t)), J<4) = i ((a,)4 " GM2y(t) + 3y\t)) , jf = i (to)5 " 10to)33/(i) + 15xty\t)) , jf = i (to)6 - 15(xt)4y(t) + 45(xt)2y2(t) - 15y3(t)) ,

jf} = i (to)7 - 21to)53/(i) + 105to)V(*) - 105xty3(t)) ,

48) = ^ (to)8 - 28(xtfy(t) + 210(xt)4y2(t) - 420(xt)Y(t) + 105y4(i))

w. p. 1. As follows from the results of Sect. 1.1.6, for the case fa(r),..., (t) = ) and i1 = ... = = 1,..., m the formula (1.54) transforms into (6.73).

6.7 Representation of Iterated Stratonovich Stochastic Integrals of Multiplicity k with Respect to the Scalar Standard Wiener Process

Let us prove the following relation for iterated Stratonovich stochastic integrals (see, for example, [79])

* ¿2

(AT

n!

dftl • • • dftn = —— w. p. 1, (6.75)

0 0

where t G [0,T].

At first, we will consider the case n = 2. Using Theorem 2.12, we obtain

* t * ¿2 t t2 t

J J dftldft2 = J J dftldft2 + \Jdti w. p. 1. (6.76) 0 0 0 0 0

From the relation (6.72) for n = 2 it follows that

t t2 / - ^ 2

dftAft3 = -^fr " \ [ dh w. p. 1. (6.77)

0 0

Substituting (6.77) into (6.76), we have

st *t2 2

dftldfh = -^jp w. p. 1.

0 0

So, the formula (6.75) is correct for n = 2. Using the induction assumption and (2.4), we obtain

*t * t2 *t t t 1

0 0 0 0 0 w. p. 1. From the other hand, using the Ito formula, we get

(/t)B+1 - ^ (/t)"_1 [{-^dfT w.p. 1. (6.79)

(n + 1)! J 2(n — 1)! J n! 0 0

From (6.78) and (6.79) we obtain (6.75). It is easy to see that the formula (6.75) admits the following generalization

* t * t2 / t \ n

J Wn)... I Wi )dftl...dftn = ±ljifj(T)dfT\ w. p. 1, (6.80) 0 0 \0 /

where t e [0,T] and ^(r) is a continuous nonrandom function at the interval [0,T ].

To prove the equality (6.80), first consider the case n = 2. Using Theorem 2.12, we get

*t * t2 t t2 t

J m)J m)dfhdft2 = f wo) JmwtM, + \ J^ds w. P. 1. 0 0 0 0 0

(6.81)

From the relation (6.73) for n = 2 it follows that

t2 / t \ 2 t 2 1 1

faU) J m)dftldft2 = ^ I J Ms)dfs I --J f(s)ds w. p. 1. (6.82) 0 0 0 0

Substituting (6.82) into (6.81), we obtain

*t * t2 / t \ 2 J m)J m)dftldft2 = 11Ji/j{s)dfs J w. P. i. 0 0 0

Thus the formula (6.80) is proved for n = 2. Applying the induction assumption and (2.4), we have

* t * t2 * t / T \n

J '0(Wi) ...J m)dftl ■ ■ ■ dftn+1 = J '0(r)i I J fas)dfs J dfr = 0 0 0 \0 /

t / T \ n t / T \ n-1

= J Hr)^ J mdfs dfT+l- J J mdfs dr 0 0 0 0

(6.83)

w. p. 1. Applying the Ito formula, we obtain

1 (} V+ } 1 fr V-1

+ I (jiKs)df)j dfr W. p. 1. (6.84)

From (6.83) and (6.84) we get (6.80).

6.8 Weak Approximation of Iterated Ito Stochastic Integrals of Multiplicity 1 to 4

In the previous chapters of the book and previous sections of this chapter we analyzed in detail the methods of so-called strong or mean-square approximation of iterated stochastic integrals. For numerical integration of Ito SDEs

the so-called weak approximations of iterated Ito stochastic integrals from the Taylor-Ito expansions (see Chapter 4) are also interesting.

Let F, p) be a complete probability space, let {Ft,t e [0,T]} be a non-decreasing right-continuous family of a-algebras of F, and let ft be a standard m-dimensional Wiener process, which is Ft-measurable for all t e [0,T]. We suppose that the components ft(i) (i = 1,... ,m) of this process are independent.

Let us consider an Ito SDE in the integral form

t

+ y a(xT, r )dr + I B (x7 0

t t

xt = x0 + J a(xT,r)dr + J B(xT,r)dfT, x0 = x(0,w), (6.85)

where xt is some n-dimensional stochastic process satisfying to the Ito SDE (6.85), the nonrandom functions a : Rn x [0,T] ^ Rn, B : Rn x [0,T] ^ Rnxm guarantee the existence and uniqueness up to stochastic equivalence of a solution of (16.851) [95], x0 is an n-dimensional random variable, which is F0-measurable and m{|x0|2} < to, x0 and ft — f0 are independent for t > 0.

Let us consider the iterated Ito stochastic integrals from the classical Taylor-Ito expansion (see Chapter 4)

s T2

= / ■../dwti'' (k > 1),

tt

where wTi) = f((i) for i = 1,..., m and w[0) = r, i/ = 0 if A/ = 0 and i/ = 1,..., m if A/ = 1 (/ = 1,..., k). Moreover, let

Mk ^(A1,...,Ak) : A/ = 0 or 1, l = 1,...,^.

Weak approximations of iterated Ito stochastic stochastic integrals are formed or selected from the specific moment conditions [77], [79], [80], [87], 88] (see below) and they are significantly simpler than their mean-square ana-

logues. However, weak approximations are focused on the numerical solution of other problems [77], [79], [80], [87], [88] connected with Ito SDEs in comparison

with mean-square approximations.

We will say that the set of weak approximations

j(«l---ifc ) (Ai...Afc )s,t

of the iterated Ito stochastic integrals

J

(Ai...Afc )s,t

from the Taylor-Itô expansion (14.22) has the order r, if [77], [79] for t G [t0,T] and r G n there exists a constant K G (0, to) such that the condition

m^n J

<7=1

(A1g)--- Akg))t,to

Jô

< =1

(iig)--- ^ )

(Alg)--- Ak9g )t,t0

F

to

< K(t-to)r+1 w. p. 1 (6.86)

is satisfied for all (A^ ...AjJ) G Mjg, i1g),..., ijg = 0,1,...,m, k< < r, g = 1,...,l, l = 1, 2,..., 2r + 1.

If we talk about the unified Taylor-Ito expansion (14.27), then we will say that the set of weak approximations

ô

(il-.-ifc)

Z1...Z.

1---lfcs,i

of the iterated Ito stochastic integrals

t2

I

(H---ifc )

h---hs,t

= /(t - tk )'

(t - ti)11 df^ ..f (ii,...,ik = 1,...,m)

;(ifc)

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has the order r, if for t G [t0, T] and r G n there exists a constant K G (0, to) such that the condition

m m(t

< =1

Jg!

(g) ,(g)

'1 ---

jj (t ~ to)*' ff...

B t,to

< =1

< K(t - to)r+1 w. p. 1

Jg!

(b) ,(b)

1(g) 1 '1 ... 'k

g t,to

F

to

<

(6.87)

is satisfied for all , j, l(g),..., ljg^ G Aqg, i 1<),..., ijg) = 1,..., m, qg < r,

g = 1,..., l, l = 1, 2,..., 2r + 1, where

Aq = j(k,j, ¿1,... ,lk) : k + j + ]T] lp = q; k,j,..., lj = 0,1,... j .

The theory of weak approximations of iterated Ito stochastic integrals is not so rich as the theory of mean-square approximations. On the one hand, it is

s

k

connected with the sufficiency for practical needs of already found approximations [77], [79], [87], and on the other hand, it is connected with the complexity of their formation owing to the necessity to satisfy a lot of moment conditions.

Let us consider the basic results in this area.

In [79] (also see [77]) the authors found the weak approximations with the orders r = 1, 2 when m,n > 1 as well as with the order r = 3 when m = 1, n > 1 for iterated Ito stochastic integrals

J (¿1 ---ife)

(Ai...Afc )t,to'

Recall that n is a dimension of the Ito process xt, which is a solution of the Ito SDE (6.85) and m is a dimension of the Wiener process in (6.85).

Further, we will consider the mentioned weak approximations as well as weak approximations with the order r = 4 when m = 1, n > 1 [152] (2000) for iterated Ito stochastic integrals

I

(¿i ...¿k)

Zi...Z

kt,to

In order to shorten the record let us write

m

(¿1s)- ¿kg) (Alg)... A kg >o+A,io

m d=f m'

n

.5=1

J

As)

¿(a)

ks

)

(AlS) - Aki )

(6.88)

where A e [0,T - to], (a^ ... A£gS0 G , kg - r, g = 1,---,1-

Further in this section, equalities and inequalities for conditional expectations are understood w. p. 1. As before, 1A means the indicator of the set A.

Let us consider the exact values of conditional expectations (6.88) calculated

in

77], [79] and necessary to form weak approximations

J

(¿1...¿k)

(Ai...Ak )to+A,to of the orders r = 1 , 2 when m, n > 1

m'{ jj} =^=¿2} ,

M/ i jin) j(0i2) \ _ M/ r jdi) 7(i30) 1 _ 1 V (!) (01) J ~ I (!) (10) J ~ 2

1

= -AHu

{¿^¿2}:

u1 { ]{nn) l{viH)\ --A2lr -il IVI \'J(ii) J(ii) j ~ 2 A'«<=«a-L

{¿1=¿з}1{¿2 = ¿4};

(6.89)

(6.90)

(6.91)

1

A2 when ¿1 = ... = ¿4

m'{ JiJJ4'} ^a2/2

when = i4, ii = ¿2 = ¿4

or ¿3 = ¿4, ¿1 = ¿4, ¿2 = ¿3

0 otherwise

(6.92)

3A2

when ¿1 = ... = ¿4

r{JJJJ;)} A A2 : if ^gf1--^ *heumbe._, (6.93)

0

two pairs of identical numbers'

otherwise

M \J(1°) J(01) j

= -A3li,-

6

{il = i2} '

m'{ J((1°°)J((1°°)} = m'{ J™ J,

(0il) (0i2)

»2)\ _ ¿A 3-1 (01) "(01) J ~ J-{«i=«2}'

(6.94)

(6.95)

(01) (1) (1) (1)

M <! JL°1)) J((1)) J((1)) J(^} = m' { J((1°0) J((1)) J((1)) J((1))}

'(1°) J(1) J(1) J(1)

3A3/2

=

A3/2

0

when i1 = . . . = i4

if among ¿1,..., i4 there are two pairs of identical numbers

otherwise

(6.96)

M/ i /(0»i) 7(i2) T^M)! _ I A3-, -,

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IVI \J(01) j(l) j(ll) J — J-{«l = i3}J-{i2=M}:

M/ i 7(n0) 7(i2) j(isk)\ _ I A3-, . -,

I (10) (1) (11) / ~~ 3

{ii=i3} a{«2=m} ;

(6.97)

(6.98)

il5A3

3A3

m'{ J(

(il)

(1) •••"(1)

J (¿6)1

J1) J

= <

A3

0

when i1 = ... = i6

if among i1,..., ¿6 there is a pair and a quad of identical numbers

if among ¿1,... ,¿6 there are three pairs of identical numbers

otherwise

(6.99)

m' j 7(i1i2) T(i3i4) T(i5i6) 1

ivi s j(n) j(n) j(n) j

6

AM 1ii2=i4| (1

{¿1=i5} 1{i3 = i6} + 1{«1=i6} 1{«3=«5^ +

+ 1{i2 = i6} ( 1{«1=«3> 1{«4 = i5> + 1{i1=i4> 1{i3 = i5^ +

+ 1{i4 = i6} (1{«1=»3} 1{«2 = «5} + 1{i2=«3} 1{«1 = i5^ I ,

(6.100)

KAf( T(i1i2) T(i3i4) T(i5) T= M \J(11) J(11) J(1) J(1) / =

- 1 a3

+ 6A3 ' 1{n = n] i1^^}1^^} + 1{i2=i6}1{i4=is}) +

+ 1{i2 = i3} (1{«1=«5} 1{«4 = «6} + 1{«1=«6} 1{«4 = «5^ + + 1{i1 = i4} (1{i3=«5}1{i2 = «6} + 1{«3=«6}1{i2 = i5^ +

+ 2 • 1{i2 = «4} (1{«1 = «5} 1{«3 = «6} + 1{«3=«5} 1{«1 = «6^ ) '

(6.l0l)

M' / T(i1i2) T(i3) T(i6)l

M \J(11) J(1) ••• J(1) /

i (m ... J((;f } - Al{il=ia}M' { ... J$}) . (6.102)

l

Let us explain the formula (6.101). From the following equality

7(i5) T(i6) _ T(i5i6) I T(i6i5) | A I „r „ 1

J(1) J(1) = J(11) + J(11) + A1{i5=i6} w. P. 1

we obtain

M' j J^(i1i)2) J-((i1«4) J((i5) J(i6)

(11) (11) (1) (1)

= M^ J(i1i2) J(i1i4) J(i1i6) ^ +

I |\/|' f T(i1i2) T(i3i4) T(i6i5)\ I A 1 + M \J(11) J(11) J(11) / +A1{i5 = i6}

(11) (11) (11) m

¿5 ¿6 ) \ 1)

'/ 7 (i1i2) T fe^l

\J(11) J(11) J'

(6.103)

Applying (6.91), (6.100) to the right-hand side of (6.103) gives (6.101). It is necessary to note [77], [79] that

m' n J

.<7=1

¿1»>... №

1 kg

(Alg)... A<£)

)

=0

if the number of units included in all multi-indices ^A1g)... is odd (kg — r, g = 1,...,/). In addition [77], [79]

m' fl J((i1 •••iko)

< =1

'(A10)... a£)

< KAYl,

where yz = £Z/2 + pZ, is a number of units and pZ is a number of zeros included in all multi-indices ^A1g)... Ajg^ , kg — r, g = 1,...,/, K e (0, to) is a constant.

In the case n, m > 1 and r = 1 we can put [77], [79

JW = Af(i) (ï = 1,...,m),

(1)

where Af(¿), i = 1,..., m are independent discrete random variables for which

p{Afw = ±\/A} = i

It is not difficult to see that the approximation

-^Uv^ (i = 1, • • •, m)

.(i)

(1) °

also satisfies the condition (6.86) when r = 1. Here are independent standard Gaussian random variables.

(

In the case n,m > 1 and r = 2 as the approximations J((1)), J((1T)2), J((i10)

(1) ' "(11) > (10)

are taken the following ones [79]

j((;;} = = = ^a • af^, (6.io4)

2

Jô(

(¿1«2)

= i (Af(n)Af(i2) + , (6.105)

p{Afw = ±\/3Â} = i, p{Af(i) = o} = ^,

(n) ~ 2

where Af(i) are independent Gaussian random variables with zero expectation and variance A or independent discrete random variables for which the following conditions are fulfilled

1

6:

2 3

V(i1i2) are independent discrete random variables satisfying the conditions

P {V(Ht2) = ±a} = i when io < il,

V(i1i1) = -A, V(i1i2) = -V(i2i1) when i1 < i2,

where i1, i2 = 1, . . . , m.

Let us consider the case r = 3 and m = 1,n > 1. In this situation in addition to the formulas (I6.89I)-(I6.10,'3) we need a number of formulas for the conditional expectations (6.88) when m = 1.

We have [77], [79]

m'{ J(1) J(m)} = m'{ J(01) J(m)} = m'{ J(10) J(m)} = 0,

M'{J(oii) J(n)} = M'M ioi)^(ii)} = M;{ J(no) J(ii)} = ^A3,

m'{ J(ooi)«/(i)} = m'{ J(oio) J(i)} = m'{ J(ioo) J(i)} = ^a3,

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m;{ J(ioo)^(io)} = J(ooi)^(oi)} = ^a4, m'{ j(in) j(n)} = 0,

m;{ J(oio) J(io)} = m'{ J(oio) J(oi)} = ^a4, m' {(J(iii))2} = ^a3,

M'{ -7(100)^(01)} = M'{ J(ooi)^(io)} = http://doi.org/10.21638/11701/spbu35.2023.110 Electronic Journal. http://diffjournal.spbu.ru/ A.874

M'{ J(iio) J(i0)} = M/{J(110)J(01)} = M'{ J(ioi) J(io)} = 0, M'{J(101)J(01)} = M'{ J(011) J(10)} = M'{ J(011) J(01)} =

M' {4om (J(1))2} = M' {J(1„D (J(1))2} = M' {./,„„, (,/(1))2} = 1A3, M' {./(in) (./(i))3} = A3, M' {./(in)J(ii)J(i)} = ¿A3,

where

io+A t2

J(A,...Ak ) = / ' ../>,

to to

/t(0) = t, /t(1) = /t is standard scalar Wiener process, A/ = 0 or A/ = 1, l = 1,..., k.

In [77], [79] using the given moment relations the authors proposed the

following weak approximations of iterated Ito stochastic integrals for r = 3 when m = 1, n > 1

J(i) = A/ (6.106)

J(i0) = A/ , J(0i) = A • A/ - A/ , (6.107)

J(ii) = 2 (i/^f) ~ ' = = = g^2 '

1 . (/. r\2

where

J{uo) = J{ ioi) = J{ oil) = g A ^A/J - A J , %ii) = ^/((A/)2-3AV

A/~N(O,A), A/~N(O,1A3), M {A/A/} = 1A2.

3 7 ' l " " J 2

Here N(0, a2) is a Gaussian distribution with zero expectation and variance a2.

Finally, we will form the weak approximations of iterated Ito stochastic integrals for r = 4 when m = 1, n > 1 [1]-[16 .

The truncated Taylor-Ito expansion (14221) when r = 4 and m = 1 includes 26 various iterated Ito stochastic integrals. The formation of weak approximations for these stochastic integrals satisfying the condition (6.87) when r = 4 is extremely difficult due to the necessity to consider a lot of moment conditions. However, this problem can be simplified if we consider the truncated

unified Taylor-Ito expansion (4.27) when r = 4 and m = 1, since this expansion includes only 15 various iterated Ito stochastic integrals

I0, I1, I00, I000, I2, I10, I01, I3, I11, I20, I02, I100, I010, I001, I

0000,

where

to+A

t2

r def Il1-1k =

(t0 - tj )lk ... (t0 - t1 )l1 dft1 ... dftk (k > 1)

to to

and ft is standard scalar Wiener process.

It is not difficult to notice that the condition (6.87) will be satisfied for r = 4 and i1 = ... = i4 if the following more strong condition is fulfilled

m

nv... ikB) -n0

.g=1

<7=1

1kg

F

to

< K(t - t0)5 w. p. 1 (6.108)

for all lig)... ljg) G A, kg < 4, g = 1,...,l, l = 1, 2,..., 9, where K G (0, to)

and

A = ^ 0,1, 00,000, 2,10, 01,3,11, 20,02,100,010,001,0000

is the set of multi-indices.

Let (see Sect. 5.1 and 6.6) [14]-[16], [152]

Î0 = VÂ(0, 70o = ^a((CO)2-I),

(6.109)

A3/V 1 \ A3/V 3 \

h = — ( Co + J , Iooo = — ( (Co)3 - 3Co ), (6.110)

2

/oooo = ^((Co)4-6(Co)2 + 3).

(6.111)

Here and further

io+A i0+A

def 1 def 2 3

Co = —F= / dfs, Cl =

y/A

A3/2

S-to- y j dfs,

to

where fs is scalar standard Wiener process.

It is not difficult to see that Co, Z1 are independent standard Gaussian random variables. In addition, the approximations (I6.109l)-(l6.111l) equal w. p. 1

to the iterated Ito stochastic integrals corresponding to these approximations. This implies that all products

/

n

g=1

'l ••• 'kg

which contain only the approximations (I6.K)9I)-(I6.111) will convert the left-hand side of (6.108) to zero w. p. 1, i.e. the condition (6.108) will be fulfilled automatically.

For forming the approximations

/100, I010, I001, /10, I01, /11, /20, /02, /2, I3 it is necessary to calculate several conditional expectations

M , (6.112)

m m i

.5=1

/(g) /(g)

11 ••• kg

(g) /(ff) 1 ••• /kg

We will denote (6.112) (as before) as follows

where /fa ... /%> G A.

n I

.5=1

/(g)... /(g)

11 ••• kg

We have

m'{ /3} = m'{ 13( i0)2} = m'{ /3 100} = 0, m'{ /3 /0} =

A4

4

m'{ /2 ( /0)2} = m'{ /2 /00} = m'{ /2/000} = m'{ /2 /0000} = 0, m'{ /2 ( /00)2} = m'{ /2 ( /0)4} = m'{ /2/000 /0} = 0,

m'{ /2 /00 ( /0)2} = m'{ /2 /10} = m'{ /2 /01} = m'{ /2/1/0} = m'{ /2} = 0,

A3 A 4

M'iVo} = -3-, M'iWof} = A4, M'Woolo} = —,

A4

M '{I2h} = -—, M '{Ifa = M'fafao} = M'{ Vooo} = M'{fa(I0f} = 0,

A4 A4

M'{Ifaoolo} = M'{fah} = 0, W\'{I20(I0)2} = —, M'{/2o/oo} = ,

A4 A4 A4

M'j/nM2} = —, M'{/nM = —, M'iUlo)2} =

A4

MV02I00} = —, M'{/A} = W\f{IxI0} = M/{/A(/o)2} = M'{/A/oo} = 0, M/{/A/1} = M/{/A/0000} = M/{/A(/00)2} = M,{/A(/0)4} = 0, M/{/Ai000i0} = M/{/Ai00(i0)2} = M^/aM = 0,

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M^/01} = M^/AW = 0, A4 A4 A4

M'{Wooo} = --¿J, M'j/iool/o)3} = —, M'jWoolo} = —g-, A4 A4 A4

M'{Wooo} = M'{/oio(/o)3} = -—, M'jWoo/o} = ,

A 4 3A4 3A4

M'{Wooo} = —M'{/ooi(/o)3} =---r, M'{IooiIooIo} =

o 4 o

M'{//0} = M'{/p /000} = M/{/p(/0)3} = M/{/p/00/0} = 0,

M/{/p/1} = M'{/p/0000} = M'{/p(/0)5} = M'{/p(/00)2/0} = 0,

M'{/p/00(/0)3} = M'{/p W/0)2} = M'{/p/0000/0} = 0,

M'{/p/000/00} = M'{/p/100} = M'{/p/010} = 0,

M'{/p/001} = M'{/p/2} = M'{(/p)2/0} = M'{/p/00/1} = 0,

M^/10/01/0} = M'{/p} = M'{/p/1(/0)2} = 0,

A3 A3 A4

M/{710(/0)2} = M'j/io/oo} = —M/{/1o(/oo)2} =

3 6 3

A4

M/{710(/0)4} = -2A4, M'{/ioWo} =

6

5A4

M'{I10I00(I0)2} =

M'K/io)2} = t7t5 M'l/ioM = M'{hohIo} =

6

A4 A4 5A4

12 o 24

2A3 A3 2A4

M'{/oi(lo)2} = -—, M'j/oi/oo} = -—, Mf{I01(I00)2} = -—,

A4

M'j/oii/o)4} = -4A4, M'j/oi Wo} = -—,

5A4 A4 3A4

M'{I0iIoo(Io)2} = —M'K/qi)2} = M'j/oi/i/o} =

3 4 o

where

^ = 02, 11, 20, A = 100, 010, 001, p = 10, 01

(these recordings should be understood as sequences of digits).

The above relations are obtained using the standard properties of the Ito stochastic integral and the following equalities resulting from the Ito formula

A2

(/o)4 = 24/qooo + 12A/00 + 3A2, (70 o)2 = 6/0ooo + 2A/0o + —,

/00 ( /0)2 = 12 /0000 + 5A /00 + A2, /1/0 = /10 + /01 -/00 ( /0)3 = 60 /00000 + 27A /000 + 6A2 /0,

A2

2 '

( /0)5 = 120 /00000 + 60A /000 + 15A2 /0,

/000( /0)2

( /00)2/0 = 30/00000 + 12A /000 +

2

10A2

/0,

4

20 /00000 + 7A /000 + A2 /0, /0000 /0 = 5 /00000 + A /000,

A2 A2

^000^00 = IO/00000 + 3A/000 + hoh = ^001 + ^010 + hoo--

( /0)3 = 6 /000 + 3A /0, /00/0 = 3 /000 + A /0, /10 /0 = /010 + /100 + A /1 + /2, /000 /0 = 4 /0000 + A /00, ( /0)2 = 2 /00 + A,

hih = 2/ooi + /010 — + A^/o)

w. p. 1.

Using the given before moment relations, we can form the weak approximations /100, /010, /001, /10, /01, /11, /20, /02, /2, /3 [14]-[16], [152]

100 =

A5/2

(C0)3 - 3(0 , /

010

A5/2 ~Y2~

(C0)3 - 3(0 , (6.113)

/001 =

A5/V ,^3 , A-,,» \ 2

20 =

8 v(Co) -3Coj, /ii = T^(io)--lJ,

(6.114)

(6.115)

a7/'2 - a5/'2 , v ^ , /3 = -—Co, /2 = —(Co + ^-Ci), (6.116)

710 = A2 ( -I ((Co)2 - 1) - ¿=0><i ± ^ ( (CO - - 1 ) ). (6.117)

/01 = A2 ((Co)

K(Co)2 ■ 0 ■ ¿iCoCi T ii^((Ci)2 ■ v J ■ (6'118)

where (0, Z1 are the same random variables as in (6.109)—(6.111).

It is easy to check that the approximations (6.109)—(6.111). (6.113) (6.118) satisfy the condition (6.108) for r = 4 and m = 1, n > 1, i.e. they are weak approximations of the order r = 4 for the case m = 1, n > 1.

Chapter 7

Approximation of Iterated Stochastic Integrals with Respect to the Q-Wiener Process. Application to the High-Order Strong Numerical Methods for Non-Commutative Semilinear SPDEs with Nonliear Multiplicative Trace Class Noise

7.1 Introduction

There exists a lot of publications on the subject of numerical integration of stochastic partial differential equations (SPDEs) (see, for example [153]-[177]).

One of the perspective approaches to the construction of high-order strong numerical methods (with respect to the temporal discretization) for semilinear SPDEs is based on the Taylor formula in Banach spaces and exponential formula for the mild solution of SPDEs [159], [161]-[164]. A significant step in this

direction was made in [163] (2015), [164] (2016), where the exponential Milstein and Wagner-Platen methods for semilinear SPDEs with nonlinear multiplicative trace class noise were constructed. Under the appropriate conditions

164] these methods have strong orders of convergence 1.0 — £ and 1.5 — £ corre-

spondingly with respect to the temporal variable (where £ is an arbitrary small posilive real number). It should be noted that in [168] (2007) the convergence with strong order 1.0 of the exponential Milstein scheme for semilinear SPDEs was proved under additional smoothness assumptions.

An important feature of the mentioned numerical methods is the presence in them the so-called iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process [170]. Approximation of these stochastic in-

tegrals is a complex problem. The problem of numerical modeling of these stochastic integrals with multiplicities 1 to 3 was solved in [163], [164] for the case when special commutativity conditions for semilinear SPDE with nonlinear multiplicative trace class noise are fulfilled.

If the mentioned commutativity conditions are not fulfilled, which often corresponds to SPDEs in numerous applications, the numerical modeling of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process becomes much more difficult. Note that the exponential Milstein scheme [163] contains the iterated stochastic integrals of multiplicities 1 and 2 with respect to the infinite-dimensional Q-Wiener process and the exponential Wagner-Platen scheme [164] contains the mentioned stochastic integrals of multiplicities 1 to 3 (see Sect. 7.2).

In [176] (2017), [177] (2018) two methods of the mean-square approximation of simplest iterated (double) stochastic integrals from the exponential Milstein scheme for semilinear SPDEs with nonlinear multiplicative trace class noise and without the commutativity conditions are considered and theorems on the convergence of these methods are given. At that, the basic idea (first of the mentioned methods [176], [177]) about the Karhunen-Loeve expansion of the Brownian bridge process was taken from the monograph [77] (Milstein approach, see Sect. 6.2). The second of the mentioned methods [176], [177] is based on the results of Wiktorsson M. [83], [84] (2001).

Note that the mean-square error of approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process consists of two components [176], [177]. The first component is related with the finite-dimentional approximation of the infinite-dimentional Q-Wiener process while the second one is connected with the approximation of iterated Ito stochastic integrals with respect to the scalar standard Brownian motions.

It is important to note that the approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process can be reduced to the approximation of iterated Ito stochastic integrals with respect to the finite-dimensional Wiener process. In a lot of author's publications

(see Chapters 1,2, and 5) an effective method of the mean-square approximation of iterated Ito (and Stratonovich) stochastic integrals with respect to the finite-dimensional Wiener process was proposed and developed. This method is

based on the generalized multiple Fourier series, in particular, on the multiple Fourier-Legendre series (see Sect. 5.1).

The purpose of this chapter is an adaptation of the method [1]-[62] for the mean-square approximation of iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process. In the author's publications [23], 47] (see Sect. 7.3) the problem of the mean-square approximation of iterated

stochastic integrals with respect to the infinite-dimensional Q-Wiener process in the sense of the second component of approximation error (see above) has been solved for arbitraty multiplicity k (k £ N) of stochastic integrals and without the assumptions of commutativity for SPDE. More precisely, in [23], 47] the method of generalized multiple Fourier series (Theorems 1.1, 1.2, 1.16)

for the approximation of iterated Ito stochastic integrals with respect to the scalar standard Brownian motions was adapted for iterated stochastic integrals with respect to the infinite-dimensional Q-Wiener process (in the sense of the second component of approximation error).

In Sect. 7.4 (also see [24], [48]), we extend the method [176], [177] and esti-

mate the first component of approximation error for iterated stochastic integrals of multiplicities 1 to 3 with respect to the infinite-dimensional Q-Wiener process. In addition, we combine the obtained results with the results from [23], 47] (see Sect. 7.3). Thus, the results of this chapter can be applied to the

implementation of exponential Milstein and Wagner-Platen schemes for semilinear SPDEs with nonlinear multiplicative trace class noise and without the commutativity conditions.

Let U, H be separable R-Hilbert spaces and LHS (U, H) be a space of Hilbert-Schmidt operators mapping from U to H. Let F, p) be a probability space with a normal filtration {Ft,t £ [0,TT]} [170], let Wt be an U-valued Q-Wiener process with respect to {Ft,t £ [0,T]}, which has a covariance trace class operator Q £ L(U). Here and further L(U) denotes all bounded linear operators on U. Let U0 be an R-Hilbert space defined as U0 = Q1/2(U). At that, a scalar product in U0 is given by the relation [164

<u,w>Uo = (Q—1/2u,Q—1/2w) u

for all u, w £ U0.

Consider the semilinear parabolic SPDE with nonlinear multiplicative trace class noise

dXt = (AXt + F(Xt)) dt + B(Xt)dWt, X0 = t £ [0,T], (7.1)

where nonlinear operators F, B (F : H ^ H, B : H ^ LHS(U0, H)), the linear operator A : D(A) C H ^ H as well as the initial value £ are assumed to satisfy the conditions of existence and uniqueness of the SPDE mild solution (see [164], Assumptions A1-A4).

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It is well known [173] that Assumptions A1-A4 [164] guarantee the existence and uniqueness (up to modifications) of the mild solution Xt : [0,T] x Q ^ H of the SPDE (7.1)

X = exp(At)£ + exp(A(t-t))F(Xt)dr + / exp(A(t-t))B(Xt)dWT (7.2)

w. p. 1 for all t G [0,T], where exp(At), t > 0 is the semigroup generated by the operator A.

As we mentioned earlier, numerical methods of high orders of accuracy (with respect to the temporal discretization) for approximating the mild solution of the SPDE (7.1), which are based on the Taylor formula for operators and an exponential formula for the mild solution of SPDEs, contain iterated stochastic integrals with respect to the Q-Wiener process [159], [161]-[164

Note that the exponential Milstein type numerical scheme [163] and the exponential Wagner-Platen type numerical scheme [164] contain, for example, the following iterated stochastic integrals (see Sect. 7.2)

T

t2

J B(Z)dWtl, J B'(Z) J B(Z)dWtl dWt2, (7.3)

t t \t T / t2 \ T / t2

i B'(Z) i F(Z)dt A dWt2, i F'(Z)\ i B(Z)dWti I dt2, (7.4)

J B'(Z^ j B'(Z) I j B(Z)dWti j dWtJ dWt3,

T

t2

J B''(ZW J B(Z)dWti ,J B(Z)dWti dWt2,

(7.5)

(7.6)

where 0 < t < T < T, Z ^ H is an Ft/B(H )-measurable mapping and F', B', B'' denote Frechet derivatives. At that, the exponential Milstein type

t

scheme [163] contains integrals (17.31) while the exponential Wagner-Platen type scheme [164] contains integrals (17.3)—(776) (see Sect. 7.2).

It is easy to notice that the numerical schemes for SPDEs with higher orders of convergence (with respect to the temporal discretization) in contrast with the numerical schemes from [163], [164] will include iterated stochastic integrals

(with respect to the Q-Wiener process) with multiplicities k > 3 [162] (2011). So, this chapter is partially devoted to the approximation of iterated stochastic integrals of the form

/[$(k>(Z)]r,t = J (Z) |... | y3$,(Z) | j )dW(l j dW(2 j .. .j dW(l,

(7.7)

where 0 < t < T < T, Z : ^ ^ H is an Ft/B(H )-measurable mapping and (v)( ... ($2(v)(^i(v))... )) is a k-linear Hilbert-Schmidt operator mapping from U0 x ... x U0 to H for all v e H.

k times

In Sect. 7.3.1 we consider the approximation of more general iterated stochastic integrals than (7.7). In Sect. 7.3.2 and 7.3.3 some other types of iterated stochastic integrals of multiplicities 2—4 with respect to the Q-Wiener process will be considered.

Note that the stochastic integral (7.6) is not a special case of the stochastic integral (7.7) for k = 3. Nevertheless, the extended representation for approximation of the stochastic integral (7.6) is similar to (7.12) (see below) for k = 3. Moreover, the mentioned representation for approximation of the stochastic integral (7.6) contains the same iterated Ito stochastic integrals of third multiplicity as in (7.12) for k = 3 (see Sect. 7.3.2). These conclusions mean that one of the main results of this chapter (Theorem 7.1, Sect. 7.3.1) for k = 3 can be reformulated naturally for the stochastic integral (7.6) (see Sect. 7.3.2).

It should be noted that by developing the approach from the work [164

which uses the Taylor formula for operators and a formula for the mild solution of the SPDE (7.1), we obviously obtain a number of other iterated stochastic integrals. For example, the following stochastic integrals

T / t2 t2 t2 \

J B'''(Z) J B(Z)dWtl J B(Z)dWtl J B(Z)dWtl dWt2!

T /is / t2 t2

J B'(Z) J B''(Z) J B(Z)dWtl ,J B(Z)dWtJ dWtJ dWts,

T /is ts / t2

J B''(Z) J B(Z)dWti J B'(Z) J B(Z)dWti dWtJ dWts,

tl ■

T /is / t2

J F'(Z) ^ J B'(Z) ^ J B(Z)dWti j dWt2 j dt3,

T / t2 t2 \

J F''(Z) J B(Z)dWti J B(Z)dWti dt2, t \t t / T / t2 t2 \

J B''(Z) J F(Z)dti,y B(Z)dWtJ dWt2

t t t

will be considered in Sect. 7.3.3. Here Z : ^ ^ H is an Ft/B(H)-measurable mapping and B', B'', B''', F', F'' are Frechet derivatives.

Consider eigenvalues A, and eigenfunctions e,(x) of the covariance operator Q, where i = (i1,..., id) G J, x = (x1,..., xd), and J = {i : i G Nd and A, > 0}.

The series representation of the Q-Wiener process has the form [170

W(t.,x) = t G [0,T]

¿gJ

or in the shorter notations

A)

wt = 5>>/W>, ie [o,f],

t

¿e J

where wt(i), i G J are independent standard Wiener processes.

Note that eigenfunctions e,, i G J form an orthonormal basis of U [170 Consider the finite-dimensional approximation of Wt [170

Wf = ^v^wf, t G [0, T], (7.8)

¿gJm

where

JM = {i : 1 < i1,...,id < M and A, > 0}. (7.9)

Using (7.8) and the relation [170

wf) = ^=(e,,w t)u, ieJ, (7.10)

we obtain

WM = S e (e, Wt>u, t e [0,T], (7.11)

¿gJm

where (-, ^ is a scalar product in U.

Taking into account (7.10) and (7.11), we note that the approximation /[$(k)(Z)]M of the iterated stochastic integral /[$(k)(Z)]T,t (see (7.7)) can be written w. p. 1 in the following form

/[$(k)(Z )]

M T,t

T / / is / t2

= J (Z ) ..J J ^(Z ) J $!(Z )dWMM dWMM ..J dW.

M

= ^ (Z)(... ($2(Z )($x(Z )e ri) er2 ) • • •) erfc x

ri,...,rfc gJm

T is t2

X/ ."J I d(eri' Wti >u d(er2, Wt2 >u ••• d(erfc, Wifc >u = t t t

r1 Ar2 • • • Arfc X

ri,...,rfc gJm

T is t2

xy .../ J dw«ri)dw<r2 >... dwir' >, (7.12)

t t t

where 0 < t < T < T

Remark 7.1. Obviously, without loss of generality, we can write JM = {1, 2,..., M}.

As we mentioned before, when special conditions of commutativity for the SPDE (7.1) be fulfilled, it is proposed to simulate numerically the stochastic

integrals (I7.3I)-(I7\6) using the simple formulas [163], [164]. In this case, the numerical simulation of the mentioned stochastic integrals requires the use of increments of the Q-Wiener process only. However, if these commutativity conditions are not fulfilled (which often corresponds to SPDEs in numerous applications), the numerical simulation of the stochastic integrals (I7.3I)-(17T6) becomes much more difficult. Recall that in [176], [177] two methods for the mean-square approximation of simplest iterated (double) stochastic integrals defined by (7.3) are proposed. In this chapter, we consider a substantially more general and effective method (based on the results of Chapters 1 and 5) for the mean-square approximation of iterated stochastic integrals of multiplicity k (k E N) with respect to the Q-Wiener process. The convergence analysis in the transition from JM to J, i.e., from the finite-dimensional Wiener process to the infinite-dimensional one will be carried out in Sect. 7.4 for integrals of multiplicities 1 to 3 similar to the proof of Theorem 1 [177 .

7.2 Exponential Milstein and Wagner—Platen Numerical Schemes for Non-Commutative Semilinear SPDEs

Let assumptions of Sect. 7.1 are fulfilled. Let A > 0, tp = pA (p = 0,1,..., N),

and N A = T. Consider the exponential Milstein numerical scheme

(

Yp+1 = exp (AA)

tp+i

Yp + AF(Yp)+y B(Yp)dWs+

V tp

rp+1

+ / B'(Yp)

/

V

\

B (Yp)dWT

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dWs

!

(7.13)

and the exponential Wagner-Platen numerical scheme [164

^P+i

= exp

/

( AA

exp . , ,p

V

rp+1

^Yp + AF(Y>)+y B(Yp)dWs+

tp+i

+ B ( Yp )

/

\

B (Yp)dWT

dWs+

!

s

TP+i

+^F'(YP)(AYP + F(Yp)\+ I F\YP)

(

V

\

B (Yp)dWT

ds+

!

f T S ^F'\Yv) (B(Yp)e,, B(Yp)e^j + ieJ ^ '

i tp+

+A

Tp+i s Tp+i

J J B(Yp)dWTds J B(Yp)dWs

+

\ tp tp

J

Tp+i

Tp+i s

+A ^ B' (Yp^ AYp + F (Yp^ dWs -J J B' (Yp^ AYp + F (Yp ^ dWT ds+

Tp Tp Tp

/

Tp+i / s s

J B"(YP) J B(Yp)dWT,J B(Yp)dWT

pp tp \p

dW,+

/

Tp+i

+ B' (Yp)

/

V

B ( Yp )

/

V

\

B (Yp)dW6

dW7

dWs

(7.14)

/ / y

for the SPDE (7.1), where Yp is an approximation of XTp (mild solution (7.2) at the time moment tp), p = 0,1,..., N, and B', B'', F', F'' are Frechet derivatives.

Note that in addition to the temporal discretization, the implementation of numerical schemes (7.13) and (7.14) also requires a discretization of the infinite-dimensional Hilbert space H (approximation with respect to the space domain) and a finite-dimensional approximation of the Q-Wiener process. Let us focus on the approximation connected with the Q-Wiener process.

Consider the following iterated Ito stochastic integrals

T

T t2

J(i)T = dw

(ri) j(ri0)

(10)T,t

dwtri)dt2, J(°r)T ,t

dtidwt(2r2),

t t

T t2

T ts t2

J

(ri22) (11)T,t

dwt(ri)dwt(22),

J

(7.15)

J J J dwt(ri)dwt(r2)dwt(2s), (7.16)

tt

(ri22rs)

(111)T,t _ I I I ""'"ti ttt

s

1

where n,r2,r3 E JM, 0 < t < T < T, and JM is defined by (7.9).

Let us replace the infinite-dimensional Q-Wiener process in the iterated stochastic integrals from (7.13), (7.14) by its finite-dimensional approximation 7.8). Then we have w. p. 1

tp+i

J B(Yp)dWf = ]T £(>geTlv^J,

T riG Jm

(ri) (1)7"p+1>7

(7.17)

/ Tp+1 s Tp+1

J j B{Yp)(Wfds-^ J B(Yp)dW,

y Tp Tp Tp

A

TM -p)d W s

/

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rP+1

= A J B (Yp) (

hGJm

J

(0r1)

2 (1)tp+1,tP (01)Tp+1,Tp y '

(7.18)

Tp+1 rp+1 s

A J B' (Yp^ AYp + F (Yp)) dWM -J J B' (Yp) ( AYp + F (Yp)) dWM ds

pp

Tp+1 s

J B'(Yp^ ^AYp + F(Yp)) dTdW

sM

E

B'(yp) ( AY, + F(Y„) ) e,, v^J,(0ri)

(01)Tp+1,Tp'

r1 GJ

M

(7.19)

tp+1

F ( Yp )

\Tp

B (Yp)dW

M

T

ds =

/

E mwnK^Hlw, - -/{oX«,,.

r1 GJ

M

(7.20)

p

s

Tp+i

B' (Yp)

/

V

B (Yp)dW

M

T

dWM =

/

£ B'(Yp) (B(Yp)e 2i ) er2 21 ^22 J(

(ri22) (11)7"p+i>

(7.21)

ri^eJ

M

Tp+i

B' (Yp)

\TP

B' (Yp)

\TP

B (Yp )dWM

dW.

M

dWM =

y y

£ (B'(YP) (B(i;)eri)er2)er3v/AriAr2Ar3ji[;^) (7.22)

ri,r2 ,rs GJm

Tp+i

B' '(Yp)

\TP

s

B (^,)dWM Jb (^

dWM =

y

ri,r2,rsG Jm

X

TP+i /

tp \tp

\

dw<ri> J dwTr2>

tp

dw(rs) •

y

(7.23)

Note that in (r7.18l)-(r7T20) we used the Ito formula. Moreover, using the Ito formula we obtain

s s

f dw^ f dw(r2> = J'rir:,' + J((2I2::'„+i^^(s - Tp) w. p. 1, (7.24)

where 1A is the indicator of the set A. From (7.24) we have w. p. 1

TP+i / s s \

i dw(ri) / dwir2)

T / T

Tp \ Tp T,

V

p

/

dw(rs) = 7 (rir2 rS) + T (r2 rirS) + T (0rS)

dws = J(111)Tp+i,Tp + J(111)Tp+i,Tp + 1{ri=r2} J(01)Tp+i,Tp .

(7.25)

s

After substituting (7.25) into (7.23), we obtain w. p. 1

Tp+1 is s \

B' '(Yp)

B(Yp)dWM,y B(Yp)dW.

\Tp Tp )

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dWM =

^ B"{YP) (B(Yp)eri,B(Yp)eT2)er3y/XnXr2Xr3 x

p

r1,r2,r3GJM

/ t(r1r2r3) + T(r2r1r3) + -t T(0r3) \ (7 26)

X J(111)Tp+1,Tp + J(111)Tp+1,Tp + 1{r1=r2} J(01)Tp+1,rJ • (7.26)

Thus, for the implementation of numerical schemes (7.13) and (7.14) we need to approximate the following collection of iterated Ito stochastic integrals

t(r1) T(0r1) T(r1 r2) T(r1 r2r3) (7 27)

J(1)T,t, J(01)T,t, J(11)T,t, J(111)T,t, ('.2,)

where r1 ,r2,r3 G JM, 0 < t < T < T.

The problem of the mean-square approximation of iterated Ito stochastic integrals (7.27) is considered completely in Chapters 1 and 5.

7.3 Approximation of Iterated Stochastic Integrals of Multiplicity k (k E N) with Respect to the Finite-Dimensional Approximation WM of the Q-Wiener Process

In this section, we consider a method for the approximation of iterated stochastic integrals of multiplicity k (k E N) with respect to the finite-dimensional approximation WM of the Q-Wiener process Wt using the mean-square approximation method of iterated Ito stochastic integrals based on Theorems 1.1, 1.2, 1.16.

7.3.1 Theorem on the Mean-Square Approximation of Iterated Stochastic Integrals of Multiplicity k (k E N) with Respect to the Finite-Dimensional Approximation WM of the Q-Wiener

Process

Consider the iterated stochastic integral with respect to the Q-Wiener process in the following form

I [$(k)(Z = J (Z) ^ J )x

t2

x I i $x(Z )^i(ti )dWtJ ^2(i2)dWtJ ..J ^ (tk )dWtk, (7.28)

where Z : Q ^ H is an Ft/B(H)-measurable mapping, WT is the Q-Wiener process, (v)( ... ($2(v)($1(v)))... ) is a k-linear Hilbert-Schmidt operator mapping from Up x ... x to H for all v e H, and 0i(r),..., € L2{[t, T}).

k times

Let I[$(k)(Zbe the approximation of the iterated stochastic integral (7.28) '

I [$(k)(Z = J (Z) ^ | $2(Z )x

x I i $i(Z)^i(ti)dWi1 I ^2(t2)dW^ I . J ^(tk)dWtk

.i

/ k \ i/2

ri ) er2 ) . . .) erfc ^ x

ri,r2,...,rfc gJm \/=i /

(ri r2...rfc)

x J I ^ ' 'I

where 0 < t < T < T and

t t 3 t 2

(ri r2..-rfc) _ / , /, N / , /, \ / , /, N 7 i22) ,7„r(rfc)

x J[^(k)]Trtr2-rfc), (7.29)

J Mf"') = ^ (tk)... ^2) ^i(ti)dwt(ri)dwj2r2) ...dW

tk

is the iterated Ito stochastic integral (1.5).

Let I[$(k)(Z), ^(k)]Mtp1'""'pfc be the approximation of the iterated stochastic integral (7.2

/ k x1/2

Y, (Z}(... ($2(Z }($i(Z )e

ri } er2} . . 0 erfc ^ X

ri,r2,...,rfc gJm \/=1 /

i(rir2 ...rfc )pi,...,pfc

x J[#fe)]T;, (7.30)

where J[^(k)]TT)pi'...'pfc is defined as the expression before passing to the limit on the right-hand side of (1.312)

Pi Pfc / k [k/2]

j ^o™-. )pi,..,pk = ^... ^ Ci,.J n j ) + £(-i}

ji=0 j=0 \/=1 m=1

m k—2m

X S n 1{rg2s-i = rg2s =0} 1{jg2s-i = j»2. ^ C

({{Si >32}v>{32m-i>32m}},{9iv>9fc-2m}) s = 1 1=1

{3i,32>'">32m-i>32m>9i>'">9fc-2m}={i>2>'">fc}

(7.31)

Let U, H be separable R-Hilbert spaces, U0 = Q1/2(U), and L(U, H) be the space of linear and bounded operators mapping from U to H. Let

L(U,H}o = {T|uo : T G L(U,H}},

where T |Uo is the restriction of operator T to the space U0. It is known [170

that L(U, H)o is a dense subset of the space of Hilbert-Schmidt operators

Lhs (Uo,H).

Theorem 7.1 [14]-[16], [23], [47]. Suppose that ^(t),.. (t) e ¿2([t,T]) and (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Furthermore, let the following conditions be satisfied:

1. Q e L(U) is a nonnegative and symmetric trace class operator (A« and e« (i e J) are its eigenvalues and eigenfunctions (which form an orthonormal basis of U) correspondingly) and WT, t e [0,T] is an U-valued Q -Wiener process.

2. Z : Q ^ H is an Ft/B(H)-measurable mapping.

3. e L(U, H)0, $2 e L(H, L(U, H)0), and $k(v)( ... ($2(v)($1(v)))... )

is a k-linear Hilbert-Schmidt operator mapping from Uo x ... x Uo to H for all

k times

v G H such that

)(... ($2(Z ) ($i(Z )er1 ) er2 )...) e

rfc

< Lk <00

H

w. p. 1 for all ri; r2,..., rk G JM, M G N. Then

m

<

H

where

pi pk

< L (k!)2 (tr Q)k Zk

ji=0 j=0

tr Q = ^^ ^ <

¿e J

(7.32)

Ik - llK Hi2([i,T ]k) = j K2

[t,T

K 2(ti,...,tk )dti ...dtk,

C —

Cjk ..ji —

/k

K (ti,...,tk )JJ (ti )dti ...dtk

7=1

[t,T ]k 7=i

is the Fourier coefficient,

^i(ti).. .^k(tk), ti < ... < tk

K(ti,..., tk) —

k k-i n>(t7) n ^<ti+i} '

0,

otherwise

7=i

where t1 ,...,tk E [t,T] (k > 2) and K (t1) = ^ (t1) for t1 E [t,T] (1A denotes the indicator of the set A).

Remark 7.2. It should be noted that the right-hand side of the inequality (7.32) is independent of M and tends to zero if p1,...,pk ^ to due to the Parseval equality.

2

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2

Remark 7.3. Recall the estimate (1.323), which we will use in the proof of Theorem 7.1

m {( j [^Tt^) — J [^TT^ )pi'...,p^ % <

pi pk

< fcM 4 — £ ...ECU

j i=0 jfc=0

where J[^(k)]

(ri r2...rfc) •

T,t

is defined by (1.5) and J[^(k)]

(rir2...rfc)pi,...,pfc •

T,t

is defined by

(7.31).

Proof. Using (1.127), we obtain

m

i[$(k) (z ), ^(k)]mt — i[$(k) (z ), ^(k)]

]T,t

H

m

I ^ $k(Z)(... ($2(Z)($1(Z)e r i ) er2 ) . . .) erfc An x

^ ri,r2,...,rfc gJm \/=1 /

J

(ri r2...rfc) T,t

J

(r ir2...rfc)p i,...,pfc

T,t

2

H

(7.33)

/ k \1/2 / k X1/2

m E E IIMnArM x

r i,r2?...,rkeJM (r,,r2,...,rk): {r,,r2,...,rk}={r i,r2,...,rfc} \/=1 / \/=1

x $k(Z)(... ($2(Z) ($1(Z)eri) er2)...) e^ ,

$k(Z) (... ($2(Z)($1(Z)en) erO ..OerO x

H

x M^ J

(ri r2...rfc) T,t

J

(ri r2...rfc )p i,...,pfc T,t

X

l(r i r2 ...rk) 7L/,(k)i(ri r2...rk )p i>...»pfc

x j[^TT2^) — J[^(k)]

T,t

Ft

< (7.34)

2

l k \1/2 / k N1/2

< E E IlM (I^1 x

r1,r2EJM (r1 ,r2,...,rk): {r1,r2,...,rk}={r1,r2,...,rfc} \/=1 J

1=1

xM

(Z)(... ($2(Z) ($1(Z )en) er2)...) e

Tfc

X

H

X

(Z) (... ($2(Z) ($1(Z)en) e^) ...)erfc

x

H

X

M J

• ••rfc)

J

• ••rfc)Pl,•••,Pfc

X

xi j ^(k)]TT2--.rfc) - J [^(k)](r1r2 ^ )p1'...'pfc

JT,t

<

< Lk

Lk E

E

k x1/2 / k x1/2

II An II1 x

1=1

m

J

X

...rfc)

J

• ••rfc)P1,•••,Pfc

X

(r1r2...rk)

J [^(k)](r1r2...rk )P1,...,Pk

JT,t

<

/ k \1/2 / k x1/2 < Lk ^ ^ IlM 1 x

•r1,r2v,TfcEJM (r1,r2,...,rk): {r1,r2,...,rk}={r1,r2,...,rfc} \/=1 / \/=1

mJ

...rfc)

J

)P1,...,Pfc

1/2

X

1/2

m

J [^f^2^)

JT,t

(r1r2...rk)p1,...,pfc T,t

<

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2

f k \1/2 / k \1/2 < Lk E E (ilM ( IlArM x

r i,r2,...,rfceJM (r,,r2,...,rk): {ri,r2,...,rk}={ri,r2,...,rfc} \/=1 / \/=1

p i pk

1/2

p i pk

1/2

X

k! 4 — £ ..^CLji

k! 4 — £

C 2

jk ...j i

<

ji =0 jk=0

j =0 jk=0

ii p i pk < Lk £ k! Ari Ar2 . . . Aj k! 4 ^ . . . £ Cl..ji

2i,22 ,...,rk GJm \ \ j i=0 jk=0

p i pk

Lk (k!)2 £ Ar iAr2 ...Ar J Ik — E ... E Cj....

ri,r2,...,rk GJm \ j i=0 jk=0

, <

j <

p i pk

< Lk (k!)2 (tr Q)k 4 — £... £ j.j ,

ji=0 jk=0

where (•, ^>H is a scalar product in H, and

£

(r i,r2,...,rk): {r ,,r2v-,rk}={r i,r2v-,rk}

means the sum with respect to all possible permutations (r1, r2,..., rk) such that {r1 ,r2,... ,rk} = {r1,r2,... ,rk}.

The transition from (7.33) to (7.34) is based on the following theorem.

Theorem 7.2 [14]-[16], [23], [47]. Suppose that ^(t),.. (t) G L2([t,T]) and (x)}°=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]). Then, the following equality is true

M4 J

(r i...rk ) T,t

J

(r i ...rk )p i,...,pk T,t

X

J

(m i...mk)

T,t

J

(m i...mk )p i,...,pk T,t

Ft = 0 (7.35)

w. p. 1 for all r1,..., rk, m1,..., mk e JM (M e N) such that {r1,..., rk} = {m1,.. .,mk}.

Proof. Using the standard moment properties of the Ito stochastic integral, we obtain

M J

(ri...rfc)

T,t

J

(mi...mfc)

T,t

F^ =0

(7.36)

w. p. 1 for all ri,..., rk,mi,..., mk £ JM (M £ N) such that (ri,..., rk) = (mi,... ,m*).

Using (1.306), (1.311), (1.317), and (1.322), we obtain

pi

J

(mi...mfc)pi,...,pfc = V^ c t'u ^ l(m

T,t =2.^1... cjk...jiJ [rji... j]t,

ji=0 jk=0

(mi...mfe)

,t '

(7.37)

where

T

j [j ... j iT;;'..mk)

t2

E J j (tk) ...J j (ti)dwimi)... dw

tm) w. p. 1,

(7.38)

(jiv-jfc) t

and

E

(ji,-..,j'fc)

means the sum with respect to all possible permutations ji ..., jk). At the same time if jr swapped with in the permutation (ji,...,jk), then mr swapped with mq in the permutation (mi,... ,mk). Another notations are the same as in Theorems 1.1, 1.2, 1.16 (J'[ j ... jiT™1"^ is defined by (1.299)).

Then w. p. 1

pi Pk

M J

xm < J

(^i.-rfc ) T,t

J

(mi-mfc ^i,...^ T,t

Ft

}Pi Pk

E . . . E ^jk-j'i

n — n

ji=0 jk =0

T,t

T t2

E J j (tk) ..J j (ti)dwt(mi)... dw

(mk) tk

X

Ft

(jl,---,jk) t

From the standard moment properties of the Ito stochastic integral it follows

that

T

t2

mJ

(ri...rk) T,t

E J j (tk) ..J j (ii)dw<mi). • • dw

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(mk) tk

(jiv"Jk) t

Ft = 0

w. p. 1 for all ri,...,rk, mi,..., mk £ JM (M £ N) such that {ri; {mi,.. .,mk}.

Then

M J

(ri...rfc) T,t

J

(m i...mfc )p i,...,pfc T,t

F^ =0

w. p. 1 for all ri,...,rk, mi,..., mk £ JM (M £ N) such that (ri, {mi,.. .,mfc}.

Using (7.37), (7.38), we have

.,r*} =

(7.39) .,rk} =

M J

(r i...rfc )p 1,...,pfc

T,t

J

(m i...mfc )p i,...,pfc

T,t

Ft

P i

Pk

P i

Pk

E . . . E Cjk ...j i E . . . E Cqk...

j i=0 jk=0 q i=0 qk =0

qi

X

T

t2

m

E / j (tk) . J fai (ti)dwt(r i) ...dw

(rk) tk

X

,(j i,...,j'k) t

/ T t2

x E /faqk(tk)..^faqi(ti)dwimi)...dw

(mk) tk

.(qi,...,qk) t

Ft = 0

(7.40)

w. p. 1 for all ri,...,rk, mi,..., mk £ JM (M £ N) such that (ri,..., rk} = {mi,.. .,mk}.

From (7.36), (7.39), and (7.40) we obtain (7.35). Theorem 7.2 is proved.

Corollary 7.1 [14]-[16], [23], [47]. Suppose that {fa(x)}=0 is an arbitrary complete orthonormal system of functions in the space L2([t,T]) and ^i(r),... (t) £ L2([t,T]). Then, the following equality is true

mJ

(r i...rk ) T,t

J

(r i ...rk )p i,...,Pk T,t

X

J

(m i...mi)

T,t

J

(m i...m;)qi,...,q; T,t

Ft = 0

w. p. 1 for all l = 1, 2,..., k — 1 and ri,..., rk, mi,..., m/ £ JM, pi,... ,pk, qi,...,qi = 0,1, 2,...

7.3.2 Approximation of Some Iterated Stochastic Integrals of Multiplicities 2 and 3 with Respect to the Finite-Dimensional Approximation W^ of the Q-Wiener Process

This section is devoted to the approximation of iterated stochastic integrals of the following form (see Sect. 7.1)

/c[B(Z), F(Z)]Mf = J B'(Z) i J F(Z)di1j dWM, (7.41)

h[B(Z),F(Z)]¥j = J F'(Z) i J B(Z)dWM j dt2, (7.42)

T / t 2 t 2 \

12[B(Z)]MMt = J B"(Z) J B(Z)dWMM J B(Z)dWtM dWtM. (7.43) t t t

Let Conditions 1, 2 of Theorem 7.1 be fulfilled. Let B''(v)(B(v),B(v)) be a 3-linear Hilbert-Schmidt operator mapping from U0 x U0 x U0 to H for all v G H. Then we have w. p. 1 (see (7.

Io[B(Z),F(Z)}^t= J2 /¿'iWi/k, x/V^n /,• (7-44)

r(O'-i)

B (Z )F (Z )en V ^ri J

ri GJm

r(''iO)

/1[B(Z),F(Z)]^= E /•"(/)(/*(/)<, )>/V./ (7.45)

ri G Jm

12[B(Z)]MMt = E B''(Z)(B(Z)eri, B(Z)e ri ^r 2 Ar3 x

ri,r2,r3G Jm

x / ( / dwTri^ dw[r2M dw(r3). (7.46)

t t t

Using the Ito formula, we obtain

s s

i dw<ri) /" dwp] = J<™> + j'^) + 1{r,=r,}(s - t) w. p. 1. (7.47)

From (7.47) we have

T / s s \

f dw<ri> / dw<r3> = J™' + j(<™> + i^=„, w. p. 1.

t t t

(7.48)

Note that in (7.44), (7.45), (7.47), and (7.48) we use the notations from Sect. 7.2 (see (7.15), (7.16)). After substituting (7.48) into (7.46), we have

h[B(Z)]M = E B''(Z) (B(Z)eri,B(Z)e r2 ) er3 \A r i Ar2 Ar3 X

ri,r2,r3£JM

/ t(r i r2r3) + T(r2r ir3) + 1 T(0r3^ w p 1 (7 49)

X J(iii)T,t + J(iii)T,t + 1{ri=r2}J(0i)T,^ w. p . 1 . (7 49)

Taking into account (5.137), (5.138), we put for q = 1

T(0r3)q _ j(0r3) _ (T ~ if'2 / (r3) 1 (ra)\

J(01 )T,t - (oi)T,i --2-^jr1 J P' ^ j

T(n0)q _ r(riO) _ (T - ¿)3/2 / (ri) 1 (ri)\

(io)T,i (io)T,i -2-V / P ' ^ j

where J^yrt denote the approximations of corresponding iterated Ito

stochastic integrals.

Denote by 70[B(Z),F(Z)]Mq, ii[B(Z),F(Z)]Mq, h[B(Z)]Mtq the approximations of iterated stochastic integrals (7.44), (7.45), (7.49)

7o[£(Z),F(Z)]^= £ B\Z)F(Z)eTl^K1j{(7.52)

r(0i'i )<?

B (Z )F (Z )er 1 V 1 J

ri gJm

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r(''iO)?

/1[ß(Z),F(Z)]^= £ t7"53)

r 1 gJm

/2[ß(Z)]$9 = ^ B"(Z) {B(Z)en,B(Z)eT2) eT3\JAriAr2Ar3 x

r i,r2,r3GJM

X J

T(rir2r3)q + T(r2r1r3)q + -t T(0r3)^ (7 54)

'(111)T,t + J(111)T,t + 1{r1=r2}J(01)T,^ , ('.54)

where q = 1 in (17.521). (17.5,3) and the approximations Jlll)^, Jui)?^ are defined by (1.106) for some q > 1.

From (7.44), (7.45), (7.49), (7.52)-(7.54) we have

/q[B(Z), F(Z)]MMt - 10[B(Z), F(Z)]$* = 0 w. p. 1,

h[B(Z), F(Z)]MMt - Ii[B(Z), F(Z)]£* = 0 w. p. 1,

i2[b (z )]MMt - I2[B (Z =

E B"(Z) (B(Z)eri,B(Z)er2)er3y/^>^X

n,T2,r3 GJm

/ / T(r1r2r3) _ T(r1 r2r3M + ( J(r2r1 r3) _ T(r2r1r3w p -i

^ J(iii)T,t J(iii)T,t ^ + J(iii)T,t J(iii)T,t J J w. p. 1.

Repeating with an insignificant modification the proof of Theorem 7.1 for the case k = 3, we obtain

m

h [b (Z )]MM - h[B (Z

2

1 <

H

<4C(3!)2(tr E

V jl,j2,j 3 =0

where here and further constant C has the same meaning as constant Lk in Theorem 7.1 (k is the multiplicity of the iterated stochastic integral), and

_ v/(2j1 + l)(2j2 + l)(2j3 + l)rT .3/2^

( ./."../:•./• g K1 L ) ( j:J:.i- '

1 z y

C333231 = J P33(z) J P32 (y) J P31 (x)dxdydz, -1 -1 -1

where P3(x) is the Legendre polynomial.

7.3.3 Approximation of Some Iterated Stochastic Integrals of Multiplicities 3 and 4 with Respect to the Finite-Dimensional Approximation WM of the Q-Wiener Process

In this section, we consider the approximation of iterated stochastic integrals of the following form (see Sect. 7.1)

T / t2 t2 t2 \

13[B(Z)]MMt = / B'''(Z) J B(Z)dWMM J B(Z)dWMM J B(Z)dWtM dWMM,

t \t t t /

14[B (Z )]£ =

T / t3 / t2 t2 \ \

= J B'(Z) J B''(Z) J B(Z)dWMM J B(Z)dWtM dWtf dWMM,

tl

.t t (Z )]M =

T f t3 t3 / t2

= j b''(z) j b(z)dwm j b'(z) j b(z)dwtm dwt2 i dwm,

tl

,t t \t

T / t3 / t2

/.[B (Z ),F (Z )]M( = J F' (Z ) J B' (Z ) J B (Z )dW(MM dW£ dis,

tl

ttt

T / t2 t2

/7[B (Z ),F (Z )]M = | F''(Z ) J B (Z )dWMM J b (Z )dw£) di2,

t t t T / t2 t2 \

/8[B (Z ),F (Z )]M = | B ''(Z ) J F (Z )dti^ B (Z )dWMf dW".

t t t

Consider the stochastic integral /3[B(Z)]Mt. Let Conditions 1, 2 of Theorem 7.1 be fulfilled. Let B'''(v)(B(v),B(v),B(v)) be a 4-linear Hilbert-Schmidt operator mapping from U0 x U0 x U0 x U0 to H for all v G H.

We have (see (7.29)) /3[B(Z)]^ = ^ B'"(Z) (B(Z)eri,B(Z)eT2,B(Z)e,i) en fa XnA,,A,-3A,-4x

ri,r2,r3,r4GJM

T / s *1

t t t t

s s \

w. p. 1. (7.55)

By analogy with (2.398) or using the Ito formula, we obtain

T (ri) T(12) T(13) = T(111213) I T(111312) I T(121113) 1 T(12I3I1) 1 T(131112) 1 T(13I2I1) 1 J(i)s,tJ(i)s,tJ(i)s,t = J(iii)s,t + J(iii)s,t + J(iii)s,t + J(iii)s,t + J(iii)s,t + J(iii)s,t +

+1 ( J(130) + J(013) \ +1 ( J(120) + J(012) \ +

+ 1{11=12} J(i0)s,t + J(0i)s,ty + A{11=13} J(i0)s,t + J(0i)s,^ +

+ 1/ x (j(n0) + J(0n) ^ =

+ 1{12=13} J(i0)s,t + J(0i)s,ty =

ET(111213) + ( _ t) f-t T(11) + 1 T(12) + 1 T(13) \

J(iii)s,t + (s l) ^{12=13} J(i)s,t + -{11=13} J(i)s,t + -{11=12} J(i)s,t)

(11,12,13)

(7.56)

w. p. 1, where

E

(11,12,13)

means the sum with respect to all possible permutations (ri, r2, r3). We also use the notations from Sect. 7.2 (see (7.15), (7.16)).

After substituting (7.56) into (7.55), we obtain

h[B(Z)]%t = J2 (B(Z)eTl,B(Z)eT2,B(Z)er3) < r:sJ\r.\,,\r:,\r. x

11,12,13,146 Jm

V I V^ T(11121314) _ -| I(1314) _ -| I(1214) _ -| I(1114M (7 57)

* I z^ J(iiii)T,t -{11=12}I(0i)T,t -{11=13}I(0i)T,t -{12=13}I(0i)T,t I ('.5i)

^(11,12,13) /

w. p. 1, where J(([112i1T*4) is defined by (15.107) and

T s

1(01^ = /(i - s) / dwT11>dw<12». (7.58)

tt

Denote by I3[B(Z)]Mtq the approximation of the iterated stochastic integral (7.57), which has the following form

h[B(Z)$f = (В(г)еТ1,В(г)еТ2,В(г)еТз) X

Г1,Г2,Г3,Г4€7М

v . V^ т(rir2r3r4)q_i i(r3r4)q_i i(r2r4)q_- r(rir4)q

X 1 J(1111)T,t -L{ri=r2}1 (01)T,t 1{ri=r3}J (01)T,t 1{r2=r3}J (01)T,t

Jri,r2,r3)

(7.59)

where the approximations Jni^T"^, /(01)Ti are based on Theorem 1.1 and Le-gendre polynomials (see (5.15) and (5.59)).

For example, from (15.151) we have (here we use the notation /^Т t from the formula (5.15))

j{rir2)q _ T — t j(rir2)q _ (T ~ ty ( _J_(a(»'i) /Ф'2) , V )T,t - 2 №)T,t 4 1 л/о^0 S.1 +

, • (/ • _ Cf^C^ ] ]

tiV + l)(2i + 5)(2i + 3) (2г-1)(2г + 3) ' ''

j(rir2)q _ ^_^ / /ФчЫЫ i Y^ ^ ЛпЫЫ _ Л'чЫ^Л _ i |

'J(ll)T,i — 2 1^0 So / ' . ^.¿2 _ I l^"1 ^г ^г J iri=r2} I '

i=1 (7.61)

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where notations are the same as in Theorem 1.1. For r1 = r2 we get (see (5.42))

M J ( ИГ1Г2) T(>'r>'2)q\2\ _ (T )4 /5 ^ 1

L(0i)T,i J(0i)T,i; Г 16 I 9 ^4i2-l

\ ¿=2

у 1 у (< + 2)3 + (t + l)3 .

^(2,-1)2(2,+ 3)2 ^(2i + l)(2i + 5)(2i + 3)=j- >

From (1.127) and (7.62) we obtain

4 /г q

1

(01)T,i Г- о In

\ ¿=2

^ 1__^ (i + 2)2 + (i + l)2

jfa (2г - l)2(2i + 3)2 ^ (2г + 1)(2г + 5)(2г + З)2

where r1, r2 = 1,..., M.

From (7.57) and (7.59) it follows that

!s[B(Z)]M - Js[B(Z)]$q =

= E B'"W {B{Z)eri,B{Z)er2,B{Z)er3)

ri ,r2,r3,r4GJM

I V^1 / T(rir2r3r4) _ T(rir2r3r4)q\ _ 1 / I(r3r4) _ I(r3r4)q\ _

X I \vJ(1111)T,t J(1111)T,t y 1{ri =r2} \vJ(01)T,t J(01)T,ty

V(ri,r2,r3)

1

( I(1214) _ I(1214)A -. f I(1114) _ I(11\ w p 1 (7 63)

{11=13} \vI(0i)T,t I(0i)T,^ -{12 =13} \vI(0i)T,t I(0i)T,^M w . p . 1 (7 63)

Repeating with an insignificant modification the proof of Theorem 7.1 for the cases k = 2 and k = 4, we obtain

m

2

!s[B(Z)]" - 1a[B(Z)]Mr;q'

<

H

<C(trQ)'(6W(ÇLJl- V C^J+SWB,

\4 / /ii\2__

I 24

j'i?j2 J3,j4=0

where is the right-hand side of (7.62) and

r V^'1 + 1)(2J2 + l)(2j3 + 1)(2j4 TT) 2

( ./ :./ î./. ./ 'V ( ./:./:■,/: ./ • V ' - O^i J

1 u z y

CHnnn = J P.,4 (u) J j (z) J Pj2 (y) J Pji (x)dxdydzdu, -1 -1 -1 -1

where Pj(x) is the Legendre polynomial.

Consider the stochastic integral /4[B(Z)]Mt. Let Conditions 1, 2 of Theorem 7.1 be fulfilled. Let B'(v)(B''(v)(B(v),B(v))) be a 4-linear Hilbert-Schmidt operator mapping from U0 x U0 x U0 x U0 to H for all v G H.

We have (see (7.29))

I4 [B (Z )]MMt = E B'(Z ) (B ''(Z ) (B (Z )eri, B (Z ) er3 ) e^ x

ri,r2 ,r3 ,r4G Jm

T s / t t \

x sj\r.\r..\r.,\r: J J [J dwul] J dw^dw™ w. p. 1. (7.65)

t t t t

From (7.48) and (7.65) we obtain

I4[B(Z)]MMt = E B'(Z) (B''(Z) (B(Z)e11, B(Z^) 613) 614 x

11,12,13 ,146jm

x \/Xr.Xr..Xr:tXr: (^(ini^f + ~~ 1{''i=r2}^((io)T,i) w. p. 1, (7.66)

where

T s

1(101^ = / /(i - T)dwT13)dw(14). (7.67)

tt

Denote by I4[B (Z^^ the approximation of the iterated stochastic integral (7.66), which has the following form

I4 [B (Z )]Mtq = £ B' (Z) (B''(Z) (B (Z )611, B (Z )612) 613) 614 x

11,12,13,146 jM

V A \ \ \ f 7-(»'l»'2»-3»-4)g 1 j{r2rir3ri)q r(r3n)q\ 1 (7 Ro\

XV AHA12 A13 A14 J(iiii)T,t + J(iiii)T,t - ±{11=12}I(i0)T,^ w. p. 1, (7.68)

where the approximations J^n^Ti^, I"(10)'7)i are based on Theorem 1.1 and Legendre polynomials.

For example, from (15.16) we have (here we use the notation I(([0)*T')t from the formula (5.16))

Arir2)q _ T ~ t Jrir2)q _ (T ~ t-Y I 1 Aro) Ari) J(10)T,t. - 2 AAT,t 4 I ^/3^0 c.1 +

, y/i- • 1K;W (/ • \\ f769,

til V(2i + l)(2i + 5)(2i + 3) — l)(2i + 3)y y ' j

where the

approximation J"(i"1 yT^t is defined by (17.611).

Moreover,

m{ ( — = Eq (ri = r2), (7.70)

where Eq is the right-hand side of (7.62) (see (5.42)). From (7.66), (7.68) we have

14[B (Z )]Mt — 14[B (Z )]Mtq =

= ^ B'(Z) (B"(Z) (B(Z)eri,B(Z)eT2) eT3) eT4 faXriA,,A,-3A,-4x

ri,r2,r3,r4£jM

I | 7(rir2r3r4) _ T(rir2r3^)q\ + / T(r2rir3r4) _ J(r2rir3r4)q\ _

X 1 1 J(iiii)T,t J(iiii)T,t I + J(iiii)T,t J(iiii)T,t I

1 r ! i r(r3r4) _ /(r3r4\ w p 1

1{ri=r2} ^ (10)T,t 1 (10)T,t H w. p. 1.

Repeating with an insignificant modification the proof of Theorem 7.1 for the cases k = 2 and k = 4, we obtain

m

/4 [b (Z )]Mt - /4 [b (Z )]Mtq

<

H

< C (tr Q)4 ( 22(4!)2 ( fafa - £ C^, ) + (2!)% j,

V V jl,j2,j3,j4=0 / /

where is the right-hand side of (7.62) and Cj-j-j is defined by (7.64).

Consider the stochastic integral /5[B (Z )]M. Let Conditions 1, 2 of Theorem 7.1 be fulfilled. Let B''(v)(B(v),B'(v)(B(v))) be a 4-linear Hilbert-Schmidt operator mapping from U0 x U0 x U0 x U0 to H for all v G H.

We have (see (7.29)) I:, I HZ)')1 j = J2 B"(Z)(B(Z) eT3, B'(Z)(B(Z) eT2 ) eTl ) eT4 fa A,, A,, A,-3 A,-4 x

ri,r2 ,r3 ,r4GJM

T / s s t

x / ( / dwTr3^ J dwir2)dwTri^ dw(r4) w. p. 1. (7.71)

t \t t t

2

Using the theorem on replacement of the integration order in iterated Ito stochastic integrals (see Theorem 3.1 and Example 3.1) or the Ito formula, we obtain

T

'dwi13W I dwi12)dw^ I dw(,14) =

t

t \t t t

= T(12111314) + T(12131114) + T(13121114)+ = J(iiii)T,t + J(iiii)T,t + J(iiii)T,t +

+ 1 ( I(1214) _ I(1214^ _ 1 I(1114) w p 1 (7 72)

+ -{11=13} lvI(i0)T,t I(0i)T,^ -{12=13}I(i0)T,t w. p. 1, ('.'2)

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where we use the notations from Sect. 7.2 (see (17.16)) and I((d11TT)>t, 1(10)1^ are defined by (7.58), (7.67). ' '

After substituting (7.72) into (7.71), we obtain

I5 [b (Z )]MMt = e B''(Z )(B (Z )613 ,B '(Z )(B (Z )612 )611 )614 x 11,12,13,146 Jm

V /\ \ \ \ I 7(r2'rl'r3'r4) . T(r2i-3i'ii-4) , T(r3r2rir4)

x y A11A12 A13 Ar-4 I J(iiii)T,t + J(iiii)T,t + J(iiii)T,t + + 1{11=13} (I(10}T,t - I(01}T,t) - 1{12=13}I(10}TT,^ w. p. 1 (7.73)

Denote by I5[B (Z^^ the approximation of the iterated stochastic integral (7.73), which has the following form

I5[b(Z)]MMtq = e B''(Z)(B(Z)613, B'(Z)(B(Z)612)611 )614 x

11,12,13,14 6 JM

V /\ \ \ \ I 7(r2?"lr3r4)'? 7-(i'2i,3i,li,4)'? I tO^'^'I^ )<?_,_

XV A11A12 A13A14 I J(iiii)T,t + J(iiii)T,t + J(iiii)T,t + + 1 r , i I(1214)f _ I(1214)A -. I(1114)^ w p 1 (7 74)

+ 1{11=13} \vI(i0)T,t I(0i)T,^ -{12 =13}I(i0)T,H w . p . 1, ('.'4)

where the approximations J(([11i1!14')q, I(o1)T)f, and I(([0)r)i are based on Theorem 1.1 and Legendre polynomials.

From (7.73), (7.74) it follows that

I5[B(Z)]M - I5[B(Z)]MMtq = = ^ B"(Z)(B(Z)eT3, B'(Z)(B(Z)eT2)eTl)eT4y^A,,A,-3A,-4 x

r1,r2 ,r3,r4GJM

X| IJ (r2r1 r3r4) _ J (r2r1 + / J (r2 r3r1r4) _ J (r2 ^r^qA + / J (r3r2r1r4) _ J (r3r2nr4)q\ +

(iiii)T,t J(iiii)T,t y + J(mi)T,t J(iiii)T,t J(iiii)T,t J(iiii)T,t y"

+ 1 /Y I(r2r4) _ I(r2r4)^A _ f I(r2r4) _ r(r2r4)q^ _

+ 1{ri=r3} (i0)T,t 1 (i0)T,ty ^ (0i)T,t 1 (0i)T,^y

— 1 r ! f f(rir4) _ r(rir4^ U w p 1

1{r2=r3} ^1 (i0)T,t 1 (i0)T,ty I w. p. 1.

Repeating with an insignificant modification the proof of Theorem 7.1 for the cases k = 2 and k = 4 and taking into account (7.70), we obtain

m

I5 [B (Z )]MMt -15 [B (Z )]Mtq

2

1 <

H

< C (tr Q)4 f3=(4!)= f il^ - £ C^,) + 3=(2!)X>),

j1 j2j3 j4=0

where is the right-hand side of (7.62) and Cjj-j is defined by (7.64).

Consider the stochastic integral 16[B(Z),F(Z)]Mt. Let Conditions 1, 2 of Theorem 7.1 be fulfilled. We have (see (7.29)) '

I6[B(Z),F(Z)]%t= J2 ^(Z)(JB/(Z)(JB(Z)eTl)eT2)v/Ä^:x

r1,r2GJM

T s t

X / / / dwir1)dwTr2)ds w. p. 1. (7.75)

ttt

Using the theorem on replacement of the integration order in iterated Ito stochastic integrals (see Theorem 3.1 and Example 3.1) or the Ito formula, we obtain

T s t

J J |dwir1 )dwTr2)ds = (T - t)J((1r;rT)t + I((0r1 rTt w.p. 1. (7.76) ttt

After substituting (7.76) into (7.75), we have

11,126 Jm

x ((t - +1(11 w.p.1. (7.77)

Denote by I6[B(Z),F(Z)]Mtf the approximation of the iterated stochastic integral (7.77), which has the following form

I6[B(Z),F(Z)]*ff= J2 F'(Z)(B\Z)(B(Z)eri)er2)y/>^x

ri ,r2GJj

M

X f(T - t)« + 1iorirTl) , (7.78)

(ii)T,t "(0i)T,t;

where the approximations l((111iT))q, J((111T)t are defined by (17.60), (17.61). From (7.77), (7.78) we get

I6[B(Z), F(Z)]Mt - I6[B(Z), F(Z)]ftq =

£ F'(Z)(B'(Z)(B(Z)eri)er2)y/^x

ri,r2Gj

M

V f(T-t) ( J(1112) _ 7(1112)A + f I(1112) _ I(1112)^ w p 1

x 6) \vJ(ii)T,t J(ii)T,^ + \vI(0i)T,t I(0i)T,^ w. p. 1.

Repeating with an insignificant modification the proof of Theorem 7.1 for the case k = 2, we obtain

m

2

Je [B (Z ),F (Z )]M - Je [B (Z ),F (Z )]#*"

<

H

< 2C(2!)2 (tr Q)2^(T - t)2Gq + Eqj ,

where and Eq are the right-hand sides of (5.41) and (7.62) correspondingly.

Consider the stochastic integral J7[B(Z),F(Z^^ Let Conditions 1, 2 of Theorem 7.1 be fulfilled.

Then we have (see (7.29))

I7[B(Z),F(Z)]%t= J2 F'\Z)(B(Z)eri,B(Z)er2)^X^x

ri,r2 £Jm

X / ( / dwTri^ dwTr2) | ds w. p. 1. (7.79)

t t t

From (7.47) and (7.76) we get w. p. 1

T / s

dwTri) J dwTr2) I ds = t t t T T 2

- f /(rir2)ds+ f /(r2ri)ds + li ^ _

tt

(ri r2) i T(r2 ri) , -I

(T — t)

= (T - t) f /(rir2) + T(r2ri^ + T(rir2) + r(r2ri) + "I — ^ A yJ(ll)T,t ~r y j(01)t,i j(01)t,i a{»'1=»'2}

= (T — t) (J(i):r,t J((^):r,t — 1{ri=r2}(T — O) + i r(rir2) , r(r2ri) (T — t)2

"rJ(oi)T,i ^ J(oi)T,i ~r~ J-{n=i-2} 2

_ (T +\ r(ri) j(r2) , T-(rir2) . r(r2ri) (T — t)2

— ^ t) J^tjJ(i)T,t ^ 1{oi)T,t ^ 1{oi)T,t J-{n=r2} 2 ' y'-ou>

After substituting (7.80) into (7.79), we obtain

I7[B(Z),F(Z)]%t= J2 F"(Z)(B(Z)eri,B(Z)er2)-s/X^x

ri,r2£JM

Denote by 17[B(Z),F(Z)]Mtq the approximation of the iterated stochastic integral (7.81), which has the following form

I7[B(Z),F(Z)$f= £ F',(Z)(B(Z)eri,B(Z)er2)^K^x

ri,r2eJM

SX ( (T +\ r(ri) r(r2) , r(rir2)^ j-(r2ri)q -, (T - i)2\

XIU J(i)T,t J(i)Tj "T" J(oi)T,i "T" J(oi)T,i J-{'-i=r2} 2 i'

where the approximation J^i^t is defined by (17.60). From (7.81), (7.82) it follows that

J7[B(Z),F(Z)]M - J7[B(Z),F(Z)]Mq = £ F"(Z)(B(Z)en,B(Z) x

H^GJm

V A ( ( r(?'l?'2) _ r(rl''2)A , ( Arori) _ 7-(r2ri)çr\\ 1

XV AriAr2 ^\vJ(01)T,t J(01)T,^ + VJ(01)T,t J(01)T,V7 w. p. 1.

Repeating with an insignificant modification the proof of Theorem 7.1 for the case k = 2, we obtain

m

J7[B(Z),F(Z)]MM - J7[B(Z),F(Z)]Mq

< 4C(2!)2 (tr Q)2 Eq

H

where Eq is the right-hand side of (7.62).

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Consider the stochastic integral J8[B(Z),F(Z^^ Let Conditions 1, 2 of Theorem 7.1 be fulfilled.

Then we have w. p. 1 (see (7.29)) Is[B(Z),F(Z)]^ = - J2 B'\Z)(F(Z),B(Z)en)er2^K,I^t. (7.83)

ri,r2G Jm

Denote by J8[B(Z),F(Z)]Miq the approximation of the iterated stochastic integral (7.83), which has the following form

h[B(Z),F(Z)]%> = - Y. B"(Z)(F(Z),B(Z)erJen/X^J^

ri,r2G Jm

(7.84)

where the approximation J^Tt is defined by (17.60). From (7.83), (7.84) we get

2

I8[B(Z),F(Z)]£ - ft[B(Z),F(Z)]?f =

r1,r2GJ,

M

Repeating with an insignificant modification the proof of Theorem 7.1 for the case k = 2, we obtain

m

I8[B(Z),F(Z)]Mt - I8[B(Z),F(Z

< C(2!)2 (tr Q)2 E

H

where Eq is the right-hand side of (7.62).

2

7.4 Approximation of Iterated Stochastic Integrals of Miltiplicities 1 to 3 with Respect to the Infinite-Dimensional Q-Wiener Process

This section is devoted to the application of Theorem 1.1 and multiple Fourier-Legendre series for the approximation of iterated stochastic integrals of multiplicities 1 to 3 with respect to the infinite-dimensional Q-Wiener process. These iterated stochastic integrals are part of the exponential Milstein and Wagner-Platen numerical methods for semilinear SPDEs with nonlinear multiplicative trace class noise (see Sect. 7.2). Theorem 7.3 (see below) on the mean-square convergence of approximations of iterated stochastic integrals of multiplicities 2 and 3 with respect to the infinite-dimensional Q-Wiener process is formulated and proved. The results of this section can be applied to the implementation of high-order strong numerical methods for non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise.

7.4.1 Formulas for the Numerical Modeling of Iterated Stochastic Integrals of Miltiplicities 1 to 3 with Respect to the Infinite-Dimensional Q-Wiener Process Based on Theorem 1.1 and Legendre Polynomials

This section is devoted to the formulas for numerical modeling of iterated stochastic integrals from the Milstein type scheme (17.13) and the Wagner-Platen type scheme (7.14) for non-commutative semilinear SPDEs. These inte-

grals have the following form (below we introduce new notations for the stochastic integrals (7.88)-(7.91) and their approximations)

T

Ji[B(Z)]T,t = J B(Z)dWti, (7.85)

T ¿2 t

J2[B(Z)]T,t = A\I I B(Z)dWtldto - S-Jl J B(Z)dWtl , (7.86)

¿ ¿ ¿

T

J3[B(Z),F(Z)]T,t = (T — t)J B'(Z)(AZ + F(Z^dWti

(7.87)

T ¿2

J J B'(Z)^AZ + F(ZdWtidt2, ¿¿

T / ¿2

J4[B(Z), F(Z)]T,t = J F'(Z) J B(Z)dWtJ dt2, (7.88)

¿¿

T / ¿2

1i[B(Z)]T,t = J B'(Z) J B(Z)dWt^ dWt2, (7.89)

¿¿

T / ¿3 / ¿2

12[B(Zfc = J B'(Z) J B'(Z) J B(Z)dW^ ) dW¿J dWt3, (7.90)

T / ¿2 ¿2 \

13[B(Zfc = J B''(Z) J B(ZjdWii J B(Z)dW^ I dW¿2, (7.91)

¿

where Z : ^ ^ H is an F¿/B(H)-measurable mapping, 0 < t < T < T.

Note that according to (03)-(l720), (53), (5~T37): and (5im we can write the following relatively simple formulas for numerical modeling [24], [48

T

Ji[B(Z)]£ = J B(Z)dWM

s

116Jm

/ T t2 T \

J2[B(Z)]« =aIJJ B(Z)iIW^dt> - J B(Z)dW?! =

t t t

(T2v|i/2 E ZtBiZJe.-.v^Cr'', (7.92)

116Jm T

J3[B(Z), F(Z)]Mt = (T - t)J B'(Z)(AZ + F(Z)jdW:

M t1 -

T t2

-J J B '(Z ) ^ AZ + F (Z) J dWMM dt2 =

tt

2

116JM

(7.93)

t / ti

MM

J4[B(Z),F(Z)]« = J F'(Z) I J B(Z)dW« I dt2 =

= ^^ E F'(Z)B(Z)e,, ^ - ^Cin>) , (7.94)

116J

where (011), (i11) (ri 6 Jm) are independent standard Gaussian random variables.

Further, consider the stochastic integrals (I7.89)-([7T91), which are more complicate, in detail.

Let Ii [B(Z)]Mit, I2[B(Z)]Mit, I3[B(Z^ be approximations of the stochastic integrals (7.89)—, which have the following form (see (7.21), (722), and (7.26))

Ii[B(Z)]Mt = / B'(Z) f J B(ZdWM =

B'{Z) {B{Z)eri)er2y/X ri ^r2 ^11)^' (7.95)

ri ,r2 GJm

T / ¿3 / ¿2

12[B(Z^ = / B'(Z ) J B'(Z ) J B(Z)dWM dW^ dW;

M

¿3

ri ^r2 ^r3 J((]L11)T3,Í, (7.96)

ri,r2,r3GJM

T / ¿2 ¿2 \

13[B(Z)]MMi = / B''(Z) J B(Z)dW¿M J B(Z)dWM dWMM =

¿

^ (B(Z)eri,B(Z)eT2)er3y/Kj^X

ri,i2 ,13GJm

(r1r2r3) (r2r1r3)

I n-'+ J(r2rir3) + 1 J(0r3M (7 97)

X Jaii)^ + ^m^ + 1{ri=i2}j(0i)T,^ • ('.97)

Let 1i[B (Z ^¿^[B (Z)] m/, /3[B(Z)]r/ be approximations of the stochastic integrals (I7.95I)-(7T97), which look as follows

ri,l2 GJm

ri ,12,13GJm

(7.98)

= (B{Z)en,B(Z)eT2) eT3\JAriA,.2A,.3 x

ri,r2,r3GJM

/ T(rir2r3)q + T(r2rir3)q + 1 T(0r3)

X ^(111)^ + J(111)T,¿ + 1{ri=r2}J(01)T,¿

where the approximations J^yf, J^m)^, J(mjT ¿f of the stochastic integrals

(OS) are defined by (57136), (5139) and J^ has the form (57137), q

1.

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7.4.2 Theorem on the Mean-Square Approximation of Iterated Stochastic Integrals of Multiplicities 2 and 3 with Respect to the Ininite-Dimensional Q-Wiener Process

Recall that LHS(U0, H) is a space of Hilbert-Schmidt operators mapping from

(2) (3)

U0 to H. Moreover, let LHS(U0,H) and L^(U0,H) be spaces of bilinear and 3-linear Hilbert-Schmidt operators mapping from U0 x U0 to H and from U0 x U0 x U0 to H correspondingly. Furthermore, let

INILhs(Uo,H), II' be operator norms in these spaces.

'¿hs (uo,h )

'lSIS (UO ,H )

Theorem 7.3 [14]-[16

48], [56], [57]. Let Conditions 1,2 of Theorem

7.1 be fulfilled. Furthermore, let

B(v) G Lhs(Uo,H), B'(v)(B(v)) G (Uo,H),

B'(v)(B'(v)(B(v))), B''(v)(B(v), B(v)) G LHS(Uo,H)

for all v G H (we suppose that Frechet derivatives B', B'' exist; see Sect. 7.1). Moreover, let there exists a constant C such that w. p. 1

B (Z )Q"

<C,

Lhs (Uo,H )

B' (Z )(B (Z ))Q-

LHS (uo,h )

<C,

B '(Z )(B '(Z )(B (Z )))Q-B'' (Z )(B (Z ),B (Z ))Q-

lh^s (Uo,H )

LHS (uo,h )

<C,

< C

for some a > 0. Then

m

Ii[B(Z)]T,t - 1i[B(Z)]Mp

<

H

p

<(T-t)2(Cb(trQ)2(i-£ 1

2aN

2 ^ 4j2 - 1

+ Kg ( sup A»

¿G J \Jm

, (7.99)

2

m

12[B(Z)]r>i - 12[B(Z)]

T>t

<(r-i)3|cl(trO)3(J- v CJSM1

j1>j2>j3=0

<

H

2

2a

+ Lq ( sup A»

¿GJ \Jm

, (7.100)

m

13[B(Z)]T>t - 13[B(Z)]Mq

< (T-i)3 (c2(tr Q)3 ; g

<

H

j1 >j2 >j3=0

2aN

E + Mq.sup A

GJ \JM

(7.101)

where p, q G N, C0, C1, C2, Kq, Lq, Mq < to, and

C

v/(2j1 + 1)(2j2 + 1)(2j3 + 1)

j3j2 jl

8

C ■

Cj3j2j1 ,

1 z y

jj = J pj3 (z) J pn (y) J pn (x)dxdydZ -1 -1 -1

where Pj(x) (j = 0,1, 2,...) is the Legendre polynomial.

Remark 7.4. Note that the estimate similar to (7.99) has been derived in

[1761

177] (also see [163]) with the difference connected with the first term on

In [177] the authors used the Karhunen-Loeve

the right-hand side of expansion of the Brownian bridge process for the approximation of iterated Ito stochastic integrals with respect to the finite-dimensional Wiener process (Mil-stein approach, see Sect. 6.2). In this section, we apply Theorem 1.1 and the system of Legendre polynomials to obtain the first term on the right-hand side of (7.99).

Remark 7.5. If we assume that A« < C'i-7 (7 > 1,C' < to) for i G J,

then the parameter a > 0 obviously increases with decreasing of 7

Proof. The estimate (7.99) follows directly from (7.32) for k = 2 (the first term on the right-hand side of (7.99)) and Theorem 1 from [177] (the second

2

term on the right-hand side of (7.99)). Further C3, C4, ... denote various constants.

Let us prove the estimates (7.100), (7.101). Using Theorem 7.1, we obtain

m

12[B (Z fc — /2 [B (Z)]

M,q T^

2m

H

M

/2 [B (Z fc — /2[B (Z )]M

+2m

/2[B (Z — /2[B (Z )]T;

M,q ¿

<

H

+

H

2m

/2[B (Z )]t, — /2[B (Z

+

H

+ C3(T — t)3 (tr Q)3

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6

E Cj . (7.102)

ji J2 J3=0

m

/3[B (Z fc — /3 [B (Z)]

+ 2m

M,q T,¿

2m

H

M

/3 [B (Z fc — /3[B (Z )]M

+

H

/3 [B (Z )]MMi — /3[B (Z )}Mf

(7.103)

H

Repeating with an insignificant modification the proof of Theorem 7.1 for the case k = 3, we have (also see Sect. 7.3.2)

m

/3[B (Z — /3[B (Z )]M;q

<

H

ji,j2,j3=0

1

q

(7.104)

where constant C has the same meaning as constant Lk in Theorem 7.1 (k is the multiplicity of the iterated stochastic integral).

Combining (7.103) and (7.104), we obtain

2

1

2

m

/3[B (Z fc — /3[B (Z)]

M,q T,¿

2m

H

M

/3 [B (Z fc — /3[B (Z )]M

+

H

+ C4(T - i)3 (tr Qf 1 1

q

E C

2

g / v ~ J3j2jl

jij2 j3=0

(7.105)

Let us estimate the right-hand sides of mentary inequality (a + b + c)2 < 3(a2 + b2 + c2), we obtain

and (7.105). Using the ele-

m

/2 [B (Z fc — /2 [B (Z

< 3 ( ET'M + ET'M + ETf) , (7.106)

H

2

m

M

/3[B (Z fc — /3[B (Z )]M

3

^1,M . r>2,M , n3M Gt ¿ + G^ + G

, (7.107)

H

2

where

E i,M =

m

T

¿3

¿2

B'(Z) / B'(Z) / B(Z)d (W¿1 — WM dW¿ J dW-

¿3

H

E2,M = ^ =

m

T

¿3

B (Z) B (Z)

J B(Z)dWMM d (W(2 — WfMf) J dW<3

H

E3,M = Eт,¿ =

m

T

¿3

B (Z) B (Z)

j b(z)dWMM dWMMM d (w<3 — wMM)

H

2

gtm = m

T

t2

B ''(Z )

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J B(Z)dWti J B(Z)d (Wti - WMM) dWt2

H

gtm = m

T

t2

B ''(Z)

B(Z)d (Wti - WM) J B(Z)dWtM dW^

H

gf = m

T

B ''(Z)

' t2 t2 \ IB(Z)dWMMJ B(Zd (Wt2 - WMM)

H

We have

E 1,M =

ET,t =

T

m

is / t2

B'(Z) JB'(Z) N B(Z)d (Wti - WMM) dWi2

.t \t

dta <

Lhs (Uo,H )

T

< C5 / m

ts / t2

B '(Z)

J B(Z)d (Wti - WMM) dWi2

dt.3 =

H

T is

= c5 / i m

t t

B '(Z)

' i2

J B (Z )d (Wti - w«)

dt2dt3 <

Lhs (Uo,H )

T ts < m

tt

t2

J B (Z )d (Wti - WMM)

dt2dt. < (7.108)

H

2a T ts t2

< C^ sup Ai

¿GJ \Jm

m

ttt

B (Z )Q-

dt1 dt2dt3 < (7.109)

Lhs (Uo,H )

2

2a

< C7 ( sup A,) (T - t)

¿GJ \JM

(7.110)

Note that the transition from (7.108) to (7.109) was made by analogy with the proof of Theorem 1 in [177] (also see [163]). More precisely, taking into account the relation Qae, = Aaei, we have (see [177], Sect. 3.1)

m

t2

J B(Z)d (Wti - WM)

H

m

t2

£ V^/B(Z)eidw-

(¿) ti

¿gJ\Jm t

H

t2

£ AJ m

¿gJ\Jm t

B (Z )Q-aQae,

dti =

H

t2

e aj+2a/ m

¿gJ\Jm t

B (Z )Q-ae,

dti =

H

sup A

¿GJ \JM

\ 2a t2 (

; /m £ A.,

/ t ^¿gJ\Jm

2a t2 ✓

< | sup A,) i m^ A,

¿GJ\JM / J [ ¿J

B (Z )Q-ae,

dt1 <

H

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dti =

H

2a t2

sup Ai

¿GJ \Jm

m

B (Z )Q-

dti

Lhs (Uo,H )

(7.111)

Further, we also will use the estimate similar to (7.111). We have

E 2,M = ET,t =

2

T

m

¿3 / ¿2

B(Z) JB(Z) JB(Z)dWM I d (W^ — WiM)

dt3 <

Lhs (Uo,H )

T

< C8 / m

¿3

B (Z)

¿2 \

J b(Z)dwjf d (W¿2 — wmM)

dt3 <

H

2a T ¿3

< Cg ( sup Ai

ieJ\Jm

m

¿¿

¿2

b'(z) jb(z)dwm iq

dt2dt3 <

Lhs (Uo,H )

2

2a T ¿3

< C^ sup Ai

ieJ \Jm

m

¿¿

B'(Z) (B(Z)) Q-

Lhs (Uo,H )

2a

< C9( sup Ai) (T — t)3.

ieJ \Jm

(t2 — t)dt2dt3 <

(7.112)

2

Moreover,

2a

E3M < I sup Ai | x

ieJ \Jm

T

Xm

¿

B' (Z ) ( J B' (Z) ( J B (Z )dWMj dW^ j Q-

dt3 <

Lhs (Uo,H )

2a

< Ci0 sup Ai X

ieJ \Jm

T

m

B'(Z) (B'(Z) (B(Z))) Q-

(t3 — t)2

lh^s (Uo,H )

dt3 <

2

2a

< C11 ( sup A, ) (T - t)3

¿GJ \Jm

(7.113)

Combining (7.102), (7.106), (7.110), (7.112), and (7.113), we obtain (7.100). We have

T

m

G1,M =

GT,t =

t2

B ''(Z)

' t2 t2

J B(Z)dWti J B(Z)d (Wti - WM)

dt3 <

Lhs (Uo,H )

T

< c1w m

t2

B (Z )dWt

H

t2

J B(Z)d (Wti - WM)

dt3 <

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H

T

< C12

m

V

t2

B (Z )dWt

4 U

1/2

X

H

/

X

T t2

< cjj i m

tt

X

T

< C14 y (t2 - t)

t

m

V

B (Z) /

t2

J B(Z)d (Wti - WM)

4

1/2

dt3 <

H

y

1/2

dti x

m

V /

Lhs (Uo,H ) t2

J B(Z)d (Wti - WM)

m

V

t

t2

4

1/2

dt3 <

H

y

J B (Z )d (Wti - WMM)

4 1/2

dt3. (7.114)

H

/

Let us estimate the right-hand side of (7.114). If s > t, then for fixed M G N and for some N > M (N G N) we have

2

i

m

B(Z)d W — WM)

H

m

{( E y/%B(Z)ej(wP-wU

^ \jGJn\Jm

j B (Z )ej \WS w /5

j 'e Jn \Jm

H

£ fafafahXv M¡/B(Z)eJ,B(Z)

j,j',l,l'ejw\jm v

ej, b(Z j (B(Z)ei,B(Z)er ) x

H

Xm 4 ( wj) — w^ fwj') — wij'A fw(l) — wf) fw(l') — wf')

F

¿=

= 3(s — i)2 £ A2m[

jeJw\ Jm v

jeJn\Jm

B (Z )ej

+

H

+(s — t)2 £ Aj Aj' ( m

j,j'eJN \JM (j=j')

B (Z )ej

H

B (Z )ej

+

H

+2( B(Z)ej, B(Z)ej') <

H

< 3(s — t)2 £ A2M

, jeJn\Jm

B (Z )ej

+

H

+ E Aj Aj'm

j,j' eJN\JM (j =j')

= 3(s — t)2m ] ( £ Aj

\j gjn\jm

B (Z )ej

H

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B (Z )ej

<

H

4a

< 3(s — t)2 sup A j m <m £ Aj

ieJn\Jm

, jeJn\Jm

B (Z )Q—aej

<

H

4

s

2

4

2

4

2

4a

< C15(s - t)2 ( sup A^ m

¿gJn\JM

B (Z )Q-

(7.115)

Lhs (Uo,H )

Carrying out the passage to the limit lim in (7.115) and using (7.114), we

N ^TO

obtain

2a

gTM < C16 ( sup A^ (T - t)

¿GJ \Jm

(7.116)

Absolutely analogously we get

2a

GTM < C17 | sup A^ (T - t)

¿GJ \JM

(7.117)

Let us estimate . We have

2a

GTM < ( sup A^ x

¿GJ \JM

T

XIm

t

t2

B''(Z) J B(Z)dWMM J B(Z)dWM I Q

tt

< I sup A^ ^ £ E AiAjA/X V^-7^ / ¿G J j,/G JM

dt2 <

Lhs (Uo,H )

T

X^(t2 - t)2 ( M

B ''(Z )(B (Z )ej,B (Z ^Q"^

+

H

+m

B ''(Z )(B (Z )ej ,B (Z )ej )Q"aei

H

B ''(Z )(B (Z )e/,B (Z ^Q-^

+

H

+m

B ''(Z )(B (Z )ej ,B (Z )e/)Q"aei

x

H

x

B ''(Z )(B (Z )e/,B (Z )ej )Q"aei

dt2 <

H

4

2

2a

< Ci8( sup Ai) (T - t)3

¿GJ \JM

(7.118)

Combining (7.105), (7.107), and (7.116)-(7.118), we obtain (7.101). Theorem 7.3 is proved.

Let us consider the convergence analysis for the stochastic integrals (17.86)-(7.88) (convergence for the stochastic integral (7.85) follows from (7.111) (see Theorem 1 in [177] or [163])).

Using the Ito formula, we obtain w. p. 1 [164

T

J2[B (Z )]T,t =

\ - s + 0 AB(Z)dWs,

T

Ja[B (Z ),F (Z )]T,t =

J (s - t)B'(Z) ^AZ + F(ZdWs

Suppose that

m

B' (Z )( AZ + F (Z ^ Q-

< oo,

Lhs (Uo,H )

m

AB (Z )Q"

<

lhs (uo,h )

for some a > 0.

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Then by analogy with (7.111) we get

m

J2[B(Z)]T,t - J2[B(Z)]M

M t

<

H

2a

< Ci9(T - t)3( sup Ai

¿GJ \JM

m

M

Ja[B(Z),F(Z)]T,t - Ja[B(Z),F(Z)]M

<

H

2

2a

< C2o(T - t)3 sup Ai

yeJ \Jm

where J2[B(Z)]Mt, J3[B(Z),F(Z)]M are defined by (7.92), (7.93).

Moreover, under the conditions of Theorem 7.3 we obtain for some a > 0

m

J4[B(Z),F(Z)]T,t - J4[B(Z),F(Z)]M

H

m

T

t2

J F'(Z) Jb(Z)d (Wtl - WMM) dt

< (T - t) J m J F'(Z) ^|B(Z)d (Wti - Wif)j

T t2

< C2i(T - t) j m j j B (Z )d (Wti - WMM )

<

H

2

H

dt2 <

H

2a T t2

< C2i(T - t) ( sup Ai

ieJ \Jm

m

t t

B (Z )Q"

dt1dt2 <

Lhs (Uo,H )

2a

< C22(T - t)3( sup Ai

ieJ \Jm

where J4[B(Z),F(Z)]Mt is defined by (7.94).

2

Epilogue

The results presented in this book were developed [52], [53] in the form of a software package in the Python programming language that implements the numerical methods (4.65)-(4.69), (4.74)-(4.78) (see Chapter 4) with the orders 1.0, 1.5, 2.0, 2.5, and 3.0 of strong convergence based on the unified Taylor-Ito and Taylor-Stratonovich expansions. At that for the numerical simulation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 we used the formulas from Sect. 5.1, i.e. method based on Theorem 1.1 and

we used the database

multiple Fourier-Legendre series. Note that in with 270,000 exactly calculated Fourier-Legendre coefficients.

Using computational experiments it was shown in [54], [55] (also see Sect. 5.4) that in most cases all the exact formulas from Sect. 1.2.3 for the mean-square approximation errors of iterated Ito stochastic integarls can be replaced by the formula (1.80) for k = 1,..., 5. This allows us to neglect the multiplier factor k! (see the formula (1.127)). As a result, the computational costs for the approximation of iterated Ito stochastic integrals are significantly reduced. For the same reason, we can replace the multiplier factor (k!)2 by k! in the formula (7.32) in practical calculations.

Iterated stochastic integrals are a fundamental tool for describing and studying the dynamics of various types of stochastic equations. In recent years and decades, numerical methods of high orders of accuracy have been constructed using iterated stochastic integrals not only for Ito SDEs, but also for SDEs with jumps [88], SPDEs with multiplicative trace class noise [163], [164], [168], McKean SDEs [178], SDEs with switchings [179], mean-field SDEs [180], Ito-Volterra SDEs [168], etc.

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Похожие темы научных работ по математике , автор научной работы — Kuznetsov Dmitriy Feliksovich

Текст научной работы на тему «MEAN-SQUARE APPROXIMATION OF ITERATED ITO AND STRATONOVICH STOCHASTIC INTEGRALS: METHOD OF GENERALIZED MULTIPLE FOURIER SERIES. APPLICATION TO NUMERICAL INTEGRATION OF ITO SDES AND SEMILINEAR SPDES (THIRD EDITION)»