Mean-Square Approximation of Iterated Itô and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Itô SDEs and Semilinear SPDEs Текст научной статьи по специальности «Математика»

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DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N4, 2021 Electronic Journal, reg. N&C77-39410 at 15.04.2010 ISSN 1817-2172

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

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

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

Dmitriy F. Kuznetsov

Dedicated to My Family

The first edition 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. Available at:

http: //diffjournal.spbu.ru/EN/numbers/2020.4/article.1.8.html

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 1994-2021.

The basis of this book composes on the monographs [1]-[15] and recent author's results [16]-[62].

This monograph (also see books [6]-[11], [14], [15]) 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 (Fourier-Legendre series as well as trigonometric 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 5 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).

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 e N (Chap-

ter 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 G 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.

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 stochastic integrals are subject for study in this book. The mentioned iterated Ito and Stratonovich stochastic integrals are defined by the following formulas

hk (tk)... i ^i(ii)dwt(;i)... dw(ik) (Ito integrals),

hk (tk).. .J 11(t1)dwt(|l)... dwt(ik) (Stratonovich integrals),

t

where h (t) (l = 1,..., k) are continuous nonrandom functions at the interval [t, T] (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: wTi) = f(i) for i = 1,... ,m and w[0) = t, f(i) (i = 1,... ,m) are independent standard Wiener processes, i1,..., ik = 0,1,..., m.

The above iterated stochastic integrals are the specific objects in the theory of stochastic processes. From the one side, nonrandomness of weight functions hi (t) (l = 1,..., k) is the factor simplifying their structure. From the other side, nonscalarity of the Wiener process fT with independent components f(i) (i = 1,..., m) and the fact that the functions h(t) (l = 1,..., k) are different for various I (l = 1,..., k) are essential complicating factors of the structure of iterated stochastic integrals. Taking into account features mentioned above, 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 f((i) (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" [63] 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.

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 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 ^i(s),... (s) = ^(s). 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(s),... (s) 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 ^1(s),^2(s) = 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 Itô 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. DOI: http://doi.org/10.18720/SPBPU72/s17-228 (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 Also we mention the books [6] (2010), [7] (2011), [8] (2011), [9] (2012), [10] (2013), [11] (2017) and [14] (2020), [15] (2021).

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, ^ (s) and ^2(s) = 1) for the multidimensional case: ii,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 method.

In [II] using the Fourier method [I], the attempt was made to obtain the mean-square approximation of elementary iterated Stratonovich stochastic integrals of multiplicities 1 to 3 (k = 1,..., 3, (s),..., ^3(s) = 1) for the multidimensional case: ii,..., ¿3 = 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. 2.6.2, 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). He proposed to use converging in the mean-square sense trigonometric Fourier expansion of the Brownian bridge process (version of the so-called Karhunen-Loeve expansion), which we may use to expand the iterated stochastic integrals. Using this method, the expansions of two simplest iterated Ito stochastic integrals of 1st and 2nd multiplicities into the series of products of standard Gaussian random variables were obtained and their mean-square convergence was proved in [I].

As we noted above, the attempt to develop this idea together with the Wong-Zakai approximation [64]-[66] 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 approximation for the iterated Stratonovich stochastic integrals of 3rd multiplicity in

the monograph [II] (see discussions in Sect. 2.6.2, 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 G 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. Moreover,

the noted expansion of iterated Ito stochastic integrals is reformulated using the Hermite polynomials in Sect. 1.10.

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. For the systems of Haar and Rademacher-Walsh functions the expansions of iterated stochastic integrals become too complex and ineffective for practice [IV].

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 [19] (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 the significant advantage (in comparison with the trigonometric system of functions) in approximation of iterated Ito stochastic integrals for which not all weight functions are equal to 1.

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 [64 -66] (see discussions in Sect. 2.6.2, 6.2).

7. The convergence in the mean of degree 2n (n G 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(t1)... 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 for iterated Stratonovich stochastic integrals of multiplicities 1 to 5 in Chapter 2. The rate of mean-square convergence of approximations of iterated Stratonovich stochastic integrals is found (Sect. 2.12).

11. The results of Chapter 1 (Theorems 1.1, 1.2) and Chapter 2 (Theorems 2.1-2.9) can be considered from the point of view of the Wong-Zakai approximation [64]- [66] 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. 2.6.2, 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 [51], [52] in the form of a software package in the Python programming language. The mentined 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 [51], [52] the database with 270,000 exactly calculated Fourier-Legendre

coefficients.

Dmitriy F. Kuznetsov December, 2021

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

*T

... dw«

t

Wt

J [^(fc)]

/ (¿1—ifc ) T>t' 1(li...lk)T,t

J l^kt,

J [^(fc)]

Pi,—,Pk Ah-ik )p

T,t

(li.-.lk )T,t

J *[^(fc)]

p T*(il-ik)P

Tt /(li...lk)T,t

jMT% J[^]Tiit"-ik)

J'[$]Tfc], J'[$]Trk:

Pn(x)

Hn(x), hn(x)

Hn(x,y) L2(D)

N"Nl2(D)

tr A

II'IIH

(u,v)n lhs (U,H )

I'liffS (U,H )

Stratonovich stochastic integral

Q-Wiener process

iterated Ito stochastic integrals

iterated Stratonovich stochastic integrals

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

T

,dWT

stochastic integral with respect to the Q-Wiener process

Contents

Preface

Acknowledgements

Basic Notations

15

16

1 Method of Expansion and Mean-Square Approximation of Iterated Ito Stochastic 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 1.1.2

1.1.3

1.1.4

1.1.5

1.1.6

1.1.7

1.1.8

1.2

26

27

Introduction....................................................................27

Ito Stochastic Integral........................................................28

Theorem on Expansion of Iterated Ito Stochastic Integrals of Multiplicity

k (k G N)......................................................................30

Expansions of Iterated Itô Stochastic Integrals with Multiplicities 1 to 7

Based on Theorem 1.1 ........................................................45

Expansion of Iterated Ito Stochastic Integrals of Multiplicity k (k G N) Based on Theorem 1.1............................ 56

Comparison of Theorem 1.2 with the Representations of Iterated Ito Stochastic Integrals Based on Hermite Polynomials............ 58

On Usage of Discontinuous Complete Orthonormal Systems of Functions in Theorem 1.1................................ 62

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

68

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

Ito Stochastic Integrals from Theorem 1.1 ................. 69

1.1.10 Conclusions.................................. 74

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.................................. 76

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

for Iterated Itô Stochastic integrals..................... 77

76

1.2.3 Exact Calculation of the Mean-Square Approximation Errors for the Cases k =1,...,5............................... 85

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

Stochastic Integrals Based on Theorem 1.1................. 99

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)....................103

1.4 Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson

Measures Based on Generalized Multiple Fourier Series .............. 109

1.4.1 Stochastic Integral with Respect to Martingale Poisson Measure.....109

1.4.2 Expansion of Iterated Stochastic Integrals with Respect to Martingale

Poisson Measures ............................... 112

1.5 Expansion of Iterated Stochastic Integrals with Respect to Martingales Based

on Generalized Multiple Fourier Series ..............................................11

1.5.1 Stochastic Integral with Respect to Martingale...............119

1.5.2 Expansion of Iterated Stochastic Integrals with Respect to Martingales . 121

1.6 One Modification of Theorems 1.5 and 1.8.....................128

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 . 128

1.6.2 Example on Application of Theorem 1.9 and the System of Bessel Functions 130

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

in Theorem 1.1....................................133

1.7.1 Convergence with Probability 1 of Expansions of Iterated Ito Stochastic

Integrals of Multiplicities 1 and 2 ...................... 133

1.7.2 Convergence with Probability 1 of Expansions of Iterated Itoô Stochastic

Integrals of Multiplicity k (k G N) .....................138

1.7.3 Rate of Convergence with Probability 1 of Expansions of Iterated Ito

Stochastic Integrals of Multiplicity k (k G N) ...............156

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

of Iterated Ito Stochastic Integrals..........................[15'

1.8.1 Formulation and Proof of Theorem 1.1 Modification............15

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

Based on Theorem 1.11 ............................ 166

1.9 Expansion of Multiple Wiener Stochastic Integral Based on Generalized Multiple

Fourier Series ..................................... 168

1.10 Reformulation of Theorems 1.1, 1.2, and 1.13 Using Hermite Polynomials .... 172 2 Expansions of Iterated Stratonovich Stochastic Integrals Based on General-

ized Multiple and Iterated Fourier Series 188

2.1 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicity 2 Based

on Theorem 1.1. The case p1 ,p2 ^ ^ and Smooth Weight Functions......188

2.1.1 Approach Based on Theorem 1.1 and Integration by Parts........188

2.1.2 Approach Based on Theorem 1.1 and Double Fourier-Legendre Series Summarized by Pringsheim Method.....................200

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

216

2.2

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

223

2.2.1

224

2.2.2

2.2.3

2.2.4

2.2.5

239

2.3

2.4

2.5

The Case — oo and Constant Weight Functions (The Case of Legendre Polynomials)............................

The Case p1,p2,p3 — oo, Binomial Weight Functions, and Additional Restrictive Conditions (The Case of Legendre Polynomials)........

The Case p1;p2,p3 — oo and Constant Weight Functions (The Case of Trigonometric Functions)...........................256

The Case p1 = p2 = p3 — o, Smooth Weight Functions, and Additional Restrictive Conditions (The Cases of Legendre Polynomials and Trigonometric Functions)...........................265

The Case p1 = p2 = p3 — o, 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 — o (Cases of Legendre Polynomials and Trigonometric Functions)............................28i

Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 5 Based on Theorem 1.1. The Case p1 = ... = p5 — o and Constant Weight Functions (The Cases of Legendre Polynomials and Trigonometric Functions) . . .....308

Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k G N) Based on Generalized Iterated Fourier Series Converging Pointwise . . . .....326

2.5.1 Introduction..................................326

2.5.2 Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of

273

Multiplicity k (k G N)............................32'

2.5.3

2.5.4

Further Remarks...............................361

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

2.6

Mean-Square Convergence..........................37.

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

2.6.1 Formulation of Hypotheses 2.1-2.3 ....................

38?

38?

2.6.2 2.6.3

Hypotheses 2.1-2.3 from Point of View of the Wong-Zakai Approximation^^

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 ........................... 393

2.7 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 2.7 and 2.8...................399

2.7.1 Introduction..................................399

2.7.2 Another Proof of Theorem 2.7........................400

2.7.3 Another Proof of Theorem 2.8......... ...............405

2.8 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 and Wong-Zakai Type Theorem .............. ......................418

2.9 Modification of Theorem 2.7 for the Case of Integration Interval [t, s] (s G (t, T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 and Wong-Zakai Type Theorem .............. ......................424

2.10 Modification of Theorem 2.8 for the Case of Integration Interval [t, s] (s G (t, T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 4 and Wong-Zakai Type Theorem ............ ........................E33

2.11 Modification of Theorem 2.9 for the Case of Integration Interval [t, s] (s G (t, T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 5 and Wong-Zakai

Type Theorem .................................... 44

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

Stochastic Integrals of Multiplicities 2 to 4 in Theorems 2.2, 2.7, and 2.8 .... 44' 2.12.1 Rate of the Mean-Square Convergence of Expansion of Iterated Strato-

novich Stochastic Integrals of Multiplicity 2 ................ 44

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

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

2.12.3 Rate of the Mean-Square Convergence of Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4................454

2.13 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 4 in Modifications of Theorems 2.27, 2.29, and 2.31 for the Case of Integration Interval [t, s] (s G (t, T])........466

2.14 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k. The

Case ¿1 = ... = ik = 0 and Different Weight Functions ^i(r),..., (т).....I47C

2.15 Comparison of Theorems 2.2 and 2.6 with the Representations of Iterated Stratonovich Stochastic Integrals With Respect to the Scalar Standard Wiener Process.........................................477

2.16 One Result on the Expansion of Multiple Stratonovich Stochastic Integrals of

Multiplicity k. The Case i1 = ... = ik = 1,..., m .................1479

2.17 A Different Look at Hypotheses 2.1-2.3 on the Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k......................483

2.18 Invariance of Expansions of Iterated Ito and Stratonovich Stochastic Integrals from Theorems 1.1 and 2.43 ............................. 488

3 Integration Order Replacement Technique for Iterated Ito Stochastic Inte-

grals and Iterated Stochastic Integrals with Respect to Martingales 495

3.1 Introduction......................................49

3.2 Formulation of the Theorem on Integration Order Replacement for Iterated Ito

Stochastic Integrals of Multiplicity k (k G N) ...................500

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

plicity 2........................................501

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

plicity k (k G N) ...................................506

3.5 Corollaries and Generalizations of Theorem 3.1 ..................510

3.6 Examples of Integration Order Replacement Technique for the Concrete Iterated

Ito Stochastic Integrals................................514

3.7 Integration Order Replacement Technique for Iterated Stochastic Integrals with

Respect to Martingale ................................ 518

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

tic Differential Equations 525

4.1 Introduction......................................525

4.2 Auxiliary Lemmas ..................................531

4.3 The Taylor-Ito Expansion..............................1535

4.4 The First Form of the Unified Taylor-Ito Expansion................1538

4.5 The Second Form of the Unified Taylor-Ito Expansion ..............1541

4.6 The Taylor-Stratonovich Expansion.........................1543

4.7 The First Form of the Unified Taylor-Stratonovich Expansion..........1546

4.8 The Second Form of the Unified Taylor-Stratonovich Expansion.........1550

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

the Classical Taylor-Ito and Taylor-Stratonovich Expansions...........1552

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

Strong Numerical Methods for Itô SDEs......................55'

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

High-Order Strong Numerical Methods for Itô SDEs................56

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 1.1, 2.1—2.10 570

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

Integrals of multiplicities 1 to 6 Based on Legendre Polynomials ......... 570

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

grals of multiplicities 1 to 3 Based on Trigonometric System of Functions .... 61 5.3 A Comparative Analysis of Efficiency of Using the Legendre Polynomials and

Trigonometric Functions for the Numerical Solution of Itô SDEs.........62

5.3.1 A Comparative Analysis of Efficiency of Using the Legendre Polynomials

and Trigonometric Functions for the Integral Jil)^ t............1628

5.3.2 A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Integrals t, J(11)T t, J(oi)T t,

т(»1°) т(п»2гз) 630 J( 10)T,t, J( 111)T,t................................630

5.3.3 A Comparative Analysis of Efficiency of Using the Legendre Polynomials

and Trigonometric Functions for the Integral JO^T2 t...........1638

5.3.4 Conclusions..................................640

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

Stochastic Integrals Based on Theorem 1.1 and Multiple Fourier-Legendre Series 641

5.5 Exact Calculation of the Mean-Square Approximation Errors for Iterated Stratonovich Stochastic Integrals I^Tt, 1*1)т\, 1(oo)tt, 1(ooo)t 3 ............1654

5.6 Exact Calculation of the Mean-Square Approximation Error for Iterated Stratonovich Stochastic Integral I^Ocjo02)i3 ).........................1660

6 Other Methods of Approximation of Specific Iterated Itô and Stratonovich

Stochastic Integrals of Multiplicities 1 to 4 678

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 ..................... 678

6.2 Milstein method of Expansion of Iterated Ito and Stratonovich Stochastic Integrals684

6.3 Usage of Integral Sums for Approximation of Iterated Ito Stochastic Integrals . . 692

6.4 Iterated Ito Stochastic Integrals as Solutions of Systems of Linear Ito SDEs ... 69

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

Integrals ........................................ 698

6.6 Representation of Iterated Ito Stochastic Integrals of Multiplicity k with Respect

to the Scalar Standard Wiener Process Based on Hermite Polynomials ...... 70

6.7 Representation of Iterated Stratonovich Stochastic Integrals of Multiplicity k

with Respect to the Scalar Standard Wiener Process ...............708

6.8 Weak Approximation of Iterated Ito Stochastic Integrals of Multiplicity 1 to 4 . 710

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 724

7.1 Introduction ...................................... 724

7.2 Exponential Milstein and Wagner-Platen Numerical Schemes for Non-Commutative Semilinear SPDEs...............................731

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 735

7.3.3

7.4.2

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 W^ of the Q-Wiener Process...............735

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 ............................744

Approximation of Some Iterated Stochastic Integrals of Miltiplicities 3 and 4 with Respect to the Finite-Dimensional Approximation W^ of the Q-Wiener Process ............................747

7.4 Approximation of Iterated Stochastic Integrals of Miltiplicities 1 to 3 with Respect to the Infinite-Dimensional Q-Wiener Process................758

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.........758

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

762

Epilogue Bibliography

774

775

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 G N. The method of generalized multiple Fourier series for expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity k (k G 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 discontinuous complete orthonormal systems of functions in the space L2([t,T]k), k G 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 G 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 G 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 G 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]-[61] (also see early publications [67] (1997),

68] (1998), [69] (2000), [70] (2001), [71] (1994), [72] (1996)). Note that another

approaches to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals can be found in [62], [73]-[90 .

Specifically, the approach [1]-[61] appeared for the first time in [71], 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 [71], [72

contain rather fuzzy formulations and a number of incorrect conclusions. Note that in [71], [72] 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 [71], [72] are correct for a sufficiently narrow particular case when

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 [67]-[70] (also see [1]-[62]).

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 Theorem 1.1) 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 multiplicity k of iterated Ito stochastic integrals.

On the other side, the results of Chapter 2 [6]-[21], [24], [26], [28], [30]-[37],

40], [41], [43]-[45], [50], [67]-[70] 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 2 [6]-[21], [24], [26], [28], [30]-[37], [40], [41], [43]-[45], [50], the final formulas for expansions of iterated Stratonovich stochastic integrals (of second multiplicity in the general case and of third, fourth, and fifth multiplicity in some particular cases) 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, u) : [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, u) =f 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,u) is measurable with respect to the pair of variables (t,u).

2. The function £(t,u) is Ft-measurable for all t £ [0,T] and £(t, u) 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 j < o for all t £ [0,T].

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

0 = r'N> < t<W> < ... < tNN' = T, max

0 1 N 0<j<N-1

we will define the sequence of step functions

T(N) _ _(N)

^ 0 if N ^ to (1.1)

£(N)(t,u) = £(r)JV >) w. p. 1 for t G

(N)

T(N) T(N) Tj ' Tj+1

where 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) G M2([0,T]) as the following mean-square limit [91], [92] (also see [75])

N1

l.i.m. V£(N> (V,(N),

u

f (Tj+1 -/j

T

= / £/, (1.2)

where £)(t, w) is any step function, which converges to the function £(t, w) i the following sense

in

T

lim / M

0

£(N)(t,u) - £(t,u)

dt = 0.

(1.3)

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

It is well known [91] that the Ito stochastic integral exists as the limit (1.2) and it does not depend on the selection of sequence £)(t,w). Furthermore, the Ito stochastic integral satisfies w. p. 1 to the following properties [91

M< I £t d/t 0

F0 = 0,

M

T

T

£t d/t

0

F

0

Fo > = M < I £t dt

T T

(a£t + ß^t)d/t = a £t d/t + ß fa d/t,

o

o

2

where &, fa G M2([0,T]), a, £ G R1.

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

N —1 T

Lim. £ i(N) j,") j — j') =/ (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 G N)

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 stochastic 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.

Let us consider the following iterated Ito stochastic integrals

t t2

J [^k)]T,t = i (tk)... f ^i(ti)dwt(;i).. .dwi:k), (1.5)

where every ^(t) (l = 1,..., k) is a nonrandom function on [t,T], wTi) = fTi) for i = 1,..., m and w[0) = t, i1,..., ik = 0,1,..., m.

Let us consider the approach to expansion of the iterated Ito stochastic integrals (1.5) [1]-[61] (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 G 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]. Define the following function on the hypercube [t, T]k

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

K (ti,...,tk ) =

0,

otherwise

k k—1

= n ^ (ti) II l<tl+i}, 1=1 1=1

(1.6)

where t1,...,tk G [t,T] (k > 2) and K(t1) = ^1(t1) for t1 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(t1,... ,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(t1,..., tk) G L2([t, T]k) is converging to K(t1,..., tk) in the hypercube [t, T]k in the mean-square sense, i.e.

lim

Pi

Pi Pk

K (ti,...,tk ) — ^ ^ (ti)

ji=0 jk=0 i=i

= 0, (1.7)

L2([t,T ]k )

where

C =

Cjk •••ji =

/k

K (ti,...,tk ^ (ti )dti ...dtk

i=i

[t,T ]k

is the Fourier coefficient, and

(1.8)

m

L2([t,T ]k)

k\ =

( \i/2 J f 2(ti,...,tk )dti ...dtk

\[t,T ]k

/

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

t = t0 < ... < tn = T, AN = max Arj ^ 0 if N ^ to, ATj = Tj+i — t

0<j<N—i

j

(1.9)

k

Theorem 1.1 [1] (2006) (also see [2]-[61]). Suppose that every fa(т) (l = 1,..., k) is a continuous nonrandom function on [t, T] and {fa (x)}°=0 is a complete orthonormal system of continuous functions in the space L2([t,T]). Then

Pi Pk /к

J [^(к)Ь = l.i.m4 E ■■^C^AU Ф

l.i.m. Y, j(Til)AwT;;} • • • j(TiklAwi;;' , (1.10)

where

Gk = Hk\Lk, Hk = {(/i,...,W : 1i,...,1k = 0, 1,...,N — 1}, Lk = {(/i,...,/k) : 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 = | fa (s)dw<° (1.11)

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjfc...j1 is the Fourier coefficient (11.81). AwTj = wTj++1 — wTj (i = 0,1,... ,m), {rj}N=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 fa (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. £ (Tji)Awij;) w.p. 1, (1.12)

where AwTj = wTj++1 — wTj (i = 0,1,..., m), {rj}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 = E . . . E n JW'lWi.j + ^N w. p. 1, (1.13) jk =0 ji=0 l=1

where

s

J [Ml ]s,e = j Ml (T )dwTil) e

and

N— Tjfc+1 s

= E / Mk(s) / Mk-i(T)J[M(k-2)]T,tdwTik-l)dwiik) + jk=0

Tjk Tjk

k-3

+ E G[M«-r+ilN X

r=1

jk-r+1-1 """ r

1 Tjk-r + 1 S

X ^ J Mk-r(s)/ Mk-r-i(T)J[M(k-r-2)]T,tdwTik-r-l)dwiik-r) +

jk-r =0 T. _

j3-1

+G[^<kVE J [M(2)] Tj2+1,TJ2.

j2=0

where

N-1 jk-1 jm+1-1 k

= E E • ■ E n J [Ml]

jk=0 jk-1=0 jm=0 l=m

(Mm,Mm+1, . . . ) ^ №> . . . ,Mk) = Mlk) =

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

l.i.m. = 0. (1.14)

N ^TO

Comparing (1.13) and (1.14), we get

N-1 j2-1 k

J= l.i.m. ^ ... £ II J^ ]jw. p. 1. (1.15)

jk=0 ji=0 /=1

Let us rewrite J[^/]rjl+1,rjl in the form

J №] r,+i, j = M/ (j )Aw j + / (Mt ) - M/(j ))dw^

and substitute it into (1.15). Then, due to the moment properties of stochastic integrals and continuity (which means uniform continuity) of the functions fa (s) (/ = 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 (11.12) 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. £ Sfa ,...,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

/[*]$d=f f...i$(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

(k)

I [<4k] = l.i.m. ^ ...£ $(Tji ,...,Tjk ^Awij.f w.p. 1. (1.19)

jk=0 ji=0 i=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, ¿3 = 1,..., m. We have

T is t2

i K! d=f|| |$(il,i2,t3)dw<;i)dwii2)dwiis) =

t t t N-1 TjS

= I /^(ti,t2,Tjs)dWi(;i)dwi;2)AwT;.s)

js=0

t t

N—i js-i Tj2+i ^

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

N^ js=0 j2=0 j { s

N —i js —i Ti2.,+i / Tj2 \

J +j $(ti,t2,Tjs^irwifAwj

V t j /

N — i js —i j2 i Tj2 + i j + i

N^œ . .

j3=0 j2=0

= l.i.m. £££ / / $(t1,t2,Tj3^w^W^Awj +

N—œ . J J

j3 =0 j2=0 ji=0 T,

N —1 j3 —1 Tj2 + 1 ^

+ l.i.m. ££ / /$(t1,t2,rj3^w^W^Awj. (1.20)

N— ~ j3=0 j2=0 j j 3

'J2 '72

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 —1 j3 —1 Tj2+1 ^ N—1 j3 —1 1

EE I J ^i-'-'-^3Ar,3 < .i/-EE2{Ar»)2}0

j3=0 j2=0 j3=0 j2=0

'72 '72

when N —y to. Here M is a constant, which restricts the module of the function ^(t1, t2, t3) due to its continuity, Arj = rj+1 — Tj.

Considering the obtained conclusions, we have

T is t2

i y y i2,i3)dw<:'»dw«:2>dw«;3> =

t t t N-1 ¿3-1 ¿2- j + 1 j + 1

= l.i.m. EEE / / $(ti,t2,Tj-3)dw( ^dw^Awj

N^œ . „ . „ . „ J J 3

¿3=0 ¿2=0 ji=0

N-1 ¿3-1 ¿2-1 j + 1 TjV+1

t1 t2 j3

= l.i.m. £££ / / ($(Î1,Î2,rJ3) - $(Î1,rJ2, Tj3 ) ) dw<;1)dwi:2)Aw<'3) +

N^œ . . . „ „ ¿3=0 ¿2=0 ¿1=0 T,

j2 j1

N-1 ¿3-1 ¿2-1 j+1 j+1

+l.i.m. £££ / / ($(Î1,ri2,Tj3) - $(ri1 ,r,2,rA))dwi11)dw<22)Aw<;3) +

N^œ . . . „

¿3=0 ¿2=0 ¿1=0 T,

j2 j1

N-1 ¿3-1 ¿2-1

+ l.i.m. Y E E ^(Tj-1,Tj2,Tj3)Aw!;;)Awi;2)Awi;3). (1.21)

N^TO . . .

¿3=0 ¿2=0 ¿1=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 ¿3-1 ¿2-1 Ty; Ty;

EEE I I W1't2,7-,3) - $(¿1,7^2 , T?3 ))2 At-,3 . (1.22)

¿3=0 ¿2 =0 ¿1=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 A-^ < (j = 0,1,...,N - 1), then the distance between points (t^t^T^), (t^T^, Tj3) is obviously less than £(e). In this case

|$(t1,t2,Tj3 ) - $(t1,Tj2 , TJ3 )| <£.

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. E «(j 'II Awj = J'WS, (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

JWTg = /E ...dw

tfc

w. p. 1, (1.24)

where

E

(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

JW?] = E /•••/.....tk

w(il)... dw

(ifc) ifc

w. p. 1, (1.25)

(]i,---,t&) ]

where permutations (ti,..., 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 $(ti5... , tk) E C(Dk) or $(ti5... , tk) is a continuous nonrandom function in the open domain Dk and bounded at its boundary. Then

T t2

M

I [*]

(k) T,t

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

t t

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

Proof. Using standard properties and estimates of stochastic integrals for E M2([t,T]), we have [92]

2'

M

T

T

= / M{|^t|2}dr, t

(1.26)

M

T

& dT

Let us denote

ti+i,---,]fc

T

< (T - t)J M{|^t|2}dT.

tl + 1 t2

$(ti,...,tk)dw];i)

. . dw

(il)

(1.27)

where l = 1,..., k - 1 and £[^..^t = ... ,tk). By induction it is easy to demonstrate that

i, .,tk,t E 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,..., ik = 1,..., m from the proof of Lemma 1.2 we obtain

M

I [*]

(k) T,t

T t2

$2(ti,...,tk )dti --.dtk.

(1.28)

t t

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

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

where

1=1

T

J[pfo = W(s)dw(il), $(ti,...,tk) = HW(ti),

1=1

and the integral J[«]Tkt is defined by the equality (11.161). Proof. Let at first i/ = 0, I = 1,..., k. Let us denote

N—1

Jb/]n = e ^(Tj)Aw(_jl). j=0

Since

IIJ[Wi]n -ft J[w]

T,t =

1=1

1=1

k /1- 1

¿(n J J [Wl]N - J M^t) 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

l=1

1/2

<

2

k

k

2

J [mi]n - J [m/]t,î

< C^ ( M '

/=1

where Ck is a constant. Note that

J[M/]N - J[M/]T>t = E J,

j=0

rj+1

J [Am /]Wj = / (m/ (Tj ) - Ml (s)) dw(il).

n- 1

(1.31)

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

M

N-1

E J] Wj j=0

N-1

M

j=0

J [Ami ]

Tj+i>

+

N-1

+ 6^ M

j=0

J [Ami]

tj+I>

2ï j-i

M

q=0

J [Ami]

Tq+1>

(1.32)

Moreover, since J[Am/]t+1>r is a Gaussian random variable, we have

M

J [Ami]

■j+i/j

rj+i

= / (Mi(Tj) - Ml(s))2ds,

M

4

J [AMl]Tj+1,Tj

/ rJ+1

\

(M/(Tj) - Mi(s))2ds

V-j )

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

I N —1 4

M

N-1

E J[AMl]Tj+1,Tj j=0

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

< 3 J>Tj )2 + 6 £ At^ AT, <

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

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

4

4

2

2

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 Aw j in (11.291) for some I G {1,..., k}

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

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

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

N-1

T

J[w ]n = E (Tj) (Awij°) , J[W1 ]T,t d=f / W1(s)ds. j=0 {

We have

/

M

V

M

V

j=0

M

N-1

J [w1]n - J [^1 ]T,t

1/4

T

¿>1 (Tj)(Awii^)2 - i (s)ds j=0 {

4 1/4

/

Tj+1

N -1 f 2

^ (Tj) (Awj) - (W1(s) - W1(Tj) + W1(Tj)) ds

V

M

N-1 , 2

(Tj) (Awj) - ATj

j=0

1/4

+

\ 4 \

>

/ J /

1/4

<

+

N-1 y

E / (W1 (Tj) - (S)) dS j=0 T

(1.33)

4

4

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

M

N-1 , 2

(Tj) (Aw") - At;

N-1 ( ,

(v (Tj ))4 mH (Aw!;

j=0 N-1

>1 - ATj \ +

N -1 / 2

+6£>i(Tj))2 M (Aw!"1 ^ - ATj

j=0 I v

X

j-1

X > (Vi(Tq q=0

N-1 j-1

22

£ (Vi(Tq))2 M ((Aw<q'^2 - At,

N-1

60 (Tj))4 (ATj)4 +

j=0

+ 24 £ (vi(Tj))2 (ATj)2 £ (vi(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 ^ (s) at the interval [t,T]. Then, taking into account (1.30), (1.31), (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-1 i2-1

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

T(;k) =

h

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

i.i.m. E... E k (Tii,..., Tik )Aw!;i)... AwT;k) =

ik

N

ik =0 ii=0 N 1 N 1

lfm. £... £ K(Tii,..., Tik)Aw<;i>... Aw<;k> =

N

= l.i.m.

N

ik =0 ii =0

N1

E K (Tii,..., Tik )AwT;i)... Aw!;;) = (1.35)

¡i,...,ik=0

Iq =lr; q=r; q,r=i,...,k

4

2

2

T t2

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

t t )

where permutations (tl5... ) 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[M(k)]T,t = £ J.. ^K(ti,...,tk)dwill) ...dw(::) w.p.i, (1.37) (ti,...,t:) t t

where permutations (tl5..., ) 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 in the permutation (t1,..., ), then ir swapped with in the permutation (i1,..., ).

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

T t2

J G(ti,...,tk)dti ...dtk = £ J .. ^G(ti,...,tk)dti ...dtk, (1.38)

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

where permutations (ti5... ) when summing are performed only in the values dti.. . . At the same time the indices near upper limits of integration in the iterated integrals are changed correspondently and G(ti;... ) is the integrable function on the hypercube [t,T]k.

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

J

T,t =

pi

T

t2

E-E

ji=0 jk=0

C

Cjk •••ji

E (j (ti) ••• j (tk)dwt(;i)... dw

(ik) tk

+

t (ti,-,tfc)

T?Pi,...,Pk

+ RT,t

Pi Pk N-1

E . . . E Cjk---ji i.i.m. E j (Tii)... j (Tik )AwT;i)... AwT;k)+

ji=0 jk=0 ii,...,ik=0

lq =lr; q=r; q,r=i,...,k

+ R?it---'Pk = (1.39)

Pi Pk / N-1

E ... E Cjk---ji i.i.m. E j (Tii)... j (Tik )AwT;i)... AwTj)-

—n —n \ n^TO i k

ji=0 jk=0 \ ii,---,ik=0

i.i.m. £ j (Tii )AwT;i)... j (Tik )AwT;k M + (ti,---,tk )GGk

+R?it---'Pk

Pi Pk

E . . . E Cjk---ji X

j'i=0 jk=0

k

XI 11 j) - i.i.m. £ j (Tii )AwT;i)... j (Tik )Aw!;k ^ i=1 (ii,---,ik )eGk

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

where

T / p i Pk k \

E ... k (t1.....tk)-E ..^Cjk-ji n^jl (ti)

i,---,tk) t i V j i=0 jk=0 i=1 /

R?t---'Pk = V I... I IK (ti.....tk) -> '...VCj Ujl (ti )| X

(t 1,...,tk) t t ^ j 1=0 =0

x dwti 1) ...dw(ik), (1.41)

where permutations (ti,..., tk) when summing are performed only in the values dwt(i 1)... 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 (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< (RTr^ % <

T T / Pi Pk k \

< Ck £ ... K (ti,...,tk) ...j i (*<) x

(t k )t t ^ j i=0 jk=0 i=i /

xdt1... dtk =

2

/. / P i Pk k \

= Ck J K(ti,...,tk) ..^Cjk...ji n^ji (tiH dti ...dtk ^ 0

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

(1.42)

if p1,... — oo, where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral J[M(k)]T,t. Theorem 1.1 is proved.

It is not difficult to see that for the case of pairwise different numbers i1,..., ik = 0,1,..., m from Theorem 1.1 we obtain

p i pk

J = l.i.m- E i Zji 1 > ...j'. (1.43)

p l,...,pk —O z-' z-' 7 1 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]-[61]

p1

- (n)

/ . Ci z

p1 —yoO

T,t = l.i.m. £ CjiCji'), (1.44)

p ^ra ' * J i

j =0

P i P2

J^ = Ä £ E Cj2ji ( Zji ')Zj22) - 1{. , = .2=0}l(ji =j2} ) , (1-45)

j i =0 j2=0

Pi P2 P3

=„ £ £ £ j ci:°cj33)-

ji=0 j2=0 j3=0

1{i1=i2=0}1{j1=j2}C]33) - 1 { i 2=i3=0} 1 {j2 = j3 } Cjl ^ - l{ii = i3=0}l{ji=j3}C)2^ I ' (1'46)

(ii)

-(i2)

Pi

P4

(ii)

ji=0 j4=0

J=1

1{ii = i2=0}1{ji=j2}Zj3 Zj4 - 1{ii=i3=0}1{ji=j3}Zj2 Zj4

-1{ii = i4=0}1{ji=j4}Zj2 Zj3 - 1{i2=i3=0}1{j2=j3}Zji Zj4 -

-1{i2=i4=0}1{j2=j4}Zji j - 1{i3 = i4=0}1{j3=j4}Zji j + + 1{ii = i2=0}1{ji=j2}1{i3=i4=0}1{j3=j4} + 1{ii = i3=0}1{ji=j3}1{i2 = i4=0}1{j2=j4} +

+ 1{ii=i4=0}1{ji=j4}1{i2 = i3=0}1{j2=j3} (1'47)

Pi

P5

J [#>]T,( = l.i.m £

ji=0 j5=0

■ji

n zj;

j=I

{ii=i2=0} {ii=i4=0} {i2=i3=0} {i2=i5=0} {i3=i5=0}

+1{ii= i2=0} {ji =j2} {i3 i4=0}

+1{ii= i2=0} {ji =j2} {i4 i5=0}

+1{ii= i3=0} {ji =j3} {i2 i5=0}

+1{ii= i4=0} {ji =j4} {i2 i3=0}

+1{ii= i4=0} {ji =j4} {i3 i5=0}

Z (i3)Z (i4) Z (i5

{ji=j2} Zj3 Z j4 Zj5

Z (i2) Z (i3)Z (i5

{ji=j4}Zj2 Z j3 Zj5

Z (ii)Z (i4) Z (i5

{ j2=j3} Z ji Z j4 Zj5

Z (ii)Z (i3)Z (i4

{j2=j5} Zji Z j3 Zj4

Z (ii)Z (i2 ) Z (i4

{j3=j5 } Zji Zj2 Z j4

Z(i5 { j3=j4} Z j5

Z(i3 {j4=j5} Zj3

Z(i4

{ j2 = j5 } Z j4

Z(i5 { j2 = j3 } Z j5

Z(i2 {j3=j5} Zj2

+ + + + +

1

1

1

1

1

{ii=i3=0}

{ii=i5=0}

{i2=i4=0}

{i3=i4=0}

{i4=i5=0}

1

{ii=i2=0}

1

{ii=i3=0}

1

{ii=i3=0}

1

{ii=i4=0}

1

{ii=i5=0}

1{ji = 1 { j2 1 { j3

1 {j4 = j

1{ji = j

j j

j Z(i2 j3} Zj2

(i2 j5 } ^ j2

(ii j4}S ji

(ii j4} Zji

j5 } Z j4 } Z

j4 } Z

i;)

Z(i4)Z(i5

Zj4 Zj5

Z (i3)Z (i4

Zj3 Zj4

Z (i3)Z (i5

Zj3 Zj5

Z (i2 ) Z (i5

Zj2 Zj5

Z (ii)Z (i2) Z (i3) +

=j5} Zji Z j2 Zj3 +

=j2} j3} j3} j4} =j5}

{i3=i5=0} {i2=i4=0} {i4=i5=0} {i2=i5=0} {i2=i3=0}

Z(i4 { j3 =j5 } Z j'4

Z(i5 { j2 = j4 } Z j'5

Z(i2 { j4 = j5 } Z j2

Z(i3

{ j2 = j5 } Z j'3

Z(i4 { j2 =j3 } Z j'4

) + ) + ) + ) + ) +

4

+ 1 {i1 = i5=0} 1 {j1=j5} 1 {i2=i4=0} 1 {j2=j4}Cj33 + 1 {i1=i5=0} 1 {ji=js} 1 {l3=l4=0} 1 {¿3=74} j + + 1 {i2=i3=0} 1 {j2=j3}1 {i4 = i5=0} 1 {j4=j5}Cj1il) + 1 {l2 = l4=0} 1 {¿2=4} 1{i3 = «5=0} 1 {j3=j5}Cj 1'l) +

+ 1{l2 = l5=0}1{j2=j5}1{l3=l4=0}1{j3=j4}C]111^ , i1-48)

Pi P6 /6

J ^ 6>]T,' = Z ■■• E C^II <f-

j1=0 j6=0 \/=1

1{11= =16=0} j Z ( l2 =j6}Zj2 Z (l3)Z (l4) Z (l5 ) _ Zj3 Zj4 Zj5 {l2 = =l6=0} {j2 = Z(l1 =j6}Zj1 Z (l3)Z (l4 Zj3 Zj4 Z(l5) Zj5 -

1 {«3 = =l6=0} {j3 = ■ iZ( l1 =j6}Zj1 Z(l2)Z(l4)Z(l5) _ Z j2 Zj4 Zj5 {l4 = =l6=0} { j4 = ■ T,z (l1 =j6}Zj1 Z(l2)Z(l3) Zj2 Zj3 Z(l5) Zj5

1{«5 = =l6=0} {j5 = Z (l1 =j6}Zj1 Z (l2) Z (l3)Z (l4) _ Z j2 Zj3 Zj4 {l1 = =l2=0} j Z(l3 =j2 } Z j3 Z(l4)Z(l5 Zj4 Z j5 Z(l6) Zj6

1{l1 = =l3=0} j Z(l2 =j3 } Z j2 Z(l4)Z(l5)Z(l6) _ Zj4 Zj5 Zj6 {l1 = =l4=0} j Z(l2 =j4 } Z j2 Z (l3)Z (l5 Zj3 Z j5 Z(l6) Zj6

1{l1 = =l5=0} j ■ iZ(l2 =j5 } Z j2 Z (l3)Z (l4) Z (l6) _ Zj3 Zj4 Zj6 {l2 =l3=0} {j2 = ■ T,z (l1 =j3}Zj1 Z(l4)Z(l5 Zj4 Z j5 Z(l6) Zj6

A{l2 = =l4=0} {j2 = Z(l1 =j4} Zj1 Z (l3)Z (l5 ) Z (l6) _ Zj3 Zj5 Zj6 {l2 =l5=0} {j2 = Z(l1 =j5 } Zj1 Z (l3)Z (l4 Zj3 Zj4 Z(l6) Zj6

1 {«3z =l4=0} {j3 = ■ t,z (l1 =j4 } Z j 1 Z(l2)Z(l5)Z(l6) _ Z j2 Z j5 Zj6 {l3 =l5=0} {j3 = ■ T,z (l1 =j5 } Z j 1 Z(l2)Z(l4) Zj2 Zj4 Z(l6) Zj6

— 1 {l4 = l5=0} {j4 Z (l1)Z (l2 ) z =j5} Z j1 Z j2 Z (l3)Z (l6) 73 Zj6 +

+ 1 {l1 = = l2=0} {j1 = =j2} {l3 =l4=0} {j3 Z(l5 =j4}Zj5 Z(l6) Zj6 +

+ 1 {l1 = = l2=0} {j1 = j2} {l3 =l5=0} {j3 ■ iZ(l4 =j5}Z74 Z(l6) Zj6 +

+ 1 {l1 = = l2=0} {j1 = =j2} {l4 =l5=0} { j4 = Z(l3 =j5}Zj3 Z(l6) Zj6 +

+ 1 {l1 = = l3=0} {j1 = =j3} {l2 =l4=0} {j2 Z(l5 =j4}Zj5 Z(l6) Zj6 +

+ 1 {l1 = = l3=0} {j1 = =j3} {l2 =l5=0} {j2 Z(l4 =j5}Zj4 Z(l6) Zj6 +

+ 1 {l1 = = l3=0} {j1 = =j3} {l4 =l5=0} {j4 ■ T,z (l2 =j5 } Zj2 Z(l6) Zj6 +

+ 1 {l1 = = l4=0} {j1 = j4} {l2 =l3=0} {j2 Z(l5 =j3}Zj5 Z(l6) Zj6 +

+ 1 {l1 = = l4=0} {j1 = j4} {l2 =l5=0} {j2 Z(l3 =j5}Zj3 Z(l6) Zj6 +

+ 1 {l1 = = l4=0} {j1 = j4} {l3 =l5=0} {j3 ■ T,z (l2 =j5 } Z j2 Z(l6) Zj6 +

+ 1 {l1 = = l5=0} {j1 = =j5} {l2 =l3=0} {j2 Z(l4 =j3}Zj4 Z(l6) Zj6 +

+ 1{H = =¿5=0} =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} j Z(¿1)Z^6) i =j5}Zj1 Zj6 +

+ I{i2 = = ¿4=0} {j2 = =j4} {¿3 = =¿5=0} {j3 = z ^V (¿6)+ =j5} Zj1 Zj6 +

+ I{i2 = = ¿5=0} {j2 = =j5} {¿3 = =¿4=0} {j3 = Z (¿1)Z ^6) + =j4}Zj1 Zj6 +

+!{ie= =¿1=0} {j6 = =j1} {¿3 = =¿4=0} {j3 = Z(¿2)Z(¿5) 1 =j4}Zj2 Zj5 +

+1{i6= =¿1=0} {j6 = =j1} {¿3 = =¿5=0} {j3 = Z(¿2)Z(¿4) 1 =j5}Zj2 Zj4 +

+ !{ie= =¿1=0} {j6 = =j1} {¿2 =¿5=0} {j2 = Z (¿3)Z ^4^) 1 =j5 } Z j3 Z j4 +

+ 1{i6= =¿1=0} {j6 = =j1} {¿2 =¿4=0} {j2 = Z (¿3)Z (¿5) 1 =j4}Zj3 Zj5 +

+ 1{»6= =¿1=0} {j6 = =j1} {¿4 = =¿5=0} { j4 = Z(¿2)Z(¿3)+ =j5 } Zj2 Zj3 +

+ 1{i6= =¿1=0} {j6 = =j1} {¿2 =¿3=0} {j2 = Z^4) Z(¿5) 1 =j3}Zj4 Zj5 +

+ 1{»6= = ¿2=0} {j6 = =j2} {¿3 = =¿5=0} {j3 = Z (¿1)Z (¿4) 1 =j5} Zj1 Z j4 +

+ 1{i6= = ¿2=0} {j6 = =j2} {¿4 = =¿5=0} { j4 = Z (¿1)Z (¿3) 1 =j5}Zj1 Zj3 +

+ 1{i6= = ¿2=0} {j6 = =j2} {¿3 = =¿4=0} {j3 = Z (¿1)Z (¿5) 1 =j4}Zj1 Z j5 +

+ 1{i6= = ¿2=0} {j6 = =j2} {¿1 =¿5=0} j Z ^V (¿4) 1 =j5}Zj3 Zj4 +

+ 1{i6= = ¿2=0} {j6 = =j2} {¿1 =¿4=0} j Z (¿3)Z (¿5) 1 =j4}Zj3 Zj5 +

+ 1{i6= = ¿2=0} {j6 = =j2} {¿1 =¿3=0} j Z(¿4) Z^5) 1 =j3}Zj4 Zj5 +

+ 1{i6= =¿3=0} {j6 = =j3} {¿2 =¿5=0} {j2 = Z (¿1)Z (¿4) 1 =j5} Zj1 Z j4 +

+ 1{i6= =¿3=0} {j6 = =j3} {¿4 = =¿5=0} { j4 = Z (¿1)Z (¿2) 1 =j5}Zj1 Z j2 +

+ 1{i6= =¿3=0} {j6 = =j3} {¿2 =¿4=0} {j2 = Z (¿1)Z (¿5) 1 =j4}Zj1 Z j5 +

+ 1{i6= =¿3=0} {j6 = =j3} {¿1 =¿5=0} j Z(¿2)Z(¿4) 1 =j5}Zj2 Zj4 +

+ 1{i6= =¿3=0} {j6 = =j3} {¿1 =¿4=0} j Z(^¿2 ) Z(¿5) 1 =j4}Zj2 Zj5 +

+ 1{i6= =¿3=0} {j6 = =j3} {¿1 =¿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) 1 =j5}Zj1 Z j2 +

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

+1 {¿6= =¿4=0} {j6 = =j4} {¿2 = =¿3=0} {j2 = Z (¿1)z (¿5)1 =j3}Zj1 zj5 +

+1 {¿6= = ¿4=0} {j6 = =j4} {¿1 = =¿5=0} j Z (»2)z (»3) + =j5}Zj2 Zj3 +

+1 {¿6= = ¿4=0} {j6 = =j4} {¿1 = =¿3=0} j Z (»2)z (»5) + =j3}Zj2 Zj5 +

+1 {¿6= = ¿4=0} {j6 = =j4} {¿1 = =«2=0} j Z (»3)z (¿5) 1 =j2}Zj3 Zj5 +

+1 {¿6= = ¿5=0} {j6 = =j5} {¿3 = =«4=0} {j3 = Z (»1)z (*2) 1 =j4}Zj1 Z j2 +

+1 {¿6= = ¿5=0} {j6 = =j5} {¿2 = =¿4=0} {j2 = Z (*1)Z (;3) 1 =j4}Zj1 Zj3 +

+1 {¿6= = ¿5=0} {j6 = =j5} {¿2 = =¿3=0} {j2 = Z (»1)Z (;4) 1 =j3}Zj1 Z j4 +

+1 {¿6= = ¿5=0} {j6 = =j5} {¿1 = =¿4=0} j Z(¿2)Z(»3)+ =j4}Zj2 Zj3 +

+1 {¿6= = ¿5=0} {j6 = =j5} {¿1 = =¿3=0} j Z (*2)Z (;4) , =j3}Zj2 Zj4 +

+1 {¿6= =¿5=0} {j6 = =j5} {¿1 = = ¿2=0} Z (»3)Z M =j2 } Z j3 Zj4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(1.49)

P1 P7 /7

j[^<7>]T,( = um £c7.JnZ*1'

j1=0 j7=0 \1=1

777

— 1{¿1 = ¿6=0J'l=J6} n Zj — 1{¿2=¿6=0J2=J6} n Zj — 1{¿3=¿6=0J3=J6} n Zj —

1 = 1 1 = 1 1=1 1=1,6 i=2,6 i=3,6

777

— 1{¿4 = ¿6=0J4=J6} n Zj — 1{¿5=¿6=0J5=J6} n Zj — 1{¿1=¿2=0J'l=J2} n Zj —

1=1 1 = 1 1=1 i=4,6 i=5,6 ¿ = 1,2

777

1{¿1 = ¿3=0,j1=jз} Ü '^j^i81) - ^¿^^J!^} Ü '^^¿¿^) - ^¿^^J!^} II j -

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

777

"1{¿2 = ¿3=0J2=J3} jji Zj — 1{¿2 = ¿4=0J2=J4} Zj — 1 {¿2 =¿5 =0,j2 =j5^ Zj —

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

777

"1{¿3 = ¿4=0J3=J4} Zj — 1{¿3 = ¿5=0J3=J5} Zj — 1{¿4=¿5=0J4=J5} Zj

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

-^¿U-1}!! i?'^1 - "^{¿^ = ¿2=0,^7=^2^^^ Cj'^1 - 1{¿7=¿3=0,j7=jз} II Z "

^¿1) i , TT /^¿1) i , , TT Z(¿1)

j1

1=1 1=1 1=1

1=1,7 1=2,7 1=3,7

777

(¿1)

TT ^(¿1) 1 ,

(¿1)

-1{¿7=¿4=0J7=j4} Ii j - 1{¿7=¿5=0,j7=j5} Ü j - ^ = ¿6=0j7=6} Ü Cj'^1 +

1 = 1 1 = 1 1=1

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

+ 1{¿1=¿2=0jl=j2,¿3=¿4=0jз=j4} Zj + 1{¿1=¿2=0,jl=j2,¿3 = ¿5=0,jз=j5} Zj +

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

+ 1{¿1=¿2=0J1=J2,¿4=¿5=0J4=J5} n Zj + 1{¿1=¿3=0J1=J3,¿2 = ¿4=0J2=J4} Zj +

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

+ 1{¿1=¿3=0J1=J3,¿2=¿5=0J2=J5} Zj + 1{¿1=¿3=0,J1=J3,¿4 = ¿5=0,J4=J5} Zj +

+ 1{¿1=¿4=0J1=J4,¿2=¿3=0J2=J3} Zj + 1{¿1=¿4=0,J1=J4,¿2 = ¿5=0,J2=J5} Zj +

-{¿1=¿4=0,jl=j4,¿2 = ¿5=0,j2=j5} 1=5,6,7 1=3,6,7

+ 1{¿1=¿4=0J1=J4,¿3=¿5=0J3=J5} n Zj + 1{¿1=¿5=0J1=J5,¿2 = ¿3=0J2=J3} Zj +

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

7

7

7

+ 1{H = =«5=0,j'1 = =j5,i2 = =«4=0,j2 = =j4} n zj;' 1 /=3,6,7 + 1{;1= =;5=0J1 = =i5,;3 = = ;4=0,j3 = =i4} n zi;'1+ /=2,6,7

+ 1{»2 = =«3=0,j2 = =j3,i4 = =«5=0,j4 = =j5} n j 1 /=1,6,7 + 1{i2 = =;4=0i2= =i4,;3 = = ¿5=0 J3 = j5} n zf+ /=1,6,7

+ I{i2 = =«5=0,j2 = =j5,i3 = =«4=0,j3 = =j4} n j 1 /=1,6,7 + 1{i6 = =;1=0j6= 1 ,;3 =;4=0,i3= j4} n z|'1+ /=2,5,7

+ 1{i6 = =ii=0,j6= =j1,i3 = =«5=0,j3 = =j5} n j 1 /=2,4,7 + 1{i6 = =;1=0j6= 1 ,;2 =;5=0,i2= j5} n zi;"+ /=3,4,7

+ 1{i6 = =ii=0,j6= =j1,i2 = = «4=0,j2 = =j4} n zf /=3,5,7 + 1{i6 = =;1=0j6= 1 ,;4 =;5=0,i4= j5} n zir,+ /=2,3,7

+ 1{i6 = =ii=0,j6= =j1,i2 = = «3=0,j2 = =j3} n i /=4,5,7 + 1{i6 = =;2=0j6= 2 ,;3 =;5=0,i3= j5} n zi; /=1,4,7

+ 1{i6 = = «2=0,j6 = =j2,i4 = = «5=0,j4 = =j5} n Zi ;1 /=1,3,7 + 1{i6 = =;2=0j6= 2 ,;3 =;4=0,i3= j4} n z.i;'1+ /=1,5,7

+ 1{i6 = = «2=0,j6 = =j2,i1 = = «5=0,j'1 = =j5} n zj;1 /=3,4,7 + 1{i6 = =;2=0j6= =j2,;1 = =;4=0,i1= j4} n zi;^ /=3,5,7

+ 1{i6 = = «2=0,j6 = =j2,i1 = =«3=0,j1 = =j3} n zf /=4,5,7 + 1{i6 = =;3=0j6= =i3,;2= =;5=0,i2= j5} n zi;,1+ /=1,4,7

+ 1{i6 = = «3=0,j6 = =j3,i4 = = «5=0,j4 = =j5} n zj /=1,2,7 + 1 { ;6 =;3=0j6= =i3,;2= =;4=0,i2= j4} n zf+ /=1,5,7

+ 1{i6 = =«3=0,j6= =j3,i1 = = «5=0,j1 = =j5} n zj;) /=2,4,7 + 1 { ;6 =;3=0j6= =i3,;1 = =;4=0,i1= j4} n zi; 5+ /=2,5,7

+ 1{i6 = =«3=0,j6= =j3,H = = «2=0,j1 = =j2} n zj;1 /=4,5,7 + 1 { ;6 =;4=0j6= =i4,;3 = =;5=0j3= j5} n zj;^ /=1,2,7

+ 1{i6 = = «4=0,j6 = =j4,i2 = = «5=0,j2 = =j5} n zji'1 /=1,3,7 + 1 { ;6 =;4=0j6= =j4,;2 = =;3=0,i2= j3} n zf+ /=1,5,7

+ 1{i6 = = «4=0,j6 = =j4,i1 = = «5=0,j1 = =j5} n zi;' /=2,3,7 + 1 { ;6 =;4=0j6= =i'4,;1 = =;3=0,i1= j3} n zir1+ /=2,5,7

+ 1{i6 = = «4=0,j6 = =j4,i1 = = «2=0,j1 = =j2} n /=3,5,7 + 1 { ;6 =;5=0j6= =i'5,;3 = =;4=0,i3= j4} n zf+ /=1,2,7

+ 1{i6 = = «5=0,j6 = =j5,i2 = = «4=0,j2 = =j4} n /=1,3,7 + 1 { ;6 =;5=0j6= =i'5,;2 = =;3=0,i2= j3} n zf+ /=1,4,7

+1{i6= =«5=0,j6 = =j5,i1 = =¿4=0 jl = =j4} n zir > /=2,3,7 + 1{»6= =i5=0,i6 = =i5,il = =i3=0,il= =i3} n zii »+ /=2,4,7

+1{i6= =«5=0,j6 = =j5,H = =«2=0,j'l = =j2} n zii' /=3,4,7 + 1{»7 = =H=0J7 = =il,i2 = = i3=0,i2 = =i3} n zii »+ /=4,5,6

+ 1{ir= =il=0j7 = =j'l,i2 = =«4=0,j2 = =j4} n zii'» /=3,5,6 + 1{»7 = =H=0J7 = =il,i2 = = i5=0,i2 = =i5} n zii' »+ /=3,4,6

+ 1{ir= =ii=0,jr= =j'l,i2 = =«6=0,j2 = =j6} n zii'» /=3,4,5 + 1{»7 = = il=0,i7 = =il,i3 = = «4=0,i3= =i4} n zii »+ /=2,5,6

+ 1{ir= =ii=0,jr= =j'l,i3 = = «5=0,j3 = =j5} n zii" /=2,4,6 + 1{»7 = = il=0,i7 = =il,i3 = =i6=0,i3= =i6} n zii »+ /=2,4,5

+ 1{ir= =ii=0,jr= =j1,i4 = = «5=0,j4 = =j5} n zii'» /=2,3,6 + 1{»7 = = il=0,i7 = =il,i4 = =i6=0,i4 = =i6} n zii »+ /=2,3,5

+1{il= = ¿2=0,^7 = =j'l,i7 = = il=0,j5 = =j6} n zii'» /=2,3,4 + 1{»7 = = i2=0,i7 = =i2,il = =«3=0,il= =i3} n zii »+ /=4,5,6

+ 1{ir= = «2=0jr = =j2,i1 = = «4=0,j'l = =j4} n zji» /=3,5,6 + 1{»7 = = i2=0,i7 = =i2,il = =i5=0,il = =i5} n zii »+ /=3,4,6

+ 1{ir= = «2=0,j7 = =j2,il = = «6=0,j'l = =j6} n zji" /=3,4,5 + 1{»7 = = i2=0,i7 = =i2,i3 = =«4=0,i3 = =i4} n zii »+ /=1,5,6

+ 1{ir= = «2=0,j7 = =j2,«3 = = «5=0,j3 = =j5} n zji» /=1,4,6 + 1{»7 = = i2=0,i7 = =i2,i3 = =i6=0,i3= =i6} n zii »+ /=1,4,5

+ 1{ir= = «2=0,j7 = =j2,i4 = = «5=0,j4 = =j5} n zii» /=1,3,6 + 1{»7 = = i2=0,i7 = =i2,i4 = =i6=0,i4 = =i6} n zii »+ /=1,3,5

+ 1{ir= = «2=0,j7 = =j2,i5 = = «6=0,j5 = =j6} n zii» /=1,3,4 + 1{»7 = = *3=0j7 = =i3,il = =«2=0,il = =i2} n zii»+ /=4,5,6

+ 1{ir= = «3=0,j7 = =j3,il = = «4=0,j'l = =j4} n zii» /=2,3,5 + 1{»7 = = «3=0,i7 = =i3,il = =i5=0,il = =i5} n zii»+ /=2,4,6

+ 1{ir= = «3=0,j7 = =j3,il = = «6=0,j'l = =j6} n zii» /=4,2,5 + 1{»7 = = «3=0,i7 = =i3,i2 = =i4=0,i2 = =i4} n zii»+ /=3,5,6

+ 1{ir= = «3=0,j7 = =j3,«2 = = «5=0,j2 = =j5} n zii» /=1,4,6 + 1{»7 = = «3=0,i7 = =i3,i2 = =i6=0,i2 = =i6} n zii»+ /=1,4,5

+ 1{ir= = «3=0,j7 = =j3,i4 = = «5=0,j4 = =j5} n zii» /=1,2,6 + 1{»7 = = «3=0,i7 = =i3,i4 = =i6=0,i4 = =i6} n zii»+ /=1,2,5

+ 1{i7=i3=0,j7=j3,i5=i6=0,j5=j6} ü j + 1{i7=i4=0,j7=j4,H = i2=0,i1=j2} ü j +

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

+ 1{i7=i4=0,j7=j4,i1=i3=0,j1=j3^ Ü j + ^^M^jV^^H^^j'^M ü j +

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

+ 1{гr=г4=0,jr=j4,гl=г6=0,il=j6} H j + 1{гr=г4=0,jr=j4,г2=гз=0,j2=jз} H j +

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

= 1 Ii, , TT

+ 1{í7=i4=0,j7=j4,i2=i5=0,j2=j5} Cj; + ü j +

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

+ 1{i7=i4=0,j7=j4,i3 = i5=0,j3=j5^ Ü Cj, + ü j +

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

+ 1{i7 = i4=0,j7=j4,i5 = i6=0,j5=j6^ Ü j + 1{i7 = i5=0,j7=j5,H=i2=0,i1=j2} ü Cj ; +

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

+ 1{i7 = i5=0,j7=j5,H = i3=0,i1=j3} ü j + 1{i7 = i5=0,j7=j5,H=i4=0,i1=j4} j +

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

^«i) + 1 TT 1

j + M^^^jV^S^^^J^^m X! /=2,3,4 /=1,4,6

(; 1 (; 1

+ 1{i7 = i5=0,j7=j5,H = i6=0,i1=j6} ü j + ü Cj ; +

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

+ 1{i7 = i5=0,j'7=j'5 ,i2 = i4=0,j'2=j'4} H Cj ; + ü Cj ; +

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

+ 1{i7 = i5=0,j7=j5,i3 = i4=0,j3=j4} Cj ; + Cj ; +

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

+ 1{¿7 = «5=0,i7=i5,«4 = «6=0,i4=i6^ H j + 1{«7 = «6=0,i7=i6,«1=«2=0,i1=i2^ H Cj ; +

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

+ 1{i7 = i6=0,j'7=j'6,i1 = i3=0,j'l=j'з^ H j + 1{i7 = i6=0,j7=j6,H=i4=0,i1=j4} ü Cj ; +

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

+ 1{¿7 = «6=0,i7=i6,«1 = «5=0,i1=i5^ H Cj ; + 1{i7 = «6=0,i7=i6,«2=«3=0,i2=i3^ H Cj ; +

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

+ 1{¿7 = «6=0,i7=i6,«2 = «4=0,i2=i4^ H j + 1{«7 = «6=0,i7=i6,«2=«5=0,i2=i5^ H j +

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

+ 1{i7 = i6=0,j7=j6,i3 = i5=0,j3=j5} ü j + ^^^^jV^^M^^J^M ü j +

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

+ 1{¿7=«6=0,i7=i6,«3=«4=0,i3=i4^ H C

(; 1 j

/=1,2,5

^ 1{¿2=¿3=0,j2=j3,¿4=¿5=0,j4=j5,¿6=¿7=0,j6=j7} + 1{i2=i3=0,i^ +

+ 1{г2=гз=0,j2=jз,г4=г7=0,j4=j7,г5=г6=0,j5=j6} + ^^^O^^,^^^^^,^^^^^} + + 1{г2=г4=0,j2=j4,гз=г6=0,jз=j6,г5=г7=0,j5=j7} + ^^^O^^,^^^^^,^^^^^} + + 1{i2=i5=0,j2=j5,iз=i4=0,jз=j4,i6=i7=0,j6=j7} + ^^^O^^^^^^^^^^j^jV} + + 1{i2=i5=0,j2=j5,iз=i7=0,jз=j7,i4=i6=0,j4=j6} + ^^^O^^^^^^^^^^j^jV} + + 1{г2=г6=0,j2=j6,гз=г5=0,jз=j5,г4=г7=0,j4=j7} + ^^^O^^,^^^^^,^^^^^} +

+ 1 iz(il» —

^M^^O^^,^^^^^,^^^^^} J Sil

" ^1{¿l=¿з=0,jl=jз,i4=i7=0,j4=j7,i5=i6=0,j5=j6} + 1{гl=гз=0,jl=jз,г4=г5=0,j4=j5,г6=г7=0,j6=j7} +

+ 1{il=iз=0,jl=jз,i4=i6=0,j4=j6,i5=i7=0,j5=j7} + +

+ 1{¿l=¿4=0,jl=j4,¿з=¿6=0,jз=j6,¿5=¿7=0,j5=j7} + +

+ 1{il=i5=0,jl=j5,iз=i4=0,jз=j4,i6=i7=0,j6=j7} + +

+ 1{il=i5=0,jl=j5,iз=i7=0,jз=j7,i4=i6=0,j4=j6} + +

+1 {¿6 = il =0^6 = 1 ,i3 = i5 =0^3=5 ,i4 = i7=0,j4=j7 } + +

+ 1{il=i7=0,jl=j7,iз=i4=0,jз=j4,i5=i6=0,j5=j6} + +

+ 1 lz (i2»_

+ 1{i1 = i7=0,j1=j7,i3 = i6=0,j3=j6,i4=i5=0,j4=j5^ I i

" ^1{¿1=¿2=0,j1=j2,i4=i5=0,j4=j5,i6=i7=0,j6=j7} + 1{гl=г2=0,jl=j2,г4 = г6=0,j4=j6,г5 = г7=0,j5=j7} +

+ 1{¿1 = ¿2=0,j1=j2,¿4 = ¿7=0,j4=j7,¿5 = ¿6=0,j5=j6} + +

+ 1{il=i4=0,jl=j4,i2=i6=0,j2=j6,i5=i7=0,j5=j7} + +

+ 1{il=i5=0,jl=j5,i2=i4=0,j2=j4,i6=i7=0,j6=j7} + +

+ 1{il=i5=0,jl=j5,i2=i7=0,j2=j7,i4=i6=0,j4=j6} + +

+ 1{i6=il=0,j6=jl,i2=i5=0,j2=j5,i4=i7=0,j4=j7} + +

+ 1{¿1 = ¿7=0,j1=j7,¿2 = ¿4=0,j2=j4,¿5 = ¿6=0,j5=j6} + +

+ 1 iz (i3)_

+ 1{i1 = i7=0,j1=j7,i2 = i6=0,j2=j6,i4=i5=0,j4=j5} I i

^ 1{«1=«2=0,i1=i2,iз=«5=0,i3=i5,«6=«7=0,i6=i7} + 1{«1=«2=0,i1=i2,iз=«6=0,i3=i6,«5=«7=0,i5=i7} +

+ 1{¿l=¿2=0,il=i2,iз=«7=0,iз=i7,«5=«6=0,i5=i6} + ^H^^,^^,^^^^^,^^^^^} + + 1{¿l=¿з=0,il=iз,i2=«6=0,i2=i6,«5=«7=0,i5=i7} + ^H^^,^^,^^^^^,^^^^^} + + 1{¿l=¿5=0,il=i5,i2=«з=0,i2=iз,«6=«7=0,i6=i7} + ^H^^,^^,^^^^^,^^^^^} + + 1{¿l=¿5=0,il=i5,i2=«7=0,i2=i7,«з=«6=0,iз=i6} + 1{г6=гl=0,j6=j^г2=гз=0,j2=jз,г5=г7=0,j5=j7} + + 1{«6=«l=0,i6=il,i2=«5=0,i2=i5,«з=«7=0,iз=i7} + 1{г6=гl=0,j6=j^г2=г7=0,j2=j7,гз=г5=0,jз=j5} + + 1{¿7=«l=0,i7=il,i2=«з=0,i2=iз,«5=«6=0,i5=i6} + 1{г7=гl=0,j7=il,г2=г5=0,j2=j5,гз=г6=0,jз=j6} +

+ 1 IC (;41_

+ 1{¿7 = «1=0,i7=i1,i2 = «6=0,i2=i6,«3=«5=0,i3=i5^ I j

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

+ 1{¿l=¿2=0,il=i2,iз=«7=0,iз=i7,«4=«6=0,i4=i6} + ^n^^j^^,^^^^^,^^^^^} + + 1{¿1 = ¿3=0,i1=i3,i2 = «6=0,i2=i6,«4 = «7=0,i4=i7} + ^H^^jl^,^^^^^,^^^^^} + + 1{¿l=¿4=0,il=i4,i2=«з=0,i2=iз,«6=«7=0,i6=i7} + 1{н=г4=0,j\=j4,г2=г6=0,j2=j6,гз=г7=0,jз=j7} + + 1{¿l=¿4=0,il=i4,i2=«7=0,i2=i7,«з=«6=0,iз=i6} + 1{г6=гl=0,j6=j^г2=гз=0,j2=jз,г4=г7=0,j4=j7} + + 1{«6=«l=0,i6=il,i2=«4=0,i2=i4,«з=«7=0,iз=i7} + 1{г6=гl=0,j6=j^г2=г7=0,j2=j7,гз=г4=0,jз=j4} + + 1{¿l=¿7=0,il=i7,i2=«з=0,i2=iз,«4=«6=0,i4=i6} + ^H^^,^^,^^^^^,^^^^^} +

+ 1 1C (;51_

+ 1{¿7 = «1=0,i7=i1,i2 = «6=0,i2=i6,«3=«4=0,i3=i4^ I j

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

+ 1{¿l=¿2=0,il=i2,iз=«7=0,iз=i7,«4=«5=0,i4=i5} + ^n^^j^^,^^^^^,^^^^^} + + 1{¿l=¿з=0,il=iз,i2=«5=0,i2=i5,«4=«7=0,i4=i7} + ^H^^,^^,^^^^^,^^^^^} + + 1{¿l=¿4=0,il=i4,i2=«з=0,i2=iз,«5=«7=0,i5=i7} + 1{н=г4=0,j\=j4,г2=г5=0,j2=j5,гз=г7=0,jз=j7} + + 1{¿l=¿4=0,il=i4,i2=«7=0,i2=i7,«з=«5=0,iз=i5} + ^H^^,^^,^^^^^,^^^^^} + + 1{¿l=¿5=0,il=i5,i2=«4=0,i2=i4,«з=«7=0,iз=i7} + ^H^^,^^,^^^^^,^^^^^} + + 1{¿7=«l=0,i7=il,i2=«з=0,i2=iз,«4=«5=0,i4=i5} + 1{г7=гl=0,j7=j^г2=г4=0,j2=j4,гз=г5=0,jз=j5} +

+ 1 IC (;61_

+ 1{¿7 = «1=0,i7=i1,i2 = «5=0,i2=i5,«3=«4=0,i3=i4} I j

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 ^ , (1.50)

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

Consider a generalization of the formulas (1.44)-(lTT50) for the case of arbitrary multiplicity k for J)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)), (1-51)

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 (1.51) is a partition and consider the sum with respect to all possible partitions

y ^ g2,---,g2r-1g2r,91---9fc-2r . (1.52)

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

Below there are several examples of sums in the form (1.52)

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

({S1>S2},{91>92})

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

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

E

a

Ö,1Ö,2,<M2q3

({31>32}>{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 1.2 [4] (2009) (also see [5]-[15], [22], [27], [37], [46], [47]). In the conditions of Theorem 1.1 the following expansion

J

l.i.m.

P1 1

i=0

[k/2]

-i)rx

EC-,.., IIj)+E(-D

i=0 \Z=1

r=1

X

E

II

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

=0}l{i32s-1 = i32s }

k-2r

nc

i=1

)

(i.53)

k

r

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

In particular, from (1.53) for k = 5 we obtain

P1 P5 /5

J [^ = l.i.m £ ...£C,5..Jn .'

j1=0 j5=0 \/=1

3

)

iis1 = »g2 =0}1{js1 = } H .

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

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

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

+ E 1{is1 = is2 =0}1ijs1 = js2 }1iis3 = ig4 =0}1ijs3 = jg4

(i91 )

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

The last equality obviously agrees with (1.48).

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! .44l)-(irT5Pl) can be verified in the following way. If ii = ... = i7 = i = 1,..., m and ^i(s),..., ^7(s) = ^(s), then we can derive from (1.441) (150) [2]-[15], [27] the well known equalities

t,t = j; st,t,

■Wi3)kt = i - S6t,At,) , = I - 6$,*AT>i + 3A|,) ,

J[^]T, = i (6% - 104,iAT)i + ,

J^V = i - + lOÖ^A^ - 105^A^)

w. p. 1, where

T T

¿T>t = ^(sf AT,t =

r2'

t t

which can be independently obtained using the Ito formula and Hermite polynomials [99].

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 (1.45)) [2]-[15], [27

(; 1 (; 1

j=i.i.m. ( £ jz};} -E c

jljl

}i,}2=0

jl=0

( P jl-1 / \ P /

'is EE j + ^zi:1zi2) + Ecjljl (

\jl=0 j2=0 7 jl=0 v

/ p jl-1 p / \jl=0 j2=0 jl=0 v

<f) -1

-1 =

= l.i.m.

pis

jl=0 j2=0 1

jl=0 p

1 p 1 p /

o- o„-n n V

2

I J1>J2~ \ j1=j2

1

jl=0 2

/

= £5- [¡{i^n-lie]

1

jl=0

2

2 j l

jl=0

^ - w. p. l.

(1-54)

Let us explain the last step in (1.54). For the Ito stochastic integral the following estimate [94] is valid

q/2

M

T

< M

^ |2dT

(1.55)

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

T

J |£r|2dT < oo w. p. 1,

t

T x q/2 ■

M

^ |2dT

< oo.

p

p

2

2

Since

T

^T>t - E C.1. = / (^(s) - E C.10.1 (s)) f(i)

.1 =0 *t v .1=0

then applying the estimate (1.55) to the right-hand side of this expression and considering that

T 2

J (V(s) -E C.10.1 (s)) ds ^ 0

,sl - Cj(sM ds —

t ji=0

if p —y to, we obtain

T

p

sjf = q - l.i.m. x ' z

i ^(s)dfs(i) = q - l.i.m. E C.1 (f, q > 0. (1.56)

J .^TO „

t .1 =0

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

p—TO

easy to conclude that w. p. 1

^ (g6.1 c«)2 =

This equality as well as Parseval's equality were used in the last step of the formula (1.54).

When k = 3 and p1 = p2 = p3 = p we obtain (see (1.46)) [2]-[15], [27]

J[^(3)]t,< = l^ro- E 6.3.2.1 cj:,c«2)cj:

.1 >.2 >33=0

p

(:)Z (:)Z (:) .3

ppp

E6 Z(:) _ V^ 6 Z(:) _ V^ 6 Z(:)

63331.1 . / v 6.2.2.1 . / v 6.1.2.1 . 31,33=0 31,32=0 31,32=0

= ^ E 63332.1... - E i6. + + C313331 j C^ ) =

\31 >.2 >.3=0 31,33 =0 ^ /

/ P 31-1 .2 1 , \

= ^ EE y V ( + + + + 63132.3 + 6313332 j X

P TO \31=0 32=0 33=0 ^ '

>(:)z(:)z(:) i Xz31 z32 z33 +

p jl-1 / \ + y ^ ^A ( Cj3j1 j3 + Cjlj3j3 + ^^jl j f

A. — n ---n V /

z(;A z(;)+

zj3 / zjl +

jl =0 j3=0 p j l - 1

p jl / \ 2 + £ £ (jj. + CW + j j zj31+

jl =0 j3=0 J

+ £ Cj

p o p

/7 (^3 ^ljljl \ zjl

jl =0 jl,j3 =0

y ^ (Cj3jljl + Cjljlj3 + Cjlj3jl) C

j3

p j l - 1 j 2 - 1

^ E EE Cj, c}2 Q3 zj;1 zj

\jl=0 j2=0 j3=0

p j l - 1

jl=0 j3=0

p jl-1 2 Ji Vsi3 j V ~r 2 Z^ Z^ ./'• •'••< v / /

jl=0 j3=0

1 p X 3 1 p

1 / 3 1

,3 (Ai) g ^A wA jl=0

2

E C}i Cj3 C

j3

j1,j3=0

l.i.m.

pis

6

E

( ; 1 ( ; 1 ( ; 1 Cj1 Cj2 Cj3 j zj2 Cj3 +

j1 j2 j3 = 0 j1=j2 j2=j3 j1=j3

p j1-1 2 p j1-1 2

Ai) < 1st sr r2 r /7WwW+

u V ^ ) ' 2 Z^ Z^ ./'• v / / "r

jl=0 j3=0 jl=0 j3=0

1 p X 3 1 p

- Vr3

g Z^ ii v ^ jl=0

E Cjl Cj3 Cj(

j3

j1,j3 =0

jl,j2,j3=0

p j l - 1

p j l - 1

fi ( 3 E E (^i J Cjf + 3 E E ^ V Sji

«V z (;)+ zj3 +

jl=0 j3=0

+C

jl=0

3 /z(;) jl Vzjl

jl=0 j3=0 3

+

1 p ji-1 2 -, p ji-1 2 ji=0 j3=0 ji=0 j3=0

1 P 3 1 p

+ 6 E (4l)) "2 E ChCh.

ji=0 ji,j3=0

, 3

p \ 3 p p

li- UlE^c: -¿E^IXJ

ji=0 / ji=0 j3=0

= w. p. 1. (1.57)

The last step in (1.57) follows from Parseval's equality, Theorem 1.1 for k = 1, and the equality

lpi.m. Ci <<?] = w. p. 1,

which can be obtained easily when q = 8 (see (1.56)).

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

Cjij2 + Cj2ji = Cji Cj2 , 2Cjiji = Cji , (L58)

Cjij2j3 + Cjij3j2 + Cj2j3ji + Cj2 jij3 + Cj3j2ji + Cj3ji j2 = Cji Cj2 Cj3 , (l-59)

2 (Cjijij3 + Cjij3ji + Cj3ji ji ) = Cji Cj3 , (1-60)

6Cjijiji = Cji. (1.61)

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,...?

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 ].

Furthermore, let us suppose that (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 (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]

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

(1.62)

Suppose that the functions (s) (l = 1,...,k) satisfy the condition (*) and the partition {t^ j^Lo 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

4

2

2

the integrand function is continuous at the interval [Tj , Tj+1], except possibly the point Tj+1 of finite discontinuity.

Let ^ E (0, Arj) be fixed. Due to continuity (which means uniform continuity) of the functions ^(s) (/ = 1,... ) at the interval [Tj, Tj+1 — p] we have

rj+i

(<#(Tj) - (s))2ds =

rj+i-M "j+i

\2 d ^ 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 —^

(1.63)

When obtaining the inequality (1.63) we supposed that ATj < £(e) for all j = 0,1,..., N — 1 (here £(e) > 0 exists for any £ > 0 and it does not depend on s),

b(Tj) — p/(s)| <£

for s E [Tj ,Tj+1 — p.] (due to uniform continuity of the functions ^/(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 (s) is the point Tj+1).

Performing the passage to the limit in the inequality (1.63) when ^ ^ +0, we get

rj+i

(<£z(Tj) - (s))2ds < e2Ar,.

(1.64)

Using (1.64) to estimate the right-hand side of (1.62), we obtain

( N—1

M >

\

E J[A^/j j=0

tj+I>

n- 1

N-1 j-1

< e4( ^(ATj)2 + AtJ <

j=0

j=0 q=0

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

This implies that

M

N-1

E J[A^/j

Tj+1>

j=0

0

(1.65)

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 ^ E (0, Arj2) and p E (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 + , .

\ Tj rJ2 + l-M /

J+J,

V j "ji+i-p /

(^(¿1,^2, Tj3 ) - Tj2 , Tj3))2 dt1dt2 =

/ 22 + i-M22 + i-p 22+i-M 2i + i 2'2 + i Tj2 + i-p TJ2 + i 2i + i \

/ hi hi I + / /.

\ Tj2 2i Tj 2i + i-P TJ2 + i-M 2i Tj2 + i-^Th + i-P J

X

X ($(Î1,Î2,Tj3) - $(Î1,Tj2 ,Tj3)) dtxdÎ2 <

< e2 (At, - m) (At, , - p) + M2p (At, - m) + m2m (At, , - p) + M2mp, (1.66)

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,rj3), (t1 , Tj2, Tj3)). We suppose here that the partition {t,}N=0 contains all discontinuity points of the

function , t2, as points Tj (for each variable with fixed remaining two variables). When obtaining the inequality (1.66) we also supposed that potential discontinuity points of this function (for each variable with fixed remaining two variables) are contained among the points t?1+1, Tj2+1, t?3+1.

Let us explain in detail how we obtained the inequality (1.66). Since the function $(t1;t2,t3) is continuous at the closed bounded set

Q3 = j(t1,i2 ,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, ATj3), 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 , t?2,Tj3) E Q3 is obviously less than 6(e) (ATj < 6(e) for j = 0,1,..., N — 1), then

|$(*1,i2, Tjg ) — $(i1,Tj2 , Tjg )| <e.

This inequality was used to estimate the first double integral in (1.66). Estimating the three remaining double integrals in (1.66) we used the boundedness property for the function $(t1,t2,t3) in the form of inequality

|$(*1,i2, Tjg ) — $(*1,Tj2 , T?^ )| <M.

Performing the passage to the limit in the inequality (1.66) when m, p ^ +0, we obtain the estimate

TJ2 + 1 TJ1 + 1

j ($(*!, *2, Tjg) — $(i1,Tj-2 , Tjg))2 dt1dt2 < e2At?2 At? . Tj2 Tj1 This estimate provides

N — 1 j3 —1 j2 1 Tj2 + 1 j + 1

EEE J J (^(t1,t2,Tj-3 ) — $(t1,Tj-2 , Tjg ))2 dM^Aj <

j3 =0 j2=0 j1=0

j2 j1

N — 1 j3 — 1 j2 1 (T _ t)3

E E E a^J2atjs <

j3=0 j2=0 j1=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 E [(j - 1)/2n, (j - 1)/2n + 1/2n+1)

Vnj(x) = < —2n/2, x E [(j - 1)/2n + 1/2n+1, j/2n)

0,

otherwise

V

T-t

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, (1.67) \/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], ...

and Hilbert spaces

L2([T0,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

CgU = / 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 E 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 E N) of expansions from Theorem 1.1.

According to the notations of Theorem 1.1 (see (1.41)), we have

Rt f",pfc = J[0(k)]T,t - j[^(k)]T;r,pfc =

T t2

= E /.../Rpi...pk(t1,...,tkf 1)...dft;k), (1.68)

(t i,-..,tfc) t t

where

P1 Pk k

Rp1...pfc (t1,... ,tk) d=f k (t1,... ,tk) — £... £ Cjk...j1 j] (ti),

j1=0 jfc=0 1=1

J [^(k)] T,t is the stochastic integral (O), J[^(k)]T1f"'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|(RTF**)2} < Ck J (t1,...,tk)dt1 ...dtk,

[t,T ]k

where Ck is a constant. Assume that

ti t2

nfc11 = / (t1,..., tk)df«;;')... dfti-;1', i = 2, 3,..., k + 1,

t t

nTk) = J... J Rp1..,k(t1,..., tk)df«:')... df«:k), n!k+„( d=f

tt

Using the Ito formula it is easy to demonstrate that

2A t ( / s X 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 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)

M

£t d/r

X

V

/

x / (M{if})1/nds.

to

After raising to power n the obtained inequality and dividing the result by

n— 1

/

M

t \2nl\ i ir d/

V

we get the following estimate

/

2n'

M

iTd/J ^ < (n(2n - 1))n / (M {£s2n})1/nds . (1.69)

Using the estimate (1.69) repeatedly, we have

T

M

(k)

2n

< (n(2n - 1))'

M

n ('-1) '/tk ,t

2n^ \ 1/n

dt J <

< (n(2n - 1))nx

X

/ T / / tk

/ (n(2n - 1))n \t \ \t

m {(£;?)

2n"

1/n

1/n \

dtk- 1

dtk

/

= (n(2n - 1))

2n

t tk

,t t

2n 1 \ 1/n

M (nt(k--20 1 dtk-1dt J < ...

< (n(2n - 1))n(k-1)

T tk ts

t t t

m { (n(

(1) 2,t

2n

1/n

dt3 ... dtk- A =

t

t

t

n

t

t

n

n

n

n

n

T t2

(n(2n - 1))n(k-1)(2n - 1)!!

R 1 (ti,..., tk)dti... dtk I <

p i...pk

,t t

< (n(2n - 1))n(k-1)(2n - 1)!!x

x

/

V[t,T ]k

\

RP 1 ...Pk (t1) . . . 5 tk)dt1 • • • dtk

/

The penultimate step was obtained using the formula

m{ (nsf

t2

= (2n - 1)!! I I r2 1 ...Pk (t1 ,...,tk )dt1

'p i...pk '

which follows from Gaussianity of

t2

nt2,)t = I RPi...Pk (t1,

.,tk )dfi;i).

Similarly, we estimate each summand on the right-hand side of (1.68). Then, from (1.68) using the Minkowski inequality, we finally get

Ml ( RT,r'Pk) 2nj <

<

(

!

k ! V V

(n(2n - 1))n(k-1)(2n - 1)!!

/

\t,T ]k

\ n\ 1/2n\

2n

RP 1 ...Pk (t1' • • • 5 tk)dt1 • • • dtk

y y

/

= (k!)2n(n(2n - 1))n(k-1)(2n - 1)!!x

x

/

\[t,T ]k

\

RP 1 ...Pk (t15 . . . 5 tk)dt1 • • • dtk

(1.70)

/

n

n

n

n

Using the orthonormality of the functions (s) (j = 0,1, 2,...), we obtain

J Rpi...Pfc (ti,..., tk )dti ...dtk =

[t,T ]k

Pi Pk k \ 2

K(ti,..., tk) - ^ ... ^ Cj...ji n (ti) dti... dtk =

[tT]k V ji=0 jk=0 1=1 /

= J K 2(ti,...,tk )dti ...dtk -

[t,T ]k

~ Pi Pk k

2 I K (ti,...,tk ...jij] (ti )dti ...dtk+

[i;T]fc ji=0 =0 1=i

^ /Pi Pk k \ 2

+ M E^ECk ...j^n ^ (ti ) dti ...dtk =

[t T]k Vi=0 jk=0 i=i '

= J K 2(ti,..., tk )dti ...dtk -

[t , t ]k

Pi Pk ~ k

^ ..^Cjk ...ji J K (ti,...,tk (ti )dti ...dtk+

ji=0 jk=0 [t,T]k 1=i

T

Pi Pi Pk Pk

f i i-'i i-'k i'k 'V /»

+EE-EEc, ...ji Cjk...jUl / ^(t(t)dti

' _A •/ rv ' _A ./ rv 7 1

ji=0 ji=0 jk=0 jk=0 i=i t

~ Pi Pk Pi Pk

= I k2(ti,..., tk)dti... dtk -2 £... E cl.ji + E ■ ■. E CI..ji =

[t T]k ji=0 jk=0 ji =0 jk =0

!> Pi Pk

= J K 2(ti,...,tk )dti ...dtk ...Ed ...ji. (1.71)

[t , T ]k ji=0 jk =0

Let us substitute (1.71) into (1.70)

M

J[0(k)]T,t - J

\ 2n Pl,-;Pk \

T,t

<

< (k!)2n(n(2n - 1))n(k-1)(2n - 1)!! x

i p p. \

x

/pi Pk

k 2(t1.....tk № ...dtk •■•£cjk ..ji

~ _A ' _A

\t,T ]k

ji=0 jk =0

J

Due to Parseval's equality

Rpi...pk(t1, ■ ■ ■,tk)dt1.. .dtk =

[t,T

/Pi Pk

K2(t1,...,tk )dt1 ...dtk ..^C2 ...ji ^ 0

' _A ' _A

[t,T

ji=0 jk=0

(1.72)

(1.73)

if pi,... ,pk ^ oo. Therefore, the inequality (1.70) (or (1.72)) 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.

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]-[16], [29] (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 [73]-[76], [83], [84], [87], [88], but the Legendre polynomials.

n

4. As it turned out [1]-[61], 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]-[61]. Another advantages of the application of Legendre polynomials in the framework of the

mentioned problem are considered in [19], [38] (see Sect. 5.3).

5. The Milstein approach [73] (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 [74]-[76], [83], [84], [87], [88]) 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 multiplicity (in the general case) and third multiplicity (for the case ^1(s),^2(s), ^3(s) = 1; i1,i2,i3 = 1,..., m) of the iterated Ito stochastic integrals (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, in practice, we usually choose p1 = ... = pk = p —y to. For iterated series, the condition p1 = ... = pk = p — to obviously does not guarantee the convergence of this series. However, in [74]-[76], [84 the authors use (without rigorous proof) the condition p1 = p2 = p3 = p — to within the frames of the Milstein approach [73] together with the Wong-Zakai approximation [64]-[66] (see discussions in Sect. 2.6.2, 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]-[15], [30] (see Sect. 1.7.1) and for the general case of iterated Ito stochastic integrals of multiplicity k (k E N) in [14], [15], [27],

(see Sect. 1.7.2).

7. The generalizations of Theorem 1.1 for complete orthonormal with weight r(ti)... r(tk) > 0 systems of functions in the space L2([t,T]k) (k G N)

as well as for some other types of iterated stochastic integrals (iterated stochastic integrals with respect to martingale Poisson measures and iterated stochastic integrals with respect to martingales) [1]-[15], [39] are presented in Sect. 1.3-1.6.

8. The adaptation of Theorem 1.1 for iterated Stratonovich stochastic inte-

grals of multiplicities 1 to 5 was carried out in [6]-[21], [26], [28], [30]-[37], [41],

43]-[45], [48], [50] (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], [15], [22], [23], [46], [47] (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 G 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 multidimensional Wiener process, because of we need to take into account all possible combinations of components of the 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 [73]-[78], [83]-[85], [87], [88].

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 1.3 [12]-[16], [29]. Suppose that 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

m{ J [^(k)]T,t - J [^(%) ^ = J K 2(ti,...,tk )dti ...dtk -

[t,T ]k

p p « T t2

?(ifc)

£...£ Cj-k^.j, M J W'(k)]T,f E f j (tk)... i j (ii)dft(il)... f

_A _n I / \ J J

* (1.74)

ji=0 jfc=° ^ (ji,-,jfc) t where

T t2

J i^(k)]T,t = / ^ (tk) ..J ^i(ti)dft(;i)... df(kik),

P P / k \

J i^(k)]T,( = £ .. ^ Cjk .ji n j1 - jj' . (1.75)

ji=0 jfc=0 \l=1 J

^ji^-vj = E j(1.f(T'k^ (1.76)

N4M i k

(ii,...,ifc )€Gfc

the Fourier coefficient Cjk...j has the form (11.81),

T

j = / fy (s)f (1.77)

are independent standard Gaussian random variables for various i or j (i =

E

(jlv-jfc)

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 (ii5..., ik) (see (1.74)); another notations are the same as in Theorem 1.1.

Remark 1.3. Note that

T t2

J[0(kW j(tk)... j(ii)dfT...df,

ti tk

t t T t2 T t2

= ^ J 0k (tk) ..J 0i(ti)dft(;i)... <ifc) J j (tk) ..J 0, (ti)dfi;i)... )

t t t t T t2

= J 0k (tk )0jfc (tk) .J 0i(ti)0ji (ti)dti.. .dtk = Cjk ...ji. tt

Therefore, in the case of pairwise different numbers ii,..., ik from Theorem 1.3 we obtain

M<! (J [0(kj]T,t - J [0(kj]Tt

^ p p : J K2(ti,...,tk )dti ...dtk . (1.78)

[t,T]k j1=0 jk=0

Moreover, if ii = ... = ik, then from Theorem 1.3 we get

M< (J [0(k)]T,t - J [0(k)]Tt

^ P P /

: J K 2(tl, . . . , tk )dtl ...dtk ^ ..^Cjfc...ji £ Cj ...j

[i;T]fc ji=0 j'fc=0 V(ji,-,jfc)

where

E

(j'lv-Jfc)

2

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[0(3)]t,î - J[0(%) ^ = j 022(Î2)| 0?(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 ii,..., ik = 1,..., m and pi = ... = pk = p, we obtain

p p

J

T,t =

-(;l) jl

°jlvJk

(1.79)

jl=0 jk =o

,i=i

For n > p we can write

p n

J

n

T,t

E+E

jl=0 jl=p+b

(E + E K^n zf - sj.;:-)

jk=o jk =p+i.

jk

J=i

=J

T,t

+

(1.80)

-Y(;l ...;k )

Let us prove that due to the special structure of random variables Sj (see (Il .44l)-(fTT50). (I! .531), (I! .76)) the following relations are correct

M II z

(;i) _ n(;l-;kh =o jl jl ,--;jk ' '

1=i

(1.81)

M

n

J=i

A;i) c(;l-;k )

n

j=i

z (;i) — s (il-;*) ji j l,—k

= 0,

(1.82)

where

and

(ji,...,jk) G Kp, (ji) e Kn\Kp

Kn = {(ji,..., jk) : 0 < ji,..., jk < n}

k

k

p

n

p

Kp = {(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=i

N-1

An) c(ii...ifc)

ji,...,jfc

l.i.m.

Nico

£ j(Tii) •••0jk(Tikf ... AfTk) =

=0

iq = ir ; q=r; q,r=i,...,fc

T

t2

E / j(tk)...¡j(ii)dfi:')...ff> w.p. 1

?(ifc )

(1.83)

(ji,...,jfc) t

where

£

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); another notations are the same as in Theorem 1.1.

So, we obtain (1.81) from (1.83) due to the moment property of the Ito stochastic integral.

Let us prove (1.82). From (1.83) we have

0

M

" - CiMi IK:J - j::

/(:i...ifc) Jfc

(:)

J = i

,i=i

( T t2

M\ E E / j(tk) •••/ j(ti)dft

i,u i™ . . . ff )X

(j'i,...,j'fc) (j'i,...,j'k) t T t2

x /j(tk) .. jtf...<:fc)

<

tt

T T

< E / j (tk)^jk (tk)dtk . . J i (ti)0j(ti)dti

(j'i,...,j'k) t t

k

k

= E 1{j 1=j '}... 1{jk=jk}, (1.84)

(j iv-JD

where 1A is the indicator of the set A. From (1.84) we obtain (1.82).

Let us explain (1.84) for the cases k = 2 and k = 3 in detail. We have ( T t2

m E E J j&)/fyj(ti)dfi;1 )df(;2)x

T t2

x / fy-2 (t2^ fyj i(ti)dfi; 1 f tt T T

= J fy2 (s)fyj2 fyj1 (5)fyj'(5)d5+

tt

T T

+ 1{i 1 = i2^ fy2 fyj1 (s)fyj2 (s)ds = tt

= 1{j 1 =ji}1{j2=j'2} + 1{i 1 = i2} • 1{j2=j '}1{j 1 =j'2}, (1.85) T t3 t2

M £ £ f fyj3(is)/ j(t2)/ fyj1 (iif 1 'df^'df

T t3 t2

x| j №>)/ j te)/ fyj ¡(ti)f 1 ff

ttt T T T

= J fy3 (s)fyj3 fyj2 (s)fyj2 (s)dsy fyj1 (5)fyji(s)d5 +

ttt

T T T

+1{i 1=i2} y fyj3(s)fyj31 (s)fyj2fyj2 (s)fy i(s)ds+ ttt T T T

+ 1{i2 = i3^ 1 (s)fy fy2 (s)fyj3 (s)dsy fy3 (s)fyj2 (s)ds +

ttt

T T T

/ j(s) j(s)dW j(s) j (s)dW j(s) j(s)ds+ t t t T T T

+I{ii=i2=is^ y j (s) j(s)d^ j (s)0j2(s)d^ j(s) j (s)ds+ ttt T T T

+ I{ii=i2 = i3 } / j (s) j (s)ds / j (s) j (s)dW j (s) j (s)ds =

1

{j3 =j'3}1{j2=j2}J-{ji=ji} ^ J-{ii = i2K J-{j3 =jH{ji=j2H{j2

+ 1{i2=i3} ^ 1{ji=ji}1{j2=j'}1{j3 =j'2} + 1{ii = i3} ^ 1{ji=j'} 1 {j2 =j'2} 1 {j3 =ji} + + 1{ii=i2=i3} • 1 {j2 = j3} 1 {ji = j2} 1 {j3= ji} + + 1{ii=i2=i3} ^ 1 {ji =j'3} 1 {j3 = j'2} 1 {j2 = ji} . (1.86)

1

+ 1

1

1

1

From (1.85) and (1.86) we get

IEE

M< y > I j (h)J hi (ti)d^i)d^2)X

lv(j'i,j2) (j'i,j'2) t t

T t2

xy j fe)/ j (ti)dC f

tt

< 1{ji=ji}1{j2=j2} + 1{j2=ji}1{ji=j2}

<

E 1 {j' i = j' i} 1 {j2 = j'2 } ' (j'i,j'2)

M

T t3 t2

E E Jjjfe)/j(ti)df';i)df;(;2)df(:3)x

(j'i,j2,j3) (j'i,j'2,j'3) t T t3

t2

x / j(tsW 0j'2(t2W j(ti)dft(iii)dft(2i2)dft(3i3)

<

^ 1 {j3 = j3} 1 {j2 = j'2} 1 {ji =ji} + 1 {j3 = j3} 1 {ji = j'2}1 {j2=ji} +

+ 1{j2=j3}1{j 1 =j2}1{j3=j + 1{j 1 =j3}1{j3=j2}1{j2=j il

+ 1{j 1 =j i} 1{j2=j3} 1{j3=j2} + 1{j 1 = } 1 {j2 = j2 } 1 {j3 = j i} + 1{j3=j '} + 1{

E 1{j 1 =j 1} 1 {j2 = j'2 } 1 {j3 = j'3 } ' (j1 ,j2,j3)

where we used the relation

T

(s)ds = 1{g=q}, g, q = 0,1, 2 ...

From (1.82) we obtain

M

= 0.

Due to (1.75), (1.79), and (1.80) we can write

e[^e1^=J- J

l.i.m. e[^ft1'" = JW^kt - Jfo^Tt = e^f/-

n—>00 ' ' '

We have

0

M

p+1 T,t

J

M

{(e W'fö1 - e^'%1"*+ew^fö1'") j №-<%}

<

M

{(e^sy - e №(k)Cn) J [^(%}| + |m {e [^(k)CnJ [^(k)iT,t}

MJ

T,t

J

T,t ' J

<

<t /lM (J [#k)]T,t - J [^(k)]"t

M J

2

< ^ m| (j[0(k)]T,t - J[0(k)]T,O }x

<

< K^M | - ^ 0 if ^^oo, (1.87)

where K is a constant.

From (1.87) it follows that

m{ £ [0(k)]T+iJ [0(%} =0

or

m{ (j[0(k)]T,t - J[0(k)IT,) j[0'%} = 0.

The last equality means that

M { J[0(k>]T,f J[0(%} = M { (j[0<%)^ . (1.88)

Taking into account (1.88), we obtain

M { (J[0(k)]T,t - J[0(%) ^ = M { (J[0(k)]T,) ^ +

+M { (J[0(k)]^)^ - 2M { J[0(k)]T,tJ[0(k)]^} = M { (J[0(k)]t,)^ -

-M { J [0(k)]T,tJ [0(%} = = J K2(ti,...,tk )dti ...dtk - m{ J [0(k)]T,tJ [0(k)IT,t} . (1.89)

[t,T ]k

Let us consider the value

M { J[^'k»]T,( J[V<%} .

The relations (1.75) and (1.83) imply that

T t2

f f t> t> j= E... Ec,., E j.(tk)... I,(ii)df'; 1 >...df«:'».

j 1=0 jfc=0 (, 1,...,jfc) t t

(1.90)

After substituting (1.90) into (1.89), we finally get

m{ J [^(k)]T,t - J [^(%) ^ = J K2(ii,...,ik )dii ...dtk -

[t,T ]'

p p f T t2

- E • •. £ C:, M J[#>]„£ J fyj' (ik) - . J j (ii)df;(: 1'... dft:*

j 1=0 jfc=0 [ (j 1,-,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)]r,( - J№<%)2} d=f Ep,

lK llL2([t,T]k) = J ^2(ti5 . . . 5 tk )dti . . . dtk d=f ik [t,T ]k

The case k = 1

In this case from Theorem 1.3 we obtain

EP = ii-£ c2 .

j =0

p

The case k = 2

In this case from Theorem 1.3 we have

(I). ii = «2:

Ep = 12 j, (1.91)

ji ,j2=0

(II). ii = «2 :

p

Ep = ^2 - Y Cj2ji - ^ Cj2 ji Cjij2 . (1.92) ji,j2 =0 ji,j2=0

Note that from (1.75), (1.83), (1.85), (1.88), and (1.89) we obtain

p p

Ep = ^2 - Y Cj2ji - 1{ii=i2} Y Cj2ji Cjij2. (1.93) ji ,j2=0 ji,j2 =0

Obviously, the relation (1.93) is consistent with (1.91) and (1.92).

Example 1.1. Let us consider the following iterated Ito stochastic integral

T t2

= / / fif (1.94)

tt

where ii, i2 = 1,..., m.

Approximation of the iterated Ito stochastic integral (1.94) based on the expansion (1.10) (Theorem 1.1, the case of Legendre polynomials) has the following form

t_t I ^ 1

jinh)'P ' •-(i-i^ Jin) X > -L

An)Aii) i v^ (A^A^ — A^A^A _ i 1

So SO _ ] Sj Si —1 y } I •

(1.95)

(00)T,t- 2 \S0 So ^ Jlw^l

i=i -

Note that (1.95) has been derived for the first time in [67] (1997) (also see 68]-[70]) with using the another approach. This approach will be considered in

Sect. 2.5. Later (1.95) was obtained [1] (2006), [2]-[61] on the base of Theorem 1.1.

p

Using (1.91), we get

- «5)s} = ^ (5 - g tJ ■ (L96)

where ¿1 = ¿2 •

It should be noted that the formula (1.96) also has been obtained for the first time in [67] (1997) by direct calculation.

The case k = 3 In this case from Theorem 1.3 we obtain

(I). ii = ¿2, ¿1 = ¿3, ¿2 = ¿3 :

<1 = J~ ^ C 2

EP = /3 - £ Cj. (1.97)

j 1 ,j2,j3=0

(II). ii = ¿2 = ¿3 :

Ep = /3 - ^ Cj3j2j 1 ( E Cj3j2j 1 ) , (l-98)

j1,j2,j3=0 V(ji,j2 ,j3)

(III).1. ¿1 = ¿2 = ¿3 :

p p

Ep = /3 - Cj3j'2ji - E jl j2 Cj3j2j1 , (l-99)

j1,j2 ,j3=0 j1,j2,j3=0

(III).2. ¿1 = ¿2 = ¿3 :

pp Ep = /3 - ^ Cj3j'2j'1 - ^ Cj2j3j1 Cj3j2j1 > (1-100)

j1,j2 J3=° j1j2,j3=0

(III).3. ¿1 = ¿3 = ¿2 :

pp

Ep = /3 - ^ Cj3 j2j1 - E Cj3j2 j1 Cj1j2j3 • (1-101)

j1,j2 J3=° j1j2,j3=0

p

It should be noted that from (1.75), (1.83), (1.86), (1.88), and (1.89) we obtain

p

Ep = - Y Cj23j2ji -

ji J2,j3=0 p

-1{i i = i 2} y ^ Cj3j2 ji Cj3j'ij2 -

j'i,j2,j3=0 p

j2j3ji

"1{»2 = »3} Cj3j2 jiC

j'i,j2,j3=0

p

"1{»i = »3} ^ ^ Cj3j2 ji Cj'ij2j3" j'i,j2,j3=0

p

1{»i=»2=»3} ^ y Cj3j2j'i (Cj2j'ij3 + Cj'ij3j2) . (1.102)

ji J2,j3=°

Obviously, the relation (1.102) is consistent with (1.97)-(1.101).

Note that the cases k = 2 and k = 3 (excepting the formula (1.98)) 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 <*>, (1.103)

t t t

where ii, i2, i3 = 1,..., m.

Approximation of the iterated Ito stochastic integral (1.103) based on Theorem 1.1 (the case of Legendre polynomials and pi = p2 = p3 = p) has the following form [1] (2006), [2]-[61]

ji

p

EC 1 z (ii)Z (i2 )z(i

Cj3j2j'i I zj'i zj2 zj3 J2,j3 =0 V

1 (iii2i3)p = V"^ C I z(ii)Z(i2)z^ _ 1 1 z(i3)_

J(000)T,t Z^ Cj3j2j'H Sj'i z j2 zj3 1{ii = i2} 1{ji =j2}zj3

-1{i2=i3} 1{j2=j3} zj(ii) - 1{ii = i3} 1{ji=j3}zj(22) ) ' (1.104)

where

VM±ME±JMä±Jl{T _ tf'2chhh, (i,ü5)

C- ■ ■ =

WJ3J2J1 g

1 z y

j = j Pj3 (z) J Pj2 (y) J Pj1 (x)dxdydz, -1 -1 -1

where Pj(x) is the Legendre polynomial (f = 0,1, 2,...). For example, using (1.99) and (1.100), we obtain

21 (T - t)3 ^2 v- ^ ^

lvl | (000)T,t J(000)T,t J \ 6 A' ./:;./: / Z^ ( ./':t./'\/Vt ./:•./: ./ "

j1,j2,j3=0 j1,j2,J3=0

where ¿1 = ¿2 = ¿3,

2| (T - t)3 ^2 v- ^ ^

lvl | ^J(000)T,i J(000)T,i J I ~ 6 Z^ ./:;./: / Z^ * ./: ./:■,/ ( ./:■,/: ./ •

j1j2,j3=0 j1,j2,J3=0

where ¿1 = ¿2 = ¿3.

The exact values of Fourier-Legendre coefficients Cjjj can be calculated for example using computer algebra system Derive [1]-[15], [30] (see Sect. 5.1,

Tables 5.4-5.36). For more details on calculating of Cjjj using Python programming language see [51], [52].

For the case ¿1 = ¿2 = ¿3 it is convenient to use the following well known formula

¿0^ = ^-i)3/2((^1,)3-3C«")) W.p.l.

The case k = 4 In this case from Theorem 1.3 we have (I). ¿1,..., ¿4 are pairwise different:

p

= ^ - Cj4...ji,

j1,...j4=0

(II). ii = «2 = «3 = «4:

Ep = ^4 - Y, Cj4-j'i ( Cj4-j'i

j'iv",j4=0 V(j'i,-,j4)

(III).1. «i = «2 = «3, «4; «3 = «4 :

ep

^4 - ^ Cj4...ji ( Yj Cj4-

j'iv",j4=0 V(j'i,j2)

(1.106)

(III).2. «i = «3 = «2, «4; «2 = «4 :

Ep

E4

^4 - ^ Cj4...ji ( Yj Cj4-

j'iv",j4=0 V(j'i,j3)

(1.107)

(III).3. «i = «4 = «2, ¿3; «2 = «3 :

E4p

^4 - ^ Cj4...ji ( Yj Cj4...

j'i,...,j'4=0 V(j'i,j4)

ji

(1.108)

(III).4. «2 = «3 = «i,«4; «i = «4 :

Ep E4

^4 - Y, Cj4.. ji ( X] Cj4...

j'iv",j4=0 V(j2,j3)

ji

(1.109)

(III).5. «2 = «4 = «i, «3; «i = «3 :

E4p

^4 - ^ Cj4...ji ( Yj Cj4...

j'i,...,j'4=0 V(j2,j4)

(1.110)

(III).6. «3 = «4 = «i, «2; «i = «2 :

Ep E4

^4 - ^ Cj4...ji ( ^ Cj4...j'i

j'i,...,j'4=0 V(j3,j4)

(1.111)

p

p

p

(IV).1. «i = «2 = «3 = «4:

Ep = I4 - E Cj4...ji E Cj4...ji ), (1.112)

ji,...,j'4=0 V(ji ,j2,j3)

(IV).2. «2 = «3 = «4 = «i:

ep = I4 - e Cj4...ji e Cj4...ji), (1.113)

ji,...,j'4=0 V(j2 ,j3,j4)

(IV).3. «i = «2 = «4 = «3:

Ep = I4 - E Cj4...ji E Cj4...ji ), (1.114)

jiv",j4=0 V(ji ,j2,j4)

(IV).4. «i = «3 = «4 = «2:

Ep = I4 - E Cj4...ji E Cj4...ji ), (1.115)

jiv",j4=0 V(ji,j3,j4)

(V).1. «i = «2 = «3 = «4:

ep = 14 - e Cj4...ji E e Cj4...ji)), (1.116)

j'iv">j'4=0 V(ji,j2A(j3,j4)

(V).2. «i = «3 = «2 = «4:

ep = 14 - E Cj4...jJ E E Cj4...ji)), (1.117)

j'iv">j'4=0 \(j'i ,j3A(j2,j4)

(V).3. «i = «4 = «2 = «3:

ep = 14 - E Cj4...jJ E E Cj4...ji)). (1.118)

j'iv",j4=0 \(j'i ,j4A(j2,j3)

p

p

p

p

p

p

p

The case k = 5

In this case from Theorem 1.3 we obtain

(I). ¿1,..., ¿5 are pairwise different:

p

EP = /5 - £ C|...ji,

j1,"-,j5=0

(II). ¿1 = ¿2 = ¿3 = ¿4 = ¿5:

p

Ep = /5 - ^ Cj5...j1 ( E C35-3 1

j1,...j5=° V(j1,...,j5)

(III). 1. ¿1 = ¿2 = ¿3, ¿4, ¿5 (¿3, ¿4, ¿5 are pairwise different):

p

Ep = /5 - ^ Cj5...j1 ( E Cj5...j1

j1,...,j5=0 V(j1,j2)

^(jU2 )

(III).2. ¿1 = ¿3 = ¿2, ¿4, ¿5 (¿2, ¿4, ¿5 are pairwise different):

p

Ep = /5 - E Cj5...ji ( E Cj5...ji

j1,---,j5=0 V(jl,j3)

(III).3. ¿1 = ¿4 = ¿2, ¿3, ¿5 (¿2, ¿3, ¿5 are pairwise different):

Ep = - E Cj5...j1 ( E Cj5...

j1,...,j5=0 V(j1,j4)

(III).4. ¿1 = ¿5 = ¿2, ¿3, ¿4 (¿2, ¿3, ¿4 are pairwise different):

p

Ep = /5 - ^ Cj5.. j1 ( E Cj5...j1

j1,...,j5=0 V(j1,j5)

(III).5. ¿2 = ¿3 = ¿1, ¿4, ¿5 (¿1, ¿4, ¿5 are pairwise different):

Ep = /5 - ^ Cj5...j1 ( E Cj5... j1,...,j5=0 V(j2,j3)

p

p

(III).6. «2 = «4 = «i, «3, «5 («i, «3, «5 are pairwise different):

Ep = ^5 - E Cj5.. ji ( E Cj'5... ji?...?j5=0 V(j2,j'4)

(III).7. «2 = «5 = «i, «3, «4 («i, «3, «4 are pairwise different):

p

Ep = ^5 - E Cj5.. ji ( E Cj's..^i

j'iv"j5=0 V(j2,j5)

(III).8. «3 = «4 = «i, «2, «5 («i, «2, «5 are pairwise different):

Ep = ^5 - E Cj5.. ji ( E Cj5... ji?...?j5=0 V(j3,j4)

(III).9. «3 = «5 = «i, «2, «4 («i, «2, «4 are pairwise different):

p

Ep = ^5 - E Cj5.. ji ( E Cj5...ji

j'i,...,j'5=0 V(j3,j5)

(III).10. «4 = «5 = «i, «2, «3 («i, «2, «3 are pairwise different):

Ep = ^5 - E Cj5.. ji ( E Cj5...

j'i,...,j'5=0 V(j4,j5)

(IV).1. «i = «2 = «3 = «4, «5 («4 = «5):

p

Ep = ^5 - E Cj5.. ji ( E Cj5...ji

j'i,...,j5 =0 \(j'i ,j2,j3)

(IV).2. «i = «2 = «4 = «3, «5 («3 = «5):

p

Ep = ^5 - E Cj5.. ji ( E Cj5...ji

j'i,...,j5 =0 \(j'i ,j2,j4)

p

p

p

(IV).3. ¿1 = ¿2 = ¿5 = ¿3, ¿4 (¿3 = ¿4):

p

Ep = /5 - Cj5...j1 ( E Cj5...j^ ,

j1,...,j5 =0 V(j1 ,j2 ,j5 ) '

(IV).4. ¿2 = ¿3 = ¿4 = ¿1, ¿5 (¿1 = ¿5):

p

Ep = /5 - ^ Cj5...j1 ( E Cj5...j^ , j1,...,j5 =0 V(j2 ,j3 ,j4 ) '

(IV).5. ¿2 = ¿3 = ¿5 = ¿1, ¿4 (¿1 = ¿4):

p

Ep = /5 - ^ Cj5...j1 ( E Cj5...j1 ) , j1,...,j5 =0 V(j2 ,j3 ,j5 ) /

(IV).6. ¿2 = ¿4 = ¿5 = ¿1, ¿3 (¿1 = ¿3):

p

Ep = /5 - ^ Cj5...j1 ( E Cj5...j1 ) ,

j1,...j5 =0 V(j2 ,j4 ,j5 ) /

(IV).7. ¿3 = ¿4 = ¿5 = ¿1, ¿2 (¿1 = ¿2):

p

Ep = /5 - ^ Cj5...j1 ( E Cj5...j1 ) , j1,...,j5 =0 V(j3 ,j4 ,j5 ) '

(IV).8. ¿1 = ¿3 = ¿5 = ¿2, ¿4 (¿2 = ¿4):

p

Ep = /5 - ^ Cj5...j1 ( E Cj5...j1 ) ,

j1,...,j5 =0 \(j1 ,j3 ,j5 ) /

(IV).9. ¿1 = ¿3 = ¿4 = ¿2, ¿5 (¿2 = ¿5):

p

Ep = /5 - ^ Cj5...j1 ( E Cj5...j1 ) ,

j1,...,j5 =0 V(j1 ,j3 ,j4 ) '

(IV).10. «i = «4 = «5 = «2, «3 («2 = «3):

Ep = ^5 - E Cj5.. ji ( E Cj5...

j'i,...,j'5 =0 V(j'i ,j4,j5)

(V).1. «i = «2 = «3 = «4 = «5:

Ep = ^5 - E Cj5.. ji ( E Cj5...ji

ji ,...,j5=0 \(j'i,j2,j3 ,j4)

(V).2. «i = «2 = «3 = «5 = «4:

Ep = ^5 - E Cj5.. ji ( E Cj5...

ji ,...,j5=0 \(j'i,j2,j3 ,j5)

(V).3. «i = «2 = «4 = «5 = «3:

Ep = ^5 - E Cj5.. ji ( E Cj5...ji

ji v"j5=0 Mj'i,j2,j4 ,j5)

(V).4. «i = «3 = «4 = «5 = «2:

Ep = ^5 - E Cj5.. ji ( E Cj5...

ji ,...,j5=0 \(j'i,j3,j4 ,j5)

(V).5. «2 = «3 = «4 = «5 = «i:

Ep = ^5 - E Cj5.. ji ( E Cj5...j'i

ji v"j5=0 \(j2,j3,j4 ,j5)

(VI).1. «5 = «i = «2 = «3 = «4 = «5:

ep = i5 - £ ¿v., £ £Cj,..

ji?...?j5=0 \(j'i ,j2A(j3,j4)

p

p

p

p

p

p

p

(VI).2. ¿5 = ¿1 = ¿3 = ¿2 = ¿4 = ¿5:

ep = /5 - £ Cj5 J£ £j

j1v",j5=° V(j1 ,j3) ^(j2,j4)

(VI).3. ¿5 = ¿1 = ¿4 = ¿2 = ¿3 = ¿5:

Ep = /5 - ^ Cj5...j1 ( E ( E Cj5...j1

j1,...,j5=0 v(j1 ,j4A(j2,j3)

(VI).4. ¿4 = ¿1 = ¿2 = ¿3 = ¿5 = ¿4:

Ep = /5 - £ £ £j

j1,...,j5=0 v(j1 ,j2A(j3,j5)

(VI).5. ¿4 = ¿1 = ¿5 = ¿2 = ¿3 = ¿4:

Ep = /5 - E Cj5...j1 ( E I E Cj5...j1 j1,...,j5=0 v(j1 ,j5) V(j2 ,j3)

(VI).6. ¿4 = ¿2 = ¿5 = ¿1 = ¿3 = ¿4:

ep = /5 - £ cj j£ £j

j1,...,j5=0 V(j2 JsAjua)

(VI).7. ¿3 = ¿2 = ¿5 = ¿1 = ¿4 = ¿3:

Ep = /5 - E Cj5...j1 ( E I E Cj5...j1 j1,...,j5=0 v(j2 ,j5A(j1,j4)

(VI).8. ¿3 = ¿1 = ¿2 = ¿4 = ¿5 = ¿3:

ep = /5 - £ e,5 J£ £c,...

j1,...,j5=0 v(j1 ,j2A(j4,j5)

p

p

p

p

p

p

p

(VI).9. «3 = «2 = «4 = «i = «5 = «3:

Ep E5

^5 - E Cj5 . . j i E E Cj5 . . j i

j'i,...,j'5=0

j ,j4) \(j'i,j5)

(VI).10. «2 = «i = «4 = «3 = «5 = «2:

Ep = ^5 - E Cj5...ji E E Cj5...ji

j'i,...,j'5=0

j ,j4^(j3,j5)

(VI).11. «2 = «i = «3 = «4 = «5 = «2:

Ep E5

^5 - E Cj5 . . j i E E Cj5 . . j i

j'i,...,j'5=0

j ,j3^(j4,j5)

(VI).12. «2 = «i = «5 = «3 = «4 = «2:

Ep = ^5 - E Cj5...ji E E Cj5...ji

j'iv"j5=0

j ,j5) \(j3,j4)

(VI).13. «i = «2 = «3 = «4 = «5 = «i:

Ep = ^5 - E Cj5...ji E E Cj5...ji

j'i,."j5=0

j ,j3) \(j4,j5)

(VI).14. «i = «2 = «4 = «3 = «5 = «i:

Ep = ^5 - E Cj5...ji E E Cj5...ji

j'iv"j5=0

j ,j4) \(j3,j5)

(VI).15. «i = «2 = «5 = «3 = «4 = «i:

E5p

^5 - E Cj5 . . j i E E Cj5 . . j i

j'i,...,j5=0

j ,j5^(j3,j4)

p

p

p

p

(VII).1. ¿1 = ¿2 = ¿3 = ¿4 = ¿5:

ep = /5 - £ cj w1 £ £ j

j1,...,j5 =0 V(j4,j5A(j1,j2 ,j3)

(VII).2. ¿1 = ¿2 = ¿4 = ¿3 = ¿5:

Ep = /5 - E Cj5...j1 ( E ( E Cj5...j1 j1,...,j5 =0 V(j3,j5A(j1,j2 ,j4)

(VII).3. ¿1 = ¿2 = ¿5 = ¿3 = ¿4:

Ep = / - E Cj5 ...j1 ( E I E Cj5...j1 j1 v",j5=0 V(j3,j4A(j1 ,j2 ,j5 )

(VII).4. ¿2 = ¿3 = ¿4 = ¿1 = ¿5:

Ep = /5 - E Cj5...j1 ( E I E Cj5...j1

j1,...,j5 =0 V(j1,j5A(j2,j3 ,j4)

(VII).5. ¿2 = ¿3 = ¿5 = ¿1 = ¿4:

ep = /5 - £ jJ^I Z j

j1,...,j5 =0 V(j1,j4A(j2,j3 ,j5 )

(VII).6. ¿2 = ¿4 = ¿5 = ¿1 = ¿3:

Ep = /5 - E Cj5...j1 ( E I E Cj5...j1

j1,...,j5 =0 V(j1,j3A(j2,j4 ,j5 )

(VII).7. ¿3 = ¿4 = ¿5 = ¿1 = ¿2:

ep = /5 - £ C, £ j

j1,...,j5 =0 V(j1,j2A(j3,j4 ,j5 )

p

p

p

p

p

p

(VII).8. = ¿3 = ¿5 = ¿2 = i4:

Ep

E5

j1,-,j5 =0

^(j2j4) \(j1,j3 ,j5 )

(VII).9. ¿1 = ¿3 = ¿4 = ¿2 = ¿5:

Ep = ^5 - E Cj5 .j1 ( E I E Cj5...j1

j1,-,j5 =0 \(j2,j5 A(j1,j3 ,j4)

(VII).10. ¿1 = ¿4 = ¿5 = ¿2 = ¿3:

Ep = ^5 - E Cj5 .j1 ( E ( E Cj5...j1 j1,-,j5 =0 Mj^j^ Mj1>j4 ,j5)

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]-[15], [29]. Suppose that every ^(t) (l = 1,...,k) is a continuous nonrandom function on [t,T] and {fa(x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fa(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the estimate

M

J

- J

/

< k!

P1,-",Pfc

/p1 rs

k 2(ti,...,ik )dti ...rftk --.ecu

' _A ' _A

<

Ps

(1.119)

\[t,T ]k

j1=0 js =0

/

is valid for the following cases:

1. = 1,...,m and 0 <T — t< to,

2. i1,...,ik = 0,1,..., m, if + ... + > 0, and 0 <T — t< 1,

where Jis the iterated Ito stochastic integral (11.5), J[^^yt'"^ 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.

p

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[^(k)]T!t'"'Pk is the expression on the right-hand side of (11.10) before passing to the limit l.i.m. and

T r / Pi Pk k \

= e "i (k(ti.....tk)-en^j i(**)x

(tfc)i t V j i=0 jk=0 1=1 /

x dwt(; i) •••dw(ik), (1.120)

where

£

(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 (1.120) 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 •

From the orthonormality of the system (x)}°=0

(see (1.71)), (1.26),

(1.27), (1.120), and the elementary inequality

(«1 + «2 + ••• + ftp )2 < p(a? + + ••• + «P) , P £ N (1.121)

M^ J[#k)]rt- J№(k)lPi r'Pk ^ <

we obtain the following estimate

iTt-^rr'TTt

T t2

< "

r r ( P1 Pk k \ 2

k! E ... K ) ...ji n^i (*i) dti ...dtk

(ii,...,ifc) t t V ji=0 j =0 1=1 /

// Pi Pk k \ 2

K(ti,..., tk) - E • •. E Cjk...ji n ^fa) dti... dtk

u^ik V ji=0 jk=0 1=1 /

[t,T

= k!

/

/Pi Pk

K 2(ti.....tk № ...rfik

~ _A ' _A

(1.122)

\[t,T

ji=0 jk =o

/

where T — t G (0, to) and ii,..., ik = 1,..., m.

From the orthonormality of the system {fa(x)}TO=0, (H2E), (OH), (0201), and (1.121) we obtain

T t2

<

/

M

K (ti

T,t

- J

Pi

Tit""PM <

Pk

(tij-'-j^k) t t

tk) - E • • • E Cjk---ji n (ti) ) dti... dtk

ji=0 jk=0

1=1

// Pi Pk k \ 2

K(ti,..., tk) - E ... E Cjk---ji n (ti) dti... dtk

u^k V ji=0 jk=0 1=1 /

[t,T

= Ck

/Pi Pk

K 2(ti.....tk № ...dtk .--ECU

~ _n ~ _n

\[t,T

j1=0 jk=0

/

where ii,..., ik = 0,1,... ,m, ii + ... + i| > 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)ai, (T — t)a2, ..., (T — t)afc!},

where ai,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

2

k

It means that for the case i1, • • • , ik = 0,1, • • • ,m, i2 + • • • + ¿k > 0, and 0 < T — t < 1 we get (1.119). Theorem 1.4 is proved.

Example 1.3. The particular case of the estimate (1.119) for the iterated Ito stochastic integral loOojT^ (see (1.103)) has the following form

2\ ^r f (T — t)3 V- ^2

1 V (000)T,i J(000)T,i / J - I 6 ( ./:■,/:•./•

j1,j2,j3=0

where i1, i2, ¿3 = 1, • • •

, m and Cj3j2ji is defined by the formula (1.105).

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;f-pk

!> Pi Pk

= J K 2(t1, • • •, tk )dt1 •••dtk — ^ ••^C2 j, (1.123)

[t,T]k ji=0 jk=0

where notations are the same as in Theorem 1.4.

The stochastic integrals on the right-hand side of (1.120) will be independent in a stochastic sense for the case of pairwise different i1, • • •, ik = 1, • • •, m- Then

m ((J [^(k)]T,t — J [^(k)]T1t""Pk>

T H / P 1 Pk k \

m<! E ••• K (ti—tk) -E ..^Cjk^n (tin x

v(ti,...,tk) t t V ji=0 jk=0 1=1 /

2'

x<;i) ..f

r / T ^ / pi Pk k \

£ mH ... K (ti-----tk)^ (tiH x

(ti,...,tk) l Vi { \ ji=0 jk=0 1=1 J

2

xdf«!i) ..f

T ^ / Pi Pk k \ 2

= E ■■■ k(ti,...,tk)-E..•ECk...j1n^(tin dti...dtk = (ii,...,ifc) t t \ ji=0 jk=0 1=1 /

^ / Pi Pk k \ 2

= J (K(ti,...,tk) "E ..^Cj-k ..,n j (tin dti ...dtk =

[t T]k j1=0 jk=0 1 = 1 /

^ Pi Pk

= J K2(ti,...,tk)dti...dtk-E...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.

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 L2([i,T])) with respect to the system

converges to the function f (x) in the mean-square sense with weight r(x), i.e.

t P 2

lim i f (x) ^E Cj^(xM r(x)dx = 0, (1.124)

{ \ j=0 /

where

T

Cj = J f (x)tfj(x)r(x)dx (1.125)

t

is the Fourier coefficient.

The relations (1.124), (1.125) can be obtained if we will expand the function

f(x)\/r(x) G L2([i,T]) into a, usual Fourier series with respect to the complete orthonormal with weight 1 system of functions

{^•(xjv^)}0

j=0

in the space L2([t,T]). Then

T

lim / ( f{x)\Jr{x) — y^ Cj^j(x)\/r(x) ) dx

p—>00

t j =0

t p 2

= lim j (f (x) — ^ Cj^j (x)J r(x)dx = 0, (1.126)

{ v j=0 /

where Cj is defined by (11.1251).

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 /•• \

A-(i!,. / II yAti) "£■••£ ¿^.j, || ^(i,) v^fe) X

[iiT]A <=i j=i i=0 f=l ;

xdt1... dtk =

// Pi Pk k \ 2

K (t1-----tk) ..^C^jk ...jill (ti)

[t,T]k V ji=0 jk=0 1=1 /

lim / |K (t1,----tk) -> ...> Cj \ I*,-, (ti ) I x

pi,. . . . . ]k ji=0 jk=0

k

x (J! r(ti) dt1 ...dtk = 0, (1.127)

,1=1 )

where

Cjfc...j = J K(ti,... ^ (ti))dti ...dtk.

[t,T]k 1=1

Let us consider the following iterated Ito stochastic integrals

t t2

t t

where every ^(t) (l = 1,..., k) is a nonrandom function on [t,T], wT^ = f-^ for i = 1,..., m and wT0) = t, i1,..., ik = 0,1,..., m. So, we obtain the following version of Theorem 1.1.

Theorem 1.5 [13]-[15], [27], [39]. Suppose that every ^/(t) (l = 1,...,k)

is a continuous nonrandom function on [t,T]. Moreover, let (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

J[0(k)]T,( = V ...E^in C<

Pi'-'Pfc. n . n . , 1

ji=0 jfc=0 \Z=1

-l.i.m. E , (1.129)

(ti,...,tfc )GGfc

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, ii,..., ik = 0,1,... ,m,

T

cf = / ^(s)v^dwW t

are independent standard Gaussian random variables for various i or j (in the

j - wj (i = 0,1,...,m), {Tj }N=0

case when i = 0), AwTj = wTj++1 — wTj (i = 0,1,..., m), {Tj}N=0 is a partition

of [t, T], which satisfies the condition (1.9),

k

) n I (t ldtl

= / K (ti,...,tk ) n (ti W^) dti ...dtk (1.130)

[t,T ]k 1=1

is the Fourier coefficient,

^i(ti).. .^k(tk), ti < ... < tk K (ti,...,tk)H , ti,... ,tk G [t,T], k > 2,

0, otherwise

and K(ti) = ^i(ti) for ti 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,t =

t t2 k

E /•••// v^dwl;0• • • ^ (ti,"-,tfe) t t 1=1

pi t t2

ri rs r T /

E - Er- ' /•••/ E n(^^)v^)

j'i=o jk =o t t (ti,...,tfc) \/=i

DPi,"-,Pk

pi Pk

= E • • • E Cjk-jix ji=0 jk=0

N-l

xi.i.m. £ ... v^Aw(j>+

N^œ— i k

ii,...,ik = 0

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

+ RRTiv>Pk

LT,t

Pi Pk

= E . . . E ^Ji x

ji=0 jk=0

N-l

x| l.i.m. E

l.i.m. E ^i(-ni) v^)Aw(;;)... V^JAw^) +

('i,..-i'k)6Gk

RPi,...,Pk

Pi Pk

E . . . E Cjk...ji x

ji=0 jk=0

k

XI n ci;° - l.i.m. E **(^J v7^)Aw[;;)... ^(.Jv^jAwS;)) + 1=1 (li,...,1k)GGk

+RT1;...,Pk w. p. i,

where

T t2 / * i ,...,tk) + + \ 1 = 1

RPiv",Pk

rt,;

(;iv->;k) ; ; \ ¿=1

Pi Pk k

E-E ^ ...ji

ji=0 jk=0 1=1

r(t1 " 1 dw;i .. .dw;k

where permutations (ti,..., 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 tq in the permutation (ti,... ,tk), then ir swapped with iq in the permutation (ii,..., ik).

Let us estimate the remainder Rpt" of the series. According t° Lemma 1.2 and (1.38), we have

M

T

t2

DPivvPk

RT,t

< Ck E J

(ilv,ifc) t

i=i

pi Pk

E • • • EII(v1'.'v^)) <//!•••^ ji=0 j =0 l=i /

(1.131)

// Pi Pk k \

K (ti,...,tk) ...EC* .^n (ti)

^k V ji=0 jk=0 i=i J

Ck J I ^v

[t,T ]k

X

X I n r(ti )) dti ...dtk ^ 0

,i=i /

(1.132)

if pi,... ^ ro, where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral (1.128). Theorem 1.5 is proved.

Let us formulate the version of Theorem 1.4.

Theorem 1.6 [14], [15], [27], [39]. Suppose that every hi (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)\/r(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the estimate

m (J[^(k)]T,t - J#(k)]T1t""Pk

<

< k!

^ !> / k \ p l Pk ^

I k2(ti,...,tk) nr(t,)dti...rftk- £... £Cl..ji

y[t,T]k \l=i J j i =0 jk =0 y

/

(1.133)

is valid for the following cases:

1. ii,...,ik = 1,...,m and 0 <T — t< ro,

2. ii,..., ik = 0,1,..., m, if + ... + ik > 0, and 0 < T - t < 1

,-2

2

2

k

2

2

where J[^(k)]T,t is the stochastic integral (11.1281), J[^(k)JTt'"'^ is the expression on the right-hand side of (1.129) before passing to the limit l.i.m. ; another

PlvvPfc ^^

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

i>(A, A) = v(A, A) - |A|n(A).

Let (Q, F, P) be a fixed probability space, let {Ft, t £ [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 £ [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 £ [0,T], y £ 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 p(t, y) £ H2(n, [0,T]) as the following mean-square limit [91]

T T

i /V(t, y)z/(dt,dy) = l.i.mj /VN)(t, y)z>(dt,dy), (1.134) J J n^to J J

0 Y 0 Y

where p(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

y) - >(i, y) ¡>n(dy)dt ^ 0

It is well known [91] that the stochastic integral (1.134) exists, it does not depend on selection of the sequence ) (t, y) and it satisfies w. p. 1 to the following properties

M | J J p(t, y)i>(dt, dy) F01 =0,

T

J yWi(t, y) + ^(t, y))z>(dt,dy) =

0Y

T T

= a/ / Pi (t, y)i>(dt,dy) + W / <^(t, y)i>(dt,dy),

0Y

0Y

M

T

0Y

p(t, y)z>(dt, dy)

T

M = // M

0Y

{l^(t, y)|2

Fo

where a, ^ £ R1 and (t,y), p2(t,y), p(t, y) from the class H2(n, [0,T]).

2

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 [91]

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, (1.135)

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 [91]

t

(zt)p = J J{(zT— + Y (t, y))p — (zT—)p) v(dT,dy) w. p. 1, (1.136)

0Y

where p £ N, zT — means the value of the process zT at the point t from the left,

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 (1.136) [91 .

Let us consider the useful estimate for moments of stochastic integrals with respect to the Poisson measure [91]

ap(T) < jmaxi} ^ J J (((6p(t, y))i/p + 1)' — 1) n(dy)drj J , (1.137)

where

ttp(t) = sup m{|zT|p), bp(T,y) = m{|y(t,y)|p).

0<t<t ^ } ^ }

We suppose that the right-hand side of (1.137) exists. According to (see (1.135))

t t t J J Y(T y)i>(dT,dy) = J J Yy)v(dT,dy) — ^ J Y(T y)n(dy)dT

0 Y 0 Y 0 Y

and the Minkowski inequality, we obtain

M

{iz|2»})1/2p < (m{n2»})1/2p + (m{l-t!2p>"1/2p

(1.138)

where

t

¿t = J J Y(t, y)i>(dr,dy) 0Y

and

t

def

^ = J J Y (T' y)n(dy)dr.

0Y

The value M j |iT|2p| can be estimated using the well known inequality [91]

M{< t2p"W M

p(r, y)n(dy)

2p

dr, (1.139)

where we suppose that

M

Y (r, y)n(dy)

2p '

dr < 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 Xi(ti, yi)^(i 1 )(dti, dyi) ...^(dtk,dyk), (1.140)

t X t X

where ii,..., ik = 0,1,...,m, Rn =f X, X/ (t, y) = (t )p/ (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

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,

¿>«(dt,dy) = v(i)(dt,dy) - n(dy)dt (i = 1,... ,m)

are independent martingale Poisson measures for various i, ¿>(0)(dt,dy) =f n(dy)dt.

Let us formulate the analogue of Theorem 1.1 for the iterated stochastic integrals (1.140).

Theorem 1.7 [1]-[15], [39]. Suppose that the following conditions are hold:

1. Every (t) (/ = 1,...,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. For I = 1,..., k and q = 2k+i the following condition is satisfied

J |pi(y)|qn(dy) < ro.

X

Then, for the iterated stochastic integral with respect to martingale Poisson measures P[x(k)]T,t defined by (1.140) the following expansion

pi pk / k P [x(k)]T,t = l.i.m. £ ...^j i m

ji =0 jk =0 \g=i

^ E H j(Tig)/ ^(y)i>(ig)([Tig''Tg+iMy})

(l )eGfc X /

(1.141)

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) : = 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, random variables

T

nfg) = / fa (T) J Pg(y)z>(ig}(dr,dy)

t X

are independent for various (if = 0) and uncorrected for various j,

k

C —

^ j, j 1 —

J K (ti,...,tk)n j (ti)dti ...dtk

[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].

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. £ / »(Tji,y)'^(,,)([Ti,,Tj1+i),dy) (1.142)

j =0 j 1=0 l=1

is valid w. p. 1, where {Tj}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 (1.140), z/(i|)(dtl, dyl) is used in the integral P[x(k)]T,t instead of ¿>(i|)(dtl,dyl) (/ = 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 (1.142) 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) (/ = 1,... ) 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. £ <B(Tj,,..., j)n I W(y^'fe+i),dy) =f P[<4k>,

j i,---,jfc =0 l=1 X

N-^oo

T t2

def

J ...J tk )J ^(y)^ 1 j(dii,dy) ...y ^ (y)i>(ifc)(dtk ,dy) =

t t X X

=

where $(i1,...,ik): [t,T|k ^ R1 is a bounded nonrandom function and the sense of notations of the formula (1 . 142) is remaining .

Note that if the functions ^(y) (l = 1,...,k) satisfy the conditions of Lemma 1.4 and the function $(t1,... ,tk) is continuous in the domain of integration, then for the integral P^ly] the equality similar to (11.142) is valid w. p. 1.

Lemma 1.5. Assume that the following representation takes place:

g/y) = h/(tM(y) (1 ^..^A^

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) < oo for p = 2k+1.

X

Then

T

II I I gi(s, y)^(ii)(ds,dy) — P[$]Tk] w. p. 1,

(1.143)

i=i

t X

where i\ — 0,1,..., m (l — 1,..., k) and

$(ti,...,tk ) —JJ h (ti).

1=1

Proof. Let us introduce the following notations

N-1 ,,

J [gl]N = E / gl (Tj ' y)^(il)([rj ,Tj+1),dy)'

j=0 X

T

J ]T,t =

t X

def ' / gi(s, y)^(ii)(ds,dy),

where {Tj}N=0 is a partition of the interval [t,T] satisfying the condition (1.9). It is easy to see that

nj [gi ]n -II J [gi ]

T,t —

i=i

i=i

k /l—i \ / k = E II J(Jfe]N — J[gl]T,t) n J^]n

l=i \q=i / \q=l+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

IIJ [gi ]n -n J [gi ]

T,t

i=i

i=i

2\\ i/2

k

< C*E ( M

i=i

J [^]n - J fe]T,7

i/4

(1.144)

where Ck < to. We have

N-1

J fe ]n - j ^ ]T,t— EJ [Agi]

1

q=0

k

4

where

rg+ 1

J[AgL+ 1 = (h(Tq) - hi(s)) / 0i (y)z>(i1 )(ds,dy).

JTq+ 1

Let us introduce the notation

h(N)(s) = hl (Tq), s G [Tq ,Tq+i), q = 0,1,...,N - 1.

Then

N-1

T[A

JTg+l >

J [gi]n - J [gi]T,t = E j [Agl

q=0

T

= / (hiN)(s) - hi(s)) y hi(y)g(i}(ds,dy).

t X

Applying the estimates (1.137) (for p = 4) and (1.138), (1.139) (for p = 2) to the value

4'

M

T

(hi

t X

h(N)(s) - hi(s)) / 0i (y)z>(i1 )(ds,dy)

taking into account (1.144), the conditions of Lemma 1.5, and the estimate

|hi (Tq) - hi (s)| <£, s G [Tq ,Tq+i], q = 0,1,...,N - 1, (1.145)

where £ is an arbitrary small positive real number and |Tq+1 — Tq| < we obtain that the right-hand side of (1.144) converges to zero when N ^ to. Therefore, we come to the affirmation of Lemma 1.5.

It should be noted that (1.145) 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| < 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 i = 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 estimates for moments of stochastic integrals with respect to the Poisson measure, we obtain

2 ^ k

,2

M^ (RTW-) Ckn/ ^2(y)n(dy)x

T t2

x [■■■I 1K (ti'-.-'tk) ■■■2^Cj ...j 11J^^ >' x

(ti,...,ifc) t t \ j 1=0 jk=0 1=1

xdt1 ■ ■ ■ dtk =

T / Pi Pk k \

E /■■■/ (k(ti.....tk)-E-Ec,...j,n^(ti)

.....tfc){ i \ ji=o jk=0 1=1 )

k „ „ / pi Pk k \

C*n/^2(y)n(dyW K(t1,_,tk) -E ■■^Cj-k ^ (ti)

1=1 X [t,T]k ^ ji=0 jk=0 1=1 /

2

X

xdt1 ■ ■ ■ dtk <

/. / Pi Pk k \ 2

< Ck J ( K(t1, ■ ■ ■,tk) ■■^Cj-k.^n(tin dt1 ■ ■■dtk ^ 0

[t T]k j1=0 jk =0 1 = 1 '

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, RTV''^ has the following form

T ^ / pi Pk k \

E /■■■/ k(t1—tk)-E■■^Cjk...jin^fa) x

(ti,...,tk) t t \ ji=0 jk=0 1=1 /

x / P1(y)i>(ii)(dt1,dy) ■■■ i Pk (y)v>(ik) (dtk, dy), (1.146)

where permutations (ti,... ,tk) when summing in (1.146) are performed only in the values pi(y)i>(il)(dti, dy)... pk(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 iq 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} = j(T) / }(dr,dy) (l = 1, 2)

j _ / j

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, (1.147)

t

m{ |Ms - M/} < Cp | s - t|, p = 3,4,...,

where 0 < t < s < T, p(t) is a non-negative and continuously differentiable nonrandom function at the interval [0,T], Cp < œ 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

{l£t|2} P(t)dt

<.

For

any partition {t(NH of the interval [0,T] such that j j=0

0 — Tn(N) < t(N) < ... < t(N) — T, max

0 i N 0<j<N -i

t(N) _ t(N) Tj+i Tj

^ 0 if N ^ o (1.148)

we will define the sequence of step functions £(N) (t,w) by the following relation

£(N)(t,w) = £ (t(nU) w. p. 1 for t G [t(n),t(N!

where 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 [91]

Ni

T

Nim. S £(N) (t]N),7(m(t^^NU) — m(t]N),^ d=f / £TdMT, (1.149)

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

N^œ J 0

£(N)(t,^) - £(t,w) p(t)dt — 0.

It is well known [91] that the stochastic integral (1.149) 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

M — 0,

m

T

ÉtdMt

T

Fr

F^ = M < J £P(t)dt

T TT

j(a& + ß^t)dMt = aj ÉtdMt + ßj ^dMt, 0 0 0 where ^t G ^(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

< Kj |g(s)|ads,

(1.150)

where 0 < t < t < T, g(s) is a bounded nonrandom function at the interval [0,T], K4 < oo 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

< oo,

where 0 < t < t < T, n G N, g(s) is the same function as in the definition of the class Q4(p, [0,T]).

Let us remind that if (£t)n G H2(p, [0,T]) with p(t) = 1, then the following estimate is correct [91

M

£sds

2n

< (T - t)

2n— 1

M

{|6|2n} ds, 0 < t<T < T. (1.151)

2

n

Let us consider the iterated stochastic integral with respect to martingales

t t2

J[^Mt = / ^k(tk) . . .j ^i(ti)dMt(i1'il).. \ (1.152)

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) =f s. Now we can formulate the following theorem.

Theorem 1.8 [1]-[15], [39]. Suppose that the following conditions are hold:

1. Every ^(t) (l = 1,...,k) is a continuous nonrandom function at the interval [t,T].

2. {fa(x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fa(x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7).

3. Mp1) e Q4(p, [t,T ]), Gn (p, [t,T]) with n = 2k+1, i/ = 1,...,m, l = 1,..., k (k e N).

Then, for the iterated stochastic integral J[^(k)]Mt with respect to martingales defined by (1.152) the following expansion

Pi Pk / k

J^ = J.^ £if

ji=0 jfc=0 \/=1

-l.i.m. E fa*)AMi!'il) • • • j(Tik)AMTf;i

(tl,...,tfc)GGfc

ifc)

that converges in the mean-square sense is valid, where i1,..., 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},

{t,}N=0 is a partition of the interval [t,T] satisfying the condition similar to

(mm AmT;^ = m^ - m^ (i = 0,1,...,m, r = 1,...,k),

Lk ={(/i,...,/k): 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,

T

j1) = J fa (s)dMf'il) t

are independent for various i (if i = 0) and uncorrelated for various j (if p(t) is a constant, i/ = 0) random variables,

C —

/k

K (ti,...,tk )H (ti )dti ...dtk

= J K (^1,..., ) 11 ^ (t/)dti [t,T ]k /=1

is the Fourier coefficient,

0i(ti) ...0k (tk), ti <...<tk K(ti,...,tk)=< , ti,...,tk G [t,T], k > 2,

0, otherwise

and K(ti) = 0i(ti) for ti G [t, T].

Remark 1.4. Note that from Theorem 1.8 for the case p(t) = 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 MSM G M2(p, [t,T]) (i = 1,... ,m), Ms(r'0) = s (r = 1,..., k), and every 0/(t) (/ = 1,..., k) is a continuous nonrandom function at the interval [t,T]. Then

N —i j2 i k

J[0(k)]M( = l.i.m. £ (Tji)AM(;") w.p. 1, (1.153)

N^to . -nii '

jk =0 ji=0 /=i

where {Tj}NL0 is a partition of the interval [t,T] satisfying the condition similar to (1.148), i/ = 0,1,..., m, I = 1,..., k; another notations are the same as in Theorem 1.8.

Proof. According to properties of stochastic integral with respect to martingale, we have [91

M dMs(MiM M{ |£s|2} p(s)ds, (1.154)

M

6ds) l< (t - t) / m{&|2} ds, (1.155)

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 (1.153) and the value, which converges to zero in the mean-square sense if N —y oo. 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

Um. £ $(TJi,...,T)jnAÄi<;r) = Wj, (1.156)

.i,-Jfc =o 1=1

where (t^}N=0 is a partition of the interval [t, T] satisfying the condition similar to (1.148) and ... , 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

n / g/(s)dMs(/'il) = I[$]$ w. p. 1, /=1 t

where i/ = 0,1,...,m, l = 1,...,k,

k

$(t1,...,tk ) = n g/ (t/).

/=1

Proof. Let us denote

T

N-1

J [gl in = E gl (j )AM|'il ), J [gi I T,t = j 9i (s)dMs^), 7=0

where (t^}N=0 is a partition of the interval [t, T] satisfying the condition similar to (1.148).

Note that

IIJ [gi ]n -n J [gi ]

=

1=1

1=1

k /i- 1

E II J(J[gi]N - J[gi]T,t) n J[gq]N

1=1 \q=1 / \ q=i+1

Using the Minkowski inequality and the inequality of Cauchy-Bunyakovsky as well as the conditions of Lemma 1.7, we obtain

M

n J [gi]

N —

n J [gi ]

T,t

l=1

l=1

1/2

<

<

M

l=1

J [gi ]N — J [gi ]T,t

1/4

where Ck < to is a constant. We have

N1

J [gi]N — J [gi]T,t = E J [Agi]

Tq+1

q=0

Tq+1

J [Agi]

(gi(Tq) — gi (S)) dM^

il )

(1.157)

Let us introduce the notation

g(N)(s)= gi (Tq), S G [Tq ,Tq+1), Q = 0, 1,...,N — 1

Then

N1

J [gi]N — J [gi]T,t = E J [Agi]

Tq+1 ,Tq

q=0

T

g(N)(s) — gi (s)) dM^ 1 ).

k

2

Applying the estimate (1.150), we obtain

M

T

g((N) (s) - gl(s)) dMp1 )

<

T

< K4

g((N)(s) - gl(s)

ds =

N -1 Tq+1

/ |gi(Tq) - gi(s)|a ds<

__n «/

= K4

q=0

N1

< ^ £ (Tq+1 - Tq) = K^T - t). q=0

(1.158)

Note that we used the estimate

|g/ (Tq) - g/ (s)| <£, s G [Tq,Tq+i], q = 0,1,...,N - 1 (1.159)

to derive (1.158), where |rq+1 — Tq | < 5(e) and £ is an arbitrary small positive real number.

The inequality (1.159) is valid if the functions g/(s) are continuous at the interval [t,T], i.e. these functions are uniformly continuous at this interval. So, |g/(Tq) — g/(s)| < £ if s G [Tq,Tq+i], where |Tq+i — Tq| < 5(e), q = 0,1,... ,N — 1 (5(e) > 0 exists for any £ > 0 and it does not depend on points of the interval

[t.T ]).

Thus, taking into account (1.158), we obtain that the right-hand side of (1.157) converges to zero when N ^ to. Hence, we come to the affirmation of Lemma 1.7.

In the case when the functions g/(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 g/(s) (l = 1,..., k). So, we can apply the argumentation as in Sect. 1.1.7.

Obviously if i/ = 0 for some I = 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 estimates for moments of stochastic integrals with respect to martingales

4

a

(see (1.154), (1.155)), we obtain

M ^ i ) j, <

2'

T ll / Pi Pk k \

<Ck E /••7 U(t1,...,tk)-E..^Cjk..j,n^(tin x

(ii,...,ifc) t t V ji=0 jk=0 1=1 /

x pi(ti)dti . ..pk (tk )dtk < (1.160)

T "If Pi Pk k

< CkE ... K (ti,...,tk)-E ..^Cjk.^n ^ (tin dti ...dtk

(ti,...,tk) t t V ji=0 jk=0 i=i /

/. / Pi Pk k \ 2

= Ck J K(ti,...,tk) ..^Cjk...jin ^ (tin dti ...dtk ^ 0

[t T]k ji=0 jk=0 i=i

if pi,... — 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, R^ t'' 'Pk has the following form

T / Pi Pk k \

RrPfc = E /.../*(ti.....tk)-E-E^...j in(t/nx

(ti,''',tk) t t \ j i =0 jk=0 /=i /

x dMt(i'i 1)... dMt(kk'ik), (1.161)

where permutations (ti,... ,tk) when summing in (1.161) are performed only in the values dM^ 1)... 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 (1.131) and (1.160). If we suppose that r(x) > 0 and

<- f< ^ -

—— < c < to, r(x)

where p(x) as in (1.147), then

2

I \ -*- / I >v / s j s j JK"'JLWWJL\V/ I X

\ A__H A__H 7_1 /

]

// Pi Pk k \

K (ti-----tk ) ...jill ** (ti )

V ji=0 jk=0 1=1 /

xp(ti)dti.. .p(tk)dtk =

// Pi Pk k

K(ti—tk)-E...£<j...jill^ji(ti)) x

V ji=0 jk =0 1=1

[t,T

r(ti) r(tk )

// Pi Pk k

K(ti,..., tk) - £ ... £ Cj...ji J] **(ti) ) x

^ ji=0 jk=0 1=i

[t,T ]

k

X I n r(ti M dti ...dtk ^ 0

l=i

if pi,... ,pk ^ to (see (1.132)), 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 (Cjfc...j1 has the form (11.130).

So, we obtain the following modification of Theorems 1.5 and 1.8.

Theorem 1.9 [13]-[15], [39]. Suppose that the following conditions are

fulfilled:

1. Every 0(t) (/ = 1,... ) 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)

3. Ms(Ml} 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 (1.152) the following expansion

Pl Pk / k

J [«=pi,1...^ £ .•• £ M nif)

j 1=0 jfc =0 \/=1

l.i.m. E 1 1 )AMT1;i 1)... K)AMT*'i

(ti,...,tfc JGGfc

ifc )

that converges in the mean-square sense is valid, where ii,... ,ik = 1,... ,m,

iN j }j=0

A AX(r'i) — A/,

'"j+i

{rj}N=0 is a partition of the interval [t,T] satisfying the condition similar to (1.148), AM™ = M™ - 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 tfj (s)dMs(/'il) 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 = / K (ti,...,tk) n (t/)r(t/)) dti ...dtk J /=i ^ '

k

is the Fourier coefficient,

01 (ti) ...0k (tk ), ti <...<tk

K (ti,...,tk)

t1,..., tk G [t,T], k > 2,

0,

otherwise

and K(ti) = 0i (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, y, J, À 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 o(x), i(x),... converges in the mean-square sense to the func-

tion ^/r(x)f(x) at the interval [a, &]. 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). (1.163)

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, y $(&) + ¿$'(&) = 0,

b

(1.162)

a

(1.164)

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 (1.163) for n = 0, the eigenfunctions of which are the functions (1.164) when n = 0.

Let us consider the system of functions where

TO

Ux) = J>i r (f)

2/ r(m + 1)r(m + n + 1)

m=0 \ / \ /

is the Bessel function of the first kind,

C

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)}°=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 (1.165) 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 y/idfW (¿ = 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

Pi P2

//> fl 1J2 /dMW = u.^ 0,2ji,

0 0 jl=0 j2=0

where

T

> = | « (T )dM« 0

are independent standard Gaussian random variables for various i or j (i = 1, 2, j = 0,1, 2,...),

T s

=/ ^2 («)/T «>1 (T

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

jV7sj y/ïd^df®, 00

and as a system of functions (s)}°=0 in Theorem 1.1 we can take

y/2s (ßj

As a result, we obtain

T s

r r m P2

0 0 ji=0 j2=0

where

T

j = j h(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~s<p,m j ^n(t)dtds

00

is the Fourier coefficient. Obviously that Cj2j1 = (7j2j1. 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.

t t j1=0 j2 =0

where

T

cf = 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^-,4n(r)drds t t

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 Ito 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 Ito stochastic integral (1.94) and its expansion, which is corresponds to (1.95) for the case ii = i2

Ank) _T — t{ (i1) (j2) y- 1 ( Ali A''A Ah)Ah)\ \ n

aoo)t,t - I so SO ^-V )• AAbb)

First, note the well known fact [95].

Lemma 1.8. If for the sequence of random variables and for some a > 0 the number series

to

EM ia}

p=I

converges, then the sequence converges to zero w. p. 1. In our specific case (ii = i2)

t — t to 1 / \

jim2) _ T{iii2)p , ¿r t _^ V^ 1 (An)Ai2)

J(00)T,i ~~ J(00)T,i "T" Sp; Sp — 2 Z^ /^,-2 _ >>i Sj-1 J '

i=p+1

where

T(i1i2)p_T_A'i-^A'12) 1 Y^ 1 /V(nM»2) ^(¿IM^A \ H

7(00)T,i - I so So + = ^-l) }■ U-LO'J

. , V4i2 - 1

Furthermore,

.. r,, ,21 (T -1)2 ^ 1 (T -t)2 r 1 ,

M {l&l-} = ^ E ^ * ^ J =

i=p+1 p

i^iln

2 4

1- 2

2p +1

< (1-168) 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

to

£m{ |2}

p=i

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 (1.72) for k = 2, n = 2 and (1.168) we obtain

M{|^|4}<3 (1-169)

p

and

to to

EMH&i 1}<*"Ep<00- <L170>

p=i p=i

where constant K is independent of p.

Since the series on the right-hand side of (1.170) converges, then according to Lemma 1.8, we obtain that ^ 0 when p ^ to w. p. 1. Then

7(00)2T!t ^ loO)2^ when P ^ w. p. 1

Let us analyze the following iterated Ito stochastic integrals

T

T s

(i1i2) (01)T,t

(t - sW df^il)dfs(i2), I

(i1i2) (10)T,t

(t - T)df^il)dfs(i2)

t t

whose expansions based on Theorem 1.1 and Legendre polynomials have the following form (also see Sect. 5.1)

I

(i 1 i2) _ T t r(iii2)p (T t) / Co )ci )

(01)T,t

2 ^(oo)T,i

A (a^ a-iK^r

til \/ (2i + l)(2i + 5)(2i + 3) (2i-l)(2i + 3)

z (ii)z (i2) zi zi

+cp01),

I

(ili2) (10)T,t

_t M^p

2

2 / /-(i2^(i1)

(T -t)2 / z0i2)Z1

+

z(i1)z(i2) zi zi

I y/(2i + l)(2i + 5)(2i + 3) ~(2i - 1)(2i + 3)

+ i«10),

where

c (01)

{T-rfl ^ 1 ^(hM^) >(il)>(i2)\

/ v /7TÖ-TVSi-lSi Sh

^i=p+1

V4i2 - 1

+

i=p+1

\J(2i + l)(2i + 5)(2i + 3) (2i - l)(2i + 3)

(i1) (i2)

Z (i1)z (i2) zi zi

c (10) Sp

(T - t)

1

^i=p+1

V4i2 - 1

E1 (Ah) Aii) _ Aii) Ah)

HTÖ-T Sj Sj-1 |T

s

f

+ E

¿=p+i

s/(2i ■ 1 )(2/ • 5)(2/ + 3) (2г - 1)(2г + 3)

Z (i1)Z (i2) Si Si

Then for the case i1 = i2 we obtain

M j

с (0i) Sp

(т-ty 16

X

x Г ( } I 'i + 2)2 + '' + 1)2 I

^ ( 4i

i=p+1 \

4i2 - 1 (2i + 1)(2i + 5)(2i + 3)2 (2i - 1)2(2i + 3)2

oo

^ v- 1 K

<K } -<—, ¿=p+i ^

<

(1.171)

where constant K is independent of p. Analogously, we get

M

с (10)

<

P :

where constant K does not depend on p.

From (1.72) for k = 2, n = 2 and (1.171), (1.172) we have

M

C(0i)

+ M

с (10)

p2

and

where constant K1 is independent of p.

According to (1.173) and Lemma 1.8, we obtain that ^>01), ^P>10) — 0 when

(1.172)

(1.173)

p —у то w. p. 1. Then

I (Н«2)Р I («1«2) I («1«2)Р I («1«2)

J(01)T,t — J(01)T,t, J(10)T,t — J(10)T,t

where i1 = i2 •

Let us consider the case i1 = i2

when p —то w. p. 1,

I

(ilil) (01)T,t

(T - t)2 (T - t)2

с0г1))

2 Aii)^(ii) ^ + So Si

эО si

л/5

2

1

2

4

4

z («i)z(il) Z (ii)z («i)

Si Si+2 Si Si

+ V ___Si Si (01)

+ ti I V(2i + l)(2i + 5)(2i + 3) (2i — l)(2i + 3) I I ^^

rinn) _(T~t)2 (T-tf ((An)\> ,

V3

rWHj _ - > _ - > / Mn; \ I -su M I

J(10)T,t ~ A , I SO ) + /7T +

z(ii V(ii) z(iiV(ii)

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(il)c£2 Zi(il)Zi(i1)

..(01) _ ^ V

4 \ \/(4 l)(2i 4 5)(2i 4 3) (2i-l)(2i + 3)

(T t)2 to / Z(ii)Z(ii) Z(ii)Z(ii) \

(10) = _S±JZlL X^ I__S» S»+2__, Si Si I

4 \J(2i 4 l)(2i 4 5)(2i 4 3) (2* - l)(2i 4 3)^ '

Then

and

to 1 to 2

x 1 ^ (2i + l)(2i + 5)(2i + 3)* + (2i - 1)2(2/ + 3)2 +

£ (M {K01)f} + M {Hi}) < * ££ < d-174)

4 M , , , ^ _ , .

1 V v y v y / P2

p=i p=i

where constant K is independent of p.

According to Lemma 1.8 and (11. 174), we obtain that ^P01), ^ 0 when p ^ to w. p. 1. Then

I(i1il)p V I(^h) I(i1il)p v I(i1i1) when p oo w p 1

J(01)T,t ^ J(01)T,t, J(10)T,t ^ J(10)T,t when p ^ to W. p. 1.

p

p

Analogously, we have

I(n^p I(^2) I(il:2)p I(i1 »2) I(il:2)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 ^ ^ 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 (527H529I) in Chapter 5). Let us denote

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 ijojTi, I(i)T t, I(2)T t, I(3)T t, I(0Tt 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 E 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 E N). This section is written on the base of Sect. 1.7.2 from [14], [15] as well as on Sect. 6 from [29] and Sect. 9 from

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 < oo for any m, lim xn,m < to for any n.

Then there exist the iterated limits

and moreover,

lim lim xn,m, lim lim xn,m

n—yoo m—yoo ' m—yoo n—?>oo '

lim lim xn,m = lim lim xn,m = a.

n m m n

Theorem 1.10 [14], [15], [25], [27], [29], [30]. Let ^(t) (l =

are continuously differentiate nonrandom functions on the interval [t,T] and {fa(x)}o=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then

J[^j]T,'t ^ J[^j]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

PlvvPk

Theorem 1.1)

p p / k

j= £■■.£cWi nj'-

ji=0 jk=0 \/=1

- l.i.m. E faji (Til)AwTi;)... j (Tik)AwTik) ) , (il.....ik)EGk l k J

where i1,..., ik = 1,..., m, rest notations are the same as in Theorem 1.1. Proof. Let us consider the Parseval equality

n Pi Pk

j k2(il,...,)dti...=• • . £ Ck...ji, (1-175)

]

where

k ji=0 jk=o

^i (ti) ...^k (tk), ti <...<tk k k-i K (ti,...,tk ) = { = J] ^ (ti )]! 1{ti <ti+i},

0, otherwise 1=i 1=i

(1.176)

where t1,... e [t,T] for k > 2 and K(t1) = 01(t1) for t1 e [t,T], 1A denotes the indicator of the set A,

k

C =

J K(ti,..., tk) J} (ti)dti... dtk (1.177)

[t,T ]k 1=1

is the Fourier coefficient. Using (1.176), we obtain

T t2

Cj-fc ...>i = J > (tk (tk).. .J > (ti)0i (ti)dti.. . t t

Further, we denote

Pi Pk œ

72 def

lim V ... V C2 . = V C2 . .

,1=0 jfc =0 jivJfc =0

If p1 = ... = pk = p, then we also write

p p to

lim V^... y^ c2 ■ = Y^ c2 ■.

P ,to ' V / V 1 / V •••j1

j1 =0 jk =0 j1v,j'k=0

From the other hand, for iterated limits we write

P1 Pk to to

p1iiTO... pi^^kto E... E Cjk •j1 E... E Cjk-j1, j1=0 jk =0 j1=0 jk=0

P1 Pk to to

lim lim V .. .V C2 j1 d=f V V Cj j1

p1^TO p2,• • • ,pk^TO Z-' Z-' J^-J1 £-/ Z-/ -71

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

2

,i / v ' ' ' / v • •

ji, • • • jfc =0 ji=0 jfc=0

C,fc • • . . . ^jk •

œ œ œ œ

22

E^E^ , • =E • , (1.178)

Jfc =0 ji=0 jqi =0 =0

for any permutation (q1,..., qk) such that {q1,..., qk} = {1,..., k}.

Proof. Let us consider the value

p p

£•..£ Cjk'jl (1.179)

jq, =0 jqk =0

for any permutation (q/,..., qk), where l = 1, 2,..., k, {q1,..., qk} = {1,..., k}.

Obviously, the expression (1.179) defines the non-decreasing sequence with respect to p. Moreover,

p p p p p

r C2

ji —

53 .. .53 Cjk- ji — ^^ .. . Cj'k •••.-' — j'qj =0 jqk =0 jqi =0 j92 =0 jqk =0

OO

— £ CU <

j'iv-Jk =0

Then the following limit

jk ...j1 jk ...j1

jq, =0 jqk =0 jq, ''"'jqk =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 p, pk r r

.. .53 Cjk '''jl — 53.. .53 Cjk '''jl — 53.. .53 Cjk '''jl.

jq, =0 jqk =0 jq, =0 jqk =0 jq, =0 jqk =

This means that the existence of the limit

pp

C 2

jq,= 0 jqk =0

implies the existence of the limit

n pk

2

p, 53... 53Cjk-ji (1,181)

jqi =0 jqk =0

and equality of the limits (1.180) and (1.181).

0

Taking into account the above reasoning, we have

q p p p p

lim y^ y^ . . . y^ C2 ■ = lim y^ .. . =

p,q^o ' ' ' jk '' 'jl p^o ' jk '' 'jl

jq, =0 jq,+l =0 jqk =0 jq, =0 jqk =0

p, pk

p,, '' Jjpk^o £ . . . £ Cjk '' 'jl. (1.182)

jq, =0 jqk =0

Since the limit

oo

V^ C2

/ v j'fc- • • ji

ji, • • • jfc=o

exists (see the Parseval equality (1.175)), then from Proposition 1.1 we have

œ œ q p p

V V C2 ■ = lim lim V y .. . =

jqi =0 jq2, • • • jqfc =0 jqi =0 jq2 =0 jqfc =0

q p p œ

= • • j. = £ Ck • • ji■ (1183)

jq1 =0 jq2 =0 j'qfc =0 ji, • • • Jfc =0

Using (1.182) and Proposition 1.1, we get

œ œ q p p

E E <

jfc^ • ji

jq2 =0 jq3, • • • ,jqk =0 jq2 =0 jq3 =0 jqk =0

q p p œ

■ Ji

= Jpm EE- 1= Ck • ji = E ■ ("84)

jq2 =0 jq3 =0 jqk =0 jq2, • • • ,jqk =0

Combining (1.184) and (1.183), we obtain

W V C2 = V C2

jk jl jk jl

jql =0 jq2 =0 jq3 , ' ' ' jqk =0 jl, ' ' ' ,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

Ji

ji=0 js-i=0 js=p+1 js+i=0 j=0

E E E •••ECJ2fc-. J

œ œ œ œ œ

2

= E E-£E-£c£. (1.185)

js=p+i js-i=0 >1=0 js+i=0 j'fc=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 Cjk •ji

C 2

ji =0 js-i=0 js=0 js+i=0 jk=0

p to p TO TO

= E E Cu= EE c^ (1.186)

js=0 j1, •••js-1 ,js + 1,-,j'k =0 js =0 j91 =0 jqk-1 =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 TO TO TO p TO

53 53.. .53 Cjk • • j1 = 53.. .53.. .53 Cjk • • j1 =... =

js=0 jq1 =0 jqk-1 =0 jq1 =0 js=0 jqk-1 =0

TO TO p

= E ..T E^»• • j1 ■ (1.187)

j91 =0 j9k-1 =0 js=0

Using (1.186), (1.187) and Lemma 1.9, we get

TO TOTOTO TO TO TOTOTO TO

... ... Cj2k j1 = ... ... Cj2k j1

œ œ p œ œ

■2

j i

ji=0 js-i=0 js=0 js+i=0 jk =0

53... 53 53 53 ...53Ci • • j

œ œ œ œ œ p œ œ œ œ

22

j i

js=0 js-i=0 ji=0 js+i=0 jk=0 js=0 js-i=0 ji=0 js+i=0 jk=0

53 53... 53 53.. .53 Cjk-• ji 53 53... 53 53.. .53 Cjk • • ^

œ œ œ œ

2

.j i

js =P+1 js-i=0 ji=0 js+i=0 jk =0

... ... Cj2k ... j i

So, the equality (1.185) is proved.

Using the Parseval equality and Lemma 1.9, we obtain

/p p

k 2(ti,..., tk )dti... dtk - ^... ^ c? ...j

_A _n

b • • • 5 "k/ i — k / , • • • / ,

Ji=0 Jk=0

to p p

2

Ji / v ' * * / v "jk...ji

ji,...,jfc=0 ji=0 jfc=0

...ji 53.. .53 ...J

to to p p

v Y^ c 2 - V^ Y^ c 2

/ V • • • / V jk...ji / j' ' • / V jk...Ji

ji =0 j =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 j=0 ji=p+i J2=0 j=0 j =0 jk =0

pp TO to p toto to

2

n

ji=0 J2=0 J3=0 j =0 j =0 J2=p+i J3=0 j=0

EEE-E^.+ E E E-E +

TOTOTO p p

^ _ y^ y^ c 2

Ji / J' ' • / V jk...Ji

ji=p+i j =0 j =0 ji=0 j =0

+ ^ v 53.. .53 CJk...ji 53.. .53 CJk...j

TOTOTO p TOTOTO

22

n

ji=p+i j2=0 j=0 ji=0 j =p+i J2 =0 j =0

53 53...53cjk...ji+ E 53 53..^53c?k...j+

pp TOTOTO p p TO

1 c2

Ji —

+EE E E-Ecu +...+E... E E c|..j —

ji=0 J2=0 j =p+i j =0 j =0 ji=0 jk-i =0 j =p+i

TOTOTO TO TO TO TO

— 53 53...53ci...ji + E E 53...53cj...J+

j =p+i j =0 j =0 ji=0 J2=p+i J2=0 j=0

TOTOTOTOTO TO TOTO

2

Ji

ji =0 J2=0 J3=p+i J4=0 j=0 j =0 jk-i=0 j =p+i

+EE E E-Ecu + - + E... E E cjk..

k /TO TOTOTO TO

2

E E- E E E .■■Ecu). (1.188)

s=i \j =0 Js—i =0 js=p+i js+i=0 jk =0

Note that we use the following

p p to to to

53... 53 53 53.. .53 Cjk '' jl —

jl=0 js-l=0 js=p+1 js+l=0 jk=0

ml ms-l to to to

2

.j l

— ... ... Cj2k ... j —

ji=0 js-i=0 js=p+i js+i=0 jk=0

mi ms-i to to to

C2 j

ji

— m.-^^TO 53... 53 53 53.. .53Cjk-j'

jl=0 js-l=0 js=p+1 js+l=0 jk =0

ml ms-2 to to to to

= 53... ^3 53 53.. .53 Cjk 'jl—

jl=0 js-2=0 js-l=0 js=p+1 js+l=0 jk=0

—... —

TO TO TO TO TO

— 53... 53 53 53.. .53 Cjk 'jl

jl =0 js-l=0 js=p+1 js+l =0 jk=0

to derive (11.188), where m1,..., ms-1 > p. Denote

T t2

Cjs'"jl fajs (tS)^S(tS) ...j j (t1 )^1(t1)dt1 ...d^

tt

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 (1.185) we obtain

TO TO TO TO TO

2

.j i

ji=0 js-i=0 js=p+i js+i=0 jk =0

53... 53 53 53.. .53 Cjk - ji

TOTO TOTO TO

2

.j i

js=p+i js-i=0 ji=0 js+i=0 jk =0

53 53. 53.. ^ 53 Cjk...j

T

oo oo oo oo oo

£ £■-.££-■■£ (tk)(Cjk-i ..ji(tk))2dtk

js =p+i js-i=0 ji=0 js+i=0 jk-i=0

T

TOTO TOTO TO „ TO

£ £ --■£ £ ■■. £ /^2(tk) £ (Cjk-i. . .ji(tk))2dtk

__I 1 _A _A _A ' _A J A _A

■.2U \ V^ U \)2

i.

js=p+i js-i=0 ji=0 js + i=0 jk-2 =0 jk-i=0

T t

CO CO CO CO CO k

TOTO TOTO TOTO TOTO TOTO n n

£ £-■■££.-■£ ) ^2-i(tk-i)(Cji_2...ji(tk-i))2x

____i 1 ; _n ; _n ; _n __n

js=p+i js-i=0 ji=0 js+i =0 jk-2=0 't t

xdtk_ idtk —

T

TOTO TOTO TO

— C £ £ -£ £ ... £ / (Cjk-2...ji(t))2dT =

__i 1 _A _A _A ~ _A J

-2-..J! ( T 1 ^

js =p+i js-i=0 j'i=0 js + i=0 jk-2=0 t

T

TOTO TOTO TO „ TO

C £ £.■■£ £■.. £ / £ (Cjk-=...ji (T ))2 dT

1T1 '2

js =p+i js-i=0 ji=0 js + i=0 jk-3=0 t jk-2 =0

T T

TOTO TOTO TO „ „

C £ £"■££■"£ / /^2-2(9) (Cji-S...ji (902 d^dT —

js=p+i js-i=0 ji=0 js+i=0 jk-3 =0 t t

T

TOTO TOTO TO „

— K £ £ ...£ £ £ J(Cjt-,..ji(T))2dT —

js=p+i js-i =0 ji=0 js+i =0 jk-3=0 t

—... —

T

TOTO TO „

— Ck £ £-■■£/ ji(T))2dT

js=p+i js-i=0 ji=0 t

T

TO TO TO „ TO

TOTO TOTO TOTO /» TOTO

Ck £ E / £(Cj',..ji(T))2dT, (1.189)

js=p+i js-i=0 j2=0 t ji=0

where constants C, K depend on T -1 and constant Ck depends on k and T -1.

Let us explane more precisely how we obtain (1.189). For any function g(s) E L2([t,T]) we have the following Parseval equality

t \ - / T

00 / „ \ 00 '

œœ

50 ( J &(s)g(s)ds ) = 50 ( J !{s<T}& (s)g(s)ds I =

T t

2 2 2

(1{s<t}) g2(s)ds = g2(s)ds. (1.190)

is

t t

The equality (1.190) has been applied repeatedly when we obtaining (1.189). Using the replacement of integration order in Riemann integrals, we have

T t2

(T ) = j (tS )0S(tS) ... j (ti )0i(ti)dti =

= y j (¿1)^1^1 )J j (¿2)^2^2) ...J j (ts)0s(ts )dts ...dt2 dt1 =

t ¿1 ts-1

= <?j•,..•jl(T).

For l = 1,..., s we will use the following notation

C^» M) =

T T T

= J j (t W>z(tz) y +1 (¿1+1)^+1(^+1) ...J j (ts)0s(ts)dts... dt/+1di/.

0 t ts-1

Using the Parseval equality and Dini Theorem, from (1.189) we obtain

TO TO TO TO TO

... 53 53 53.. .53 Cjk ••j1 —

j1=0 js-1=0 js=p+1 js+1=0 jk=0

T

00 00 00 „ 00

œ œ œ œ

^c'k e e •..£/e(C.,(t))2dr

js =p+i js-i=0 ,2=0 t ji=0

T

œ œ œ „ œ 2

ck E E ...E/E^,(r0 dr

js=p+i js-i =0 ,2=0 t ji=0

2

T

to to to „ t 2

js=p+i js-i=0 j2=0

Ck E E -E / / ^2(ti) (<j...j2Mi)) dtidT = (1.191)

t t

T t

= Ck E E .^j/^(¿OE (T,t1)) dt1dT = (1.192)

js=p+1 js-l =0 j3=0 t t j2=0

T t t

= Ck E E J ^2(i1)y (t2) ((7js„'j3 M2)) dt2dt1 dT —

js=p+1 js-l=0 j3=0 t t tl

T t t

o o o „ „ „ 2

— C^ E -E / / ^2(t1 ) / ^I(t2) ((7js'„j3(t^)) dt2dt1dT —

js=p+1 js-l=0 j3=0 { { {

T t

— Ck E E -E/ / ^i(t2) (C^js'"j3(T,t2^ dt2dT —

js =p+1 js-l=0 j3 =0 { {

—... —

oo T T

— Ck E J J ^-1^-1) (<j (T,ts-1^ dts-1 dT —

js =p+1 t t

o T Tf i } V

— Ck E / / / jWs(0)d0 dudT, (1.193)

js=p+1 t t \ U /

where constants Ck, Ck', Ck depend on k and T -1.

Let us explane more precisely how we obtain (1.193). For any function g(s) E L2([t,T]) we have the following Parseval equality

2 T 2

OO I T \ oo

E / ^j'(s)g(s)d^ = E / 1{0<«<T}0j(s)ds I = j=0 V { / j=0 V {

T T

2 2 2

;i{,<s<T}) g2(s)ds = J g2(s)ds. (1.194)

e

The equality (1.194) has been applied repeatedly when we obtaining (1.193).

Let us explane more precisely the passing from (1.191) to (1.192) (the same steps have been used when we derive (1.193)).

We have

T t

/ 02(t1)E (Cj-s-j-2 (T,t1^ dt1dT—

Ji (6i) lC,s t t j2=0 T t

E II ^2 (ti ) (Cs, (r,ti^ 2 dtidr =

j2=0 t t T t

02(ti) E ^C^js •••,2 (r,ti)) dtidr

tt

,2=n+i

N-i Tj œ 2

NimE / ^2(ti^ E (C., (r, ,ti)) dti Ar,, (1.195)

_A __I 1

j=0 t ,2=n+i

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 (C» •••j2 (Tj ,t1^ , j2=0

Tj

to 2 r 2

U(Tj ,t1)^E (C»•••j2 (Tj = 02 (t2) (Cjsj (Tj ,t2^ dt2, 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

to 2

E (Csj(Tj,t1)) <£, t1 e [t,Tj] (1.196)

j2 =n+1

for n > N(e) e N (N(e) exists for any £ > 0 and it does not depend on t1). From (1.195) and (1.196) we obtain

N—1 ^ to 2

NsroE 102(t1^ E (Cj-.-j-2(Tj,t1^ dt1ATj —

j=0 t j2=n+1

N — i Tj T t

.2/, A___ I I „/,21

- £ Niin E / ^2(ti)diiATj = sj J ^2(ti)dtidr. (1.197)

j=0 t t t From (1.197) we get

T t

(6D / .

12 lim / / ^2(ti) E (CsjMi)) dtidr = 0.

nn J

t i j2=n+i

This fact completes the proof of passing from (1.191) to (1.192). Let us estimate the integral

T

J fas (0)^s(0)d0 (1.198)

u

from (1.193) for the cases when {fa(s)}°=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

T

J fa(0)^(0)d0, j > p +1, (1.199)

t

where ^(0) is a continuously differentiable function on the interval [t, T], have been obtained in [6]-[15], [20], [31] (also see Sect. 2.2.5).

Let us estimate the integral (1.198) using the approach from [20], [31 . First, consider the case of Legendre polynomials. Then fa(s) is defined as

follows

where Pj (x) (j = 0,1, 2 ...) is the Legendre polynomial. Further, we have

i fa(e)mde = ^^tfITT j p.j{y}ii){u{y))dy =

v z (v)

JT — t

■ ' (pJ+i(z(X)) - p^Mxmix) - (pJ+i(z(v)) - Pj.M^mv)-

2V2JTI v

z(x)

T f [ ((Pj+i(y) - Pj-iiyWuiyVdy I, (1.200)

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 — t'

0' is a derivative of the function 0(0) with respect to the variable u(y).

Note that in (1.200) we used the following well known property of the Legendre polynomials

- d~^r(x) = (2j + 1)Pj{x)' = 2' • ■ ■

From (1.200) and the well known estimate for the Legendre polynomials 101] (also see

K

]PM<VJ+Ki-v*)'/" »e(-1-1»'

where constant K does not depend on y and j, it follows that

x

f to (0)0(0)d0

< j [ (l - (z(x)Y)^ + (l - + Cl 1 • (1'201)

where z(x),z(v) e (—1,1), e (t,T) and constants C, C1 do not depend on j.

From (1.201) we obtain

2

where constants C2, C3 do not depend on j.

x

Let us apply (1.202) for the estimate of the right-hand side of (1.193). We

have

<

K

T t

t t \ u 1

fas(0)^s(0)d0 dudr <

1 x

1

dy

+

dy

j \Ji (1 - y2)1/2 -1-1

K3 j2

(1 - y2)1/2

dx + K2 <

(1.203)

where constants K1; 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

fa (0) =

1,

j = 0

y/T^t.

\/2sin (2tit(0 - i)/(T - t)), j = 2r - 1, (1.204) \/2cos (27tt(0 — t)/(T — t)), j = 2r

where r = 1, 2,...

Using the system of functions (1.204), we have

far-1(0)^(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

Tt

(1.205)

far (0)^(0)d0 =

I 2 f 2vr(0-t) t(û,ÂÛ T~t. / cos T _ f '0(6>)d0 =

2

1

x

x

x

IT -t if,,,. 2?rr(x -t) ,, , . 2ttr(v - t) 1 'ip(x) sin——--W\v)sm~

2 nr \ r v y T -1 r w T -1

- isin2wrT{0_tt)^(e)d0\, (1.206)

where 0'(0) is a derivative of the function 0(0) with respect to the variable 0. Combining (1.205) and (1.206), we obtain for the trigonometric case

<f>j{e№{e)(wj < jr, (1.207)

where constant C4 is independent of j. From (1.207) we finally have

T t / t \ 2

(f)Js{0)hs{0)d0 j dudr<(1.208)

I js

t t \ u /

where constant K4 is independent of js.

Combining (1.193), (1.203), and (1.208), we obtain

oo oo oo oo oo

E... E E E ... E Cjk •••j1 —

»1=0 js-1 =0 js=p+1 js+1=0 jk =0

OO TO

< L, V i < L, / ^ = (1.209)

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 (1.188) and (1.209) we obtain

p p n

Ä^, • • •, i*)^ ^..J! ^ y. (1-210)

[t,T ]k Ji=0 Jk=0

where constant Gk depends on k and T — t.

For the further consideration we will use the estimate (1.72). Using (1.210) and the estimate (1.72) for the case pi = ... = pk = p and n = 2, we obtain

M< J

T,t

- J

T,t I > -

— C2,k

p p

/r r

K 2(ti.....tk № ...«ft* — £ ...£cj .j

~ _A _A

<

\t,T

ji=0 j =0

/

H

2,k

< , p2

where

Cn,* = (k!)2n(n(2n — 1))n(k—i)(2n — 1)!!

(1.211)

and = GC2,k.

Let a and in Lemma 1.8 be chosen as follows

a = 4 & =

J

T,t

J

p,...,p T,t

From (1.211) we obtain

E 4 (J

p=i i v

T,t

J

\41 1 1

.

(1.212)

Using Lemma 1.8 and the estimate (1.212), we have

T,

T,t if P

w. p. 1, where (see Theorem 1.1)

pp

j w'(k)]!trp =

£...£CjWi m c

ji=0 jfc=0 \/=i

(il) j'i

l.i.m. E j (Tii )AwT;i) ...0jfc (Tik )AwT;fcfcM (1.213)

(li.....ik№ i k J

4

2

p

k

or (see Theorem 1.2)

p p / k [k/2] j [#>]-p = £ ..^c,,]!! j' + B-D

ji=0 j=0 \/=i r=i

r k-2r

X

E II 1{iS2s-i = Ss =0}1{jS2s-i = J32s } Ü J '

({{31>32}>'">{32r-1>32r }}>{91>--->9k-2r}) S = 1 l=1

{S1,S2>'">S2r-1>S2r>91>'">9k-2r } = {1>2>'">k}

(1.214)

where i1,..., = 1,..., m in (1.213) and (1.214). 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 \ <

p91 iTO pq2 iTO pqk iTO ' ' / J

/ pi pk ^

— k! • lim ... lim

p9i ^0 pgk ^TO

/^i rk

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

\[i,T]k Ji=0 Jk=0 y

= k!

^ P TO TO ^

J K 2(ii,...,tk № ...dtk - I=cLji

\[t,T ]k =0 jqk =0 y

= 0

for the following cases:

1. i1,...,ik = 1,...,m and 0 <T — t< ro,

2. i1,..., ik = 0,1,..., m, ¿2 + ... + ik > 0, and 0 < T — t < 1.

At that, (q1,...,qk) is any permutation such that {q1,...,qk} = {1,...,k}, J[0(k)]T,t is the stochastic integral (11.51), J^^yf"^ is the expression on the right-hand side of (11.101) before passing to the limit l.i.m. , lim means

p1v,pk iro

lim sup; another notations are the .same as in Theorem 1.1.

Remark 1.7. Taking into account Theorem 1.4 and the estimate (1.210), we obtain the following inequality

^ m,(T t)

M|(J[r>]r,f- Jbn'ir) j < p ■ (1-215)

where i1,..., ik = 1,..., m and constant Pk depends only on k.

Remark 1.8. The estimates (1.72) and (1.210) imply the following inequal-

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 Ito 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 (1.216), we obtain

< (k!)2n(n(2n - 1))n(k-1)(2n - 1)!!

(Pk)n (T - t)

(1.216)

pn

(1.217)

where n G N and

Qn,k = k\ (n(2n - i))(*-D/2 ((2n - 1)!!)1/2" y/Pk {T — tf'2.

According to the Lyapunov inequality, we have

(1.218)

for all n G N. Following [96] (Lemma 2.1), we get

J[^(k)]T,t - J[^(k)]

J[^(k)]T,t - J[^(k)]?;rp <

1

(V2- J[^(k)]T,t - J[^(k)]^"p )

w. p. 1, where

ne = sup ( p

peN

1/2-£

J

-J

T,t

and £ > 0 is fixed.

For q > 1/e, q G N we obtain (see (1.218)) [96]

M {|n£|q} = M i (sup ( p1/2-£

^peN

J

T,t

J

p,...,p

M{ sup I p(1/2-£)q IpeN V

J

J

p,...,p T,t

<

oo

<

Mp

(1/2-£)q

. p=1

J[^(k)]T,t - Jfc*

oo

ep(1/2-£)<?m p=1

j[^(k)]T,t - Jty^fc*

<

p=1

p

oo.

p=1

p

(1.220)

From (1.219) we obtain that for all £ > 0 there exists a random variable % such that the inequality (1.219) is fulfilled w. p. 1 for all p G N. Moreover, from the Lyapunov inequality and (1.220), we obtain M {||q} < ^ for all q > 1.

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

q

q

q

KT(tl, . . . ,s) = 1{tfc<s}K(ti, . . . ),

(1.221)

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

i?(ti,...,tk, s) = 1{t1<...<tfc <s} 0i(ti) ...0k (tk) =

0l(ti) ...0k (tk ), ti <...<tk <S

(1.222)

0,

otherwise

where k > 1, ti,... G [t,T], and s G (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 (1.222) 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) G L2([t, T]k) is converging to this function in the hypercube [t,T]k in the mean-square sense, i.e.

lim

Piv,Pk ^^

Pi Pk

Ki(ti,... ,tk, s)

"£ ..^Cjk ...ji (s^ (ti )

ji=0 jk =0

1=1

L2([t,T ]k )

where

Cjk...ji (s) =

/k

KT(ti,...,tk ,s)n fai (ti )dti ...dtk

i=i

[t,T ]k

S t2

= J 0k (tk )j (tk ) ...J 0i (ti )0ji (ti )dti ...dtk

t t

is the Fourier coefficient, and

l/l

L2([t,T ]k)

k=

/

\

i/2

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

y

= 0, (1.223)

(1.224)

k

Note that

S ¿2

J = / ^k (tk) ...J ^i(ti)dwt(;i) ...dw£:) = (1.225)

t t T ¿2

1{t:<s}^k (tk) ... ^l(tl)dwt;i) . . .dwt::) W. p. 1,

f -Mtkoy^w ••• J w )

t t

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 = To < ... < rN = T, AN = max Arj i 0 if N i to, Arj = Tj+1 — Tj.

o<j<N—1

(1.226)

Theorem 1.11 [15], [27]. Suppose that every -fa(t) (l = 1,..., k) is a continuous nonrandom function on [t,T] and {fa(x)}°=0 is a complete orthonormal system of continuous functions in the space L2([t,T]). Then

pi pk / k

J[^(k)]s,t = l.i.m E ...£ Cj) n 4

(ii)

l.i.m. E j (Til )AwT;;) ...fa: (ri: ^w^) , (1.227)

(ii,...,i:)GG: 1 : J

where J[^(k)]s,t is defined by (II .225), s G (t,T] (s is fixed),

Gk = Hk\Lk, Hk = {(/i,...,4) : 1i,...,1k = 0, 1,...,N - 1},

Lk = {(/i,...,4): 1i,...,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,

T

* -

j = j (T )dw<->

are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjfc...j1 (s) is the Fourier coefficient (I! .2241). AwTj = wTj++1 — wTj (i = 0, 1,...,m), {Tj}N0 is a partition of [t,T], which satisfies the condition (1.226).

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 $(tb ..., tk) in (1.161) and (1.231) is replaced by 1{ti)...)tfc<s}$(t1,... ,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)

(1.228)

where

E

(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 (1.228) 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 (1.228) as

T t2

= /.../ £ (i{tfc<S}^(t1,...,tk)dwi;i)...dwi:k)) w.p.1, t t (ti,--,tfc)

(1.229)

where permutations (t1,... ,tk) when summing are performed in (1.229) only in the expression $(t1,..., tk)dwt(;i)... dwt(;k).

It is not difficult to notice that (1.228), (1.229) can be rewritten in the form (see (1.25))

T t2

J' = £ /•••/)1{tfc<s}dwt(;i) ...dw£k) w.p.1, (1.230)

where permutations (ti,... ,tk) when summing are performed only in the values 1{tk<s}dwt(|l)... 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 l2-i

J[^(k)]s,t = l.i.m. £ ... E 1{rifc<s}^i(Tii)... ^k(Tik^w^... Awjj) =

ik=o ii=o 1 k

N-i N-i

= fm. £. ■ ■ E 1(T.k<«iK(Til,.. ■ -T<k)Aw<ii>...AwT;k> =

ik =0 ii=0 1 k

N-i

= y.m. £ 1{„, <»}K (Til,..., T,k )Aw«;;)... Aw«;;: ) =

N k 1 k il,...,ifc = o

iq =ir; q=r; q,r=l,...,fc

T t2

= / ■It, (1{tk <s}K (ti,...,tk )dwt(;l} ...dwt:k ]) w.p. 1, (1.231)

t t (tl,-,tfc)

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(;l) ...dwt(:k).

According to Lemmas 1.1, 1.3 and (1.24), (1.25), (1.229)-(1.231), we get the following representation

J

s,t

Pi P:

T ¿2

E..^Cj:...ji (s) / ... E (jji (tl) ...jj: (tk )dw

t;i) ...dwt::^ +

ji=0 j: =0

t t (ti,-",t:)

+ RT,t,s

P i P:

E^E^:-j i(S)X

ji=0 j: =0

N —1

X l.i.m. E jj i (Ti i)... j (Ti: )AwT;1 )... Aw(;:) + RT:t;s'P:

; i,...,;:=0

iq =ir; q=r; q,r = i,...,:

Pi P: f N—1

E^E Cj:...j i (s) l.i.m. E jj i (Ti i)... j (Ti: )AwT;1 )... Aw^ —

—n —n \ N, , n i :

ji=0 j:=0 \ /i,...,/:=0

l.i.m. e jj i (t/ i)AwT;i)... j (t/:)Aw(::) +

(/ i,...,/: )GG:

_L pP i,...,P: +RT,t,s

P i P:

E^E^: ...j i(S)X

ji=0 j: =0

k

X inj;;) — l.i.m. E jj i(r/ i)AwT;1 )... j(r/:) +

/=1 (/ i,...,/: )GG:

I pPi,.",P: „ -1

where

nP i,...,P: RT,t,s

T t2 ' Pi p:

E ... 1{t:<s}K(t1,...,tk) — E ..^Cj: j (S^ jj; (t/)| X

(t ,...,tk) t t j =0 jk=0 /=1

Xdwt(; i) ...dwt(;:) =

k

T t2

= £ /•••/* <s|dwi;i) (1.232)

(tl,...,tfe) t t

/T ^ Pi Pk k

../£..£ Cjk...ji (s) n ^ (ti )dwt;i)... dwt:k) (1.233) (ti,...,tk) t t ji=0 jk=0 1=1

w. p. 1, where permutations (ti,...,tk) when summing in (1.232) are performed only in the values 1{tfc<s}dwt(| l} ...dwt(ifc). At the same time permutations (tl5... ) when summing in (1.233) are performed only in the values dwt(|l)... dw*^). Moreover, the indices near upper limits of integration in the iterated stochastic integrals in (1.232), (1.233) are changed correspondently and if tr swapped with in the permutation (t1,..., ), then ir swapped with in the permutation (i1,..., ).

Let us estimate the remainder Rri's^ of the series.

According to Lemma 1.2 and (1.38), we have

M <! (R^t, /k > > <

2

T H / Pi Pk k \

< Ck £ ■■■ K (ti,...,tk )1{tk <s} -£ ..^Cjk ...ji (s^ (ti) x

(ti,...,tk) t t V ji=0 jk=0 1=1 /

X dti ...dtk, (1.234)

where constant Ck depends only on the multiplicity k of the iterated Ito stochastic integral J[0(k)]s,t and permutations (t1,... ) when summing in (1.2,34) 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 (1.234) are changed correspondently.

Since K(t1,..., ) = 0 if the condition t1 < ... < is not fulfilled, then

m i (r,1*-^ ^ <

<

T t2

Pi Pk

K(¿1, . . . )ljtk<s} - E *

ji=0

x dt1... dtk,

MÜ fai (t)

X

jk=0

1=1

(1.235)

where permutations (ti,... , tk) when summing in (1.235) are performed only in the values dt1... dtk. At the same time the indices near upper limits of integration in the iterated integrals in (1.235) are changed correspondently.

Then from (1.38), (1.223), and (1.235) we obtain

R

^PiwPk T,t,s

<

<

T t2

Pi Pk

(t 1 ,...,tk ) t t

K (t1, . . . )l{tk <s} - E •

ji=0

ECjk ...ji MÜ fa* (ti)

X

jk=0

1=1

xdt1... dtk =

2

k

2

2

k

// Pi Pk k \ 2

i^(ti,..., tk, s) - E ... E Cjk...ji (s) J] fa* (ti ) I dti... dtk 1 0

[t,T]k V ji=0 jk=0 1=1 /

if p1,... ,pk 1 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 {fa (x)}°=0 is a complete orthonormal system of functions in the space L2([t,T]), each function fa(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>]->=p,!:iii~ £■ ■ ■ E Ckj(sKf.. ■ j(1.236)

ji=0 jk=0

Note that the expression on the right-hand side of (1.236) coincides for the case k = 1, 01(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 (1.210) we obtain the following inequality

J /\ ~(7 i...., /,. s)dt\ ...dl,.. Cl^(s) < (1.237)

[t,T]k jl=0 jk=0 P

where ^(t1,...,tk, s) and Cjk ...j, (s) are defined by the equalities (1.221) and (1.224), respectively; constant (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 0(t) (/ = 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 estimate

m| J[0(k)]s,t — J[0(k)]fr'^^ <

< k!

( P Pi Pk ^

J K2(ti,..., tk, s)dti... dtk - £ ... E C2k...ji (s)

V [t,T]k ji=0 jk=0 )

(1.238)

is valid for the following cases:

1. i1,...,ik = 1,...,m and 0 <T — t< to,

2. i1,...,ik = 0,1,..., m, if + ... + > 0, and 0 <T — t< 1,

where J[0(k)]s,t is the stochastic integral (1225), J[0(k)]s;i'"'pfc is the expression on the right-hand side of (1.227) before passing to the limit l.i.m. ,

Ki(t1,... ,tk, s) and Cjfc...jl (s) are defined by the equalities (11.221) and (11.224), respectively; s £ (t,T] (s is fixed); another notations are the same as in Theorem 1.11.

Remark 1.12. Combining the estimates (1.237) and (1.238), we obtain

M {(j[^>]«,( - JW>(t'Kr*)2} < (1.239)

where i1,..., ik = 1,..., m, constant Pk depends only on k; another notations are the same as in (1.237) and (1.238).

Remark 1.13. The analogue of the estimate (1.72) for the iterated Ito stochastic integral (1.225) has the following form,

m{ (j[^(k)],t - J[^(k)]PrPk)2nj <

< (k!)2n(n(2n - 1))n(k-1)(2n - 1)!! x

x

/ r. Pi Pk ^

I K2(ti,..., tk, s)dti... dtk - ^ ... Ê Cjk...ji (s)

V ]k ji=0 jk=0 J

, (1.240)

where J[^(k)]pp;1t''"'Pk is the expression on the right-hand side of (1.227) before passing to the limit l.i.m. , i^(t1,... ,tk, s) and Cjk j1 (s) are defined by the

Ply-lPk

equalities (1.221) and (1.224), respectively; s G (t,T] (s is fixed); i1,...,ik = 1 , . . . , m.

Remark 1.14. The estimates (1.237) and (1.240) imply the following inequality

2n"

M (J]s,t - J[^(k)<

< (k\)2n(n(2n - 1 ))"(fc"1)(2n - 1)!!

where ii,..., = 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 Based on Theorem 1.11

Consider particular cases of Theorem 1.11 for k = 1,..., 5

n

Pi

J[#>]«,, = l.i.m. £ Cji(s)cj;i), (1.241)

ji=0

Pi P2

(;iU(;2)

J [^<2)]«.' = PÜm» E E Cj2ji (s) ( Cjr'Cj:2' - 1{:l=;2=0}1{jl=j2} ) . (1.242)

ji=0 j2=0

Pi P2 P3

s.t

Pi,P2,P3i^ „ ■ „ ■ „ ji=0 j2=0 j3=0

= EEEc,Sj2jiwlz^zjr j-

1{;i=;2=0}1{ji=j2}Cj3) - 1{;2=;3=0}1{j2=j3}Cj(;i) - ) I , (1.243)

Pi

P4

J = l.i.m E ..^Cj4..j, (s) J] Z

(; ) j

ji=0 j4=0

J=1

i{;i=;2=0}1{ji=j2}Zj3 Zj4 - i{;i=;3=0}1{ji=j3}Zj2 j -

1 I A;2)A;3) ^ i A(;iV(;4)

-1{;i = ;4=0}1{ji=j4}Zj2 j - i{;2=;3=0}1{j2=j3}Zji Zj4 -

i i A;i)A;3) ^ i A(;iM;2) i

-i{;2=;4=0}1{j2=j4}Zji j - i{;3=;4=0}1{j3=j4}Zji j + + 1{;i=;2=0}1{ji=j2}1{;3=;4=0}1{j3=j4} + 1{;i=;3=0}1{ji=j3}1{;2=;4=0}1{j2=j4} +

+ 1{;i=;4=0}1{ji=j4}1{;2=;3=0}1{j2=j3} ,

(1.244)

J [0(5)]

{;i=;2=0} {;i=;4=0} {;2=;3=0} {;2=;5=0} {;3=;5=0}

Pi

P5

s.t

= l.i.m.

Pi,...,P5iro

E-E^ wrn Z

(; ) j

ji=0

Z (;3)Z (;4)Z (;5

{ji=j2}Zj3 Zj4 Zj5

Z (;2)Z (;3)Z (;5

{ji=j4}Zj2 Z j3 Zj5

Z (;i)Z (;4)Z (;5

{j2=j3}Zji Zj4 Zj5

Z (;i)Z (;3)Z (;4

{j2=j5} Zji Z j3 Zj4

Z (;i)Z (;2)Z (;4

{j3=j5}Zji Zj2 Z j4

j5=0 M=1

{;i=;3=0} {;i=;5=0}

{;2=;4=0} {;3=;4=0} {;4=;5=0}

Z (;2 {ji=j3}Zj2

Z (;2 {ji=j5}Zj2

Z(;i {j2=j4} Zji

Z(;i {j3=j4} Zji

Z(;i {j4=j5} Zji

Z(;4)z(;5

Zj4 Zj5

Z (;3)z (;4

Zj3 Zj4

Z (;3)Z (;5

Zj3 Zj5

Z (;2)Z (;5

Zj2 Z j5

Z(;2)Z(;3 Z j2 Zj3

+

4

5

+I{i1= = ¿2=0} =j2} {¿3 =¿4=0} {j3 = =j4 } Z j5 + ^¿1 = =¿2=0} j =j2} {¿3 = =¿5=0} {j3 = j Cn ¿ +

+I{i1= =¿2=0} j =j2} {¿4 = ¿5=0} j Z ^ =j5}Zj3 + ^¿1 = = ¿3=0} j =j3} {¿2 = ¿4=0} {j2 = Z (¿5 =j4}Zj5 +

+I{i1= =¿3=0} j =j3} {¿2 = ¿5=0} {j2 = ■ (¿4) =j5} Z j4 + ^¿1 = = ¿3=0} j =j3} {¿4 = = ¿5=0} j =j5 } Z j2 +

+ I{i1 = =¿4=0} j =4} {¿2 = ¿3=0} {j2 = Z (¿5) =j3}Zj5 + ^¿1 = = ¿4=0} j =j4} {¿2 = ¿5=0} {j2 = Z fe =j5}Zj3 +

+ I{i1 = =¿4=0} j =4} {¿3 = ¿5=0} {j3 = ■ iZ (¿2) =j5} Z j2 + ^¿1 = = ¿5=0} {jl =j5} {¿2 = ¿3=0} {j2 = ■ iZ (¿4 =j3}Z j4 +

+ I{i1 = =¿5=0} =j5} {¿2 = ¿4=0} {j2 = z ^ =j4}Zj3 + ^¿1 = = ¿5=0} {jl =j5} {¿3 = = ¿4=0} {j3 = Z ^.¿2 =j4}Zj2 +

+ I{i2 = =¿3=0} {j2 = =j3} {¿4 = ¿5=0} { j4 = ■ iZ (¿l) =j5} Z j1 + ^¿2 = = ¿4=0} {j2 = =j4} {¿3 = = ¿5=0} {j3 = ■ T,Z (¿1 =j5 } Z jl +

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

(¿l)

jl

where 1A is the indicator of the set A, C^.... (s) (k = 1,..., 5) has the form (L223), s g (t,T] (s is fixed).

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

l.i.m.

N->oo

£ * (j,..., j) n Aw<j;) = J'^g, (1.245)

j1>---Jfc = 0

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

1=1

where for simplicity we assume that <&(ti,..., ) : [t, T]k i R1 is a continuous nonrandom function on [t,T]k. Moreover, {t.}f= _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 = ... = = 0) and similar to (1.245) was considered in [97] and is called the multiple Wiener stochastic integral [97].

Consider the following theorem on expansion of the multiple Wiener stochastic integral (1.245) based on generalized multiple Fourier series.

Theorem 1.13. Suppose that $(t1,... ,tk) : [t,T]k i R1 is a continuous nonrandom function on [t, T]k and {fa(x)}°==0 is a complete orthonormal system

k

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«=ä X.Cjk.Jni"

ji=0 jk=0 \/=1

1:i:m: ^ j fa ^W^ • • • ^j fc (Tfc W ^^ ^ ^ , (l:246) (/i,...,/k)GGk 1 ' J

Pi Pk / k [k/2]

J'^ = ".m. E...ECjk..Jnzi")+B-D

Pi,...,Pkiœ .

ji=0 jk=0 \/=1 r=1

k- 2r

x £ n 1{ig2s-i = ig2s=0}1{jg2s-i=jg2s } n j)

({{Sl>32 }>...>{S2r-l>S2r }}>{9l>...>9fc-2r}) S=1 l = 1

{Si >32 >...>S2r-l>32r>9l>...>9fc-2r }={l>2>.">fc}

(1.247)

converging in the mean-square sense are valid, where

Gk = Hk\Lk, Hk = {(/!,...,/*): l1,...,1k = 0, 1,...,N — 1},

Lk = {(l1,...,1k): h,...,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 j (s)dw<!>

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0),

/k

$(t!,...,tk)n (t/)dt1 ...dtk (1.248)

l=1

is the Fourier coefficient, AwTj = wTj++l — wTj (i = 0, 1,... ,m), fa}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.

r

Proof.

tation

I, (1.25), we get the following represen-

•/WS =

pi pk

£■•• E

ji=0 jk=0

T t2

C

Cjk ...ji

t t (ílv,ífc)

E (fe(ti)■•• j(tk)dw<;i)•.• dw(ik>) +

+ RPl,...,Pfc

T,t

Pi Pk N-1

£ • • • £ Cjk...ji l.i.m. E j(Tii) • • • j(Tik)AwT;;) • • • Aw(;k)+

ji=0 jk =0 Ji,...,Jfc=0 i k

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

T?Pi,...,Pk

Pi Pk / N-1

£•••£Cjk...ji l.i.m. E j(Tii)• • • j(Tik)AwT;i)• • • AwT;k)-

• n ■ n X N^TO

ji=0 jk=0 \ ii,...,ik=0

l.i.m. £ j (Tii )Aw(;;) • • • j (Tik )Aw[;fck M +

(l i, — ,ík )GGk

nP i,...,Pk

+RT,t

P i Pk

£•••£ Cjk...j ix

ji =0 jk=0

k

x in j) - l.i.m. £ i (Ti i )Awí¡;) • • • j (Tik )AwT;k ) ) +

i=i N^TO (i i,...,ik№

+RTxf.'Pk w. p. 1,

where

T ti ( Pi Pk k \

Y, /•••/ *(t!,...,tk)-£•■^Cjk.jn(ti)

(ti,...,tk) t t V ji=0 jk=0 1=1 /

RTrPk = V l -l I$(t1,...,tfc) -> \..VCjvjl Uj, (ti )| x

xdwt(il) ...dwt(ik),

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 iq in the permutation (i1,..., ik).

Let us estimate the remainder RyV'Pk of the series using Lemma 1.2 and (1.38). We have '

Ml i RT1t""P^ ^ <

T t2

<

«Z.J

(t i.....tk ) t t

P i Pk

$(t1,...,tk) ..^Cjk ...j i n^j, (ti)

j =0 jk=0

x

1=1

xdt1... dtk =

2

k

2

n / P l Pk k \

= Ck J U(t1,...,tk) — ^ ..^Cjfc...ji (t/)l dt1 ...dtk ^ 0

[t T]k V ji=0 jk=0 /=1 /

if p1,... ,pk ^ to, where constant Ck depends only on the multiplicity k of the multiple Wiener stochastic integral J'[^y]. The expansion (1.246) is proved. Using (1.246) and Remark 1.2, we get the expansion (1.247) (see Theorem 1.2). Theorem 1.13 is proved.

Note that particular cases of the expansion (1.247) are determined by the equalities (11.44l)-(ITT5ni), in which the Fourier coefficient Cjk ...j , (k = 1,..., 7) has the form (1.248).

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

In [98] it was noted that Theorem 3.1 ([97], 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 [98 using our notations. Note that we derive the formula (1.252) (see below) in two different ways. One of them is not based on Theorem 3.1 [97].

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 98] the case ¿i,..., ¿k = 1,..., m was considered.

Consider the multiple Wiener stochastic integral J/[0j1... fajk]T

(n-.-ifc)

($(ti,...,tk ) = faj (ti)... j (tk )) defined by (1.23) (also see

where

{faj (x)}°=0 is a complete orthonormal system of functions in the space L2([t, T]), each function fa (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7).

It is not difficult to see that

J '

ji • • • jjt,Î

= J'

j • • • ^jSm1

• • • • J '

mi ( i1...i1 )

• J'

m2 ( ¿2...«2 ) T,t

mfc ( ifc ...ifc )

(1.249)

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'

...

-/gmi+m2+---+mi_i + i -/gmi+m2+---+mi

( )

= 1 for m/ = 0; (1:250)

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 + ---+mi_i + i ^jgmi+m2+---+m;

( "l---"l )

(m/ > 0),

where we suppose that

J,'â,m1+m2+...+m|_1 + l ' ' ' J,S,m1+m2 + ... + m| J j J^'1'' ' ^ ' ' ^ ' ' ' ' ' ' ' Jhd^l

n1,l n2,l

(1.251)

where nx,/ + +... + ndl,/ = m/, nx,/,..., n^i,; = 1,... ,m/, d/ = 1,... ,m/, l = 1,..., k. Note that the numbers mx,..., mk, gx,..., depend on (ix,..., ) and the numbers nX;/,..., ndl;/, hX;/,..., hdl;/, d; depend on (jx,..., jk). Moreover, {j'gi .....jgk} = {jx,...,jk}.

Using Theorem 3.1 [97], we get w. p. 1

J

^jgmi+m2 + ---+mi_i+i ^jgmi+m2 + ---+m;

( ")

ku i) -h*, (zi:t), if =

0

=

(m/ > 0), (1.252)

z(on ni •1 fz(0) \

if i/ = 0

where Hn (x) is the Hermite polynomial of degree n

Hn(x) = (—1)nex

dxn

or

[n/2]

Hn(x) = n! ^

(—1)mx

m~,n—2m

m=0

m!(n — 2m)!2î

(n G N),

(1.253)

and (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 (1.250) and (1.252) we obtain w. p. 1

J'

m; ( i;...i; )

k,.,j(C,), if ii

— 1{m;=0} + l{m;>0} <

z(0) \ ni-' (0) A

if i — 0

(1.254)

where 1A denotes the indicator of the set A. Using (1.249) and (1.254), we get w. p. 1

J/ [ j • • • j ]t1

(il...ifc )

n

1=1

(

K j) j) ,

l{m;=0} + l{m;>0} <

V

rc(°Mni,;...j)

if ii — 0 if ii — 0

\

/

(1.255)

where notations are the same as in (1.251) and (1.252).

The equality (1.255) allows us to reformulate Theorems 1.1, 1.2, and 1.13 using the Hermite polynomials.

Theorem 1.14 [27] (reformulation of Theorems 1.1 and 1.2). Suppose that the condition (**) is fulfilled for the multi-index (¿i... ¿k) and the condition (1.251) is also fulfilled. Furthermore, let every ^ (t) (l = 1,..., k) is a continuous nonrandom function on [t,T] and {faj(x)}°=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). Then the following expansion

J

(il...ik)

Pi

Pk

l.i.m.

...

Cjk ...jix

ji=

jk=0

X

n

1=1

1

H

{m;=0} + 1{m;>0} <

V

z (i; ) \ Zjhl, J '

z(0) )ni';

..H

nd V jh

(0) \nd;.; . . 1 Zjhd;,;.

j) J, if ii—0

if ii — 0

/

(1.256)

converging in the mean-square sense is valid, where we denote the stochastic integral (O) as J[^(k)]Tvt'"ifcni,/ + n2,/ + ... + ndj,/ = m/, ni,/, n2,/,..., ndj,/ = 1,..., m/, d/ = 1,..., m/, I = 1,..., k; the numbers mi,..., mk, gi,..., gk depend on (¿i,..., ¿k) and the numbers n1;/,..., ndl;/, h1;/,..., , d/ depend on (ji,...,jk); moreover, {jg1 , ...,jgk} = {ji,...,jk}; another notations are the same as in Theorem 1.1.

Theorem 1.15 [27] (reformulation of Theorem 1.13). Suppose that the condition (**) is fulfilled for the multi-index (¿i... ¿k) and the condition (1.251) is also fulfilled. Furthermore, let $(ti,..., tk) : [t, T]k ^ R1 is a continuous non-random function on [t,T]k and {fa(x)}= 0 is a complete orthonormal system of functions in the space L2([t,T]), each function fa (x) of which for finite j satisfies the condition (*) (see Sect. 1.1.7). Then the following expansion

j '[*]

(il...ik)

Pi

Pk

l.i.m.

Pi,...,Pk ^^

...

ji=0 jk=0

Cjk...ji X

X

n

i=i

(

1{m;=0} + 1{m;>0} <

V

[Hni,; (Cjhi);) , if ii — 0

,

, if ii — 0

z(0) \ ni'; (0)

/

0

converging in the mean-square sense is valid, where we denote the multiple Wiener stochastic integral (1.245) as J'[$]ylt"ik); n1;/ + n2j/ + ... + ndlj/ = m/, n1;/, n2;/,..., ndl / = 1,...,m/, d/ = 1,...,m/, l = 1,...,k; the numbers m1,...,mk, g1,...,gk depend on (i1,...,ik) and the numbers n1;/, ...,ndl;/, h1,/,..., hdi,/, d/ depend on (j,..., jk); moreover, j,..., jgk} = {j,..., jk}; another notations are the same as in Theorem 1.13.

From (1.254) we have w. p. 1

Hk (cf), if i1 = 0

(k > 0). (1.257)

1 jk, if i1 = 0

k

J/[j - - - j]T;i...;i) =<

Let us show how the relation (1.257) can be obtained from Theorem 1.2. To prove (1.257) using Theorem 1.2 we choose i1 = ... = ik and j = ... = jk (i1 = 0,1,... ,m) in the following formula (see (1.39) and (1.53))

k

j' [j ... j ]?r,k) = n cf+

/=1

[k/2] r k—2r

+ S( —11)r S n 1{iS2s-i = iS2s =0} 1{jS2s —l = jS2s } h C

r=1 ({{Sl>S2}>...>{S2r-l>S2r }}>{9l>...>9k-2r}) S = 1 /=1

{Sl >S2 >...>S2r -1 >S2r >qi >...>?k —2r } = {;>2>...>k}

(1.258)

w. p. 1, where notations are the same as in Theorem 1.2.

The case i1 = 0 of (1.257) is obvious. Simple combinatorial reasoning shows that

r k—2r

Z) n 1 {iS2s —l = iS2s =0}1{jS2s-l = jS2s } H j =

({{3i,32}v>{32r-i>32r}}>{9iv>9k-2r}) s = 1 1=1

{3i,32v>32r-i>32r>9iv>9k-2r } = {i>2v>k}

r!

Z^) " r, (1.259)

where i1 = ... = ik, j = ... = jk (i1 = 1,... ,m), and

ri k\ k l\{k-l)\

is the binomial coefficient. We have

Cl • Cfc-2 • • • • • Cl-(r-1)2 = k\

r! r\(k — 2r) !2r * 1 ;

Combining (1.258), (1.259), and (1.260), we get w. p. 1

7., r= 1

— 2r)!2'* J ~ k V>

The relation (1.257) is proved using

From (1.255) and (1.258) we obtain the following equality for multiple Wiener stochastic integral

k [k/2] J'[j...jfr"> = ncf 1)

/=1 r=1

k-2r

X Z n 1{ig2s-i = "g2s =0}1{jg2s-i = j»2. }H j )

({{Si ,32},---,{S2r-i,S2r }},{9i,---,9k-2r}) s=1 / = 1

{Si ,S2,---,S2r-i,S2r,9i,---,9k-2r }={i,2,---,k}

r

n

/=1

/

if i/ = 0

\

1{m;=0} + 1{m1 >0} <

V

z (0fa ni-' (z (0)

i,1

j;

if i/ = 0

/

(1.261)

w. p. 1, where notations are the same as in Theorem 1.2 and (1.251), (1.252).

nd,,i

Let us make a remark about how it is possible to obtain the formula (1.252) without using Theorem 3.1 [97 .

Consider the set of polynomials Hn(x,y), n = 0,1, • • • defined by [99

Hn(x,y) = —eax-a^2

(Ho(x,y) =f 1),

a=0

It is well known that polynomials Hn(x,y) are connected with the Hermite polynomials (1.253) by the formula [99]

[n/2]

Hn(x,y) = yn'2Hn[4=)=n\Y

.VV,

¿=0

(-l)V7"2y

i!(n-2i)!2r

(1.262)

For example,

From

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

253) and (1.262) we get

Hn(x, 1) = Hn(x)

Obviously, without loss of generality, we can write

(j1 • • jk) = (¿1 • • • j1 ¿2 • • j • • • ¿r • • •jr) ,

(1.263)

(1.264)

mi

m2

where m1 + ... + mr = k, m1,..., mr = 1,..., k, r = 1,..., k, k > 0, and ji ..., jr are pairwise different.

Analyzing the proof of Theorem 1.1, we can notice that (we suppose that the condition (1.264) is fulfilled)

J ' fa^ ...fajk ]Tiri) =

N- 1

= l.i.m.

N->oo

£ j (Tii ) • •• j K )Aw(;;) ••• Aw(;;) =

ii,...,ifc=0 l q = lr; q=r; q,r=1,...,fc

N-1

= U.m. Y, j (r'i) .■■^j. (Timi )Aw<|;> .■■ Aw<;;> x

ii ,---,,mi =0 iq = ir; q=r; q,r=i,...,mi

N-1

X E j (Timi + i) . . . j (T1„;+.2 )Aw';:' . . . ^^ X . . .

,mi + i ,...',mi+m2 =0 ,q=,r; q=r; q,r=mi + i,... ,mi+m2

N1

- x E j (Tik-mr+i) ...j (Tik )Aw«;;-mr- Aw«:;).

,k-mr + i,...,,k=0 ,q=,r; q=r; q,r = k —mr + i,...,k

We have (see the proof of Theorem 1.1) w. p. 1 (i1 = 0)

N-1

E j (ti. ) ... j (Timi )aw«;;> ... Aw«;m»

ii,...,imi=0 iq =ir; q=r; q,r=i,...,mi

/N-1 \ mi [mi/2]

e^j, (Ti, )aw<;;M + £(-1)

\/i=0 J r=1

-1 )rx

/N-1 A r

x e e<j (t/, )(aw<;;>) x

-i>32r }},{qi>...>qm —2r}) Vi=0 /

({{si>32 }>...,{S2r— i>S2r }},{qi>...>qmi—2r }) Vi=0 {Si,S2 >...>S2r—i>S2r>qi>...>qmi—2r }={i,2>...,mi}

'N-1 \ mi-2r

x j (t/i)Aw(;;) ii=0 i

'N-1 \mi [mi/2] /N-1 A r

g^)^:1) + 5 (g^> (A<") Jx

'N-1 \ mi-2r

x j (t/i)aw(;;) ii=0 i

[mi/2] , . /N-1 2\ r /N-1 \ mi-2r

/N-1 N—1 2\

= H„1 y, faj1 (t/1 )awT;;>. E faJ,(T/,) (Aw<;;>) ,

\/;=0 /1=0 J

where notations are the same as in Theorems 1.1, 1.2. Similarly we get w. p. 1

N-1

£ fa2 (T,„; + ; ) . . . faj2 (T,„; + „2 . . .

l m; + 1?---?1 m;+m2 =0 l q = l r; q=r; q,r=m; + 1,...,mi+m2

/N-1 N—1 2\

= ^ £ faj2(t/;)Aw';;»^ fa?2(T/;) (Aw<;;^ ,

\/;=0 /i=0 J

N-1

E j (T-k-m,+i) .•• j (Tik )Aw«i'»mr+i... Aw«;;»

;q = ; q,r = k-m, + i,...,k

N 1 N 1

/N-1 N-1 2\

Hm^ j (Tii )AW<ii>^^|, (Tii )(AW<ii>) \ii=0 ii=0 /

Then

J '[ j... j]

(i1...i1) T,t

/N-1 N-1 2

l.i.m. Hm^ j(Tii^w^, £ j(Tii) (Aw^ | x

Vi i=0 i i =0

/N-1 N-1 2

xHm^ j(Ti i)Aw^ £ j(Ti i) (Aw^) I X i =0 i =0

X

'N-1 N-1

2

Hm^ j(t, i)Aw<|;>, £ j(t, i) (Aw<;;>) (1.265)

i =0 i =0

w. p. 1 for i1 = 0 and

J'fei • • • j]TV0) =

'N-1 \ mi /N- 1

N/im ( £ faji (T/i )at/J ... £ fajr (T/r )AT/r

Ji=0 / \/r=0

T \ mi / T

f fai (s)dsl "I J fajr (s)ds

= (C )m; .■■(cj0,)mr (1.266)

for ¿1 = 0, where we suppose that the condition (1.264) is fulfilled; also we use in (1.265) and (1.266) the same notations as in the proof of Theorem 1.1.

Applying (1.262), (1.263), Lemma 1.3, and Remark 1.2 to the right-hand side of (1.265), we finally obtain w. p. 1

J'fei • • • j ]

( ¿i...ii) T,t

T T

= HmJ / j (s)dwiii)^ j (s)d^ X ,t t T T

xHmJ i j (s)dwSii) J j (s)ds| X .t t T T

•• X Hm I i j (s)'iw<!1>, / j (s)ds| =

= Hmi j ^ ff„,2 j, ^ ...Hm, j 1

= Hmi (cf) (cj2l) ) ...Hmr j)

for i1 = 0, where we suppose that the condition (1.264) is fulfilled. Thus, an equality similar to (1.252) is proved without using Theorem 3.1

m

r

Consider particular cases of the equality (1.261) for k = 1,..., 4 and ii,..., i4 = 1,..., m (see (144) (147)). We have w. p. 1

ji

= <

J[j]Ti1t) = cjii) = h (c

j [ j j #1=cjij - i{.i=.2}i{,i=,J}

H2 v.

Ho icf) , if ii = ¿2, j = j2

ji

Hi j j

otherwise

(1.267)

j ' [ j j j ßf0 = zfzfzf -1 jj zjii) - i{j2=j3}Cj;i) - 1 j=*}ci

-(ii) j:

H3 V(;i)" , ji Ho V (;i) fa Ho ;ci3i0 ■ if ji = j2 = j3

H2 V (;i)" > Ho V (;i)" fa Hi ( ;o3"), if ji = j2 = j3

< Hi V(;i)" > H2 V(;i)" fa Ho fa, if j2 = j3 = ji

Ho V (;i)" > Hi ' z(;i)" fa H2( ;o3"), if ji = j3 = j2

Hi V(;i)" > Hi V(;i)" fa Hi ( fa, if ji = j2, j2 = j3, ji = j3

; (1.268)

J1 U M M l^i^ = z (ii)z (i2V(i2) = H /V (il)

J I0jl Mj2 Mj3 JT,i = Zji Z j2 Zj3 = H1 ^Z

where i1, ¿3 are pairwise different;

,Hi izf) Hi j

J[ jjj^f^ = C]i')Cji)Cj3) - 1 j^j = = (jj' -1 {ji=j:}) cj33) = j [ j j iTr^ [^j-3 ]T;,3i) =

H2

(ii)

ji

—

H1

(ii)

ji

where i1 — i2 — i3;

j j ]?r2)—c!:l)c)22»cí:2» - 1{j2=jS}C

z(ii) (z(i2 ) Z(i2)

Zji \VZJ2 Zj3

H1

—

H1

(ii) ji

(ii) ji

where i1 — i2 — i3;

J'[ j j j]

z(i2) z(ii)z(ii) Zj2 Zj3

H2

(ii)

ji

—

H1

(ii)

ji

H0

H1

(ii)

j2

(ii) j2

H1

H1

(i3)

j3

(i3)

j3

1{j2=j3^ — J '[ j

H2

H1

(i2)

j2

(i2)

j2

H0

H1

(i2) j3

(i2) j3

(iii2ii) _ /-(ii^(i2^(ii)

T,t

1{ji=j3^ — J '[ j

H1

H1

(i2) j2

(i2) j2

H0

H1

(ii)

j3

(ii) j3

if j1 — j2 if j — j2

(ii) ji

T1,1^) j [ j j fe2) if j2 — j

if j1 — j2

— z^i2)_ I z(i2) —

Zjl z j2 zj3 1 {j 1 = j3} z j2

(i2) 7'T^ A l(ilil)

T,t

J /[0ji j]

T,t

if j1 — j3 if j1 — j3

where i1 — i3 — i2;

T'\A A A A, l(iiiiiiii) _ A:i V(iiV(iiV(ii)

J [ j r j2 j j ]T,t — j j j j -

n z(ii)z^ _ 1 z(ii)z^ _ i z(ii)z(ii)

1{j1=j2} zj3 zj4 1 {j 1 = j3} zj2 zj4 1 {j 1 = j4} zj2 j

1 z (ii)z (ii ) — 1 z (ii)z ^ _ 1 z (ii )z (ii)+

1 {j2 = j3} zj 1 zj4 1{j2=j4} zji zj3 1 {j3 = j4} zj 1 zj2 +

+ 1 {j 1 = j2 } 1 {j3 = j4 } + 1 {j 1 = j3 } 1 {j2 = j4 } + 1 {j 1 = j4 } 1 {j2 = j3 }

'H4 z(ii) H0 z(ii) H0 z(ii) ^zj3 H0 z(ii) , zj4 , if (I)

H1 z(ii) H1 z(ii) H1 z(ii) zj3 H1 (z(ii)) , zj4 , if (II)

H2 z(ii) H0 z(ii) H1 z(ii) zj3 H1 (z(ii)) , zj4 , if (III)

H0 z(ii) H1 z(ii) H2 z(ii) ^zj3 H1 (z(ii)) , zj4 , if (IV)

H0 z(ii) H1 z(ii) H1 z(ii) ^zj3 H2 (z(ii)) , zj4 , if (V)

H1 z(ii) H0 z(ii) ,j2 H2 z(ii) ^zj3 H1 (z(ii)) , zj4 , if (VI)

H1 z(ii) H0 z(ii) ,j2 H1 z(ii) zj3 H2 (z(ii)) , zj4 , if (VII)

H1 z(ii) H1 z(ii) ,j2 H0 z(ii) > H2 (z(ii)) , zj4 , if (VIII)

H3 z(ii) H0 z(ii) ,j2 H0 z(ii) > H1 (z(ii)) , zj4 , if (IX)

H1 z(ii) H3 z(ii) ,j2 H0 z(ii) > H0 (z(ii)) , zj4 , if (X)

H0 z(ii) H0 z(ii) ,j2 H1 z(ii) zj3 H3 (z(ii)) , zj4 , if (XI)

H0 z(ii) H1 z(ii) ,j2 H0 z(ii) zj3 H3 (z(ii)) , zj4 , if (XII)

H2 z(ii) H0 z(ii) ,j2 H0 z(ii) zj3 H2 (z(ii)) , zj4 , if (XIII)

H2 z(ii) H2 z(ii) ,j2 H0 z(ii) zj3 H0 (z(ii)) , zj4 , if (XIV)

H2 V z(ii) H0 z(ii) ,j2 H2 z(ii) zj3 H0 (z(ii)) , zj4 , if (XV)

where Hn(x) is the Hermite polynomial (11.25,3) of degree n and (I)-(XV) are the following conditions

(I). j1 = j2 = j3 = j4,

(II). jb j2, j3, j4 are pairwise different,

(III). ji = j2 = J3,J4; j3 = j4,

(IV). j1 = j3 = j2 5 j4; j2 = j4,

(V). j1 = j4 = j2,j3; j2 = j3,

(VI). j2 = j3 = j1, j4; j1 = 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,

(XII). j1 = j3 = j4 = j2,

(XIII). j1 = j2 = j3 = j4,

(XIV). j1 = j3 = j2 = j4,

(XV). j1 = j4 = j2 = j3.

Moreover, from (1.249) we have w. p. 1

j' [Mji Mj, j M,4]T,f*3,4) = H1 j) H1 (zj:20 H1 (z<::)) h (zj?).

where i1, i2, i3, i4 are pairwise different;

j' [Mji Mj2 Mj: Mj4 ]Tf:!4) = J' [Mji MgTr^ (zf) H1 (z]44)), (1.269)

where ¿1 = ¿2 = ¿3, ¿4; ¿3 = ¿4;

J' [Mji Mj2 Mj: Mj4 ]?,r*4) = J' [Mji Mj:]Ti,r)H1 (zf) h (zj440, (L270>

where ¿1 = ¿3 = ¿2, ¿4; ¿2 = ¿4;

j'[MjiMj2Mj:Mj4]Ti;2,:"i) = j'[Mjim^t^h (zf) H1 (zf), (1.271)

Electronic Journal. http://diffjournal.spbu.ru/ A.185

where ¿1 = ¿4 = ¿2, ¿3; ¿2 = ¿3;

J [fa,; fa« fajS fa* lr,4) = J[fa» fajS ]T;2t;2)Hi j) Hi j), (1.272)

where ¿2 = ¿3 = ¿1, ¿4; ¿1 = ¿4;

J' [faji fa« faj3 faj4 If 3'2) = J[faj2 faj4 ]T;;(,2)Hi j) Hi (cf), (1.273) where ¿2 = ¿4 = ¿1, ¿3; ¿1 = ¿3;

J'[faj;faj2fa*faj4]Tf3'3) = J'[faj3faj4]T;3i's)Hi j) Hi j) , (1.274)

where ¿3 = ¿4 = ¿1, ¿2; ¿1 = ¿2;

J '[faji faj2 faj3 faj4 ]?r'4) = J' [faji faj2 faj3 tff0 Hi (cj44)) , (1.275)

where ¿1 = ¿2 = ¿3 = ¿4;

J'[fajifaj2faj3faj4]Ti'2'2'2) = J[fajfaj3faj4]t?'2)Hi j) , (1.276)

where ¿2 = ¿3 = ¿4 = ¿i;

J'[faj1 faj faj3 faj4 ]Tf = J' [faji far, faj4 tff0 Hi (c]33)) , (1.277)

where ¿i = ¿2 = ¿4 = ¿3;

J'[faji faj2 faj3 faj4 ]T;;('2;;'1) = j ' [faj, faj3 faj4 ]^T;;i'i;;) Hi j), (1.278) where ¿i = ¿3 = ¿4 = ¿2;

J'[faj, faj2 faj3 faj4 ]£f ^ = J' [faji faj2 ^ J' [faj3 faj4 f^ (1.279)

Electronic Journal. http://diffjournal.spbu.ru/ A.186

where ¿1 = ¿2 = ¿3 = ¿4;

J' [Mji Mj2 Mj: Mj4 ]íгi1í!:í1!:) = J' [Mji MjJi'J [Mj2 Mj4 ®2), (1.280)

where ¿1 = ¿3 = ¿2 = ¿4;

j' [Mji Mj'2 Mj: Mj4 tfiT'0 = J' [Mji Mj4 ]Tir)J' [Mj2 Mj: If. (1.281)

where ¿1 = ¿4 = ¿2 = ¿3.

Note that the right-hand sides of (11.269l)-(l1.281) contain multiple Wiener stochastic integrals of multiplicities 2 and 3. These integrals are considered in detail in (1.267), (1.268).

It should be noted that the formulas (1.53) (Theorem 1.2) and (1.256) (Theorem 1.14) are interesting from various points of view. The formulas (1.44) (149) (these formulas are particular cases of (1.53) 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 [51] and [52], approxima-

tions of iterated Ito stochastic integrals of multiplicities 1 to 6 in the Python programming language were successfully implemented using (1.44)—(1.49) and Legendre polynomials.

On the other hand, the equality (1.256) is interesting by a number of reasons. Firstly, this equality connects Ito's results on multiple Wiener stochastic integrals ([97], Theorem 3.1) with the theory of mean-square approximation of

iterated Ito stochastic integrals presented in this book. Secondly, the equality (1.256) 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 (1.82) (see the proof of Theorem 1.3).

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 5. The mean-square convergence of the mentioned expansions for the case of multiple Fourier-Legendre series as well as for the case of multiple trigonometric Fourier series 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 a different approach to expansion of iterated Stratonovich stochastic integrals of multiplicity k (k £ N) based on generalized iterated Fourier series converging pointwise (Sect. 2.5).

2.1 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicity 2 Based on Theorem 1.1. The case pi,p2 ^ to and Smooth Weight Functions

2.1.1 Approach Based on Theorem 1.1 and Integration by Parts

def

Let F, P) be a complete probability space and let f (t,w) = f : [0,T] x ^ —y R1 be the standard Wiener process defined on the probability space F, P).

Let us consider the family of a-algebras {Ft, t £ [0, T]} defined on the prob-

ability space 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 /t+A — ft 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 , t G [0, T] of the form

T T

nr = no + J ads + J bsdf, 0 0

(2.1)

where fa)m , fa)m G M2([0,T]) and

M

{|bs - br|4} < C|s - T|

for all s,t G [0,T] and for some C, 7 G (0, to).

The second integral on the right-hand side of (2.1) is the Ito stochastic integral (see Sect. 1.1.2).

Consider a function Ffa t) : R1 x [0, T] ^ R1 for fixed t from the class C2(—to, to) consisting of twice continuously differentiable in x functions on the interval (—to, to) such that the first two derivatives are bounded.

Let t(N^, j = 0,1,..., N be a partition of the interval [t, T], t > 0 such that

,(N) ^ (N)

(N )

t = t0 <Tfa <...<tN" = T,

The mean-square limit

max

0<j<N-1

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

0 if N 00.

(2.2)

N-1 /1

l.i.m V F I - U (N) + rj,

N^ V^ Tj j =0

= T

'faJ ,Tj

(N)

/T(N) - j0 =/ F(nr,T)dfr (2.3)

is called [100] the Stratonovich stochastic integral of the process Ffa, t), t g

[t,T], where t]N^, j = 0,1,..., N is a partition of the interval [t,T] satisfying the condition (2.2).

Y

It is known [100] (also see [75]) that under proper conditions, the following relation between Stratonovich and Ito stochastic integrals holds

* T T T

J F(rjT,r)dfT = J F(rjT,r)dfT^^J^(rjT,T)bTdr w. p. 1. (2.4) t t t

If the Wiener processes in (2.1) and (2.3) are independent, then

* t t

J F(nT,T)f = J F(nT,Tf w. p. 1. (2.5)

tt

A possible variant of conditions under which the formulas (2.4) and (2.5) are correct, for example, consisits of the conditions

nr G Q4([t,T]), F(nT,T) e M2([t,T]), and F(x,t) e C2(—w, w).

Further in Chapter 2, we will denote as {faj (x)}°=0 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 [101].

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

w r,

y^ Cj fa (x), Cj = f (x)fa (x)dx j=0 {

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) laying 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

(k)]T,t = / (tk)... i ^i(ti)dwt ;i}... dw(kk), (2.6)

T t2

J ["(k)fa = / " (tk) ..J faifaw^... dwt(: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]-[20], [31]. Suppose that fajfajfa is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). At the same time "fas) 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

Jfa(2)]T,t = J fai2)| "i(ti)dft(il)dft(;2) (ii,i2 = 1,...,m) tt

the following expansion

Pi P2

J *["(2)]T,t = l.i.m. EE j W

jl=0 j2=0

that converges in the mean-square sense is valid, where

T S2

Cfa = j ^fafa (S2^ "ifa)fa (si)dsids2 tt

and

T

j fa M (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

Jfa(2)fa = Jfafa,, + il{il=i2} jUti)Uti)dh, (2.8)

t

where here and further 1a is the indicator of the set A. From the other side according to (1.45), we have

P1 P2

J № (2)kt = pi.i;^ £ £ Cj (cj ;:,)cjr - =

¿1=0 ¿2=0

pi P2 min{pi,p2>

l-i.m^ £ £j C]:i)Cj22) - 1{.1=.2> lim £ Cjiji. (2.9)

1 «o—Vno ' * ' * J1 J2 L j p p —f *

= l.i-m. > 2vCj2j1 Cj^C^ - 1{i1 = i2} ^

- -' J1 J2 L J

¿1=0 ¿2=0 ¿1=0

From (2.8) and (2.9) it follows that Theorem 2.1 will be proved if

T 1 ^

- Uti)Uti)dti = YC^. (2-10)

{ j1=0

Note that the existence of the limit on the right-hand side of (2.10) will be proved in Sect. 2.1.2 for the polynomial and trigonometric cases (Lemma 2.2).

Let us prove (2.10). Consider the function

^(ii5i2) = ^(ii5i2) + il{il=f3}^i(ii№(ii)5 (2-11)

where tbt2 G [t,T] and K(tbt2) 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)

oo

K*(ti,t2) = £ Cj1 (t2(ti) (ti = t,T), (2.12) ¿1=0

where

T t2

Cj1 (t2) = J K*(ti ,t2) j (ti)dti = ^2(t2^ ^i (ti) j (ti)dti. (2.13)

t t

The equality (2.12) is satisfied pointwise in each point of the interval (t,T) with respect to the variable ti, when t2 G [t,T] is fixed, due to a piecewise smoothness of the function K*(ti,t2) with respect to the variable ti 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) + К (to + 0, to)) = (to)ho(to) = K*(to,to).

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)

то

Cj(t2) = Ej(t2) (t2 = t,T), (2.14)

j2=0

where

T T t2

Cj2ji = J Cji (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)

то то

K *(ti ,t2) = EE Cj2ji Фп (ti)0j-2 (t2), (ti,t2) G (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 то то

-ШхШь) = E Е^^ш^)- (2-i6)

ji=0 j2 =0

From (2.16) we formally have

T T

1 г, г, то то

- I ik{h)ho{ti)dti = I =

t t Л =0 j2=0

T

тото

EE/ Cj2ji ФЛ (ti )фП2 (ti)dti = ji =0 j2=0 t

p1 p2

Hm Um V VCj2jW j(¿1)^2(ti)dt

1 p2

_n _n

T

pi—>-TO

1 ) T ¿2 (tl)dtl =

¿1=0 ¿2=0 t P1 P2

=„¡im plimXE ^ i{j1=j2} =

j1=0 ¿2=0

min{p1,p2} to

= lim lim ]T Cj2j1 = E j. (2.17)

p1—)-00 W2—»TO z-' z-'

j1=0 j1=0

Let us explain the second step in (2.17) (the fourth step in (2.17) follows from the orthonormality of functions fa(s) at the interval [t,T]).

We have

T T

/TO pi „

^Cji(ti)fai(ti)dti / Cji(ti)Tji(ti)dti t ji=0 ji=0 t

T T

<y |^2(ti)Gpi(ti)| dti < C J |G«i(ti)| dti, tt

<

where C < 00 and

(2.18)

^ J ^i(s)fa(s)dsfa(r) = Gp(r). j=p+it

Let us consider the case of Legendre polynomials. Then

00

Z (ti)

£ (2ji + 1) / ^i(u(y))Pji(y)dyPji(z(ti))

¿i =pi+i

i

where

, , T - t T +t . . / T + A 2

2

2 T-f

and Pj(s) is the Legendre polynomial.

From (2.19) and the well known formula

dPJ+\{x) _ 12ti(x) = (2j + l^x),

dx

dx

= 1, 2,

(2.19)

(2.20)

(2.21)

T

J

we obtain

\GPi(h)\ = \

E

ji =pi+i

z(ti)) i(z

ji-i(z (¿i))) ^i (¿i

T-t

z (ti)

i

oo

< C0

T-t +——

E

ji=pi+i

(Pji+i(y) - Pi-i(y)) ^i(u(y))dy Pji(z(ti))

z(ti)) - pji-i(z(ti))pji(z(ti

<

+

oo

E ^i(ti)

1

ji=pi+i

2ji + 3

(Pji+2(z(ti)) - Pi (Z(ti)))-

1

2ji - 1

(Pji (z(ti)) - Pji-2(z(ti))) -

z (ti)

T t 1

2 J \ 2ji + 3

i

(Pji+2(y) - Pji (y))-

1

2ji - 1

(Pi(y) - Pji-2(y)) Ki(u(y))dy Pji(z(ti))

(2.22)

where C0 is a constant, ^i and ^i' are derivatives of the function ^i(s) with respect to the variable u(y).

[101]

From

|Pn(y)l <

K

-y2)1/4

, y g (-1,1), n g N,

(2.23)

where constant K does not depend on y and n, we have

|Gpi(ti)| <

< C0

lim \

n—TO ^—'

ji=pi+i /

vz(ti))Pji (z(ti)) - Pji-i(z(ti))P,-i (z(ti,

Z (ti)

+

+Ci E

ji=pi+i

1

Ti

\

(1 - (z(ti))2)i/2

+

dy

1

i

(1 - y2)i/4 (1 - (z(ti

^i/4

<

)

n

1

<Cn

lim (Pn+i(z(ti))Pn(z(ti)) - Ppi(z(ti

n—T^OO

£ 4

7f V ("1 - (z(t ))2)1/2 ji=pi+ij1 \(1 (z(ti)) )

< lim I - + —

+ C2

(1 - (z(ti))2)i/4

+

<

oo

E ji

n^ln pW (1 - (z(ti))2)i/2

+ C2

ji=

p+i JfV(1 - (z(ti))2)

2 1/2

<c4i i- + E 1

1

(1 - (z(ti))2)

1

2 1/4

<

>2 I // ^ v ,i'2

Pi ii=pr+i(1 - (z(ti))2)

fai Ji (1 - (z(ti))2)

2 1/4

<

<

K

+

Pil (1 - (z(ti))2)i/2 (1 - (z(ti))2)i/4/'

where Cn, Ci,..., C4, K are constants, ti £ (t,T), and

oo

1 f dx 1

. Jl ~ J x2 pi' j1=pi+i i pi

From (2.18) and (2.24) we get

T T

«TO pi „

/ ECji(ti) j(ti)dti -E / Cji(ti) j(ti)dti

{ ji=n ji=n {

<

(2.24)

(2.25)

<

K

+

PMi (1 -y2)i/2 I (1 -y2)i/4

0

if pi ^ oo. So, we obtain

T T ^

t t ji=n

T T

to „ TO „ TO

E J Cji(ti) j(ti)dti = Ey ECj2jij(ti) j(ti)dti =

ji=n t ji=n t j2=n

1

1

1

1

1

1

1

1

l

to to „ to

= EE / fa2 = E *

ji=0 j2=0 t ji=0

In (226) we used the fact that the Fourier-Legendre series

T

TO

(2.26)

TO

E Cj2ji j ¿2=0

of the smooth function Cj (t1) converges uniformly to this function at the interval [t + £,T — e] for any £ > 0, converges to this function at the any point t1 e (t, T), and converges to Cjx (t + 0) and Cjx (T — 0) when t1 = t, T.

The relation (2.10) is proved for the case of Legendre polynomials.

Let us consider the trigonometric case and suppose that {fa(x)}°=0 is a complete orthonormal system of trigonometric functions in L2([t,T]).

Denote

Spi —

T T

f, to pi „

/ ECi(ti)0ji(ti)dti -E / Cji(ti)0ji(ti)dti t ji=0 ji=01

t ti

i

J E ^2(ti)0ji(ti)y WW*(£)*

t ji=pi+i t

We have

S2pi —

t to 2pi t

/ ECji(ti)0ji(ti)dti -E / Cji(ti)0ji(ti)dti t ji=0 ji=01

t ti

«to i

T-t

J E ^2(ti)fai(ti) y (0)d0dti

t ji=2Pi+i t

ti

. 2nji (s - t), . 2nji (ti - t)

Tt

<is sin-

yWi) E I/^i(s)sin

t ji=Pi+^ t

ti

, f I ( N 2nji(s - tK 2nji (ti - t) ,

+ / -fa(s)cos———;—as cos———;- | aii

Tt

Tt

Tt

2

1

n

T

fUmuti) E ~sin

i V ji=pi+i7i

1 . 2nji (ti - t)

T-t

T — t TO 1

ji=pi+i 7i

2n

2nji{ti~t) T-i

- /'sin2^ 'Vfav/.s- siii2"-/l(/| /!

Tt

Tt

Tt

T

< Ci

= Ci

/ '02fa E _Sin 7 ji=pi+i Ji

T-t 1 . 2nji (ti - t)

<

oo

T

1

T-t . 2nji (ti - t)

dti

pi

£ j- j MtOsin T_

ji =pi+iJi *t

dti

+

fa 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

(2.27)

(2.28)

converges uniformly at the interval [t + e, T — e] for any £ > 0 due to Dirichlet-Abel Theorem, and converges to zero at the points t and T. Moreover, the series (2.28) (with accuracy to a linear transformation) is the trigonometric Fourier series of the smooth function K(ti) = ti — t, ti £ [t,T]. Thus, (2.28) converges to the smooth function at any point ti £ (t,T).

From (2.27) we obtain

S2pi < C3

T

E l(UT)-Ut)- f cos27r^"S ^ f)Us)ds

ji=pi+i7i V i

Tt

C2 C4 + — < —, Pi Pi

(2.29)

where constants C2, C3, C4 do not depend on pi.

t

t

Further,

T ti

«00 1

J E ^(ti)fai (ti^ ^iWji

t ji=2Pi t

T ti

S2pi + / ^2(ti)02pi (ti) / (0)^2pi (<?)*

<

< S2pi +

2

T-t

T

^ i

T-t

T-t

Moreover,

T

^ i

. 2npi(ti -1)

T

T~t f lupi[ti t) J

■ip2(tljcos-——-- -0i(ii)sm

2npi

T-t

T-t

T

_ f msin2^lzÉdffldt,

. (2.30)

(2.31)

The relations

(2.31) imply that

S2pi-i <

a

Pi

where constant C5 is independent of p1. From (2.29) and (2.32) we obtain

Spi —

T 00 ti

J E ^2(ti) j (ti^ ^i (0) j (0)d0dti

t ji=pi+i t

(2.32)

K

<-->0 2.33

Pi

if p1 —y oo, where constant K does not depend on p1 (p1 e 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 (2.33) 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 the 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]-[15], [26], [45]. Suppose that [^(x)}fa 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

,/,/fU, ,,,

Jfa(2)fa =/ fafa/ fail)dft(;i)dft(;2) (il,i2 = 1,...,m)

the following expansion

Pl P2

./•[./2>]r,( = l.i.m. £ £C,, cfcf (2.34)

jl=0 j2=0

that converges in the mean-square sense is valid, where

T S2

Cj2ji = fas2)fafa) / fasifal(si)dsids2 (2.35)

and

T

j fa M (s)df<->

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 C

- /fafafafad^ECfa, (2-36)

t ji=0

where Cjj is defined by the formula (1.8) for k = 2 and j = j2. At that {fa(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/ + 1)(2¿ + lWjPiWPjíy) j =0 i=0

def

where

to 1

^ E 2 (2.37)

C% = ^(2j + l)(2i + l) J f{x,y)Pi{x)Pj{y)dxdy. (2.38)

[-1,1]2

Consider the generalization for the case of two variables [102] of the theorem

on equiconvergence for the Fourier-Legendre series [103

Proposition 2.1 [102]. 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 +

/ OO \

\j=0 ¿=0

— (1 — x2)—1/4(1 — y2)—1/4Snm(arccosx, arccosy, F) ) = 0. (2.39)

At that, the convergence in (2.39) 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],

and the Fourier coefficient Cj is defined by (2.38). Proposition 2.1 implies that the following equality

/ n m 1 \

lim EE o V^J + l)(2i + IfaJPfafafa) - f(x,y) = 0 (2.40)

n,m 2

\j=0 «=0 /

is fulfilled for all (x,y) G (-1,1)2, and convergence in (2.40) 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)1/4 (1 - y2)1/4 (2.41)

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 = (t1,... ,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 = (t1,... ,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 = (t1,... ,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 [104]).

More precisely, the following statement is correct.

Proposition 2.2 [104]. 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 < to and (x,y), (xi,yi) G [—1,1]2. Then the following inequality

is fulfilled

|g(x,y)-g(xi,yi)l < kP1/4,

(2.42)

where g(x,y) in defined by (2.41),

P = \/{x-xi)2 + (y-yi)2,

(x,y) and (xi,yi) G [-1,1]2, K < to.

Proof. First, we assume that x = x1, y = y1. In this case we have

|g(x,y) - g(xi,yi)| =

(1 - x2)1/4 (1 - vT4(/(x,y) - f (xi,yi)) +

,2\ 1/4

+f (xi, V1^(1 - x^1/4 (1 - y^1/4 - (1 - x2)1/4 (1 - y2)1/4)

+ C3

where C3 < to. Moreover,

< Ci |x - xi| + C2 |y - yi| +

(1 - x2)1/4 (1 - y2)1/4 - (1 - x2)1/4 (1 - y2)1/4

(1 - x2)1/4 (1 - y2)174- (1 - x^4 (1 - y2)

.2 ) 1/4

2) 1/4

,2) 1/4

<

(2.43)

(1 - x2)i/4((1 - y^i/4 - (1 - y?)i/4) +

+ (1 - !/?)

2\1/4

2)1/4 2)1/4

— x^l — (1 — X- >

X )

1

<

<

(1 — y2)1/4 — (1 — y2)

2) 1/4

+

(1 — x2)1/4 — (1 — x2)1/4

1

(2.44)

(1 — x2)1/4 — (1 — x2) — x)1/4 — (1 — xi)1/4

2) 1/4

<

< K

+ (1 — x1)1/4 ((1 + x)1/4 — (1 + x1)1/4) — x)1/4 — (1 — x1)1/4| + 1(1 + x)1/4 — (1 + x01/4|) , (2.45)

where K1 < to.

It is not difficult to see that

(1 ± x)1/4 — (1 ± x1)1/4 1(1 ± x) — (1 ± xx)|

((1 ± x)1/2 + (1 ± xfa2) ((1 ± x)1/4 + (1 ± x^1/4)

\XI-X\1,A-

| x1 — x| 1/2

| x1 — x| 1/4

<

(1 ± x)1/2 + (1 ± X1)1/2 (1 ± x)1/4 + (1 ± X1)1/4

< |X1 - x|1/4. (2.46)

The last inequality follows from the obvious inequalities

|x1 - x|1/2

(1 ± x)1/2 + (1 ± x1)1/2 |x1 — x|1/4

(l±x)1/4+(l±Xi)1/4

< 1,

< 1.

From (I2431)-(I2461) we obtain

|g(x,y) - g(x1,y1)| <

< C1|x - X1| + C2|y - yxj + C4 (|X1 - x|1/4 + |y1

<

<

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) - gfeyO1 < K2 ( 1 - x2)l/A f (x,y) - (1 - x2iy/V(xi,yi)

(x = x:, y = where K2 < to. 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 (2.40) holds.

Consider the auxiliary function

>2(tl)WÎ2), ti > t2

K(ti,t2)H , ti,t2 G [t,T] (2.47)

^l(tl)^2(t2), ti < t2

2\1/4

and prove that

|K'(ti, t2) - K'(t1, t2)| < L (|ti - t1| + |t2 - t2l),

(2.48)

where L < to and (ti,i2), (ti,t2) G [t,T]2.

By the Lagrange formula for the functions ^(¿i), ^2(ii) at the interval

[min{ii,ii} , max{ti, t^}]

and for the functions ^i(t2), ^2(t2) at the interval

[min {t2, ¿2} , max {t2, ¿2}]

we obtain

<

|K'(ti,t2) - K'(t2,t2)| < >2(tl)^l(Î2), ti > t2 f ^2(ti)^i(t2), t2 > t2

^i(ti)^2(t2), ti < t2 I^i(ti)^2(t2), t2< t2

+

+ l |t1 — 1 + l2 |t2 — t2l , l1,l2 < to.

(2.49)

We have

>2(^1)^1(^2), t1 > t2 f >2(^1)^1(^2), ti > t2

>1(t1)>2(t2), t1 < t2 [>1(t1)>2(t2), t*< t2 0, t1 > t2, t* > t* or t1 < t2, t* < t2

= <i >2(t1)>1(t2) - >1(t1)>2(t2), t1 > t2, t1 < t2.

>1(t1)>2(t2) - >2(t1)>1(t2), t1 < t2, t1 > t2 By Lagrange formula for the functions >1(t2), >2(t2) at the interval

[min[t1, t2}, max[t1, t2}]

we obtain the estimate

>2(t1)^1(t2), t1 > t2 (>2(t1)>1(t2), t1 > t2 |

' <

(2.50)

>1(t1)>2(t2), t1 < t2 I>1(t1)>2(t2), ti< t2

0, t1 > t2, t* > t* or t1 < t2, t* < t2 < Ls|t2 -tfa , (2.51)

1, t1 < t2, t* > t* or t1 > t2, t* < t2 where L3 < to.

Let us show that if t1 < t2, t* > t2 or t1 > t2, t* < tjj, then the following inequality is satisfied

|t2 - t11 < |t1 - t11 + |t*- t2|. First, consider the case t1 > t2, t1 < t2. For this case

(2.52)

Then

and (2.52) is satisfied.

t2 + (ti— t2) < t2 < t1.

(t1 — t1) — (ti— t2) < t2 — t1 < 0

For the case ti < t2, ¿i > ¿2 we obtain

ti + (¿2 - ¿i) < ¿i < ¿2.

Then

(¿i - ¿2) - (¿2 - ¿2) < ¿i - ¿2 < o

and also (2.52) is satisfied.

From (2.51) and (2.52) we have

>2(0^2), ¿i > ¿2 ( ^2 (¿i (¿2), ¿2 > ¿2

<

0, tl > t2, ti > ti or tl < t2, t^ < t2

< L3 (|tl - tl| + |t2 - t2|H <

1, tl < t2, t^ > t2 or tl > t2, ti < t2

1, tl > t2, ti > ti or tl < t2, ti < t2

< L3 (|tl -tl| + |t2 -12|) I

1, tl < t2, tl > t2 or tl > t2, tl < t2

= L3 (|ti — tl| + |t2 - t2|). (2.53)

From (2.49), (2.53) we obtain (2.48). Let

T -1 T +1 T -1 T +1

il = —:r + —1 h = —y + —

where x,y G [-1,1]. Then

(h(x)) ^l (%)) , x > y

,

^l (h(x)) ^2 (h(y)), x < y

where x,y G [-1,1] and

T — t T + t

Mar) = —* + -f- (2-54)

The inequality (2.48) can be rewritten in the form

|K''(x,y) - K''(x*,y*)| < L* (|x - x*| + |y - y*|), (2.55)

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 (2.42) is correct.

Due to the continuous differentiability of the functions (h(x)) and >2 (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 \ dydx+

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 [ --r—7-dydx I < TO, C < TO.

J (1 -x2)1/4,/ (1 -y2)1/4 y 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 (2.48) 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 on 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 j2j2 / K' (t1,t2)0ji (i1)j (t2)dt1dt2 • j (t1)0j2 (t2) =

ji=0 j2=0 t t

/ T t2

ni n2 / n 2

= lim EE /^(¿2 (taW W*i)j (ti)dtidt2+

ni,n2j-0 j-0 W y

ji=0 j2=0 \ t t

T T

+ ^ (¿2) j (t*^ (tl)fai (tl)dt^ dt20ji (tl)0j2 (t2) =

t t2

= lim EE^i + Cj) j (ti)0j2 (t2), (2.56)

ji=0 j2=0

where (t1 , t2) G (t,T)2. At that, the convergence of the series (2.56) 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 (2.56) converges to K'(t1,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 (2.56) on the boundary of the square [t,T]2.

In obtaining (2.56) we replaced the order of integration in the second iterated integral.

Let us substitute t1 = t2 in (2.56). After that, let us rewrite the limit on the right-hand side of (2.56) as two limits. Let us replace j with j2, j2 with j, n1 with n2, and n2 with n1 in the second limit. Thus, we get

ni n2 1

lim EE^A^1^^^ Mti)Mti), h e(t,T). (2.57) ji=0 j2 =o

According to the above reasoning, the convergence in (2.57) is uniform on the interval [t + £,T — e] for any £ > 0. Additionally, (2.57) holds at each interior point of the interval [t,T].

Lemma 2.2. Under the conditions of Theorem 2.2 the following limit

lim V Cj

ji=0

exists, where Cj is defined by (2.35) if j = j2, i.e.

n

T t2

ji = / «W* te)/ W^i fc)^.

t t

The proof of Lemma 2.2 will be given further in this section.

Let us fix £ > 0 and integrate the equality (2.57) at the interval [t + £, T — £]. Due to the uniform convergence of the series (2.57) we can swap the series and the integral

ni n2 T—'5 1 T—'5

lim Z E f/ faMthttiWi = o 'MhyHtiWh. (2.58) ni,n2—' J 2 J

ji=0 j2=0 t+5 t+5

Taking into account Lemma 2.2, consider the equality (2.58) for n1 = n2 = n

T —5 n T-5

- / '0i(ii)'02(ii)rfii = lim V / ^(^1)^(^1)^1 =

2 J n—TO z—* J

t+5 ji 'j2=0 t+5

n / T t+5

= lim E Cj2jJ / j (t1)0j2 (t1)dt1 — j (t0fe (t1)dt1 —

n

jU2=° t

— j j (t1 ) j (t1)dt1 = T—5

= iiiTO 1C Cj2ji( 1{ji=j2}— (^ji (0)j (0) + j (A)j (A)) ^ =

ji j2=0

oo

= E Cjiji — £ lim £ Cj2ji j (0) j(0) + j (A) j(A) , (2.59)

ji=0 ji,j2=0 V 7

where 0 G [t,t + £], A G [T — £,T]. In obtaining (2.59) we used the theorem on the mean value for the Riemann integral and orthonormality of the functions fa (x) for j = 0,1, 2 ...

Performing the passage to the limit lim in the equality (2.59), we get

5—+0

TT

1 f ^

{ ji=0

n

Thus, to complete the proof of Theorem 2.2, it is necessary to prove Lemma

2.2.

For proof of Lemma 2.2 as well as for further consideration we will need some well known properties of Legendre polynomials [101], [103 .

The complete orthonormal system of Legendre polynomials in the space L2([t,T]) looks as follows

= J = 0,1,2,..., (2.60)

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,

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

P№) = (Pj+" Pj-iW) > J = 1,2,...

The recurrent relation has the form

J 2j + l J

Orthogonality of Legendre polynomial Pj (x) to any polynomial (x) of lesser degree k we write in the following form

1

J Qk(x)Pj (x)dx = 0, k = 0,1, 2,..., j - 1. 1

From the property

1 r 0 if k = j

/ Pk(x)P(x)dx = I

—i [2/(2j + 1) if k = j

it follows that the orthonormal on the interval [—1,1] Legendre polynomials determined by the relation

p;(x) = \l'^r1pJ(x), 3 = 0,1,2,...

Remind that there is the following estimate [101

K

where constant K does not depend on y and j. Moreover,

|Pj(x)| < 1, x e [-1,1], j = 0, 1,... (2.62)

The Christoffel-Darboux formula has the form

±(2j + 1 )PJ(x)PJ(y) = (n + i)7''^'' (2.63)

j=o y x

Let us prove Lemma 2.2 for the case of Legendre polynomials. Below in this section we write lim instead of lim . Fix n > m (n,m e N). We have

n>m

T s

n n „ „

E = E J (s) / >1(tfa (t=

ji=m+l ji=m+l t t

n l x

T~t Y, (2ji + 1) [ ih(h(x))Pn(x) [lß1(h(y))PJ1(y)dydx =

4 ^

ji=m+1

T-t

E / ^i(h(x))^2(h(x)) (Pji+i(x)Pji(x) - Pji(x)Pji-i(x)) dx_

ji=m+1_i

n i x

(T_i)" E f Uh{x))Pn{x) [ (PJ1+M-Pn-i(y))m(y))dydx

8

Tt

i

ji=m+i_i _i

i

n

I ^i(h(x))^2(h(x)) E (Pji+i(x)Pji(x) _ Pji(x)Pji_i(x)) dx

ji=m+i

n i i

8

ji=m+i_i i

Tt

^i(h(x))^2(h(x)) (Pn+i(x)Pn(x) _ Pm+i(x)Pm(x)) dx+

i

(T _ t)2

E

1

8 . 2ji + 1 J

ji=m+i _i

(Pji+i(y) _ Pji_i(y)) ^i(%))X

x (Pji+i(y) _ Pji_i(y)) «%))+

Tt

(Pji+i(x) _ Pji_i(x)) ^2(h(x))dx dy,

(2.64)

where ^1, ^2 are derivatives of the functions ^1(s), ^2(s) with respect to the variable h(y) (see (2.54)).

Applying the estimate (2.61) and taking into account the boundedness of the functions ^1(s), ^2(s) and their derivatives, we finally obtain

Cjiji

ji=m+i 1

<0^1 + 1

dx

n mj J (1 _ x2)i/2

dy

+

_i 1

dx

ji=m+i Jf U (1 _ y2)i/2 ^ (1 _ y2)i/4 y (1 _ x2)i/4

dy <

i

i

i

i

n

i

i

<cJ- + -+ V iUo (2.65)

In m jf /

\ ji=m+1 1 /

if n, m ^ to (n > m), where constants C1, C2, C3 do not depend on n and m. The relation (2.65) completes the proof of Lemma 2.2 for the polynomial case.

Consider the trigonometric case. Fix n > m (n,m G N). Denote

T t2

n n „ i

Sn,m = E Cjiji = E / >2(¿2) j (¿2) / >1 (¿1) j (t1)dtidt2.

ji =m+1 ji=m+1 t t

By analogy with (2.64) we obtain

S2n,2m = E y>2(i2)>ji (¿2)J >1 (t1)^ji (¿1 fa^ =

ji=2m+1 t t

= t ( J MtJ^f^l J Mr) sin^^W

T t2 \

a. f I U \ f / 2TTJlfa-i) \ + / '02 (¿2) COS--- / '0i(ii)COS-^T—^-CfM*2 =

tt — ¿5 (w*) jm^f^-^dt] -

ji=m+1 j1 V V t J

} , irr, ZTrjiih-t) - / '0i(il)cos---I >2(T) - >2(ii)cOS----

t

t • \

" J f) dt2 J +

ti '

+ / '^(iOsin--I >2(ii)sm--+

t

T

+ / I , (2.66)

where V4(t), ^2(r) are derivatives of the functions (t), ^2(t) with respect to the variable t.

From (2.66) we get

n1

\S2n,2m\ (2.67)

j =m+1 j1

if n,m — to (n > m), where constant C does not depend on n and m. Further,

S2n—1,2m = S2n,2m TT Tt

2 fUtoW™}t2~t] 1 UhW-^^dUdU, (2.68)

T- t T2K2' T-t ^ T-t

S2n,2m-1 — S2n,2m +

+ ^ l uMcos^hl j (/;i(il)cos^l_l)rfil(ii2, (2.69,

tt

S2n_i,2m_i = S2n,2m_i

T t2

2 ? , , n 2nn(t2 _ t) / , , , 2nn(ti _ t) 7 7

■ipo(to)cos——- / '^i(ii)cos——-dt\dto =

T — t T — t T — t

tt

T t2

_ 2 /* 2nm(t2 — t) f . , . 2nm(t1 — t) 7 7

= ¿>2n,2m + ^ / 'W2)COS--- / '0i(ii)COS--dtidt2~

tt TT Tt

- J Mtl)cJ^l^dtldh, (2.70)

tt

Integrating by parts in (r2.68l)-(l2701), we obtain

C1

l^n-l^ml < | $2n,2m | H--, (2-71)

n

C1

ISo'ii/im—i | < |So'ii/ini| H--, (2.72)

m

|S2i?.-l,2m-l| < | So'ti/Ini | + C\ (--1--J , (2.73)

\m n)

where constant C1 does not depend on n and m. The relations (263), ^EH)-^™) imply that

lim |S2n,2m| = lim |S2n-1,2m| = lim |S2n,2m-1| = lim |S2n-1,2m-l| = 0.

(2.74)

From (2.74) we get

lim |Sn,m| =0. (2.75)

The relation (2.75) completes the proof of Lemma 2.2 for the trigonometric case. Theorem 2.2 is proved.

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]-[15], [35] 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 classical theorems of mathematical analysis. 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[f2)ht + il{il=i2} j Utiyut^dh. (2.76)

t

Let us consider the function K*(t1,t2) defined by (2.11)

K*{tut2) = Mti)h(t2) + ii {t1=t2}

= K(tut2) + \l{tl=t2yUti)Ut2), (2.77)

where

tl <t2

K(ti,t2)H , ti,t2 G [t,T]. (2.78)

0, otherwise

Lemma 2.3. In the conditions of Theorem 2.2 the following relation

J [K *]T2j = J (2.79)

(2)

is valid w. p. 1, where J[K*]y t is defined by the equality (1.16).

Proof. Substituting (2.77) 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[K% = + il{il=i2} [Mti)Mti)dh = .P[iF\T,t. (2.80)

Let us consider the following generalized double Fourier sum

Pi P2

EEC2ji j (t1 )^j2 (t2 ), j1=0 j2=0

where Cj2j1 is the Fourier coefficient defined as follows

Cj2ji = J K*(ti,t2)fai(ti)j(t2)dtidt2. (2.81)

[t,T ]2

Further, subsitute the relation

Pi P2 Pi P2

K *(ti ,t2) = EE Cj2ji j (ti)0j2 (t2) + K *(ti ,t2) "EE Cj2ji j (ti ) j (t2 ) ji=0 j2=0 ji=0 j2=0

(2)

into J[K*]Tt. At that we suppose that p1,p2 < to. Then using Lemma 1.3 (the case k = 2), we obtain

J*[^(2)]r,t = £ £ Cj2jiCj;:i)C]22' + J[RP1P2]T, t w. p. 1, (2.82)

j1=0 j2=0

(2)

where the stochastic integral J[RPlP2]T t is defined in accordance with (1.16) and

Pl P2

RP1P2 (t1,t2) = K*(t1,t2) ^E Cj2j1 j (t1 (t2), (2.83)

j1=0 j2=0

T

Cf = / h(s)dfs(;), t

T t2 T t1

J [RP1P2

]T, t = j j RP1P2 (t1,t2)dft(; f + / / (t1,t2)dft(2;2)dft(1;1) +

t t t t

T

t

Using standard moment properties of stochastic integrals [91] (see (1.26), (1.27)), we get

m{ J [Rp1p2 ]T2t) ^ =

T ¿2 T t1 x 2

tl

t t t t

/Rpip2 (tl,t2)dft(il)dft(2i2) + / /Rpip2 (il,i2)dft(2i2)dft(i1M U

+ I{i1=i2} ^ j RP1P2 (t1,t1)dt1^ <

T t2 T ti

< 2 { J J (Rpip2(ti,t2))2 dtidt2 + J J (Rpip2(ti,t2))2 «ti ) +

t t t t

T

+ I{ii = i2> ( J RPiP2 (ti,ti)dti

= 2 J (Rpip2(ti,t2))2 dtidt2 + 1{n=i2} J RpiP2(ti,ti)dtj . (2.84)

[t,T ]2 /

We have

J (Rpip2 (ti, t2))2 dtidt2 =

[t,T ]2

// Pi P2 \ 2

K*(ti,t2) - EE Cj2jij (ti)0j2 fe) dtidt2 =

[t,T]2 \ ji=0 j2=0 /

/. / Pi P2 \ 2

= J K (ti,t2) ^E Cj2ji j (ti)0j2 (t2H dtidt2.

[tjT]A ji=0 j2=0 /

The function K(ti,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(ti,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 j-ill ^ (ti) =o,

ji=0 j2=0 1=i

lim

where

l/l

L2([t,T ]2)

/

\i,T ]2

L2([t,T ]2) \ I/2

/ 2(ti,t2)dtidt2

y

So, we obtain

lim / (Rpip2(ti,t2))2 dtidt2 = 0. [t,T ]2

(2.85)

Note that

T T

^(¿1^1)^1= [ (^Mti)Mti) - EE^^i^1)^2^1) )

ji=0 j2=0

Pi P2

2

T T

1 r, Pi P2 n.

- / '01(il)'02(il)rfil E Ef •'••••'• / =

t ji=0 j2=0 t

T

1 ^ Pi P2

= 2 / '^1(^1)^2(^1)^1 "EE ^ '•'' •/ :./' i =

; _n ;__n

t ji =0 j2=0

T min{pi,p2}

2

= ^ I uti)uti)dh- E ^ (2-86)

t ji=0

From (2.86) and Lemma 2.2 we get

T T

lim lim / iifafai,iifai = lim lim / iifafai,ii=

Pi—TO P2—>-TO / Pi—TO P2—>-TO /

tt

T

1 f Pi

= 0 / 01 )02(^1 — lim V(7jm

2 j pi—TO ^—'

t ji=0

T

1

>i(ti)>2(ti)dti - E Cjiji

2

t ji=0 T

= lim / Rp1p2(ti,ti)dti, (2.87)

Pi,P2 — TO J t

where lim means lim sup.

If we prove the following relation

T

lim lim / RPiP2(ti,ti)dti = 0, (2.88)

Pi—TO P2—TO J t

then from (2.87) we obtain

T

1 « to

2

t ji=0

T

lim / Rpip2(ti,ti)dti t

— 0.

(2.90)

From (2.84), (2.85), and (2.90) we get

(2)\ 2 1 J [Rpip2 ]T,t ) f = 0

lim M

and Theorem 2.2 will be proved.

Let us prove (2.88). From (2.83) and (2.15) we obtain

lim lim Rpip2 (t1,t1) = 0, t1 G (t,T)

Pi—TO P2 — TO

(2.91)

0

and |RPiP2(t1,t1)| < oo, t1 G [t,T].

Let £ > 0 be a sufficiently small positive number. For fixed p1,p2 we have

T

J Rpip2 (t1,t1)dt1 = t

t+5 T—5 T

J Rpip2 (t1, t1 )dt1 + J Rpip2 (t1, t1 )dt1 + J Rpip2 (t1, t1 )dt1 <

t t+5 T — 5

t+5 T — 5 T

<y |Rpip2(t1 ,t1)| dt1 |Rpip2(t1,t1)| dt1 |Rpip2(t1,t1)|<

t t+5 T 5

T—£

< |Rpip2(ti,ti)| dti + 2Ce,

(2.92)

t+e

where |RPiP2(t1,t1)| < C and C is a constant.

Let us consider the partition {Tj}N=0 (N > 1) of the interval [t + £,T — £] such that

r0 = t + e, rN = T -e, Tj = r0 + jA, A = -J-(T — i — 2e).

N

We have

T—e

N-1

t+e

|RpiP2(ti,ti)| dti < V" max |Rpip2(s,s)| A

■* sG[ri,ri+i]

«=0

N-1 N-1 , x

E |RP1P2 (Ti'Ti)| A + E ( max ! (s,s)| - (Ti'Ti)0 A < t"^ t"^ V^,-i+i] J

¿=0 N1

¿=0 N1

<E|RpiP2(ri'ri)l A + E

¿=0 N1

¿=0 N1

max |Rpip2 (5,s)| - |Rpip2 (Ti,ri)| sG[rj ,ri+i ]

¿=0

¿=0

E|RP1P2(ri,ri)| A + E |Rpip2(t(pip2)'t(pip2)) - |Rpip2faT,)

A =

A <

N1

< E |RpiP2(T' t)| A + £1 (T - t - 2e),

(2.93)

¿=0

where (t(pip2) ,t(pip2)) is a point of maximum of the function |RPlP2(s,s)| at the interval [ri,ri+i].

Getting (2.92) and (2.93), we used the well known properties of integrals, the first and the second Weierstrass Theorems for the function of one variable as well as continuity (which means uniform continuity) of the function |RPlP2 (s, s)| at the interval [t + £, T — e], i.e. Ve1 > 0 35(e1) > 0, which does not depend on t1, p1, p2 and if A < 5(e1), then the following inequality takes place

R (t(PiP2) t(PiP2)) _ |R (_ _ )| RPiP2 ' ^ ) |RPiP2 (/«' /«)|

< £1.

From (2.92) and (2.93) we obtain

0 <

T

Rpi p2 (ti' ti)dti

N1

< E |RpiP2(t'T<)| A + £i(T - t - 2£) + 2C£. (2.94)

i=0

Let us implement the iterated passage to the limit lim lim lim in the

e—+0

inequality (2.94) taking into account (2.86), (2.87), and (2.9l). So, we get

0 < lim lim

Pi—TO P2—TO

T

Rpip2 (t1' t1)dt1

= lim lim

Pi — TO P2—TO

T

Rpip2 (t1' t1)dt1

< e1(T — t). (2.95)

Then from (2.95) (according to arbitrariness of £1 > 0) we have (2.88). Theorem 2.2 is proved.

Note that (2.88) can be obtained by a more simple way.

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 RPlP2 (t1,t1) = 0 when t1 £ [t,T]

Pl—TO p2—^TO

holds with accuracy up to sets of measure zero and |RPlP2 (t1,t1)| < to for ti £ [t,T].

According to (2.83), we have

Rpip2 (t1,t2)= [K*(t1,t2) - E Cji (t2fa)] + V ji=0 /

/ Pi / P2 \ \

+ E Cji (t2) - E Cj2jij fa) j fa) , (2.96)

\ji=0 V j2=0 / /

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), and (2.96), we obtain

T

lim lim / RPlP2 (t1,t1)dt1 = 0.

p1—to p2—TO J 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 ^ ^ and Constant Weight Functions (The

Case of Legendre Polynomials)

Theorem 2.3 [6]-[15], [33]. Suppose that [fa(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

tl

J J J f»df™ff' (ii.i2.ia = 1,...,m)

t t t

the following expansion

*T ^¿3 ^¿2

/,> ç Pi P2 P3

/ / f^'ff = î.i.m ££ cj;:i)cj22)cj33) (2.97)

t t t ji=0 j2=0 j3=0

that converges in the mean-square sense is valid, where

T s si

Cj3j2ji = j (s) / j (s1M j (s2)ds2ds1ds

and

T

Cf = / j (s)dfi*'

t

are independent standard Gaussian random variables for various i or j. Proof. If we prove w. p. 1 the following equalities

Pi P3 . / . x

l.i.m. = + (2.98)

n n 4 V A/3 /

ji=0 j3=0

î 1 î 3 1 1

ä ^ s - ^ - à**) - <2-"»

ji=0 j3=0

Pi P3

^ E ECjij3jiZj32) = 0, (2.100)

ji=0 j3=0

then in accordance with the formulas (12.981) (12.1001). Theorem 1.1 (see (1.46)), 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 - tf2 (>> + ^=c!*3)) 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 /^ftí-'^fíí^^^rfíti3'=^M EEECj.cfcfcf

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 (2.97) will be proved. Let us at first prove that

œ 1

E CoM = -{T-t?!\ (2.101) ji=0 œ

= —=(T-tf'\ (2.102)

ji=0

We have

r (T - t)W c^ooo —-^-,

T s si

Cojiji fa fa j (sfa j =

t t t T / s \ 2

= \ Í Ms) I I o;. (*,)</*, I ds, J! > 1, (2.103)

tt where (s) looks as follows

m = ,>0, (2,04)

where Pj (x) is the Legendre polynomial.

Let us substitute (2.104) into (2.103) 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

2ji + 1

Pi+i(y) — Pi—i(y) dy

ds —

y

(Pji+i(z(s)) — Pi—i(z(s)))2 ds,

(2.105)

where here and further

z(s) — s —

T + A 2

2 / T-t

In (2.105) we used the following well known properties of the Legendre polynomials

P (y) —

1

2j + 1

Pj+i(y) — P—i(y) , P(—1) —(—1)j, j > 1.

Also, we denote

dP

def

dy

From (2.105) using the property of orthogonality of the Legendre polynomials, we get the following relation

= &TT) / {plhv) + dy =

i

(T-t)3/2/ 1

S(2Ji + 1) \2ji + 3 2j

where we used the property

P2(y)dy —

i

2j + 1

j > 0.

2

2

1

i

2

Then

fv {T-t)W

Z^ - -Q-+

ji=0

(T - if'2 1 y. 1

oo oo

(T - tf/2 (T-tf/2f y. 1 1 y. 1

6 + 8 \^4j2-l 3 + f-4j2-l

j=1 ^ A ° j i=i -71

_ (T - tf/2 (T-tf/2 (1 _ 1 1 \ _ (T - ¿)3/2 ~ 6 + 8 \2 ~ 3 + 2J ~ 4 '

The relation (2.101) is proved.

Let us check the correctness of (2.102). Let us represent C1j1j1 in the form

T / s \ 2

Cijui = \j Ms) I j M(si)d.si ) ds = t \t

Since the functions

^JpjiiV1)dy1\ , J1>1

are even, then the functions

y x 2

y x 2

A(y) / Pi (y1)dyJ dy, j1 > 1

1

are uneven. It means that C1j1j1 = 0 (j > 1). From the other side

l

V3(T-tf/2 r f . [T-tf/2

100 = — 16 J y(y + ^ = •

1

Then

^ ^ (T — t)3/2

E CAui = Cl0° + E = —7/f—'

ji=0 ji=1 V

The relation (2.102) is proved.

Let us prove the equality (2.98). Using (2.102), we get

Pi P3 Pi im .\3/2 pi P3

V^ V^ / Ai3)_X^ri A'13) I ^ ^ /-(»3) I n t

/ , / A ./:•■■/ ./ ^Z:.. / , ( "./ ./ "+" , /ö / , / / ./:■,/ ./ ^

ji=0 j3=0 ji=0 ji=0 j3=2

Pi (t _ t)3/2 Pi 2ji+2

E + + E E • (2.106)

As) + U ~ A ' + ^ ^ ^

ji=0 4\/3 j3=2 J3—even

Since

(T-^)3/2(2j1 + 1)^TT }p(Jfp( w , ,

'!/:,/ ./ =-^-J PjM J PjAyi)dyi I dy

and degree of the polynomial

y

i pji (yi)dyi

equals to 2j + 2, then Cj3jj = 0 for j3 > 2j + 2. It explains that we put 2j + 2 instead of p3 on the right-hand side of the formula (2.106).

Moreover, the function

y2

i pji (yi)dyi

is even. It means that the function

y

Pj3(y) | / Pji(yi)dyi

is uneven for uneven j3. It means that Cj3jiji = 0 for uneven j3. That is why we summarize using even j3 on the right-hand side of the formula (2.106).

Then we have

pi 2ji+2 2pi+2 pi

Ev^ C Z (i3) = V V^ C Z (i3) =

/ , Cj3jiji Zj3 = Z^ Cj3jiji Zj3 =

ji=0 j3=2,j3—even j3=2,j3—even ji=(j3— 2)/2

2pi+2 pi

= E ECj3jiji Zj33). (2.107)

j3=2,j3—even ji =0

We replaced (j3 — 2)/2 by zero on the right-hand side of the formula (2.107), since Cj3jiji = 0 for 0 < j1 < (j'3 — 2)/2. Let us substitute (2.107) into (2.106)

pi p3 pi

V^ V^ / AiAn) 1 (T ~ tf/2 An) 1 ( h "["at +

ji=0 j3=0 ji=0 V

2pi+2 pi

+ E ECj3ji ji Zj33). (2.108)

j3=2,j3—even ji=0

It is easy to see that the right-hand side of the formula (2.108) does not depend on p3.

If we prove that

2

Ä M {(£ Ya^AA - -¿t - A2 (cA + ) ) j = 0,

(2.109)

then the relaion (2.98) will be proved.

Using (2.108) and (2.101), we can rewrite the left-hand side of (2.109) in the following form

M(f(ECo**~izAr1)^ + E Ec^cA

I \ \ji=0 / 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™ £ Ej . (2i110)

j3=2,j3-even \ji=0 /

If we prove that

pi

lim

2pi+2

E ( EC

j3jiji

= 0,

(2.111)

j3=2,j3-even \ji=0

then the relation (2.98) will be proved. We have

2pi+2 / pi Л3=2,j3-even \ji=0

4

2pi+2 E

j3=2,j3-even

T

pi

2

V'

ФЛ3(s)E / Л (s1)ds1 ) ds Ji=° \t

4

2pi+2 E

Л3=2,Л3-even

/ T

/

фЛ3(s)

V1

У

22 ds

V

(s - t) - 53 / Фл'1 (s1)ds1

ji=pi+Ai у у у

4

2pi+2 E

T

Л3=2,Л3-even

V

/ фЛ3(s) E / фл!(si)dsi

{ ji=pi+A -t

2

ds

<

1

2pi+2

T

E

Л3=2,Л3-even

oo

1Фл3 (s)| E

V'

ji=pi+1 \ t

J

22 Фл1 (s1)ds1 J ds

У

(2.112)

Obtaining (2.112), we used the Parseval equality

œ il V T 2

53 / Фл1 (s1)dsH = (l{si<s^2 ds1 = s - t ji=0 \ t t

(2.113)

and the orthogonality property of the Legendre polynomials

T

Фл3(s)(s - t)ds = 0, j3 > 2.

(2.114)

2

2

2

s

1

s

1

s

1

s

Then we have

j (si)dsi —

2 i z(s)

(T — t)(2ji + 1)' "

\

4

Pji (y)dy

V

i

/

T-t

/ z(s)

\

4(2 ji + 1) Tt

Pji+i(y) — Pji—i(y) dy

V

i

y

4(2 ji + 1)

(Pji+i (z(s)) — Pji—i (z(s)))2 <

<

Tt

(2.115)

Remind that for the Legendre polynomials the following estimate is correct

|Pn(y)| <

K

y/nTT{i - y2)1/4

, y G (—1,1), n G N, (2.116)

where constant K does not depend on y and n.

The estimate (2.116) can be rewritten for the function jn(s) in the following form

|0n(s)| <

2n + 1 K

1

<

71+1 y/T^t(l-z2{S))1/4

Ki 1

VT=t(l-z2(s))1/4'

<

(2.117)

where Kx = Ky/2, .s G (i,T).

Let us estimate the right-hand side of (2.115) using the estimate (2.116)

, / x , I T — t / K2 K2

^(.SijrfSi < . . ( —- +

<

2(2ji + 1) Vji + 2 ' jW (1 — (z(s))2)i/2

(■T-t)I<2 1

2j2 (l-(^s))2)V2'

<

(2.118)

where s G (t, T).

2

s

2

2

s

1

Substituting the estimate (2.118) into the relation (2.112) and using in (2.112) the estimate (2.117) for |j(s)|, we obtain

2pi+2 / pi \ 2

E ( E Cj3jij^ <

j3=2,j3—even \ji=0

(T - 4K2

<

2pi+2

/ T

n 7 OO

16

j3=2,j3-even

ds 1

E / / 0\ 3/4 E ;2

\1 (l - (z(s)f) ji=pi+1 (T - t)3K4K?(pi + 1)/ /* dy W ^ 1

V—1

Since

64 w, a-*2)"4; ' <2,119)

i

^ dy < oo (2.120)

-1

and

(1 - y2)3/4

oo œ

y-v 1 ^ i dx 1 ^

ï if Pi ^oo, (2.122)

Pi

then from (2.119) we find

2

c(T-tnPl +1)

9

Of

j3=2,j3-even \ji=0 / 1

where constant C does not depend on pf and T — t. The relation (2.122) implies (2.111), and the relation (2.111) implies the correctness of the formula (2.98).

Let us prove the equaity (2.99). Let us at first prove that

œ

E CnjsO = (2.123)

j3=0 œ

= - —=(T-t)W. (2.124)

j3=0 V

We have

œ œ

E Cj3j3 0 = C000 + E Cj3j3 0,

j3=0 j3=1

2

r (T- *)3/2 ^000 — -

6

^30 = 16^3 + 1) / (P3+i(2/) + Pl-M dy 3i

Then

(T - tf/2 f i i \ . >;L

yV CT-tf!2 j3=0 6

, (T -11 A 1

+ ö Z^ ro.io J- 1 -L

8 j=i (2j3 + 1)(2j3 + 3) ' j=i 4j'2 — 1

, 1__1+V

^ ö Z^ /L i2 _1 Q ^

{T-tf/2 | {T-tf/2 /jj^ 1 1 v^ 1

j3=i 4 3 j3=i 3

6 8 4j32 — 1 3 4j32 — 1

_ (T - ¿)3/2 {T-tf2 f 1 _ 1 1 \ _ (T - ¿)3/2 ~ 6 + 8 V2 ~ 3 + 2J ~ 4 '

The relation (2.123) is proved. Let us check the equality (2.124). We have

T s si

Cj3j3ji — y j (s) ^ j (si^ j (s2)ds2dsids — t t t

T T T

— ^ j (s2)ds^ j (si)ds^y j (s)ds —

t S2 Si

T / T \ 2

= \J ('^2) I J (f>j3{si)dsi ) ds2 =

t s2

= (T - f)3/2(2ja + 1)v/2j7~FT jpn{y) (jPjMi , ^ ^ j

—i \y

(2.125)

Since the functions

p3 (y1)dyj ' j3 > 1

12

y

are even, then the functions

P (y) ^J Pj3 (y1)dy1^ dy, j3 > 1

are uneven. It means that Cj3j31 = 0 (j3 > 1). Moreover,

Vs(T-tf'2 r n _ (T-tf2

Cm =---J y(l - y)-dy =

— 1

Then

(T — t)3/2

j3=0 j3=1

The relation (2.124) is proved. Using the obtained results, we get

pi p3 p3 3/2 p3 pi

V^ V^ / Ai/-(»i) A T> A'A I f

/ J / A hhJ-^j- ~ / v ^isjgOSO . p; Si / J K ./:■,/:■,/ S

ji=0 j3=0 j3=0 v j3=0 ji=2

p3 (T - t)3/2 . P3 2j3+2

= E ( './:■,/:■■'--J"y=-ClH ' E E ' (2"126)

j3=0 V j3=0 ji=2,ji-even

Since

j Pji(v) f j Ph(yi)dyi\ dy, ,3>1,

-1 y

and degree of the polynomial

/ 1

p3 (y1)dy1

12

equals to 2j3 + 2, then C]3j3ji = 0 for j > 2j3 + 2. It explains that we put 2j3 + 2 instead of pi on the right-hand side of the formula (2.126).

Moreover, the function

i2

Pj3 (yi)dy:

y

is even. It means that the function

i

i2

Pji(y) | J Pj3(yi)dyi

y

is uneven for uneven j. It means that C]3j3ji = 0 for uneven ji It explains the summation with respect to even j on the right-hand side of (2.126).

Then we have

P3 2j3+2 2p3+2 P3

Ev^ C z (ii) = v^ v^ C Z (ii) =

/ , Cj3j3 ji Zji = ^ C j3 j3ji Zji =

j3=0 ji=2,ji—even ji=2ji—even j3=(ji—2)/2

2P3+2 P3

= E E cji (2.127)

ji =2,ji—even j'3 =0

We replaced (j — 2)/2 by zero on the right-hand side of (2.127), since Cj3]3ji = 0 for 0 < j3 < (ji — 2)/2.

Let us substitute (2.127) into (2.126)

P1 P3 P3 (T 3/2

Y^Y^ An)_X^r Ail) A -1) Ah)

/ , / , ( ./:■,/:■,/ ~ / „ ( ./ -/ , /o

ji=0 ]3=0 ]3=0 V

2P3+2 P3

+ E EC]3]3], if. (2.128)

ji=2,ji—even ]3=0

It is easy to see that the right-hand side of the formula (2.128) does not depend on pi.

If we prove that

"st,M i i E E ^.cj;1' - > - ')3/2 - Vl=°.

I \ —n ;__n \ V <-»

(2.129)

then (2.99) will be proved.

Using (2.128) and (2.123), (2.124), we can rewrite the left-hand side of the formula (2.129) in the following form

lim M

p3—TO

p3

y ^ Cj3j3 0 j3=0

(T -1)3/2 I ^i,

2p3+2 p3

Z0 + ^^ ^^ Cj3j3ji C

(¿i) ji

ji=2,ji-even j3=0

pi (T t) 3/2 li,,i (E ^ —i + lim

p3

p3—TO

j3=0

p3—TO p3

2p3+2

E (EC

j3j3ji

2p3+2

= lim V ( VC

ji=2,ji-even \j3=0 2

ji=2,ji-even \j3=0

If we prove that

2p3+2

p3

j3j3ji

2

pÜ5TO E ECj3j3jj =0'

ji =2,ji-even \j3=0

(2.130)

then the relation (2.99) will be proved. From (2.125) we obtain

p3

2p3+2

E E

ji =2,ji-even \j3=0

1

4

2p3+2

3

/T

ji=2,ji-even 2p3+2 / T

p3

T

\

V1

j ME ( I j (s1)ds1 ) ds2

j3=0

VS2

/

4

E

ji=2,ji-even

/

j (s2)

V

t

V

(T - S2) - E I J j (S1)ds1 j

j3=p3+1 \s2 /

(l

4

2p3+2 i t to / t

E J j (s2) E I J j (s1)ds1

2

ds2

; 7

22

\

ji=2,ji-even 2p3+2

ds2

<

V /T

j3=p3+1

/

T

\

<

4

E

ji=2,ji-even

V'

| j (s2)| E I I j (s1)ds1 | ds2

j3=p3+1

(2.131)

VS2

J

2

2

2

2

2

2

1

1

2

2

1

In order to get (2.131) we used the Parseval equality

T \ 2 T

TT

j=0 ( / j(si)dsij — j (l{s<si})2 dsi — T — s (2.132)

s

and the orthogonality property of the Legendre polynomials

T

J j (s)(T — s)ds = 0, j3 > 2. (2.133)

t

Then we have

s2

t \ 2

(T — t)

^3(si)dsi | = + ^ (p3+i (z(s2)) - p3-i (^(«2)))2 < T — t

T-i / /\2 /\2\ 1

< 2(2J3 + 1) IjTT2 + Xj (l-i^))2)1/2 <

(T - t)K2 1

In order to get (2.134) we used the estimate (2.116).

Substituting the estimate (2.134) into the relation (2.131) and using in (2.131) the estimate (2.117) for (s2)|, we obtain

2p3+2 / p3

53 (53 Cj3j3ji' < j'i=2,j'i—even \j3=0

(T - t)K4I<2 ^ / j ds2 ^ 1_

16 ■ ^ [J (1 -Z2{.So))^ ß

ji=2,ji—even y*t V V 2" j3=P3+i J3

^^(/^)'(Ii)" *»>

i

2

2

Using (2.120) and (2.121) in (2.135), we get

If (i:^,) <C(T"^(P3 + 1> with pa —> oo, (2.136)

ji=2,ji—even \j3=0 / 3

where constant C does not depend on p3 and T — t.

The relation (2.136) implies (2.130), and the relation (2.130) implies the correctness of the formula (2.99). The relation (2.99) is proved.

Let us prove the equality (2.100). Since ^ (t), (t), (t) = 1, then the following relation for the Fourier coefficients is correct

1

—I

2

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 ■

pi5p3—^ — —' pi5p3—^ —' z—' V 2 j j3

ji=0 j3=0 ji=0 j3=0 V 7

(2.137)

Therefore, considering (2.98) and (2.99), we can write w. p. 1

Pi P3

/ . . . /-'... . r ' — -C2 r1 ■

l.i.m. EE^C^W-

i,P3—œ z—' z—' -73 2

ji=0 j3=0

Pi P3 Pi P3

-l.i.m. E E Cjijij3j' - E ECj3jijiif

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

(T - tf'2 (ct] + = 0. (2.138)

w V"0 Vo1

The relation (2.100) is proved. Theorem 2.3 is proved.

It is easy to see that the formula (2.97) can be proved for the case i1 = i2 = ¿3 using the Ito formula

*T *t3 *t2 ( T \ 3

IJ J dfAdt>dt> = J /«I"' = I (CoC^0)3 = t t t \t /

where the equality is fulfilled w. p. 1.

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 2.3.

Theorem 2.4 [6]-[15], [33]. Suppose that (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 - t2)'=/ (t - tl)'■ dfiC1,')dfi(2,2)df4<S»

i i i the following expansion

Pi P2 PS

®1=EEE jjijjj1 (2.139)

ji =0 j2=0 js=0

that converges in the mean-square sense is valid for each of the following cases

1. ¿1 = ¿2, ¿2 = ¿3, ¿1 = ¿3 and li, /2, /3 = 0,1, 2,...

2. ¿1 = ¿2 = ¿3 and /1 = /2 = /3 and /1, /2, /3 = 0,1, 2,...

3. ¿1 = ¿2 = ¿3 and /1 = /2 = /3 and /1, /2, /3 = 0,1, 2,...

4. ¿1, ¿2, ¿3 = 1,..., m; /1 = /2 = /3 = / and / = 0,1, 2,...,

where ¿1, ¿2, ¿3 = 1,..., m,

T s s 1

Cjsj2ji =J(t - s)ls j (s^(t - S1)l2 j (S1^(t - S2);ij (S2^ds^ t t t

and

T

c<:1 = 1(S)df<0

t

are independent standard Gaussian random variables for various ¿ or j.

Proof. Case 1 directly follows from (1.46). Let us consider Case 2, i.e. ¿1 = ¿2 = ¿3, /1 = /2 = / = /3, and /1, /3 = 0,1, 2,... So, we prove the following expansion

Pi P2 P3

Ä1 = p E E E Cjjjj' (ii,i2,i3 = 1,... ,m),

j1=0 j2=0 j3 =0

(2.140)

where /1, /3 = 0,1, 2,... (/1 = /) and

T s si

Cj = J j (s)(t - s)13^ (t - si)1 j (si^(t - S2 )* j (s2)ds2dsi ds. (2.141)

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 - (2.142)

ji=0 j3 =0 t t

where coefficients Cjjj are defined by (2.141), then using Theorem 1.1 and standard relations between iterated Itô and Stratonovich stochastic integrals, we obtain the expansion (2.140).

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 f^) = - E 4C]:3) w. p. 1,

j3=0

t

where

T s

t t

Then

P3 Pi 2/+/3+I

v^ v^ / ^(¿3) _ £ v^ ^ ^(¿3) _

/ , / ,( ./ ■./ ./ 2 ^ J3 sj3 —

j3=0 ji=0 j3=0

2/+/3+W Pi 1 \ P3 Pi

E E cm> - Icj, ci:s) + E E^.e

j3=0 \ji=0 / j3=2/+/3+2 ji=0

Therefore,

T

P3 Pi 1 ^

M<! I EEC»U4S) .', I < AI a - s1fds1dt<«>

t 1 1 / V Zv J3Ji:/i> j3 o

Pi,P3^œ I \ z—' z—' -73 2

,j3=0 ji=0 t t

2/+/3+I / Pi

"S. E E C

j3=0 \ji=0

2

-V I +

—-- / y I ✓ , o J3 I "T"

j3=0 \ji=0

2

s

P3 pi

+ lim M

Pi,P3—TO

53 53 Cj3jiji C

, j3=21+l3+2 ji=0

(i3) j3

(2.143)

Let us prove that

pi

lim

pi—to

Cj3j

ji=0

j3jui 2 3

= 0.

(2.144)

We have

pi

53 ^Ajiji 2 ji=0

/

I Pi T V

¿53 I I (f)J1(s1)(t-s1)ldsi ds-

\ ji=01 Vi /

1

T

- Un{s){t-s)h {t-s.fds.ds I =

1 4

/ T

/

j (s)(t - s)

l3

v

Pi

v

53 / j(si)(t - si)/dsi

ji=0 V {

i

4

/T

/

j (s)(t - s)13

\t

- J (t - Slf dsi | ds | =

oo

v

(t - si)21 dsi - E 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

y t ji=pi+^ t / y

. (2.145)

2

2

2

s

2

2

s

2

2

s

2

In order to get (2.145) we used the Parseval equality

¿1 J fa 1 (S1)(t - S1)ldsJ =J K2(s,s1)ds1, (2.146)

I I - K2

¿1=° \ t / t

where

l

K(s,s1) = (t - S1) 1{s1<s}, S,S1 G [t,T].

Taking into account the nondecreasing of the functional sequence

. s ^ 2

n

Un(S) - - " * - ' - "+ "

>(s) = 53 I j(s1)(t - S1)lds1 ji=0 \ t

continuity of its members and continuity of the limit function

s

u(s) = J(t - s1)2lds1 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 (12.145) using the inequality of Cauchy-Bunyakovsky, we obtain

53 ^ ~~ ) —

2

¿1=0

T T ( _ / s \ 2\

1/* f I ^ I f '

1 2 2l l

< 3 / - s)2,3As

t

53 J j(si)(t - si)ldsi

t / y

2

ds <

T

< ^2(T - J ojjs)<ls(T -t) = i(T - (2.147)

t

when p1 > N(e), where N(e) G N exists for any £ > 0. The relation (2.147) implies (2.144).

Further,

P1 PS P1 2(j1 +l+1)+ls

E E Cjsj1j.zjss) = E E C,sj1j1 cjss). (2.148)

j1=0 js=2l+ls+2 j1=0 js=2l+ls+2

We put 2(ji+1+ 1)+13 instead of p3, since Cjjj = 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). (2.149)

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

(2.148) and (2.149) we have

Pi P3

M{|E E Cj3j,j,zj33)

ji=0 j3=2/+/3+2

2(pi+1+1)+1s pi

E Ecwi cj;3)

j3=2/+/s+2 ji=0

/ T

. _ _ pi T

E

j3=2/+/s+2

2(pi+/+1)+/3 / i Pi

\

2 / s'

ji=01

2(pi+/+1)+/3 / ^ pi

r...........3/T

E

' i ji=0 w

4

j3=21+l3+2

V

2(pi+/+1)+/3 13 p1 2

I Cj3jiji 1

j3=2/+/3+2 \ji=0

\ 2

j(51)(t - s1)/ds1 ds

J

\ 2

j(51)(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

2

s

2

2

1 2(p1+l+i)+l3 y t

-j E J •s)/:" E (/

E ( i j(si)(t - si)ldsi

¿i=P1+i V {

/t

\

ds

7 7

2

¿3=2l+l3+2

- s1)lds1 I ds

yt ji=p1+^ t / J

(2.150)

In order to get (2.150) we used the Parseval equality (2.146) and the following relation

T

J j(s)Q2i+1+is(s)ds = 0, j3 > 2/ + 1 + /3, t

where Q2l+1+/S (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

Pj1 (y)(1 + y^ dy

i

V

(T -t)

/

2l+i

/

22l+2(2ji + 1)

z(s)

X

\

X

(1 + z(s)) Rj1 (s) - l / (Pj1+i(y) - Pj1-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

T-t

Rj (s) + l2

(Pj1+i(y) - Pj1-i(y))(1+ y)^ dy

V

V

i

/

<

/

\2l+i

KJT-t)_ ~ 22l+1(2ji + 1)

2

s

2

s

2

2

(

z (s)

\

x

V

221+1Zj1 (s) + l2 J (1+ y)21-2dy ^ (Pji+i(y) - Pj1-i(y))2 dy -i -1 21+i

<

J

<

(T -1)2

221+1(2ji + 1)

x

x

JlK J 21-1 T-i

2 i

z(s)

j2i+i(y) + P2-i(y)) dy

<

V

<

i

/

(T - ¿)21+1 {_ , , /2 2(2ji + l)

2Zji (s) +

2l 1

z(s) \

j-+i(y) + Pw(y)) dy

, (2.151)

V

i

J

where

Rji(s) =

Zji(s) = Pji+i(z(s

z(s)) - P^-i(z(s)),

ji-i

Let us

using (2.116)

j(si)(t - si)1 dsi <

<

(T - ¿)21+1 ^ K2 K

(

2(2ji + 1) Vji + 2 ji

(T - t)21+iK2

2

+

2 z(s)

l2 dy

V

<

2j2

(1 - (z(s))2)

2 l2n

+

2)i/2 2l - 1j (1 - y2)

2) i/2

<

-1

!

(1 - (z(s))2)

2) i/2 2l — 1

, s G (t,T). (2.152)

From (2.150) and (2.152) we obtain

Pi P3

Mi(E E CjSjijicjr <

ji =0 j3=21+l3+2

<

1

4

2(pi+1+i)+l3

T

2

E

j3=21+13+2

v

(s)|(t - s)1^ / j(si)(t - si)1 dsj ds

ji=Pi+A t J J

<

2

2

2

2

. 2(Pi+/+1)+/^ T œ / s \ 2 ^

<l(T-tf3 E I E [ I 0.r(.A)(l *,)',/*,) ds

j3=2/+/3+2

<

^t ji=Pi+^ t /J

<

<T _ t)4/+2/3+1K4K2 2(Pi+/+1)+/^ / T

E

16

j3=2/+/3+2

V

2ds

^ (l _ <z<s))^3/4

+

T

l2n f ds

+

21 _1 / (1 _ <z<s))^1/v j.=PT+i

\

2

El

<

2

(T - f 2Pl + l / j 2dy i2TT j dy

- 64 Vx (l-y2)3/4 + 2r^iy (l-y2)V4 1 -

< C(T - 0 when Pl 00, (2.153)

pi

where constant C does not depend on p1 and T — t.

The relations (2.143), (2.144), and (2.153) imply (2.142), and the relation (2.142) implies the correctness of the formula (2.140).

Let us consider Case 3, i.e. i2 = i3 = i1, ¿2 = /3 = / = /1, and /1,/3 = 0, 1, 2, . . . So, we prove the following expansion

Pi P2 P3

Ä) = ,,,¡-i-m. EEE«j) <ii.i2.i3 = 1,...,m),

ji=0 j2=0 j3 =0

(2.154)

where /1, /3 = 0,1, 2,... (/3 = /) and

T s si

Cj3j2ji = J j(s)(t - s)^(t - S1)1j(S1^(t - S2^j(s2)ds2ds1 ds. (2.155) t t t

If we prove w. p. 1 the formula

Ts Pi P3 1 » »

l.i.m. C^cj;1' = W (t - «)a / (t - .5i)''<ifi;'»dS, (2.156)

2

where the coefficients Cjjj are defined by (2.155), then using Theorem 1.1 and standard relations between iterated Itô and Stratonovich stochastic integrals, we obtain the expansion (2.154).

Using Theorem 1.1 and the Itô 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 2/+/i+1

' E ^.cj;1' w. p. i, ji=0

T T

ôji=f j wc - ^ f(t - s)2/^

t Si

pi p3 , 2/+/i+1

v^ v^ / An)-- V^ r

/ , / ,( ./:■,/ ',/ s / 2 V

ji=0 j3=0 ji=0

2/+/i+1 / p3 1 \ pi p3

E E E^cj

ji=0 Vj3=0 / ji=2/+/i+2 j3=0

Therefore,

T

pi p3

.(ii) 1

Mi EE^cf-JA*-^ [{t-s^df^ds

1 Vi=° j3=0 *t t

2/+/i+W p3

^ £ fe C

ji=0 \j3=0

2

- V I +

—-- / y I ✓ , o 1

p3 ^^ —' \ z—' 2

ji=0 \j3=0

pi p3 2

+ lim M^ ( Y ECj3j3jiCjii) . (2.157)

\ji=2/+/i+2 j3=0

Let us prove that

p3 1 2

lim ( E < rA- (2-158)

p3^œ \ z—' 2 /

j3=0 /

2

s

We have

'3

53 ^Ajsji A | —

¿3 =0

1

— I

2

'3

T

T

T

53 I j(s2)(t - s2)11 dsW j(si)(t - si)1 dsi / j(s)(t - s)1 ds-

¿3=0 t

s2

s1

T

T

1

- § I <f>h(si )(t - Si)'1 I (t- sfdsdsi ) =

s1

T / T

¿53 J^)(t-S2)h[ j

j3=0 t \s2

j(si)(t - si)ldsi ds2-

1

T

T

~2 / feMC* - si) sTdsds 1 ) =

1

4

/T

/

j (si)(t - si)11

t

s1

'3

T

53 / j (s) (t - s)1 ds

T

\j3=0 \S1

- I (t - s)21 ds I dsi I -

s1

1 4

/T

/T

j (si)(t - si)

l1

t

V

(t - s)2lds - E I I j (s)(t - s)ds) -

¿3=P3+i

T

-J (t - s) ds I dsi | -

s1

1

4

/ T « ( T y (si)(t - si)l1 E y

^t j3=p3+i \s1

\

0j3 (s)(t - s)1 ds dsi

. (2.159)

!

2

2

2

2

2

2

2

2

2

2

In order to get (2.159) we used the Parseval equality

to / t \ 2 T

E -H = /K2(s-s1)ds- (2.160)

j3=0 V1 / t

where

l

K(s,S1) = (t - s)1 1{s1<s}, S,S1 G [t,T]. Taking into account the nondecreasing of the functional sequence

T2

n I T

U 's ■ — —t s)1

(si)^E I I j (s)(t - s)1 ds

¿3=0

s1

continuity of its members and continuity of the limit function

T

u(s1) = J(t - s)21 ds s1

at the interval [t, T] in accordance with the Dini Theorem we have uniform convergence of the functional sequence un(s1) to the limit function u(s1) at the interval [t,T].

From (12.159) using the inequality of Cauchy-Bunyakovsky, we obtain

p3 1 _ \ 2

53 ^Ahji ~ ) —

2

j3=0

T \2\

1 n „ I tt I „ \

<lj ^(s^it - Slf-dSl J tt

53 /j (s) (t - s)1 ds ^3=p3+i Vs1 / y

dsi <

T

< - tf1 J (pl(s1)ds1(T -t) = \{T - tf-+ls2 (2.161) t

when p3 > N(e), where N(e) G N exists for any e > 0. The relation (2.158) follows from (2.161).

We have

PS P1 PS 2(js+l + 1) + l1

53 53 Cjsjsj1 j = 53 53 Cjsjsj1 j. (2.162)

js=0 j1 =2l+l1+2 js=0 j1=2l+l1+2

2

We put 2(j3 + I + 1) + 11 instead of p1, since Cj3j3j = 0 when j > 2(j3 + I + 1) + 11. This conclusion follows from the relation

T / T \ 2

< !/:,/:,/ = \J <f>hM{t ~ s2)h j J (jj.,(.si )(/ - sfadsi 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!1 = E ECj3j3ji CJ!1 * (2.163)

j3=0 ji=2/+/i+2 ji=2/+/i+2 j3=0

p3

Note that we included some zero coefficients Cjjj into the sum ^ .

J3=0

From (2.162) and (2.163) we have

p3 p1

(i1)

ji

j3=0 ji=2/+/i+2

M E E Cj3j3ji ci:

'2(p3+/+1)+/ i p3 \2| 2(p3+/+1)+/ i / p3

E EC- i Ci: i M = E E^ i

ji=2/+/ i+2 j3=0 ) I ji =2/+/ i+2 \j3=0

2(p3+/+1)+/ i I p3 T /T

E 2 S / feWi*-^1 / ^(siXi-sife ds2

j i =2/+/ i+2 \ j3=0 { Vs2 /

1 2(p3+/+1)+/ i / T p3 i T \ 2 \

4 E / fe^K* - S2)/l E / )( / - Sife ds2

j i=2/+/ i+2 j3=0 Vs2 ) )

. 2(p3+/+1)+/ i / T / T

j E / (t-S.fds,-

ji=2/+/ i+2 Vi Vs2

2

T

2

E (si) (t - si)1 dsi

j3=P3+i

\

s2

ds2 ) )

/T

4

2(p3+l+i)+lW T to / T

E J j (s2)(t - s2)1^ J j (si)(t - si)

\

j1=2l+l1+2

- si) ds^ ds2

V1

j3=P3+i

s2

/

(2.164)

In order to get (2.164) we used the Parseval equality (2.160) and the following relation

T

J j (s)Q2i+1+i1(s)ds = 0, j1 > 2/ + 1 + /1, t

where Q21+1+l1 (s) is a polynomial of degree 2/ + 1 + /1. Further, we have

T

j(si)(t - si)1 dsi -

s2

(T - t)2l+i(2j3 + 1)

/ i

\

22l+2

J Pj3 (y)(1 + y)l dy V (S2) (T - t)2l+i

!

22l+2(2j3 + 1)

x

/

\

X

(1 + z(s2)) Qj3(s2) - iy (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

(s2)+ l2

(Pj3+i(y) - Pj3-i(y))(1 + y)l-i dy

V

V (S2)

/

<

JT-tf^ - 22'+i(2j3 + l)

2

2

2

1

2

2

2

i

X

/11 221+1Hj3(S2) + l2 J (1 + y)2/-2dy | (Pj3+i(y) - Pj3-i(y))2 dy

\ z(«2) z

JT-i)

<

x

\ V 7 z(«2) /

/ 2 1

ZHj3(s2I (Pf3+i(y) + Pl-i(y)) dy

V Z(S2) 7

<

(T - i)2/+i f _ „ , . /2

2(2j3 + 1) where

, (2.165)

Qj3 (s2) = P,3-1(z(s2)) - Pj3+1(z(52)),

Hj3 (S2) = P^-1(Z(S2))+ P^+1(Z(S2)).

Let us estimate the right-hand side of (2.165) using (2.116)

t 2

/ j (S1)(t - S1)1 dsj <

VS2

(T - f)2^1 / K2 K2^ < 2(2j3 + l) U + 2+ J3,

/ 1 \

2 I2 f dy

, (1 - (z(S2))2)1/2 21 - W ) (1 - y2)1/2,

\ z(s2) /

<

(T - t)21+1K2 ( 2 l2n \ , ^ ,

From (2.164) and (2.166) we obtain

1/ P3 Pi \ 2 |

E E Cj3j3ji zj;1 > <

V'3=° ji=21+1i+2 ) I

It ~ /T \2 V

/1 V^ l X „ \/

1 2(p3+/+1)+/i

4

ji=2/+/i+2

<

\t j3=p3 + 1 \s2 / y

. 2(p3+l+i)+l1 / T to / T \2 V

<j(T-t)0Jl E / ifeM E [ hsMit-sjdsA dso

\t j3=P3+i \s2 / y

4

j1=2l+l1+2

(T-tf+^K'K* / / j 2d*,

16 11/ (l-(^2))S)8/4 +

12tt f d.so \ ^ 1

<

<

2

(T - f2ps + 1 / } l2TT f dy

~ 64 (l- l/2)3/4 +27^17 (1 - y2)i/4 I ^

< C(T - i)4/+2/l+3 ^ii ^ 0 when 00, (2.167)

p3

where constant C does not depend on p3 and T - t.

The relations (2.157), (2.158), and (2.167) imply (2.156), and the relation (2.156) implies the correctness of the expansion (2.154).

Let us consider Case 4, i.e. /1 = /2 = /3 = / = 0,1, 2,... and i1,i2,i3 = 1,... ,m. So, we will prove the following expansion for iterated Stratonovich stochastic integral of third multiplicity

P1 P2 PS

I%r's) = '.i-m. EE EC««, Ci;1)Cj22)CjSS 1 (i1,i2,i3 = 1,...,m),

j1=0 j2=0 jS =0

(2.168)

where the series converges in the mean-square sense, / = 0, 1, 2, . . . , and

T s S1

CjSj2j1 = J j(s)(t - s)1/(t - s1)1 j (s1^(t - s2j (s2)ds2ds1 ds. (2.169) t t t

If we prove w. p. 1 the following formula

p1 pS

E EC**1 C - 0, (2.170)

j1=0 j3=0

( ' . . . _L_ H. . . . ( ' ■ ■ ■ -( '-(' ■

where the coefficients Cj3j2ji are defined by (2.169), then using the formulas (2.142), (2.156) when 11 = 13 = 1, Theorem 1.1, and standard relations between iterated Ito and Stratonovich stochastic integrals, we obtain the expansion (2.168).

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 (2.169) and

T

Cji = J j (s)(t — s)/ds. t

Then w. p. 1

Pi P3

plim^E £ Cji cj32) =

ji=0 j3=0

Pi P3 /1 N

=£ £ 2°*Ca - - c(2'171) ji=0 j3=0 V 7

Taking into account (2.142) and (2.156) when 13 = 11 = I as well as the Ito formula, we have w. p. 1

pi p3 -, / /

l.i.m. y y cjlj3jlcf2) -Y<1Y

?i,p3^œ Z^Z^ jij3jiSj3 2 ^ j3 j3

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

pi p3 pi p3

-l.i.m. £ ^Cj j2' - l.i.m. £ ^Cj3jijiZj32)

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

/ T Ts

1 ^ i(i- - - I if -A1

j{t - 5 /(* - s)1 /(* " sifds^-

2 ^ ji J v y s 2 ji=0 t t t

T s

4 ¡(t-s)21 J(t-si)ldf^ds = tt / T T

i E 'fit - + 2Î27TTT /(i - s)3'+1^i,>-ji=0 < ( ' i

T T

¡(t-si)11 (t-sfdsdf™

t Si

T T

ji=o i 1 t

/ T T \

1 /,„ > 17 I 1 f , >7,„/„'-A f , , 91 I 1 I

2(27 + 1) \(T " i)2'+1 J {t ~ S)'di°2> + J {t ~ A^di^ ,

tt T T

= l[ É ■- /(* - s)2'is ) /(i - s>,,if«(!2) = \j1=0 t / t

Here the Parseval equality looks as follows

/

T

= £ c?, = /<t _ s)2/

_A _n ^

- s) ds =

<T _ t)

2/+1

2/ + 1

ji=0 ji=0 t and

T I

At - s)ldfs(i2) = £ Cj3Zf w.p.1.

t j3=0

The expansion (2.168) is proved. Theorem 2.4 is proved.

It is easy to see that using the Ito formula if i1 = i2 = i3 we obtain (see (1.59))

*T *s *si

J(t - s)lJ(t - s1 )l J(t - s2)ldf^df^dfs(ii) = t t t

66

/

E CjCj::'jj w.p.1. <2.172)

ji ?2 ?3=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 2.5 [6]-[15], [33]. Suppose that [fa(x)j 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 fi'f'f3' (M2,i3 = 1,...,m)

t t t

the following expansion

9fT s 2

r r r Pl P 2 P s

J J J dftii)dft22)d^s) = pUp^ ££E-^jjj (2.173)

t t t j 1=0 j2=0 j3=0

that converges in the mean-square sense is valid, where

T s s 1

Cj3j2ji = J j(s) J j (s1) J j (S2)ds2ds1ds t t t

and

T

j = f faj (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. V V Cnnn= - dsdil»', (2.174)

pi'ps^ ji=0 j=0 2 i i

T T

Pi Ps

1

E E=2 II (lT- ^

ji=0 js=0 t t

Pi P3

.m « » - z (>2)

'■¡•m- E E-jijjiZji = 0, (2.176)

s

pi,ps^œ

ji=0 js=0

then from the equalities (2.174) (2.176), Theorem 1.1, and standard relations between iterated Ito and Stratonovich stochastic integrals we will obtain the expansion (2.173).

We have

c /-fe) _ (T - tf12^)

°P1,P3 — / v / , ./ ■./ ./ S— g SO 1-

J3 =0 ji=0

pi pi (i3)

y0,2 ji,2ji Z0

pi

+ E C0,2ji,2jiZ0i3 + E C0,2ji-1,2ji-1C0i3 + E C2j3,0,0C2j33 +

ji=1 ji=1 j3=1

p3 pi

p3 pi

+ E T, C2j3,2ji,2jic2j33) + E E C2j3,2ji-1,2ji-1C2j33) + E C2j3-1,0,0C2j3!-1 +

p3

j3=1 ji=1

p3 pi

j3=1 ji=1

+3 )

j3=1

p3 pi

+ E E C2j3-1,2ji,2jiC2j33-1 + E E C2j3-1,2ji-1,2ji-1C2j33-1, (2.177)

j3=1 ji = 1

j3=1 ji = 1

where the summation is stopped, when 2j1; - 1 > p1 or 2j3, 2j3 - 1 > p3 and

(T — i)3/2 ^ 3 (T-tf/2 „ V2(T-tf¡2 Co 01 01 — -„ 070-, Oo,2/-l,2/-l — -~ oTo-5 ^2/,0,0 — "

8n2/2 '

C2r-1,2/,2/ = 0, C2/-1,0,0 =

8n2/2 V2(T - if'2

C2r,2/,2/ =

4n1

t)3/2/(16n2/2), r = 21

0, r = 21

t)3/2/(16n212), r = 21

4n212

(2.178)

, C2r-1,2/-1,2/-1 = 0, (2.179)

cw,21-ipj-i = I —y/2(T - Î)3/2/(4tt2/2), r = I

0,

r = 1, r = 21

Let us show that

(2.180)

(2.181)

LLm. S2pi,2p3 = l.i.m. S2pi,2p3-1

pi,p3^œ pi,p3^œ

l.i.m. S2pi-1,2p3-1 = H.m. S2pi-1,2p3 •

pi,p3^œ pi,p3^œ

(2.182)

We have

2pi

S2pi,2p3 = S2pi,2p3-1 + E C2p3,ji,jiC2P33)' (2.183)

ji=0

Using the relations (2.178), (2.180), and (2.181), 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

C2p3,0,0 + E ( C ji = 1 V

(2.184)

From (2.183), (2.184) we obtain

l.i.m. S2pi,2p3 = l.i.m. S2pi,2p3—1. (2.185)

Further, we get (see (EUSMEUSO]))

2p3-1

S2pi,2p3-1 = S2pi-1,2p3-1 + E Cj3,2pi,2pi (2.186)

j3=0 3

2p3-1 2p3

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 '

= ^ 8tt2^ + ^16TT2^ i1^2^ ~ 1{p3>2pi})dp1)- (2.187)

From (2.186), (2.187) we obtain

l.i.m. S2pi?2p3—1 = l.i.m. S2pi—1,2p3—1- (2.188)

Further, we have

2p3

S2pi ,2p3 = S2pi — 1,2p3 + E jpi ,2pi Cj33(2.189)

j3=0

2p3 2p3

En /t(i3)_n A'3 I V"^ fi A'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,2piZ2j*3-1 + C2j3,2pi,2piZ2j3 J • (2.190)

From (2.190), (2.178)-(2.180) we obtain

V^ r Ais) _ (T -f )i/2 Ais) V^jT-t)^2 (i3) f91QU MAP :>r A - g7r2^2 Co 167r2p2-i{p3>2pi}Up1 • t^Ul)

The relations (2.189), (2.191) mean that

l.i.m. S2pi,2p3 = l.i.m. S2pi-1,2p3• (2.192)

Pi,P3^œ pi,p3^œ

The equalities (2.185), (2.188), and (2.192) imply (2.182). This means that instead of (2.174) it is enough to prove the following equality

2pi 2p3 1

T t

HJSu E E = ? / / dsdfA w- p-1- (2-193)

/ J / J j3jiji"3j3 o

ji=0 j3=0 "t "t

We have

2p3 2pi

o v^ v^ / A*) _ (T ~ f)3/2 An) i

2pi,2p3 — / , / / ./:■,/ ,/ C>/;., — g SO

j3=0 ji=0

Pi Pi Pi

+ El C0,2ji,2jiC0Î3) + E C0,2ji-1,2ji-1C0i3) + E C2j3,0,0C2j33) +

ji=1 ji=1 J3=1

P3 Pi P3 Pi P3

+ y E C2j3,2ji,2jiC2j33) + E E C2j3,2ji-1,2ji-1C2j33) + E C2j3-1,0,0C2j^—1 +

J3=1 ji=1 J3=1 ji=1 J3=1

P3 Pi P3 Pi

+ E E C2j3-1,2ji,2jiCj—1 + E E C2j3-1,2ji-1,2ji-1C2j33)-1- (2.194)

J3=1 ji=1 J3=1 ji=1

After substituting (2.178)-(2.181) into (2.194), we obtain

2p3 2PI /1 PI 1

EE = v-tf** !<*»>* c

j3=0 ji=0 V ji=1 j1

P3 ^ min{pi,p3} ^ P3

47r" S "4^2 S 72C2S 72C2S 1 • (2-195)

js=1 J3 js=1 J3 js=1 J3

From (2.195) we have w. p. 1

2ps 2pi / œ .

(is) _ (rp ,\3/2 / 1 /■ (is) 1 V^ 1 /-(is)

Ä EE= + ^E J Co

js =0 ji=0 \ ji=1J1

/0 Pa ^

js = 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

Ps^œ X n z—' J3

js=1

From (2.196) and (2.197) it follows that

2Ps 2pi

EEC**zjss)

js=0 ji=0

= (t - i)3/2 (Jc,i'3) + ¿c,(,*3) - ^g !<<*>_,

T s

= UI drdfM, t t

where the equality is fulfilled w. p. 1.

So, the relations (2.193) and (2.174) are proved for the case of trigonometric system of functions.

Let us prove the relation (2.175). We have

Q' dff n Aii)

_ (T — t)3/2jH) ,

ji =0 j3=0

p3 p3 pi p3

+ E C2j3,2j3,0C0ii) + E C2j3 —1,2j3 —1,0Z0ii) + E E C2j3,2j3,2ji — 1C2ji — 1 + j3=1 j3=1 ji=1 j3=1

pi p3 pi pi p3

+ E E C2j3 — 1,2j'3 — 1,2ji — 1C2?ji—1 + E C0,0,2ji —1C2jj-i)—1 + E E C2j3,2j3,2jiC2ji) +

ji=1 j3=1 ji=1 ji=1 j3=1

pi p3 pi

+ E E C2j3 — 1,2j'3 — 1,2jiC2ji) + E C0,0,2jiC2ji), (2.198)

ji = 1 j3=1 ji=1

where the summation is stopped, when 2j3, 2j3 — 1 > p3 or 2j1; 2j1 — 1 > p1 and

(T-tf2 „ _S(T-tf2 „ _V2(T-tf/2

O2/,2/,0 ~ g7r2/2 ' ^2/ —1,2/—1,0 - --, U,0,2r - --^o-'

(2.199)

C2/-l,2/-l,2r-l = 0, Co,0,2/'—1 = -^ ^ ^-, C2/,2/,2r-l = 0, (2.200)

i)3/2/(16TT2/2), r = 21 C21.21.2r =4 , (2.201)

0, r = 21

t)3/2/(16n212), r = 21

C2/-i,2/-i,2r = < ~V2(T - i)3/2/(47T2/2), r = l . (2.202)

0, r = 1, r = 21

Let us show that

LLm. S2pi,2p3 = l.i.m. S2pi,2p3 —1 = l.i.m. S2pi —1,2p3 —1 = l.i.m. S2pi —1,2p3 •

(2.203)

We have

2P3

S2pi,2p3 = S2pi —1,2p3 + E Cj3,j3,2pi(^i • (2.204)

j3=0

Using the relations (2.199), (2.201), and (2.202), we obtain

E Cj3,j3,2pi = C0,0,2pi + E Cj3 J3,2pi = ji =0 j3=1

P3 /

= C0,0,2pi + E ( C2j3 — 1,2j3 — 1,2pi + C2j3,2j3,2pi

j3=1 ^

\/2(T - i)3/2

4n2p2

From (2.204), (2.205) we obtain

(1 — 1{p3>pi}). (2.205)

u.m. S2pi,2p3 = l.i.m. S2pi — 1,2p3 • (2.206) Further, we get (see (2.199)-(2.201))

2pi —1

S2pi — 1,2p3 = S2pi — 1,2p3-1 + E C2p3,2p3,jiCj!1 , (2.207)

ji=0

2pi-1 2pi

Er» /-(«i) _ ^ a(«I) I —

C2p3,2p3,jij = C2p3,2p3,0z0 + C2p3,2p3,jij — C2p3,2p3,2piS2p1 =

ji=0 ji=1

Pi / \

C2p3,2p3,0Z0 + ¿^ 1 C2p3,2p3,2ji —1Z2j1 —1 + C2p3,2p3,2ji J — C2p3,2p3,2pi Z2p =

ji=1

From (2.207), (2.208) we obtain

H.m. S2pi — 1,2P3 = LLm. S2pi — 1,2P3 — 1- (2.209)

Further, we have

2pi

S2pi,2p3 = S2pi,2p3 —1 + E C2p3,2p3,jiCji1 , (2.210)

ji=0

2pi 2pi

E- Ali) — n z(«i) —

C2p3,2p3,jij = C2p3,2p3,0z0 + C2p3,2p3,jij = ji=0 ji=1

( i ) P i ( i ) ( i ) = —2Ps,2Ps,0C0i + E —2ps , 2PS , 2 ji -1Z2 ji -1 + C2Ps,2Ps,2j Z2ji • (2.211)

ji = 1 X

From (2.211), (2.199)-(2.201) we obtain

2Pi

^ n An) _ A ~ L__

.2 SO ifi^J

E Cop3op3jl - ^ g^ Ci'0 - 1{pi>2p3}dj,l)- (2-2!2)

The relations (2.210), (2.212) mean that

Li.m. S2pi,2p3 = LLm. S2pi,2ps-1- (2.213)

The equalities (2.206), (2.209), and (2.213) imply (2.203). This means that instead of (2.175) it is enough to prove the following equality

2Pi 2ps

T t

1-i-H.. = 1 I I <It w. p. 1. (2.214)

We have

2P1 2Ps

c/ /-(»i) _ ^ ~ L > An) I

2pi ,2p3 ~ / , / , ( Khü^i- - 6

ji=0 js=0

Ps Ps P1 Ps

+ E C2js,2js,0C0ii) + E —2js-1,2j3-1,0C0li) + E E C2js,2js,2ji-1C2jl-1 +

js=1 js=1 ji=1 js=1

Pi Ps Pi Pi Ps

+E E —2js—1,2js—1,2ji—1 c2ji—1 + E C0,0,2ji-1c2ji)-1 + E E ^jsjjc2ji)+

ji=1 js=1 ji=1 ji=1 js=1

Pi P3 Pi

+ E E —2js — 1,2js — 1,2ji C2ji) + E —0,0,2ji C2ji) • (2.215)

ji = 1 js=1 ji=1

After substituting (2.199)-(2.202) into (2.215), we obtain

2Pl 2P3 ( -, 1 P3 -,

E E = {T - tr- ^(A + i E i <ifl,+

ji=0 j3=0 V j3=1 j3

y/2 Pl 1 y/2 min{pl'p3} i y/2 pl 1 ■ \

ji=iJi ji=i ji=1J1 /

From (2.216) we have w. p. 1

2Pi 2P3 / to

EEo^=(T-^ +¿E¿

ji=0 j3 =0 \ j3=1J3

pi^to 4n j 2ji 1 / ji=1J1 /

Using the Ito formula and Theorem 1.1 for the case of trigonometric system of functions, we obtain w. p. 1

T t /

I J Jdi^dr=1-i(T-t) t t \

= \(T- tf'2 U" + 1-i.m. ^ £ jc^-i) ■ (2-218)

4 ^ pI^to n ^-=1 ji Ji J

From (2.217) and (2.218) it follows that

2pi 2p3

'.i-m^ E cw, cj;" =

ji=0 j3=0 T t

= \j j df^dr,

tt

where the equality is fulfilled w. p. 1.

T

T

dfs(il) + /(t — s)dfs(i

(ii) =

So, the relations (2.214) and (2.175) are proved for the case of trigonometric system of functions.

Let us prove the equality (2.176). Since ^(t), (t), (t) = 1, then the following relation for the Fourier coefficients is correct

( '. . . . ( '. . . . ( '. . . Lf '-(' .

^jum * ^J1J3J1 * — 2 ii J3-

Then w. p. 1

Pi P3

+2) _

E X^jj2

O-i r)o—Vnn f * '

.m.

ji=0 j3=0 Pi P3 / 1

H E ( -fîP» ~ ~ ClM ) Cf- (2.219)

ji=0 j3 =0

Taking into account (2.174) and (2.175), we can write w. p. 1

pi p3

l.i.m. E ECji j2)

ji=0 j3=0

1 Pi P3

= +0C0 ' - 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

4 \ Pi^œ n z—' ji •yi

4 \ pi^œ n ji

From Theorem 1.1 and (I2.174l)-(l2.176l) we obtain the expansion (12.173). Theorem 2.5 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 2.4.

Theorem 2.6 [10]-[15], [33]. Assume that {fa(x)}°=0 is a complete or-

thonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]) and "fa(s), ^2(s), ^3(s) are continuously differentiate functions at the interval [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

j*[^(3)]T,t = / ^te)/ ^i(ii)dfi;i)dfi;2)fs) (¿1,^2,^3 = i,...,m)

# # #

the following expansion

p

J *[^(3']t,( = 1^ £ Cjsj2j1 C^W (2.220)

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 (s) = ^2(s),

3. ¿1 = ¿2 = ¿3 and ^2(s) = ^3(s),

4. ¿1, ¿2, ¿3 = 1,..., m and -fa (s) = ^2(s) = ^3(s),

where

T s si

Cjsj2ji = J ^3(s)fas (s) ^ ^2(s1)0j2 (s1^ Mfai (s2)ds2ds1ds t t t

and

T

j = | ^ (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 = ^3(s)fas(s) / ^(s0 j (s0 / j (s2)ds2ds1ds.

Using Theorem 1.1, we can write w. p. 1

03(s)

t t

l3

2 z

73=0

l.i.m. £ <7Cji3),

3

where

T

<573 = J 073(s)Ms) J ^2(si)dsids. tt

We have

ll

1

M { ( E ( E ( ~ )

73=0 \ji=0

p A p T if V

E 2 ^ / 0/''('s'!r:!i's'! /

73=°^ 71=0 t \t /

(is)

pp

EE cMl 1 =

73 =0 \ji=0

1

— I

2

2

73=0 \ ji=0 t

1

T

J (fij-MMs) j i>2{si)dsids I = tt

-T

4 ^

T

73=0

t

073 (s)03(s

/T

/ p ( s \2 s N

V

E I i 071 (si)^(si)dsH - / ^2(si)dsi

71=0 \ t / t

ds / /

-1 ^ /» ^

4E / E

73=0 \ t 71=l+i \ t

2

07i(si)^(si)dsi I ds

(2.221)

\t \t /J

In order to get (2.221) we used the Parseval equality

00 / s \ 2 T s

E / 071 (si)^(si)dsH = K2(s,si)dsi = ^2(si)dsi,

71=A t / t t

where

K(s,si) = ^(si)1{s1<s}, s,si G [t,T].

T s

1

1

2

s

2

2

s

2

2

We have

2

t

(T-t)(2j i + l) 4

T-t

4(2j1 + 1)

/ z(s) ^

J Ph(y№ (^Y^y + ^p) dv —1

(Pji+1(z(s)) — Pji—1(z(s))) ^(s) —

z(s) \ 2 T — t /* f^n(T — t T + t"

2 j ((Pji+i(y)-Pji-i(y))V[—y + —)dy] , (2.222) —1

where

, / T+A 2

= s -

2 / T - 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 2.4. Finally, from (2.221) and (2.222) we obtain

<r K— ( ( dy + [ dy \ <

p2 (i-y2)3/4 7 (i-y2)1/4y -

<--> 0 if p ->■ oo,

P

where constants K, K1 do not depend on p. Case 2 is proved.

Let us consider Case 3. In this case we will prove the following relation

T s

P P 1 /• r

^ EE^^;0 = 2 J ^ J Msi)di^ds w. p. 1,

ji=0 J3=0

2

where

T s si

J 0(s)fas (s) y ^1) j (s1^ V^M j (s2)ds2ds1 ds. t t t

Using the Ito formula, we obtain w. p. 1

T s T T

i | Jfa(s1)di^ds = ± J msi) J^(s)dsdi^. (2.223)

t t t si

Moreover, using Theorem 1.1, we have w. p. 1

T T

5 / = i 11m. £ qcj;'», (2.224)

t si ji=0

where

T T

C* = j^1)^1)/ ^2(s)dsds1.

t si

Further,

T s si

Cjsjsji = J 0(s)j (s) ^ 0(s1)0js (s1^ 01(s2)0ji (s2)ds2ds1ds = t t t

T T T

= J ^Mj (s2^ 0(s1)0js (s1^ 0(s)0js (s)dsds1ds2 =

t s2 si

T / T \ 2

= i J ipiMfaM J l'(.s'| )o/;.(-S'| )d.S| d.s2. (2.225)

From (2.223)-(2.225) we obtain

|/p/p \ \ 2 | p/p \ 2

M V v c^ - iq <<;■> V V - =

I \ji=0 \js=0 / / I ji =0 \js=0 /

-T

4

T

ji=0

T

j (S1)^1(S1)

V'

v

E I I j(s)^(s)ds1

j3=0

vsi

T

— ^2(s)ds ds1 I =

si

^E f faMMsi) E

ji=° t j3=p+1 \si /

2

j(s)^(s)ds | ds1 \t Vi / y

In order to get (2.226) we used the Parseval equality

T

T

T

E ( / jW(s)ds ) = I K2(s,s1)ds = I ^2(s)ds,

j3=0

vsi

si

where

K(s,s1) = ^(s)l{s>si}, S,S1 G [t,T].

(2.226)

Further consideration is similar to the proof of Case 3 from Theorem 2.4. Finally, from (2.226) we get

p / p

M { (E (E - iq ) cj

ji =0 Vj3 =0

(il) j'l

<

p2

dy

+

dy

<

/ (1 - y2)3/4 J (1 - y2)1/4 -1 -1

^ K1 n

<--> 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^m. EE Cj Zj32) = 0 (Ws),«s),«s) = ^(s)).

p^TO ' ^ ' ^ J3

ji=0 j3=0

2

2

2

2

2

2

1

1

In Case 4 we obtain w. p. 1

v

(i2)

^^ X CJiJ3ji C

jij3 =0

~~ S '"C'' ~ Cjljljs ~ Cj3jljl ) -

ji,j3=0 V y

1ppp ji=0 j3=0 ji,j3=0

V

(«2 )

-Li^m. E Cj3jiji j

ji J3=0

00 T Ts

= \T,Clj J гß%s) J 1>(8l)dfMd8

ji=0 t t t

Ts TT

i [ Ijj(s) f ^2(si)dsidfs(i2) = i f ijj2(s)ds f tjj(s)dii

T T T si

/ / - i J J ^(s)dsdi^ =

t Si t t

T T T T

= IJ i)2(s)ds J 1>(s)dfM -\j i^si) J ^(s)dsdi^ = 0, t t t t

where we used the Parseval equality

/ T \ 2 T

to to 1 „ \ t

EC = E / (s)d^ = (s)ds.

ji=0 j=0 \ t ) t

Case 4 and Theorem 2.6 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

VT^t

\/2sin (2trr(6> - t)/{T -t)), j = 2r - 1, \/2cos (2trr(6> - i)/(T - ¿)), j = 2r

where r = 1, 2,...

Integrating by parts, we have

far_i(6>)0(6>)d6> = ^ [ iß(0) sin=

y/T=t

T-t

'T - t 11 ,fs 2nr(s - t)

-0(s) cos--7-- +0(0 +

2 nr

Tt

¿5

far(6>)0(6>)d6> = [ iß(0) cos27rrJ:e~^d0 =

y/T^t

Tt

'T -1 1 / ,, , . 2nr(s - t) vs) sm-

2 nr

Tt

-fm^^dei

where r = 1, 2,... and 0'(0) is a derivative of the function ^(0) with respect to the variable 0.

Then

02r-l(0)0(0)d0

C 2C 2C < — = — <-

r 2r 2r 1

(2.227)

1

s

s

s

(0)^(0)d0

<C_2C

— r 2r '

(2.228)

where constant C does not depend on r (r = 1, 2,...). From (2.227), (2.228) we get

j (0)^(0)d0

<

(2.229)

where constant K is independent of j (j = 1, 2,...). Analogously, we obtain

T

j (0)^(0)d0

<

ji :

(2.230)

where constant K does not depend on j (j = 1, 2,...). Using (2.221), (2.226), (2.229), and (2.230), we get

p / p

"{IE Ec

j3=0 \ji=0

_ \ >(¿3)

pp

M {( E ( E ^ - ) cj

ji =0 \j3 =0

^ Ki n

<--> 0 if p ->■ oo,

P

Ki

<-->■ 0 if p ->■ oo,

P

where constant K is independent of p.

The consideration of Case 4 is similar to the case of Legendre polynomials. Theorem 2.6 is proved.

In the next section, the analogue of Theorem 2.6 will be proved without the restrictions 1-4 (see the formulation of Theorem 2.6).

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 2.7 [10]-[15], [20], [31]. Suppose that {0 (x)j 0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the

s

s

space L2([t,T]). At the same time 02(s) is a continuously differentiate non-random function on [t,T] and 0i(s), 03(s) are twice continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

■.T * is * Î2

j*[0(3)]T,t = J 03^y 02(t2)y 0i(ti}dfi(;i)dfi(;2)dfi(;s) (ii,i2sz3 = i,...,m) t t t

the following expansion

p

J *[0<3)]T,i = 1^ £ jj C'CC (2.231)

j1 j2 J3=0

that converges in the mean-square sense is valid, where

T s si

Cj3j2j1 = J 03(s)0js (s) ^ 02(sl)0j2 01 (s2)0ji (s2)ds2ds1ds t t t

and

T

j = | 0j (s)df<->

t

are independent standard Gaussian random variables for various i or j.

Proof. Let us consider the case of Legendre polynomials.

From (1.46) for the case p1 = p2 = p3 = p and standard relations between Ito and Stratonovich stochastic integrals we conclude that Theorem 2.7 will be proved if w. p. 1

Ts

^ E E ? = 2 J 03(-5) J MsiWiMdsrfM, (2.232)

ji=0 js=0 t t

Ts p p T s

E E CJsjsn(r = 2 / 03(^)02(5) / 0i(-si)dfi;i)d.s, (2.233)

p œ ji=o js=o { i

pp

'is-EEzj32) = o. (2.234)

ji=0 j3=0

Let us prove (2.232). Using Theorem 1.1 for k = 1 (also see (1.44)), we can write w. p. 1

T s p

1 ( h (s) /'02(5l)'0l(-Sl)dSldfii3) = ^l.i.m. V r.- < " .

2 2 p^^

t t j3=0

where

T s

Cj = j ^2(si)^i(si)dsids.

tt

We have

2

^ m J / V^ V^ 1 v^ n A*)

ji=0 j3=0 j3=0

v 2

p / p \ \ 2

M<!(E(E' )•

j3=0 \ji=0

p p 2

EE'' =

2

j3=0 \ji=0

p / p T S S/i

j3=0 \ji=0 t t t

i ?......;........V2

"Ö I s)0,,(s) I '0i(.si)'02(si)dsids J =

2

tt

P ( T s / P

= E / ^3(s)0j3(s) / (si)x

j3=0 \ "t { Vji=0

s i \ \ 2

X J ^i{s2)(t)Ji{so)dso - i0i(.si)'02(si) ) dsids ) • (2.235) Let us substitute ti = t2 = si into (2.12). Then for all si G (t,T)

si

rc 1 1

E ■MsiWhiSl) / '01(52)^(52)^9 = -'0l(.Sl)'02(Sl). (2.236) ji=0 t

From (2.235) and (2.236) it follows that

T

si

Ep = E / 03(s)0j3(s) / E 02(si)0ji(siW 0i(s2)fai(s2)ds2dsids j3=A t t ji=p+i t

(2.237)

From (2.237) and (2.24) we obtain

p / t / z(s)

EP<C1J2 J\<t>iM\\ J

j3=0 t

T

dy

Ks)

+

dy

\

p J (1 - y2)V2 J (1 - y2)1/4

V-1 -1

\

ds / /

<

< f E (/ ) ^ E /

T

2 rs)ds = ^ ^ 0

,3=»v; / p2 / p2

if p 5 to, where constants Ci, C2, C3 do not depend on p. The equality (2.232) is proved.

Let us prove (2.233). Using the Ito formula, we have

T s T T

i J c(s)c-As) j Usi)di^ds=l- yV(si) j Ms}Us)dsdi^ w. p. 1.

t t t si

Moreover, using Theorem 1.1 for k = 1 (also see (1.44)), we obtain w. p. 1

p

I AAs) I AAsAAAsAd.^d,iM = - l.i.m. 1

T T

^ /V(s) [ Usi)Usi)dSidi^ = I l.i.m. E^iCji1^

t

ji=0

where

T T

Ci = 0i(s)fai(s) / 03(si)02(si)dsids.

(2.238)

We have

E = M

M

p p

X^X^r An) r* Aii)

/ , / , ^hhji S' 2 ./ V

ji=0 j3=0 pp

ji=0

ji=0 \j3=0

(ii) j1

2

s

p

2

p

2

p p 1 2

E V'- , (2.239)

ji=0 Vj3=0 /

T s s1

Cj3j3ji = J ^3(s)j ^2 Mj My ^i(s2 )0ji (s2)ds2 dsids = t t t T T T

= J ^i Mj My ^2(si )0j3 (si ) J' ^3(s)0j3 (s)dsdsids2. (2.240)

t s2 si

From (2.238)-(2.240) we obtain

p i T T / p

EP = E / ^i M j M / ^^M j (si)x

ji=0 \ t «a Vj3=0

x j c:,(«)oJ:Ss)(ls ) j dsids2j . (2.241)

We will prove the following equality for all si G (t,T)

T

^ „ 1

E'02(Sl)0,3(.Sl) J c,(s)o,Js)(ls = (2.242)

j3=0 si

Let us denote

#i(ii,i2) = + il{fl=,2}'02(/:i)'03(ii), (2.243)

where

Ki(ti,t2) = ^2 (ti)^3(t2 )l{ti<t2}, ti ,t2 G [t,T].

Let us expand the function Kf(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)^E ^2(tiW ^Mj (t2)dt2 • j (t2) (¿2 = t,T). (2.244)

j3=0 ti

The equality (2.244) is fulfilled in each point of the interval (t,T) with respect to the variable t2, when t i G [t, T] is fixed, due to piecewise smoothness of the function K(t ^t2) with respect to the variable t2 G [t,T] (t i is fixed).

Obtaining (2.244), we also used the fact that the right-hand side of (2.244) converges when ti = t2 (point of a finite discontinuity of the function Ki(ti, t2)) to the value

^ {K\{ti,ti — o) + Ki(t\,ti + o)) = = I<ati,ti).

Let us substitute ti = t2 into (2.244). Then we have (2.242).

From (2.241) and (2.242) we get

/ T T T \ 2

p / T T to t

EP = E / 0i(s2)0ji M / E 02(Si)0j3 (siW 03(s)0j3 (s)dsdsids2

ji=0 \ t S2 j3=p+i si

(2.245)

Analogously with (2.24) we obtain for the twice continuously differentiable function 03(s) the following estimate

T

00 „

E j (s0 / 03(s)0j3 (s)ds

j3=p+i

si

<

< P [(I ~ №l))2)1/2 + (1 " (^l))2)1^ ' (2'246)

where si G (t,T), z(si) is defined by (2.20), and constant C does not depend on p.

Further consideration is analogously to the proof of (2.232). The relation (2.233) is proved.

Let us prove (2.234). We have

p p 2 p p 2

Ep =f M £ £ Cji Cf = £ £ cj , (2.247)

[ Vi=0 j3=0 J J j3=0 \ji=0 J

T s si

Cjij3ji = J 03(s)0ji (s) J 02(si)0j3 (Si^ 0i(s2(S2)ds2dsids = t t t T si T

= J 02 (si)0j3 (si^ 0i (s2)0ji (S2)ds^ 03(s)0ji (s)dsdsi. (2.248)

t t si

After substituting (2.248) into (2.247), we obtain

Ef = E {¡Ms 1) j(s 1) E /^ Wji^(s) j(s)dsds 1 j3=0 \ t ji=0 t si

(2.249)

Introduce the auxiliary function

K^(t1,t2) = (t1)l{ti<t2}, t1,t2 G ].

Let us expand the function i^(t1,t2) using the variable when t2 is fixed, into the Fourier-Legendre series at the interval (t,T)

to ^

K^(t1,t2) = E / WOfe(t1)dt1 • j(t1) (t1 = t2). (2.250)

ji=0 t

Using (2.250), we have

p si T

E / «0) j(0)d0 / ^(s) j(s)ds =

ji=0 i si

= / ^(s) (E j(s) / (^ ds =

si \ji=0 t /

T / TO si TO si \

= J E j(s) ^ W0) j(0)d0 - E j(s)/ W0) j(W ds =

si \ji=0 t ji=p+1 t /

T T si

/rt TO />

^3(s)^1(s)1{s<si}ds - ^$(s) E j(s) / W0) j(0)d0ds =

si si ji=p+1 t

T si

/TO i

^(s) E j(sW (0)d0ds. (2.251)

si ji=p+1 t

After substituting (2.251) into (2.249), we get

E' = (j j^3(s) E j (s) j (0) j (0)d0dsds^ =

j3=0 \ t si ji=p+1 t /

P / N-i T to U \ 2

E lim E «U?) j (u?) / 03 (s) E j (s) / 0i(0)0ji (0)d0dsAui =

r—^ N^to ^ J J

J3=^ l=0 u* ji=P+i t /

P / N-i to T U* \

E lim E «u?)j(u?) E /«s)j(s)ds / 0i(0)0ji(0)d0Aui ,

^ N^to ^ . ' J J

\ l=0 ji=p+i U* t y

(2.252)

j3=0

where t = u0 < ui < ... < = T, Au/ = u/+i — u, is a point of minimum of the function (1 — (z(s))2)-a (0 < a < 1) at the interval [u/,u/+i], l = 0,1,..., N — 1,

max Au/ 5 0 when N 5 to.

0</<N—i

The last step in (22521) is correct due to uniform convergence of the Fourier-Legendre series of the piecewise smooth function Ki(s,u^) at the interval [u^ + e, T — e] for any £ > 0 (the function if (s, u^) is continuous at the interval

K,t ]).

Let us write the following relation

X VT_V2''i l

J Ms)(pji{s)ds = ——*2 2jl + 1 J Pn{y)ilj{u{y))dy = t —i

2v/2j7TT

z(x)

I ((Pjl+i(y) - Pn-i(y))^(u(y))dy I, (2.253)

—i

where x G (t,T), j > p + 1, z(x) and u(y) are defined by (220), 0/ is a derivative of the function 0i(s) with respect to the variable u(y).

Note that in (2.253) we used the following well known property of the Legendre polynomials [103]

Pj+i(—1) = — P (—1), j = 0,1, 2,...

and (2.21).

From (2.116) and (2.253) we obtain

^(s) j (s)ds

<

C

M (1 - (z(x))2)1/4

+ C1 , x G (t,T), (2.254)

where constants C, Ci do not depend on ji Similarly to (2.254) and due to

Pj(1) = 1, j =0,1, 2,...

we obtain an analogue of (2.254) for the integral, which is similar to the integral on the left-hand side of (2.254), but with integration limits x and T.

From the formula (2.254) and its analogue for the integral with integration limits x and T we obtain

T

^1(s)0ji(s)ds / ^3(s)0ji(s)ds

<

K

1

j? I (1 - (z(x))2)1/2

+ K1 , (2.255)

where x G (t,T) and constants K, Ki do not depend on j. Let us estimate the right-hand side of (2.252) using (2.255)

n-1

E'p<LT,[lim E E -2

<

. . N ^to j3=0 \ 1=0

0 / N-1

^E lim E

p2 ¿s V n-TO 1=0

L

ji=p+1 1

j? I (1 - (z(u*))2)1/2

+ K Au? <

+

K1

(1 - (z(u*))2)3/4 (1 - (zK))2)1/4

Am? <

T

< p2 £ (; ' " (/

ds

T

+ K1

ds

(1 - (z(s))2)3/4 V (1 - (z(s))2)1/4 tt

T

L\ I f d.s

?h> v / (1 - w-s»2>3/i

T

+ Ki

ds

(1 - (z(s))2)1/4

L^T-tfj^

dy

4P? ¿=0^ (1 - y?)3/4

+ K1

dy

1

(1 - y?)1/4

<

x

1

x

2

p

1

2

2

2

1

1

< = — 0 p2 p

(2.256)

if p —y to, where constants L,Li,L2 do not depend on p and we used (2.25) and (2.116) in (2.256). The relation (2.234) is proved. Theorem 2.7 is proved for the case of Legendre polynomials.

Let us consider the trigonometric case. Analogously to (2.33) we obtain

T ^ T

„ oo „

S2

E 02(5l)0j3(51) I 03(s) j(5)dsd5i

j3=p+1

si

<

Kl p

(2.257)

where s2 is fixed and constant K does not depend on p. Using (2.33) and (2.237), we obtain

p (T s ^ 7

Ep < K^ / / E 02(si)0ji(si) / 0l(s2)0ji(s2)ds2dsi

j3=0 V t t j1=p+1 t

t j1=p+1

ds =

p

n-1

K V lim V

n5to ¿3=0^ 1=0

si

J E 02(s1)0ji 01(s2)0ji (s2)dS2dS1 t ji=p+1 t

Au/

<

/

,N5to z—' p ¿3=0 \ 1=0 ^

p2

¿3=0

p

if p — to, where constants K, Ki, K2, L do not depend on p, Au/ = u/+i — u, u* G [u/, u/+i], l = 0,1,..., N — 1, t = uo < ui < ... < uN = T,

max Au/ 5 0 when N 5 to.

0< /<N- 1

Analogously, using (2.257) and (2.245), we obtain that Ep — 0 if p — to. It is not difficult to see that in our case we have (see (2.229), (2.230))

T

01(s)fai (s)ds / 03(s)fai (s)ds

0*i 0), (2.259)

j1

where constant Ci does not depend on ji

2

2

*

x

Using (2.252) and

I, we get

N-1 CO

p (

eP<K^Y, ^im E E

j3=0 y /=0 ji=p+1

p

T ul

[ ^(s) j (s)d^ ^(0) j (0)d0 2

1/1 <

Au/

<

/

N-1 1 \ " P r

E E ^ ^E(^)2^-0 (2-26°)

j3=0\/=0 ji =p+1 jl / P j3=0 P

if p ^ 00, where constants Ki, K2, L do not depend on p, another notations are the same as in

Theorem 2.7 is proved for the trigonometric case. Theorem 2.7 is proved.

2.3 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 4 Based on Theorem 1.1. The Case pi = ... = p4 ^ o (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 2.8 [8]-[15], [20], [31]. Suppose that (x)}o=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(;i)dwt(;2)dwt(33)dwt(44) (ii, i2, ¿3,24 = 0, 1, . . . , m) t t t t the following expansion

p

•J*^(4)]T.t = E jcj;1'

j1 j2,j3j4=0

that converges in the mean-square sense is valid, where

T S4 S3 S2

Cj4j3j2ji = j (s4) j (s3) j (s2) j (5i)d5id52d53ds4

(2.261)

2

and

T

<f = J 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.

Proof. The relation (1.47) (in the case when pi = ... = p4 = p — to) implies that

p

- (hMMX^MM)

/ C 74 73 79 71 z

p—TO

lü». E jiC'CW' = J[^(4)lT,t+

j'i,j2,j3,j4=0

+ 1{n=i9=0}Ai + 1{ii=i3=0} A2 + 1{n=M=0}A3 + i{i9=«3=0}A4 +

,1 /l(«1«3) , n /l(«1«9) 1 1 R

+ 1{i9 = «4=0}A5 + 1{i3=«4=0}A6 — 1{i1=i9=0}i{i3=i4=0}Bi —

— 1{i1=i3=0}1{i9 = «4=0}B2 — 1{ii=i4=0} 1 {¿9 = 23=0} , (2.262)

where J [0(4)]T,t has the form (2.7) for 0i(s),..., 04(s) = 1 and ii,...,i4 = 0,1,..., m,

A (i3i4) = l j m C , , Z(i3'ZM

A1 =lp51TO . 2^ Cj4j3jiji j Zj4 ,

j4j3,j'i=0

p

a2'2,4)=ip5mTO . E jc]:2)cj:4).

j4,j3,j2=0

p

A (i2i3) = i j m V^ C- ■ ■ ■ Z(i2' Z(i3'

A3 =lp5TO . Cj4j3j2j4Z j2 j ,

j4j3,j2=0

p

A (iii4) = l j m V^ C- ■ ■ ■ Z(n)Z(i4' A4 = lp5TO. C j4j3 j3 ji j Zj4 ,

j4,j3,ji=0

A(iii3) = l j m C. . ■ ■ Z(ii'Z(i3'

A5 =lp51TO . C j4j3 j4 ji j j ,

j4,j3,ji=0

p

A (iii2) = l j m V^ C . . Z(ii'Z(i2'

A6 = lp5TO. 2^ C j3j3 j2 ji j Zj2 ,

j3,j2,j'i=0

p p

Bi = lim \ jji, B2 = lim \ jjj,

p^œ z—' p^œ z—'

ji j4=0 j4,j3=0

B3 = lim V C

p^œ ^^

j4 j3j3j4•

j4,j3=0

Using the integration order replacement in Riemann integrals, Theorem 1.1 for k = 2 (see (1.45)) and (2.10), Parseval's equality and the integration order replacement technique for Itô stochastic integrals (see Chapter 3) [1]-[15], 105], [106] or Ito's formula, we obtain

A

(¿3M)

p

1

T

si

2

E 2j °/;i'S'! I ! I I Mi^dso ) dsidsqfq.

j4,j3,ji=0 t

(i3)z (i4) j4

t

,t

p

= l.i.m. \

p^œ ^—' 2

j4,j3=0 t

1

T s p / si

2

(«3^(«4) j4

j (sW j (si^ / j dsidsQ^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 (sl)

t

V

(s1 - t) - E / j (s2)ds2

ji=p+A t J J

2

dsi dsx

/-(«3^(«4) _ XZj3 Zj4 =

p

1

T

= l.i.m.

p^œ ^—' 2

j4 ,j3=0 t

E ^ I I («I)(«I - i)dsidsc]:3)c]:4) - =

1

2

t

T s

tt tt

T

(si - t)dw(;3)dw(i4) +

+ 91{»3=^0} lim E / <fe3(s) / <fe3(si)(siA^4)

p

j3=0 t

T s Si T

I [[ [ ds2d™ff>d^+\l{n=H^} [(Sl-i)dsi-Ap4) w. p. 1, (2.263)

î Î Î

i

s

s

s

where

feM) = l i m V^ ap Z(i3' Z(i4'

p5TO

A1i3i4) = l.i.m. V j Zf^-T,

1 p - / ' j4 j3^ j3 j '

j3 ,j4=0

T s _ / - X 2

-in f TO

p 1

if. . = —

3433 2

j (s)/ j (S1) E ( J j (s2)dsJ ds1ds. (2.264)

t t ji=p+M t

Let us consider A

(¿2«4)

A

(¿224)

= l.i.m. >

p5TO ^—'

j4,j3 ,j2 =0

A T

1

T

2 j4

t t t

2

si

j (sW j (s1M / j (S2)dsJ ds1ds-

T

2

-Ö / ^4(5) / feM /

2

s3

/

Z(i2 ) z(i4) = Z j2 Zj4

p

T

1

l.i.m. V - / fa4(s)(s-i) / fa2(si)dsids-p

= l.i.m.

p5TO ^—' I 2

j4j2=0 \ t

1

T

T

j(s) / j(S3)(s - t + t - S3)ds3ds I Zj2i2)Zj44)-

A2i2i4) + A1i2i4) + A3i2i4) = -A2i2i4) + A1i2i4) + A3i2i4) w. p. 1, (2.265)

where

^ = ^ E j*zf Z(

.m

p5TO

(i2)Z(i4) j4 ,

j4 ,j2=0

2

2

1

2

s

s

s

s

1

2

p

p

A3'2!4) = ipig. £ cr cjr'cir»,

j4 ,j2=0

T _ / s \ 2 s

1 /> TO

= i I <f>j4(s) J2 I hsMds! J faMdsrfs, (2.266)

2

t j3=p+1 \ t / t

T S _ / s \ 2

1 n n TO

= ^ J J feM E (/ dszds- (2.267)

2

t t j3=p+1 V3

Let us consider A5

(¿1*3)

A(

A5

p

l(«1«3) _

5 _

T T T T

-(i1M«3 )

piTO E J j (s3^ j (s2^ j j (s)dsds1 ds2ds31)

j4,j3,ji=0 t s3 s2 s1

p T Ts S1

E J j (s3) J j (s) J j (s0 y j (s2)ds21 dsds3Cj(i1)Cj;*3) _

p

l.i.m.

piTO

j4,j3,j1=01 s3 s3 s3

_ 1.i.m.

piTO ^-'

j4 ,j3 ,j1=0

p / T / T \ 2 T

p1

2 7 ^^ J J Mi'^dsdss-

t s3 s3

T T / s \ 2

y feM y <fe3(s) I y <fe4(si)rfsi )

t s3 s3

T T / T \ 2 \

^ I <feiM I <fe3M I I (f>j4(si)dsi ) gLs2gLs3

z(i1)z(i3) _

s3 s2

T T

p /1

/

= l.i.m. V - / 4>h(s3)(T - S3) / <f>j3(s)dsdsr

p

piTO ^—' I 2

j3,j1 =0 \ t s3

T T

y feM y (f)j3{s){s - s3)dsds3-

t s3

T T

1

4

where

2J feM J 4>33{s2){T - s2)ds2ds, ) cii^cjs3-

t s3

A4iii3) + A5iii3) + A6iii3) = -A4iii3) + A5iii3) + A6iii3) w. p. 1, (2.268)

p

a42i23) = l.i.m. y jzrzi,

4 p -TO / j3j^ji j '

j3 jj'i=0

p

A(iii3) = l i m V ep Z(ii)Z(i3)

A5 = lp5TO. ej3 ji Zji Zj3 '

j3 jj'i=0 p

Aii<" = ^ £ j*j)z!33).

j3 ,ji=0

T / T \ 2 T

CO

dPj3jl = I J faAsi) J2 ifhMds] J fa3(s)dsds:h (2.269)

t j4=p+1

s3

T T _ / s \ 2

1 " n TO

p1

^ ^ J feM J fa3(s) y (2.270)

t s3 j4=p+1 Vs3 /

T T oo / T \ 2

fjsji = \f J feM E / feM^l ds2dS3 =

t s3 j4=p+1 Vs2 /

1 T TO / T \ 2 s2

= 9 / feM y / fa4(si)dsi / (¡>h{sz)dszds2. (2.271)

2 { j4=p+1 \?2 / i

Moreover,

A3 - -5

(¿2 ¿3) + A(i2i3) =

p

= ^.m. (Cj4j3j2j4 + Cj4j3j4j2) j Zj3 =

j4,j3,j2 =0

p T s si si

= p^TO. E / 0j4 (s) / 0j3 W/ j (s2^ j (s3)ds3ds2ds1dsZj(:2)Zj(33) =

j4,j3,j2=0 t t t t

p T s1 s1 T

(«2^(i3)

E J jMy j (s2^ j M^^y j (s)dsds1Cj22)C

j4,j3,j2=0 t t t s1

p / T s1 T T

_ Ipi^TO. S ( J j M/ j M^ j M^^ j (s)dsds2ds1-

j4,j3,j2 =0 \ t t t s1

Z(i2)z(i3) _ Zj2 Zj3

T s1 / T \ 2 \

-J j j (s2 ) I J j (5)dsJ ds2ds1

t t s1

p T ? f p V^

l^TO E J j M J j (s2) (T - s1) - ^ | J j (53)ds3j

ds2ds1 x

\

j4=0

s1

(i2^/t(i3) _ oA(i2'3)

j3 ,j2 =0 t

xZj^C _ 2Ar; w. p. 1. Then

A3i2i3) _ 2A^3) - A5i2i3) _ A^3) - A5i2i3) + Ag2i3) w. p. 1. (2.272)

Let us consider A4

(¿1*4)

A (¿1*4)

T

A4

S / ^j'4 (s) / j M / j M^ j ^OdsA^jj _

j4,j3,j1=0 t t s3 s2

p 1 T s p X s

= E If / ^WE dszdsC^C^)

p TO j4,j1=0 t t j3=0 \s3

T

p 1 T

? 2 j j °, M{S ^

j4,j1 =0 t t

T s

= IJ J(s- s3)dwMdwM +

tt

2

s

1 p T s

E / (f)j4(s) J (f>j4{s3){s - s3)ds3ds - A

2

j4=0 t t

(¿i«4) 3

T s2 si

l [ [ [ dw^dSldw^ +

+ (E / (s-t)(f)j4(s) J (f)j4(s3)ds3ds-

ttt T

- ^ V. -1 -V. A / 1 r ? ■ X It ^

2

Vj'4=0 t

T s

Ts

E / j (s^(s3 - t)j(s()ds(d^ - A3iii4)

j4=0 t t )

T s2 si

i [ f f dw^dSldw^ - w. p. 1. (2.273)

ttt

Let us consider A6

(21¿2)

p

I (i1i2) _ 6=

T T T T

A( A6

(ii0(«2 )

= lp5m. E J j(s3^ j(s2^ jMy j(s)dsds1ds2ds3Zj(ii)C

j3,j2,j1=0 t s3 s2 s1

T T / T \ 2

= E 5 /few El/^3(^1 ds2ds^C^

j1,j2=0 t s3 j3=0 \s2 /

T T

p 1 T

'¿S- £ '-. J feW I - s,)ds2ds3cf<f > - A^'

j1 ,j2=0 t s3

p T s2

= E 2 j ^M(T-S2) J M^dssds^C^ ~ A ji,j2=0 t t T s2

I f(T-s2) i

s

T s2

00 „ 2

I )(T-s2) I (¡>h{sz)dszds2-

j2=0 '

t

T s1 s2 T

= \J 11 dw^dw^dSi + jl{il=i2^0} J(T- S2)ds2 - A<ilia) w. p. 1.

t t t t

(2.274)

Let us consider B2, B3

p T s / s1 \ 2

= lim E 9 / / / dsids =

piTO ^—' 2

j1 ,j4=0 t

Ts

p 1 ff p

j4=0 t t j4=0

T 1

-/(,, ,),/*, (2.275)

J j4=0

B _

P {1 T / *

JiS) S J (pj^dsidsi

\ t \t / t

j4,j3=0 \ t

T s1

s2

j (S1M j M / j (s3)dsU ds2ds1-

t t t T s1 / s1

\

j (S1M j (s) / j Mds2 dsds1

tt T s3

/

p 1 3 p p j3=0 { j3=0

T s1

^sM J {S2 - t)(t>n{s2)ds2dsl +

j3=0 t t

2

2

1

2

1

2

p p 1 T s/1 p ^ r

+ E ann - E 2 j ^3(81) J <j>j3(s)(s 1 -t + t- s)d.sdsi +

j3=0 j3=0 t t

p

j3=0

p

+ lim V^ j ■

p j3 j3

p p p

piro ,?3j3 pümc» ,?3j3 piimro ,?3j3 ( )

j3=0 j3=0 j3=0

Moreover,

p

B2 + B3 = l_im N (Cj3j4j3 j4 + Cj3j4j4 j3) = pyro '

j4 ,j3=°

p T s si si

^ Hm E J j (s) J j (s1 )y j (s2 )y j (s3 )ds3ds2ds1ds =

j4,j3=0 t t t t

p T si si T

^ lyro E J j(s!)/ j^2)/ j(s3)ds3ds^ j(s)dsds1 =

j4,j3=0 t t t s1

p / T si T T

Urn E ( J j (s1^ j (*)/ j (s2)ds^ j (s)dsds3ds1

j4,j3=0 \ t t t si

T si / T \ 2 \

-/ j(sO/ jM ( J j(s)ds I 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 / t * j4=0

TOTO f* f* r

-Ey j(s1)(T - s1 )J j(s()ds(ds1 + 2pl^roE /

p

2 ,l™£ /«4. (2'277>

p—>•TO z

j4=0

Therefore,

p p p p

pi to fj3j3 pi_mTO ' P3j3 pi-mTO P3j3 p-i^to ^ P3j3 ( )

j3=0 j3=0 j'3 =0 j3=0

After substituting the relations (2.263)-(2.278) into (2.262), we obtain

lim. Cj4 j3j2 j1 j j Zj3 zj4

piTO

jij2 ,j3 ,j4=0

T s s1

= + J J Jds2dwMdwM +

ttt

T s2 s1 T s1 s2

+il{<3=<3^0> JJJ dwMdstfwM + JJJ dw^dw^ds^ t t t t t t

T s1

+|1{h=^0}1{*3=^0} J J d.sod.si + i? = J*['0(4)]r,i+

tt

+ R w. p. 1, (2.279)

where

R _ — 1 ¿1 =0;A + 1 ¿1 =0; —A + A + A +

-{¿1=*2=0}A1 + 1{¿1=¿3=0} ^-a2 + A1 + A3

I 1 f A (^^i) A ^^ , A^A n A(«1«4) i

+1 {^ 1^^4=0} ^A4 - A5 + A6 ) - +

1 {¿2^¿4=0} I +A5 +A6 ) - 1{¿3=^4=0} -

p p p "1 {^ 1 ^=^3=0} 1 {¿2^¿4=0^ ^ p-TOE jj + p1iiTO E Cp3j3 - p1iiTO E bp3j3

j3=0 j3=0 j3=0

pp

"1 {^ 1 1 {¿2^¿3=0^ f 2 p}im E fj3j3 - P1im S apP3j3-

pi^-' -/3-/3 piTO

j3=0 j3=0

pp

1im V^ cp ■ + 1im V^ bp ■ +

piTO j3j3 piTO j3j3

j3=0 j3=0

p

+ 1 {21=29=0}1 {23=24=0} lim E ap373. (2.280)

73=0

From (2.279) and (2.280) it follows that Theorem 2.8 will be proved if

Aj = 0 w. p. 1, (2.281)

p p p p

p

a a A =

lim V ap ■ = lim y ■ ■ = lim y ■ ■ = lim y fp7 = 0, (2.282)

p—TO ' ^ ■7373 p—TO ' ^ ■7373 p—TO ' ^ ■7373 p—TO ' ^ ■73-73 73=0 73=0 73=0 73=0

where k = 1, 2,..., 6, i, j = 0,1,..., m.

Consider the w. p. 1. We have

Consider the case of Legendre polynomials. Let us prove that Af3M) = 0

2

p2

^ ( £ ajC>Z<

j3,j4 =0

p j3-1 / 2 -A p 2

y y 2ap ■ ap,., + (aj + 2ap ., ap,. + fap, +3V

j3j3 j3j3 j3j3 j3j3 j3j3 j3j3

j'3=0 j'3=A / j3=0

' p \ 2 p j3-1 2 p 2

£ j. + EE (ap3 j3 + ap3j^ + 2 £ (j (¿3 = ¿4 = 0),

j=0 J j3=0 j3 =0 j3=0

(2.283)

p

2

p

„p Z(i3)Z (i4)

Zj4

j3,j4 =0 / I j3,j4=0

M E jjZfCM = E (aj2 (i3 = ¿4, i3 = 0, i4 = 0),

(2.284)

M

£

j3,j4=°

ap Z(i3) Z(i4)

"j4 j3 C j3 C j4

(T - t)£ (ap4^ if ¿3 = 0, ¿4 = 0

j4=0

=

(T - t)£ (apj2 if ¿4 = 0, ¿3 = 0 .

j3=0

[ (T - t)2 (ap0)2 if ¿3 = ¿4 = 0

(2.285)

2

Let us consider the case ¿3 = ¿4 = 0

p _ (T-t)2y/(2j4 + 1)(2j3 + 1) -1- "32

1 y TO / yi

xj Pj4 (y)J Pj3 fe) £ (2j1 + 1) J Pji (y2)dy^ dy1 dy

-1 -1 j1=p+1 \-1

_ (r-^)2v/(2j4 + l)(2j3 + l)^

— -X.

32

1 TO 1

x ....... - 1 — " —2

Jpjsiyi) E (Pji+i(Vi) ~ Pji-i(yi))2 J Pji(y)dydyi =

(T-t)2y/WTT

-1 j1=p+1 yi

32V2j4 + 1

TO 1

x I Pn(yi) (Pu-i(yi) - PJi+i(yi)) E _1 j'i=p+1

if j4 = 0 and

32

_ (T-tfvw+T

(Jj ; ; — TT

1

1

X

1

PjsM^-yi) E -1 ji=p+1 j1 +

if j4 = 0.

From (2.116) and the estimate |Pj (y)| < 1, y £ [-1,1] we obtain

\PM\ = • y/Wy)\ < ,u/8. JGN.

V V (j ) / (1 - y2)1/8

(2.286)

Using (2.116) and (2.286), we get

1

1

l<4I + KI < Co E J / (T^TI ^ J Ü3 * 0), (2.288)

1

<ol + Kol <C0Y. y{1_dyf)lft<J (M> 1), (2.289)

j1=p+1 1 _i

where constants C0, Ci do not depend on p.

Taking into account (£283), (E2E7HE282]), we have

\ 2 ^ / \ 2

p \2| / P \ 2 P

" E «jjj _ U0 + E «j + E (°Sj3 + j>) +

j3,j4=0 / I V 33 = 1 J j3 = 1

p j3-i 2 / p 2

+ EE j + j +2 E (j + (aoc)M <

j3=i j3=i \j'3=i /

„ 11 1 V Ki p 1 / 1 1 \2

< k I - - f —\ — — V 1 <

~~ 0 I p p J X3/4I p p {jsf/2 ~

< K I + 4 ^ 2 + + [ jhLi <

~~ 0 p p3/4 y p p 1 J a;3/2 ~~

K4 K3 / 2 \ K5 < — + — 3--< —

P P \ VP/ p

if p ^ o (i3 = i4 = 0).

The same result for the cases (2.284), (2.285) also follows from the estimates (2.287)-(2.289). Therefore,

Aii3i4) = 0 w. p. 1. (2.290)

It is not difficult to see that the formulas

A2i2i4) = 0, a4;i;3) = 0, A6;ii3) = 0 w. p. 1 (2.291)

can be proved similarly with the proof of (2.2

Moreover, from the estimates (2.287)-(2.289) we obtain

p

limE ap3j3 = ° (2.292)

p^^ ' ■* J3J3

j3=0

The relations

p p

lim V bÇ ■ = ° and lim V j ■ = ° (2.293)

p j3 j3 p j3 j3

p^œ z—' J3J3 p^œ

j3 =0 j3=0

can a1so be proved ana1ogous1y with

Let us consider A3

(«2«4,

3

A^ = A^2^ + A62M) - A?2M) = -A?2M) w. p. 1, (2.294) where

p

a?2")=iî- E gp4j2

p^œ

j2 ,j4 =0

T s _ / T T

„ „ OO

gj = j (s) / j (si) E / j (s2)ds2 / j (S2)ds2 = { { ji=p+i Vsi s /

oo T T s T

= E / j(s) / j(s2)ds^ j (si) / j (s2)ds2dsids. (2.295)

ji=p+i t s t si

The last step in (2.295) follows from the estimate

0 i y

Note that

oo ( t T \ 2

4,4 = E M [ hM [ M(s2)dsods) , (2.296)

2

ji =p+1 \t s /

TT TT

OO „ nr.

gjj + gjU = E J j(s) J j (s2)ds2ds y j (s) j j (s2)ds2ds5 (2.297)

ji=p+1 t s t s

and

^ _ (r-t)V(2j4 + l)(2j2 + l)

16

1

4,2 =--x

TO 1 r

x E Ö7T1 / P^(2/i)№i-I(2/I)-^i+I(2/I))x

ji=p+1 j1 + -1

yi

x y pj2(y) (pji-1(y) - pji+1(y)) dydy1. J4, j2 < p. -1

Due to orthogonality of the Legendre polynomials we obtain

* , » _ (r-0V(2j4 + l)(2j2 + l)

TO 11

x E 27 +1 / ^4(2/1) (^-1(2/1)-^+1(2/1))^! X ji=p+1 j1 -1

1

X / Pj2(y)(Pji-1(y) - Pj-i+1(y)) dy =

1

(T - t)2(2p +1) 1

1 \ ^ | 1 if j2 = j4 = P

P2(y1)dy1

16 2p + 3 W ~p

-1 0 otherwise

xO .1 if j2 = j4 = P

(T"f)2 ' (2.298)

4(2p + 3)(2p + 1)

0 otherwise

, = (T-t)\2J4 + 1)

Vidi IQ

TO 1 1

12

x E ^7-7-9 /^4(2/1)№1-1(2/1)-^i+ife))¿yi =

, 2j1 + 1 2

ji=p+1 J1 \_1

(r-Q2(2p+l) 1 . . 2

32 2p + 3 1 I p

t 1 if j4 = P

Pp (y1)dy1 *

1 0 otherwise

(T - t)

8(2p + 3)(2p +1)

1 if j4 = P

0 otherwise

(2.299)

From (2.283), (2.298), and (2.299) it follows that

E gLzfzj:4)X ^ \j2,j:=0

2 p j'3-1 2 P

£4« + ££ j+j + 2£ j

j3=0 / j3=0 j3=0

+ 0+2

j3=o

(T -t)

(T -1)2

8(2p + 3)(2p +1)

8(2p + 3)(2p + 1)

0

if p ^ œ (i2 = ¿4 = 0).

Let us consider the case i2 = i4, i2 = 0, i4 = 0 (see (2.284)). It is not difficult to see that

T s

gp:j2 = J j (s) ^ j (s1 )Fp(s,5i)d5ids = J Kp(5,5i)0j:(s)j (sl)dsids

t t [t,T]2

is a coefficient of the double Fourier-Legendre series of the function

Kp(s,si) = 1{si<s}Fp (s,si), (2.300)

where

™ T T

oo ,, ,,

def

E I j(s2)ds2 / j(s2)ds2 = Fp(s,si).

ji=p+1

si s

The Parseval equality in this case looks as follows

T s

22

Pi n

J™, £ (gU)2 = J (k(-.si))

j4j2=0 [i)T ]2 From (2.116) we obtain

s, si )) ds1 ds = J j (Fp(s,s1))2 ds1ds. (2.301)

t t

T

si

Vsji + iVr^

Ksi)

2

2

1

1

2

IP^Msi)) - Pj^Ms^l <----7. (2.302)

2V2JT+T ' v v " Ji (1 - z2(Sl))1/4'

where z(si) is defined by (2.20), si e (t,T). From (2.302) we have

C2 1 1

From (2.303) it follows that |Fp(s, s1)| < M£/p in the domain

D£ = {(s,s1): s e [t + e,T — e], s1 e [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)d0 (2.304)

ji=0 s si ji=0 s si

at the set D£ if p ^ to.

Because of continuity of the function on the left-hand side of (2.304) we obtain continuity of the limit function on the right-hand side of (2.304) at the set D£.

Using this fact and (2.303), 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 J (1 - z2(si))1/2 (1 - z2(s))1/2

1 y

4 djä w, w, < (2-305)

where constant K does not depend on p. From (2.305) and (2.301) we get

p pi TO

,2 V^ i v \2 Ai

p15TO ^-' ^~j4j2' z-✓ ^~j4j2' — p2

j2,j4=0 j2,j4=0 j2,j4=0

0 < V G?L)2 < M m V (g>;tjy = V (g>;t]f < ± - 0 (2.306)

if p — to. The case i2 = i4, i2 = 0, i4 = 0 is proved. The same result for the cases

1) i2 = 0, i4 = 0,

2) i4 = 0, i2 = 0, 3) i2 = 0, i4 = 0

can also be obtained. Then A^2^ = 0 and a322m) = 0 w. p. 1.

Let us consider A5

(¿123)

A5i1i3) = A^1^ + A^1^ - Ag1^ w. p. 1,

where

p

^(iliз) = l i m v^ hp Z(«3) 8 pyro

j3,j'i =0

Agiliз) = l . i . m . y j zfl)z!гз),

8 / J j3jiV> ji j '

T T

hp3ji = J 071 (53^ 073 (5)Fp(53,s)d5d53.

t S3

Analogously, we obtain that a8'i23) = 0 w . p . 1. Here we consider the function

Kp(s3,s) = 1{s3<s}Fp(s3,s)

and the relation

hpp371 = J Kp(s3,s)07i(S3)073(s)dsds3 [t,T ]2

for the case ii = i3, ii = 0, i3 = 0.

For the case i1 = i3 = 0 we use (see (2.

= \ ij <f>h(s) J fa4{si)dsid.s

j4=p+1 \ t s

2

oo

T

T

T

T

ji + hpij3 = E J j (s) / j (s2)ds2dsy j (s) J j (s2)ds2ds.

j4=p+1 t s t s

Let us prove that

lim V^ cP =0.

1-^fY! ' ^ j3j3

j3=0

We have

Moreover,

cp = f P i dP —

j3 j3 f j3j3 + j3j3 gj3j3.

!»mE fj = 0 i^E j* = °.

p—>•oo z

j3 =0

p—>•oo

j3=0

(2.307)

(2.308)

(2.309)

where the first equality in (2.309) has been proved earlier. Analogously, we can prove the second equality in (2.309).

From (2.299) we obtain

0 < lim V gj 3 < lim

33

(T - t)

p—>co *

j3=0

8(2p + 3)(2p + 1)

= 0.

So, (2.307) is proved. The relations (2.281),

are proved for the

polynomial case. Theorem 2.8 is proved for the case of Legendre polynomials. Let us consider the trigonometric case. According to (2.264), we have

T / s/ \ 2 T

t ) si

Moreover (see (2.229), (2.230)),

si T

aij3 = \ J Mi81) E i J MMdso] J (f)M{s)dsdsi. (2.310)

(s2)ds2

<

(s2)ds2

si

<

K

(2.311)

where constant K does not depend on j (j = 1, 2,...). Note that

TT

T — s1

s)ds =

y/T^t'

Si

p

p

Using (2.310) and (2.311), we obtain

c^ TO 1 c^ c^

I j4 ^=p+1 Jl Pj4 ' ^' P

Taking into account (I2^31)-(I22851) and (2312), we obtain that Af3^ = 0

where constant C1 does not depend on p.

(2 31 2) we obtain A1

w. p. 1. Analogously, we get A^2^ = 0, A^1^ = 0, A[;ii3) = 0 w. p. 1 and

p p p

lim E aL'3 = 0, lim E ^ = 0, lim E fpis = 0.

p—TO ' * j^3 p—TO ' * j^3 p—TO ' * j^3

73=0 73=0 j3=0

Let us consider a322m) for the case i2 = i4 = 0. For the values gm + g22m4 and g^-1 + gi^"1 (m G N) we have (see (2297))

oo T T T T

g^ + g^m = E / 074 (sM 07i (s2 )ds2ds / 072 (s) / 07i (s2)ds2ds = 71 =2m+1 { S i s

oo ( T T T T

= E / 074 (s) / 02r-1(s2)ds2dW 072 (s) / 02r-1 (si)dsids+

r=m+^ is is

T T T T

+ J 074 (s) y 02r (s2072 (s) / 02r (s2^ds | , (2.313)

i s i

oo T T TT

j + g^-1 = E / j (sW j (s2)ds2dW j (sW j (s2)ds2ds =

j'i=2m s { s

TT TT

= jm + j4 + f j (s) J 02m(s2)ds2d^ j (s^ 02m (s2^ds, (2.314)

t s t s

where

T T _ T T

2

0J4(.S) y (j)2r—i[so)dsods = y 0j4(s) y sin27ry^ =

t

yfty/T=t f ( 27rr(s -1) \

- / <P-l s cos——--1 as,

2tt r I \ T-t / '

T T _ T T

2

(f)j4{.s) j (f)2r{s2)ds2ds = J <fe4(s) J ^d.sod.s

t s

y/2y/T=t } , ( J . 2tvr{s-t)\,

- / <p-u{s)\ —sin——- as,

2tt r / J V 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 r-1 if j4 = 0

J (j)jA(s) J (f>2r—i{so)dsods = V2(2^ ~ • i , (2.315)

t s 0 otherwise

TT TT

J j (s) J 02r(s2)ds2ds = 0, (2.316)

ts

where 2r — 1, 2r > p + 1, and j4 = 0,1,..., p. From (2.313), (2.315), and (2.316) we obtain

g2m + g2m = y^

j72 1 ¿j2j4 / v

)2 | 1 if j2 = j4 = 0

(T -1)2

ji=m+1 0 otherwise

CO (T - t)2 I1 if j4 = 0

^2m 1 / „2m , „2m \

2 i^2 yjiji) 4K2 j-2

j2=j4 ji=m+1 1 I 0 otherwise

Therefore (see (2.25)),

j + gjS I < K1/(2m) if j2 = j4 = 0

, (2.317)

g|m2 + gj2m4 = 0 otherwise

g24™41 < K1/(2m) if j4 = 0

j4 j4

(2.318)

g2m = 0

otherwise

where constant K1 does not depend on p = 2m.

For p = 2m — 1 from (2.314) and (2.316) we have

1 or 0 if J2 = J4 = 0

g2m-1 + g2m-1 g j4 j2 + g j

'.72.74

E

ji=m+1

(T-i)2

27T2J2

0

(2.319)

otherwise

The relation (2.319) implies that

2m—1 _ i 2 m—1 , 2m-l\

"hh ~ O yyjm gi2j4 /

y, (T-t? . . ~ Z^ 47T2 /12

j2=j4 ji=m+1 J1

1 or 0 if j4 = 0

0 otherwise

(2.320)

Using (2.319) and (2.320), we obtain

ig2m-1 + g ij/in 1 y

.74.72 ^J2J4

,2m-1 , „Mi < K2/(2m - 1) if j2 = j4 = 0 2m-1 2m-1 = 0

(2.321)

g., n + g.

j4 j2 ¿j2j4

ig

otherwise

j4 j4

2m-1| < K2/(2m - 1) if j4 = 0 g2m-1 = 0

(2.322)

otherwise

where constant K2 does not depend on p = 2m — 1.

The relations (2.317), (2.318), (2.321), and (2.322) imply the following formulas

gj + gj1 < K3/P if j2 = j4 = 0

gpj4 i < K3/P if j4 = 0

gj + j j = 0

otherwise g7p 7 = 0 otherwise

(2.323)

where constant K3 does not depend on p (p G N). Moreover, gp -4 > 0 (see

(2296)). 44

From (2.283) and (2.323) it follows that = 0 and Aj;2i:) = 0 w. p. 1

for i2 = i4 = 0. Analogously to the polynomial case, we obtain A7i2i:) = 0 and A^i2i:) = 0 w. p. 1 for i2 = i4, i2 = 0, i4 = 0. The similar arguments prove that A5i1i3) = 0 w. p. 1.

Taking into account (2.308), (2.323) and the relations

pp

p-i-moo fj3j3 p—moo p3j3 ,

j3=0 j3=0

which follow from the estimates

l/J3l<|, 141 < | 0V0), l/ool < y, Kol<y, (2.324) we obtain

lim E Cp3j3 = - lim E gp3 j3,

p—o ' * j^3 p—o ' * j^3

j3=0 j3=0

p K

0< lim V//;„,„ < lim —= 0.

p—œ z—' •/3-/3 p—œ p j3=0

Note that the estimates (2.324) can be obtained by analogy with (2.312); constant Ci in (2.324) has the same meaning as constant Ci in (2.312).

Finally, we have

p

lim y cp7- =0.

j3=0

The relations (2.281), (2.282) are proved for the trigonometric case. Theorem 2.8 is proved for the trigonometric case. Theorem 2.8 is proved.

Remark 2.2. It should be noted that the proof of Theorem 2.8 can be somewhat simplified. More precisely, instead of (2283)-(2285), we can use only one and rather simple estimate.

We have

2

p \ 2

^ ( E ap4j3jj

j3 =0

P

E app4j3 ( jj - 1{i3 = i4=0}1{j3=j4} + 1{i3 = i4=0}1{j3=j4} j3 j4 =0 ^

2

\ E ap4j3 ( jj} - 1{i3 = i4=0}l{j3=j4H + W} £ ap4j4 ( I Vj3 j4 =0 J j4=0 / J

= M I (, X 0 j jj - 1{i3=!4=0!1{j3=j,^ I +

0

2

p

+ ^^(E app4j4 ) • (2.325)

j4=0

The expression

EP I /u(i3^(«4) -I -I \

0/473 I Zj3 zj4 — 1{i3 = i4=0}1{j3=j4} ) j3,j4=0 V J

can be interpreted as the multiple Wiener stochastic integral (1.245) (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{ J < Ck E J..J ^2(t1,...,tk )dt1 ...dtk =

(t i ,---,tfe ) t t

= Ck y $2(t1,...,tk)dt1 ...dtk,

[t,T ]k

where J/[$]Tk] is defined by (11 .'2,3) and Ck is a constant. Then

M E (Cj33)Cj44) - 1{.3=.,=0}1{j3=j4!) I > <

j3j4=0

2

/ p \ 2 p < C2 y E app4j3j(ts)0j4Ml dt3dt4 = C2 E (aj2 . (2.326)

U T12 V?3 j4=0 / j3 j4=0

2

From (2.325) and (2.326) we get |/p \ 2 | p / p x 2 M £ '4,3zi33)ci44M <Ci £ kj2 +1{,3=.,=0^«L

^ \P3,P4=0 / J 7374=0 V?4=0

(2.327)

Obviously, the estimate (2.327) can be used in the proof of Theorem 2.8 instead of (2.283)

The estimate (2.327) can be refined. Using (1.85), we obtain

M I (. £ 0 ^ ( Zj33) j* - 1{,з=,:=0}1(jз=j:^ j =

pp

= E j,) 2 + E ap4j3jj <

j3,j4=0 j3,j4=0

p 1 p < E • Z:-«,^ E ((«'jf • ("j:jf

2

j3,j4=0 j3 ,j4=0

p

2

(1 + l^M,,}) £ j. (2.328)

j3,j4 =0

Combining (2.325) and (2.328), we have

{/ p \ 2 | p £ «jzj33)jM < (1 +1^}) £ («j2+

Vj3 ,j4=0 / J j3,j4=0

/ p \2

+ ^U^} £ j . (2.329)

2.4 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 5 Based on Theorem 1.1. The Case pi = ... = p5 5 to and Constant Weight Functions (The Cases of Legendre Polynomials and Trigonometric Functions)

This section is devoted to the construction of expansion of iterated Stratonovich stochastic integrals of fifth multiplicity based on Theorem 1.1. The mentioned

expansion converges in the mean-square sense and contains only one operation of the limit transition. As we saw in the previous sections, the expansions of iterated Stratonovich stochastic integrals turned out much simpler than the corresponding expansions of iterated Itô stochastic integrals (see Theorem 1.1). We use the expansions of the latter as a tool for the proof of expansions for the iterated Stratonovich stochastic integrals.

The following theorem adapt Theorem 1.1 for the iterated Stratonovich stochastic integrals (2.6) (^(s) = 1, i/ = 0,1,..., m, l = 1,..., 5).

Theorem 2.9 [14], [15], [36]. Suppose that [^(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 fifth multiplicity

J*[.^>]t,( = j j J J j dw<;')dwi;2)dw<33)dwi:4)dwi:5)

t t t t t

the following expansion

p

J*^(5)]r.< = '¿S- E C' CCC C

jl j2 j3 j4 j5 =0

that converges in the mean-square sense is valid, where

T ¿5 ¿4 t3 ¿2

Cc2jl = J j (t5)J C fa) J C ^ C y C (t1)dt1dt2dt3dt4dt5,

t t t t t

ii, i2, i3, i4, i5 = 0,1,..., m, and

T

j = j (s)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. Note that we omit some details of the following proof, which can be

done by analogy with the proof of Theorem 2.8. Let us denote

p

Al = liil^lim. E C4j3jljl C C C ,

jl C3 C4 C5 =0

p

A = £ Cj5j4j2j2ji Zj(1il)Zj(:4)Zj(5iS),

pyro

j'l,j2,j'4,j5=0

p

A = 1 {13=14=0} LLm. E Cj5j3j3j2ji Zji1 Zj2l2 Zj5l5 ,

pyro

j'l,j2,j3,j5=0

p

A4 = ^^^aLm. E Cj4j4j3j2ji Zj(ll)Zj(2l2)Zj(313),

pyro

j'l,j2,j3,j4=0

p

A5 = ^li^^Lim. E Cj5j4jij2ji Zj(2l2)Zj(4l4)Zj(5l5),

j'l,j2,j4,j5=0

p

A = ^ll^lim. £ Cj5jij3j2ji Zj(2l2)Zj(3l3)Zj(5l5),

j'l,j2,j3,j5=0

p

A7 = ^ll^fe £ Cjij4j3j2ji Zj(2l2)Zj(3l3)Zj(4l4),

pyro

j'l,j2,j3,j4=0

p

Ag = 1{l2=l4=0}l?;i;m. £ Cj5j2j3j2ji Zjili Zj3l3 Zj5l5 ,

pyro

j'l ,j2,j3,j5=0

p

A = ^^LLm. £ Cj2j4j3j2ji Zjili Zj3l3 Zj44 ,

pyro

j'l ,j2,j3,j4=0

p

A10 = ^^^ILm. £ Cj3j4j3j2ji Zj(lll)Zj(2l2)Zj(44),

ji,j2,j3,j4=0

p

B1 = 1 {11 ^¿2 =0} 1 {¿3^l4=0} l.y ^ Cj5j3j3jiji j ,

ji,j3,j5=0

p

B2 = 1{i2=13=0}1{I4 = 15=0}1 . i-m. £ Cj4j4j2j2ji j^,

pyro z-'

ji,j2,j4=0

P

B3 = 1 {11 ^¿2 =0} 1 { ¿4^15=0} ^ Cj4j4j3jiji Zj33),

ji,j3,j4=0 p

B4 = 1{l1=l2=0}1{l3 = l5=0}lj;i;m^ £ Cj3j4j3jiji Zj44 ,

ji,j3,j4=0

p

B5 = I(il=i3=0}l{i2 = »4=0}l'.1.1g. y CCi.C j ,

jl j2 j5=0 p

B6 = I{il=i3=0}l{i2 = i5=0}lp.1l>m^ £ CCC j ,

il,j2,j4=0 p

B7 = I{il=i3=0}l{i4 = i5=0}lp.1.m. y CClj2jl j ,

il,j2,j4=0 p

B8 = l{»l=»4=0}1{»2 = »3=0}lp.1.m^ E Cj5jlj2j2jl C ,

jlj2j5=0 p

B9 = l{H=»4=0}l{»2 = »5=0}lpiym- E Cj2jlj3j2jl j ,

jl,j2,j3=0 p

B10 = I{il = i4=0}l{i3=i5=0}l.1.m. E CCC'i Cj(22),

jl j2 ,j3=0 p

B11 = l{»l = »5=0}1{»2 = »3=0}ll.1.m^ E j ,

jl j2 ,j4=0 p

B12 = l{»l = »5=0}l{»2 = »4=0}lp.1-m^ E CCC Cj(33),

jl,j2 ,j3=0 p

B13 = I{il = i5=0}l{i3 = i4=0}l.1.m. E CJCC'i Cj(22),

jl,j2 ,j3=0 p

B14 = l{t2=M=0}l{»3=»5=0}lpj1m- E CCC CjI1 ,

jl j2 j3=0 p

B15 = 1{i2 = »5=0}1{»3 = »4=0}lp-i-]g. E CJCC'I Cjl),

jl j2 j3=0

where 1A is the indicator of the set A.

From (1.48) for the case p1 = .. .p5 = p and ^1(s),... ,^5(s) = 1 we obtain

10 15 p

J+ £ A - £ Bi = lpLm. £ j j ZfZfC},

(2.330)

= i.i.m.

i=1 i = 1 jl,j2,j3 ,j4,j5=0

where J [0(5)]T,t is defined by (2.7) for ^1(s),..., ^5(s) = 1 and i1,...,i5 0,1,..., m.

Using the method of the proof of Theorems 2.1, 2.7, and 2.8, we obtain

p

A = 1{i1=i2=0}lp.——TO. £ CP5P4P3PlPl C^ Ci^C]^ =

^i J3J4J5 =0

Mn^o}^^- £ J Ol:(l:,) J (f)j4{t4) J OjJIA e j J (f>j1{to)dto j x

j3j4j5=0 t t t ji=0 \ t /

xdt3dt4 dt5zj13)zj44) zj5) = 1 p T / / TO /r \2

iin^O}^1;1;™- £ / bXh) / <fe4(*4) / 0;,(/:{) E / <f>j1 (to)dto X

' ^ j3 j4j5=0 { { { ji=0 \ t J

xdt3dt4 dt5z(;3)z!:4) zC5) =

1p

j3 j4 j5

T t5 t4

£ j OjAh,) j <j>j4{t 4) j o ¡.J/:>,){ I :>, l)(ll->(ll \<H:>X

j3,j4,j5=0 t t t

: T ^ t5 ^ t4

, , , I

j3 j4 j5 2

tit T t5 t4 t3

xcfcfc!:5' = [ ( [ (t, - Orfw«3'=

= il{il=il/0} 11J J dt1d^cb*<ifdw£> + tttt

T t5 t3

+ 41{n=i2^o}l{i3=i4^o} J J J dt1dt3dwif+

ttt T t5 t3

+^1{n=^o}l{*4=^o} J J J dtidw{tfdt5 w. p. 1,

ttt

p

A = ^^^piTO. E Cj5j4j2j2jl Zj(l1l)Zj(4l4)Zj(5l5) =

j'l j2 j4 j5=0

- !{i2=i3=0}X

P T ¿5 / ¿4 ¿4 ¿4 \

X^ E / j (ts) ^ j Ml J j (ti) ^ j M ^ j (ta)dt3dt2 dti x

j1j2j4,j5=0 t t \t ti t2 /

xdMt5 jjj1 -

- !{i2 = i3=0}X

P T ¿5 ¿4 p / ¿4

E //^(¿4) J<f>h(ti) £ /^2)^2 ) dilX

j1j4j5=0 t t t j2=0 \tl

x dt4dtEiciri 1 zj441 c.i551 -

- 1{«2 = «3=0} X

1 p T ¿5 r ~ V

X21ÄIS' £ / I ^^ / / ^(¿2^2 dilX

j1j4j5=0 t t t j2=0 ^ /

xaMtjjj -

P T ¿5 ¿4

l{i2=^o}^m. J (j>j5(t5) J (f)j4{t4) J ^(t^iU-t^dtiX

j1j4j5=0 t t t

1P

jl j j T ¿5 ¿4

= E J Mfa) J <t>jA(U)(U-t) j ^(t^dtix

j1,j4,j5 =0 t t t

x^jj )cj;5,+

p T ¿5 ¿4

+l{i2=^0}^i;m. £ J MXh) J <f>j4(t4) J ^(t^it-t^dtix

j1,j4,j5=0 t t t

x^, cj:l)cj44,cí;;, -

^ T ^ t5 ^ t4

t t t

;T ^ t5 ^ t4

2

t t t

T t5 t4

t t t

T t5 tt

1 T t5

tt

T t5 t4 2ttt

T t5

T t4

1 , /mir, ,

tt

tt

T t5 t4

t t t

T t5 t4

ttt

T t5

T ¿4

J J{t-ti)dw^dU =

tt T ¿5 ¿4 ¿2

= i{*2=^<4 J J J J dwi;1 Wwi;4W£6)+

t t t t

T t5

J (h-t) J dw^ dt5+

tt T ¿4

J J(t-t^dw^dU w. p. 1,

tt

p

A - 1{:3=:4=0}lj.i;m. £ Cj5j3j3j2jl : 11 Cj2:21 Cj5:51 -

jl j2 J3,j5 =0

T

p T

l1^. . E / j (t5) x

1{i3=i4=0}1

p—

jl j2 ,j3 ,j5=0 t

¿5 ¿5 ¿5 ¿5

x ( / ^ (ti^ ^ (t2)/ j j (t4)dt4dt3dt2dt J dts jj CjC5:51 -

\t t1 t2 t3 /

- 1{l3 = l4=0}x

£ / f foifa) J ^2(^2) £ iy* ^3(^3)^3) dt2dtix

j1,j2,j5=0 t t t1 j3=0 \t2 /

xdts jjcf -

- 1{l3 = l4=0}x

£ / [ Mfa) [ Mh ( Uj3(t,)dtA dt2dtxx

p ^ jlj2j5=0 { { tl j3=0 \t2 /

x-jjj =

p T t5 t5

Ife^oj^m. E J (f>j5(t5) J (f)n(ti) J <t>j2{t2){th-t2)dt2dtix

jij2 j5=0 t t t1

xdt5 j1 jj0 =

1p

jl j2 j5

T t5 t5

ife^oj^m. E j <j>js(t5) j fa.iti) j <j>h(t2)(t-t2)dt2dt1x

j1 j2 j5=0 t t t1

x dt5cj 11} zj22) z£;5 )+

1p

jl j j T t5 t5

£ j fcWft-i) j (t>n(ti) j (f)n(t2)dt2dtix

jij2 j5=0 t t t1

xdt5 jjcf =

p T t5 t2

= E J (j>j5(t5) J <f>j2(t2)(t-t2) J ^(t^dtidhx

j1 j2 j5=0 t t t

xdt5zjl1,zj2:'zjr'+

p T t5 t:

E J (¡)3rXh){h - t) J (f)j2(t2) J ^(t^dtidhx

ji j2 j5=0 t t t

xdt5 Z^Zjj =

c T ^ t5 ¡j, t2

1

t t t

c T ^ t5 >jc t2

+ }\[ (h-t)[ f dw^dw^dw™ =

t t t T t5 t:

tt

1 T t5

+ (t - ti)dtidw{tl5) +

tt T t5

J(t~h) J dw^dhA tt T ¿5 t2

1 ' ' in) 7 (i2) 1 Us)

+I{i3=u*»2 {h~t] I I d< d< d< +

tt T t5

I(h~t) I dtidw^A

tt T t5

J (h-t) J dw^dh tt

T t5 t2

I I

t t t

T t5 t2

{l3=l4=0^ /(t5 - t2)J dwt( l^W^

T t5

+ J j[t ~ t^dtidw^ +

tt

1 T t5

+^1{H=*2^0}l{i3=M^0} J (h-t) J dtidw^ =

tt T ¿5 ¿3 ¿2

= /7/7 dw^Wi;2Wwt(:s)+

tttt

T t5

tt

T t5

+i 1 {i!=¿2^0} 1 {¿3=i4^o> J (h-t) J dtidw^ w. p. 1,

tt

p

A = £ Cj4j4j3j2ji Zjl) Zj(:2)Zj3) =

pyro

jl j2,j3,j4 =0

= 1{l4 = 15=0} x TTTTT

xlj.iym^ £ /0ji 0j2 (t2^ 0j3 (*)/ 0j4 (t4^ 0j4 (t5)dt5dt4dt3dt2x

jl,j2,j3,j4=0 t t1 t2 t3 t4

xdt1 «j =

= 1{l4 = 15=0} x

p T T T p / T \ 2

£ / fe^1) / ^fo) /£ jy gM^X

j1 j2 j3=0 t t1 t2 j4=0 \t3 /

x^c^cr cj33) =

= 1{l 4=15=0}x p T T T TO / T

£ / fe^1) / / ^(¿3) £ I J )

j1 j2 j3=0 t t1 t2 j4=0 \t3

x^^cf cf =

p T T T

E J (f)n(ti) J <f>h(t2) J (pn(h)(T -t3)dt3dt2x

j1 j2j3=0 t t1 t2

xdt1 «j =

* T * t3 * t2

= iiu=,^o}y (T-H)J J <iw<:'»rfw«f<iw«:3) =

t t t

T t3 t:

1 I > I I - r ( ¿O , ( 1 2 ) d-TST( 13)

l{.4^0i2 / (r - is) I I dw{;:i1'dw™dw™+

t t t

+71{m=^o}1{h=^o} [ (T-h) f dtidw{tl3) +

T is

T t2

t t

+71{M=*5^0}l{i2=^0} [ (T-tz) [ dw^dtz

tt

T t4 ts t2

1 l{i4=i5^0} /7/7 dw^W^W^W

2

tttt

T is

J(T-h) J cMw£3) +

tt T is

+ J (T-t 3) J dwi^dh w. p. 1,

tt

A5 = Ao = A7 = Ag = A9 = A10 = 0 w. p. 1,

p

B1 = l{n = »2=0}l{»3=»4=0}lpi-;m- £ Cj5jSjSj1j1 j =

j1 JS,j5=0

= 1{i1 = i2=0}1{iS = «4=0}X

£ j <fes(*s) f 4>n{u) f (f>js(U) £ ( [Mti)dh) dt:ix P ^ jSJ5=° { { { j1=0 V t )

Xdt4 dtj =

= 1{i1=i2=0}1{iS = i4=0}X

£ j <fes(*s) f 4>n{u) f (f>js(U) J2 ( [ M^dti J di3x

P ^ jS,j5=° { { { j1=0 V t /

xdt4dt5Zj5) =

= 1{l 1=12=0} 1{ 13 = 14=0} x

p T t5 p ^

x2li;™' £ / / E^W / (f>j3(U)(U-t)dt3dUdt5C

piTO j5=0 { t j3=0 {

= 1{l 1=12=0} 1 {l3 = l4=0}x

1 p T ^ TO t4

S I I I hMiu-QdududhC^

j5=0 t t j3=0 t

1 p T t5

y J (f>j5(t5) J(¿4 - t)dUdt5(^

j5 =0 t t

tt

1 T t5 t:

71{ii=i2^o}l{i3=M^o} / / / dtidt2dw^ w. p. 1,

4

ttt

p

B2 = 1 {¿2^¿3=0} 1 {¿4^¿5 =0}I*."^ Cj4j4j2j2ji Zj(l1l) =

j'l,j2,j4=0

= 1{l2 = l3=0}1{l4 = l5=0}x p T T T T T

xlp5m- S /0ji j (t*)/ 0j2 (*)/ j (t4^ j(t5)dt5dt4dt3x

jl j2 j4=0 t ti t: t3 t4

x dt 2 dt 1 Zj? 11) =

= 1{l2 = l3=0}1{l4=l5=0}x p T T T p / T \ 2

j1 ,j2=0 t ti t2 j4=^t3 /

xdt2 dt1Zjl) =

= 1{i2=iS=0}1{i4 = i5=0}X

p T T T oo/T \2

£ /^^/J/

j1 ,j2=0 t t1 t2 j4=^ts /

Xdt2 dtiZji1) =

1p

= 1{i2=iS=0}1{i4 = i5=0}X

T T T

£ / / / - t^dhdhdt^

j1'j2 =0 t t1 t2

= 1{i2=iS=0}1{i4 = i5=0}X

TT T

1 p ^ ^ p

£ / / i^Mfa) J 4>n{h){T -t^dhdhdt^ =0 t t1 j2=0 t2

= 1{i2=iS=0}1{i4 = i5=0}X

TT T

1 p i* i* i*

(»1)

=0 t t1 j2=0 t2

1/ir,=;„=m 1

p T T

jl^m. £ J ^(¿1) J(T - t2)dt2dt1C

^ I (pnyt 1) I [1 - i2)(u2

j1=01 t1

T T

1 /* --- , , (n)

(i1)

1{i2=i3^0}l{i4=is^0}^ I I (T - t2)dt2dwti

t t1

1 T t5 t2

71{i2=i3^o}l{M=i5^o} / / / dw^dt2dt5 w. p. 1,

4

ttt

p

B3 = 1{i1 = i2=0}1{i4=i5=0}lpLm. £ Cj4j4jSj1j1 CjSS)

j1jS,j4=0

1{i1 = i2=0}1{i4 = i5=0}X

p T t5 t4 p / t3

S / / 0j/ y J (t>3l{ti)dt1 dhx

j3'j4 =0 t t t ji=0 \ t

xdt4 dt5Zj(33)

= 1{li=l2=0}1{l4 = l5=0}x

S / ^(¿5) J (f>u(U) j (f>js(t:i)J2 [J& 1(^)^1]

j3'j4 =0 t t t ji=0 \ t /

xdt4 dt5Z((!3)

1p

j3

T t5 t4

^n^oil^^l^m. X] J j (f>u(U) j (f>js(t:i)(t:i-t)dt:iX

j3,j4=0 t t t

xdt4dt5Z7(T)

1p

j3

T TT

y j (f>j3(U)(U-t) j (f>ü(U) j (¡)H{h)dhx

j3 j4=0 t t3 t4

xdt4dt3Z7(T)

j3

2

(i3)

p T P / T \

p TO j3=0 t j4=0 \t3 /

73=0 t 74=0 /

p TO j3=0 t j4=0 \t3 /

1p

/3=01 j4=0

T

1{ii=i2^o}7l{i4=is^o}l-i;m. y / (f>h(U)(U-t)(T - h)dt3Ql

p TO j3=0 {

T

I (h -t)(T- t^dw^f

Then

where

T t5 Í3

= il{il=Í2?¿o}l{¿4=¿5^o} J J J dtidw^dh w. p. L

t t t

B4 = B5 = B = B7 = B8 = B9 = B10 = B11 = B12 = B13 = B14 = B15 = 0 w. p. 1.

10 15

J[^(5)]T,t + £ Ai B¿ = J[^(5)]T,t+

¿=1 i=1

T t5 t4 t3

J J JI ¿Mwí:3Wí;4Wt(:5)+ tttt T t5 t4 t2

1111 dwMdt2dw™dw™ +

tttt T t5 t3 t2

1111 dw^dw^dUdw^h

tttt T t4 t3 t2

1111 dw^dw^dw^dt^

tttt

+ S1 + S2 + S3 w. p. 1, (2.331)

T t5 t3

Si =-1{¿1=¿2^o}1{¿3=M^O} J J J dtidtzdw^ +

ttt

1 T t5

tt

T t5

+ 41{h=í27¿o}1{í3=m^o} Í (t5-t) I dtiáw^f-

tt

T t5 t2

31

{n = i2=0}l{i3=i4=0}

t t t

1

T

t5

= -l{i2=i3^0}l{i4=^0} / (¿5 dwh dt5 +

1

1

tt T t4

J J(t - t1)dwt(il)dt4+ tt

T t3

/(T - is) / dwt(;i)dt3-

1

tt T t5 t2

^l{i2=i3^0}l{i4=i5^0} J J J /1

ttt

dw(;i)di2dt5,

S3 - -l{n=i2^0}l{i4=i5^0}

T t5 t3

dt1dwt(;3)dt5+

1

ttt t t3

+^1{n=i2^0}l{i4=i5^0} / {T-h) / dMwi3

(i3)

tt T t5 t3

_71{ii=»2^0}l{i4=is^0} III dtidw{tfdt5.

4

ttt

Usage of the theorem on integration order replacement for iterated Ito stochastic integrals (see Chapter 3) [1]-[15], [68], [105], [106] leads to

1

Si - |l{ii=i2^o}l{i3=i4^o}

1

So - -l{t2=t3^()}l{t4=t5^0}

T t5 t2

if 1

ttt

T t5 t2

/7 J

ttt

dtidt2dwt(;5) w. p. 1, (2.332)

dwt(il)dt2dt5 w. p. 1, (2.333)

T t5 ts

S3 = -l{il=i2^o}l{i4=is^o} III dtidw{tfdt5 w. p. 1. (2.334)

t t t

Let us substitute (2332H2333) into (2331)

10 15

J[^(5)]r,t + £ Ai Bi = J[^(5)]T,t+

¿=1 i=i

T t5 t4 ts ////

tttt T t5 t4 t2

+^{«=«/0} J J J J +

tttt T t5 ts t2

J J J J dw|;i)dw|;3)di3dw|;6)+ tttt

T t4 ts t2

/7/7 dwi;i}dwi;3)dw

tttt

T t5 t2

+ 41{n=i2^o}l{i3=M^o} J J J dtidtodw^ +

ttt T t5 t2

+^1{*2=*3^0}l{i4=^0} J J J dw^dtodt5+

ttt T t5 ts

+ |1{ii=M0}l{i4=is^0} J J J dtidw{tfdh w. p. 1. (2.335)

ttt

According to Theorem 2.12 (see Sect. 2.5.2) for the case k = 5, the right-hand side of (2.335) is equal w. p. 1 to the following iterated Stratonovich stochastic integral of fifth multiplicity

* T * t5 * t4 * tS * t2

J JIJ J dwii'Wifdwis^w';«,

t t t t t

where ¿1, i2, ¿3, ¿4, ¿5 = 0,1,..., m.

From the other hand, the left-hand side of (2.335) is represented (according

to (2.330)) as the following mean-square limit

p

l i m Y^ c z(il)z(i2V(i3V(i4V(is)

Jp:;m' Cj5j4j3j2j1 Sji j j j j '

jl,j2 ,j3,j4,j5=0

Thus, the following expansion

J dw^dw^dw^dw^dw^ =

t t t t t p

= l i m C Z ( )Z ( i2)z ( i3) z ( i4) Z ( i5)

= lpl4m' Cj5j4j3j2j1 zji zj2 j zj4 zj5

j'lj2 j3j4j5=0

is proved, where i1, i2, ¿3, ¿4, ¿5 = 0,1,..., m. Theorem 2.9 is proved.

2.5 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.

2.5.1 Introduction

The idea of representing of iterated 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 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 [67] (1997), [68] (1998) (also see [5]-[15], [32]). 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.

2.5.2 Theorem on Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k E N)

Consider the following iterated Stratonovich and Ito stochastic integrals

*t *t2

J*[^(k) ]T,t = / ^k (tk) ...J (ti)dwt(;i)... (2.336)

t t T t2

J [^(k)]T,t = / ^k (tk) ..J ^i(ti)dwt;i)... dwtik), (2.337)

tt

where every ^(t) (l = 1,..., k) is a continuous nonrandom function on [t, T], w[;) = fi;) for i = 1,..., m and wT0) = t, ii,..., ik = 0,1,..., m.

Let us denote as (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 [101] (see Sect. 2.1.1).

Define the following function on the hypercube [t,T]k

|^i(ti) ...^k (tk), ti <...<tk k k—i

= n ^(ti) II 1{ti<ti+1}

0, otherwise l=i l=i

(2.338)

for ti ,...,tk E [t,T ] (k > 2) and K (ti) = ^i(ti) for ti E [t,T], where 1A denotes the indicator of the set A.

Let us formulate the following theorem.

Theorem 2.10 [67] (1997), [68] (1998) (also see [5]-[15], [32]). Suppose

that every ^(t) (/ = 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, the iterated Stratonovich stochastic integral J * [^(k)]Tjt defined by (2.336) is expanded

into the converging in the mean of degree 2n (n E N) iterated series

oo o k

J *W'(k)]T,( = £ ... £ CV.ji n zf, (2.339)

ji=0 jfc=0 1=1

where

T

zji) = / & (s)dw(i) t

are independent standard Gaussian random variables for various i or j (in the case when ¿ = 0) and

C —

/k

K(ti,..., ik) J! (ti)dti... dtk (2.340)

[t,T]k 1=1

is the Fourier coefficient.

Note that (2.339) means the following

{/ Pi Pk k \2n>|

1 Z-/ Z-/ I I

\ ji=0 jfc=0 1=1 / J

__(2.341)

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 / A- (il5..../,) n +

1=1 1=1 ^

l, •• •, t-fej = H'W-U HI Hti<ti+1}

k / k-1 k-1 i k-1 r k-1 ^

Y[Mti) + S II1^^!} II MtKh+i}

, 1=1 r= 1 sr,...,si = 1 1=1 1=1 I

y sr>...>si l=s i y

1=1

(2.342)

for t1,... ,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 [67] (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), (2.343)

ji=0 jk =0 1=i

where (t1,..., tk) E (t, T)k and Cjk...j is defined by (12.340). At that, the iterated series (2.343) 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 (x)}°=0 at the interval (t, T)

to

K*(ti,t2) = £ Cji(t2) j(ti) (ti = t, T), (2.344)

j1=0

where

T T

'* I

Cji(t2) = j K*(ti,t2)0ji(ti)dti = j K(ti,t2)0ji(ti)dti =

t t t2

= ^i(ti)0ji (ti)dti. t

The equality (2.344) 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 (2.344) converges when ti = t,T (not necessarily to the function K*(ti,t2)).

Obtaining (2.344), we also used the fact that the right-hand side of (2.344) converges when ti = t2 (point of a finite discontinuity of the function K(ti, t2)) to the value

i (K(u - o, to) + K(u + o,to)) = ^(uyuto) = 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)

Ci(t2) — E Cjj(t2) (t2 — t,T), (2.345)

¿2=0

where

T T t2

Cj2ji — J Cji fo) j (t2)dt2 — J ^2^) j (t2^ (ti)^ji (t1)dt1dt2

t t t

and the equality (2.345) is satisfied pointwise at any point of the interval (t,T). Moreover, the right-hand side of (2.345) converges when t2 — t,T (not necessarily to Cj (t2)).

Let us substitute (2.345) into (2.344)

to to

K*(*1,*2) — EE Cj2jijfa) j(t2), (t1 ,t2) e (t,T)2. (2.346)

ji=0 j2=0

Note that the series on the right-hand side of (2.346) converges at the boundary of the square [t,T]2 (not necessarily to K*(t1,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 (2.344))

Mh) (l{il<i2} + il^}) (2-347)

which is satisfied pointwise at the interval (t, T), besides the series on the right-hand side of (2.347) converges when t1 — t,T.

Let us introduce the induction assumption

to to to

EE- E ^k-1(tk-1)x

ji =0 j2=0 jk-2=0

tk-i t2 k_2

//» k 2

^k-2(tk-2)0jfc-2 (tk-2) ■■■ fa) j (t1)dt1 . . .dtk-2 JJ fa) — t { 1=1

k-1 k-2 x 1 x

= + • (2-348)

1=1 1=1 ^ '

Then

oo oo oo

£ ^ (tk )x ji=0 j2=0 jfc-i=0

ifc ¿2

k- 1

x / ^k-i(tk-i)0jfc-i (tk-i) ■ ■■ ^i(ti) j (ti)dti. ..dtk-^JJ ^ (ti)

i=i

to to to ✓ 1 \

= £ ' '/,(//,) ( l{ifc_1<ifc} + ^{tk^=tk} J <7, i(//, i )x

ji=0 j2 =0 jfc-2=0 ^ '

¿fc-1 ¿2 k- 2 X / ^k-2(tk-2)0jfc-2 (tk-2) ■■■ ^i(ti)0ji (ti)dti . ..dtk-^JJ (ti)

1=i

(1 \ to to to

ife-K^} + £ 'V, |(//, i)x

' ji =0 j2=0 jfc-2 =0

¿k-i ¿2 k_2

k-2

X / ^k-2(tk-2)0jfc-2 (tk-2) ■■■ /^i (ti)0ji (ti)dti ...dtk-^JJ (ti)

1=i

( 1 \ k— k-V 1

' '/,(//,) ( l{ifc_1<ifc} + ) Y['Mti) n ( 1

V / 1 1 V

{ij<ij+i} + ö^^+i}

fe-i<tk} 1 2 {tk-i=tk} ) J_J_ r^-v J_J_ I 1 2

' 1 = i 1 = i ^

k k—i / 1 N

= n^niWo + a1^'^) • (2"349) i=i i=i ^ '

On the other hand, the left-hand side of (2.349) can be represented in the following form

to to k

£ ... £ Cjk...ji n^ji (ti) ji=0 jk =0 1=i

by expanding the function

tk t2

^k(tk) J ^k—i(tk—i)0jk-i(tk—i).. ^^i(ti)0ji(ti)dti.. .dtk-i

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

-{«spp =^ + 1=0}

j= n i(- - ■=- x

p=I

T tsl+3 tSl + 2

"0k (tk ) ...J ^s;+2 (tsj +2) J ^ s; (ts; + 1 + 1 (ts; + 1) X t t t

tS[ + 1 tsl+3 tsl + 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^dts,+1dwt(;S++2)... dwt( ;k), (2.350)

where

Ak,i = {(s/, ): s/ > si-i + 1,...,s2 > si + 1, s/,..., si = 1,...,k - 1},

(2.351)

(s/ ,...,s1) G Ak,z, 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 JJof fixed multiplicity k, k G N (see (2.336), (2.337)). ' '

Theorem 2.12 [67] (1997). Suppose that every (t) (l = 1,...,k) is a

continuously differentiate function at the interval [t, T]. Then, the following relation between iterated Stratonovich and Ito stochastic integrals

[k/2] .

r=l (sr ,...,si)GAfc,r

is correct, where Yh is supposed to be equal to zero.

Proof. Let us prove the equality (2.352) using induction. The case k = 1 is obvious. If k = 2, then from (2.352) we get

J*[^]t,t = -m^t, + \m{2)]h w. p. 1. (2.353)

Let us demonstrate that the equality (2.353) is correct w. p. 1. In order to do it let us consider the process nt2,t = ^2(t2)J[^(1)]t2,t, t2 E [t,T] and find its stochastic differential using the Ito formula

dnt2,t = J [^(1)kt #2^2) + ^ife^Mdwg0. (2.354)

From the equality (2.354) we obtain that the diffusion coefficient of the process nt2,t, ¿2 E [t,T] equals to 1{n=Q}^i (¿2)^2^2).

Further, using the standard relations between Stratonovich and Ito stochastic integrals (see (2.4), (2.5)), we obtain the relation (2.353). 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

J =

[k/2]

.T

'(ftt+iM I J[p]r,t + E JlAXr'1 1 dw^ =

r=l (sr ,...,S1 )GÄfc,r

* T

[k/2]

T

1

+ E j (2-355)

r=l (sr,...,si)GAfc,r t

Using the Ito formula and standard relations between Stratonovich and Ito stochastic integrals, we get w. p. 1

* t

f ipk+1(T)j№%,tdw^ = J[^k+1)}T,t + ljbP{k+1%, (2.356)

* t

I )J№(k)]%-'si dw?^ = t

J[^(k+1)]T7"s1 if sr = k -1

= < . (2.357)

J[^(k+1)]ft-'si + J[^(k+1)]Tî""'S1 /2 if sr < k - 1

V ' '

After substituting (2.356) and (2.357) into (2.355) and regrouping of sum-mands, we pass to the following relations, which are valid w. p. 1

[k/2] .

J*№k+V]T,t = J[^k+1)ht + J[^k+i)}ST;rSl (2-358)

r=1 (sr ,...,si)eAfc+ijr

when k is even and

[k'/2]+1 .

= E i E J[^k'+1)}irSl (2-359)

r=1 (sr ,...,si)GAfc'+ir

when k' = k + 1 is uneven.

From (2.358) and (2.359) we have w. p. 1

[(k+1)/2] .

j*№k+^,t = j№k+1)ht+ J2 ¥ £ J№{k+1)Yf;r81- (2.360)

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 [75], which are fulfilled w. p. 1

.T T

* ^(tOdwf = i ^(tOdwf,

tt *T * t2 T t2

I Mt2)J ^1(t1)dwt(;i)dw(;2) = 1 Mt2)f ^1(t1)dwt(;i)dwt(;2)+ t t t t

T

+ J >h(h)Mh)dto, (2.361)

t

*T * ¿2 T t2

I Mh) ..J ^i(ti)dwi;i).. .dw(33) =1 .. J ^i(ti)dw(;i).. .dw(33) +

t t t t

T t3

J Mh) J h{h)Mh)dtodw{tf + tt t t3

+ il{i2=i3^0} J h(h)h(h) J Mt^dw^dh, (2.362) tt

*T * t2 T t2

J uu)..J ^i(ti)dwi ii)...dw(;4) =1 ^fa)..J^i(ti)dw( ii)...dw(;4)+ t t t t

T t4 t3

+ il{i1=i2^0} [Mb) ( hih) /'0l(i2)'02(i2)«wifdwi;4) +

ttt T t4 t3

+^i{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} f ■MUWziU) I )^i{t2)dt2dtA. (2.363)

Let us consider Lemma 1.1, definition of the multiple stochastic integral (1.16) together with the formula (1.19) when the function $(t1,... ,tk) is continuous in the open domain and bounded at its boundary as well as Lemma 1.3 (see Sect. 1.1.3). Substituting (2.342) 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] 1

J*№{k)}t,t = J№{k)}T,t + E «/['0(A;)]J)i"'Sl = (2.364)

r=1 (sr ,...,si)GAfcjr

where J[KTk,) is defined by (HIE) and K*(ti,... ,tk) has the form (2342). Let us subsitute the relation

Pi Pk k

K *(ti,..., tk ) = E... E Cjk ...ji n ^ (ti) + K *(ti,...,tk )-

ji=0 jk =0 i=i

Pi Pk k

- E... E Cjk...ji n^ (ti )

ji=0 jk=0 1=1

into the right-hand side of (2.364) (here we suppose that pi5... ,pk < œ). Then using Lemma 1.3 (see Sect. 1.1.3), we obtain

Pi Pk k

J*№(k)]T,t = E ■.. E Cjk . ji n j) + J[R--*]Tkt w. p. 1, (2.365)

ji=0 jk=0 1=1

where the stochastic integral J[RPi...Pk]Ti is defined by (1.16) and

Pi Pk k

(ti,... ,tk) = K*(ti,... ,tk) - E ... E Cjk...ji n^(ti), (2.366)

ji=0 jk=0 i=i

T

j) = / j (s)dw<«).

t

In accordance with Theorem 2.11, we have

lim ... lim R,i...Pk(ti,...,tk) = 0, (ti,...,tk) G (t,T)k (2.367)

and |RPi...Pk(ti,..., tk)| < œ, (ti,...,tk) G [t,T]k.

Theorem 2.13. In the conditions of Theorem 2.10 we have

lim lim ... lim M

J [RPi...Pk ]T]

2n"ï

^=0, n G N.

Proof. At first let us analize in detail the cases k = 2, 3, 4. Using (2.410) (see below) and (1.19), we have w. p. 1

N-1 N-1

l.i.m. V V Rp!p2K, Ti2jAwii;)Aw(;22)

N^to— — 1 2

/2=0 /1=0 'N-1 /2-1 N-1 /1-1

/N-1 /2-1 N-1 /1-1\

^ E E + E E (T/1.12)Aw«;i»Aw«;2»+

N ^^ V /2=0 /1=0 /1=0 /2=0/ N1

+l i.m. E RP1P2 (t/1 ,t/1 )Aw(;;)Aw(;i2) =

/1=0

T t2 T t1

I /RP1P2 fa^dw^dw^ + J J RP1P2 (t1,t2)dwt(2^dwt(1° + t t tt

T

+ 1{;1=;2=0} / RP1P2(t1,t1)dt1. (2.368)

where we used the same notations as in the formulas (1.16), (1.19) and Lemma 1.1 (see Sect. 1.1.3). Moreover,

P1 P2

Rpip2 (ti,t2) = K *(ti,t2) - EE Cj2ji j (ti) j (t2), Pi, P2 < rc. (2.369)

ji=Q j2=Q

Let us consider the following well known estimates for moments of stochastic integrals [91

M

T

2n'

T

< (T - t)n-1 (n(2n - 1))n / M {j^Tj2n} dT, (2.370)

M

T

^T dT

2n'

T

< (T - t)2n-1^ m{ j^T |2n}

dT,

(2.371)

where the process such that )n E 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 (2.370) and (2.371), we obtain

M

J [RPiP2 ]T\t

2n"

T t2

^ cn

'n I / / (Rpip2 (ti,t2))2n dtidt2 +

tt

T ti T

\2n

+ J J (Rpip2(ti,t2))2ndt2dti + l{ii=i2=0^ (Rpip2(ti,ti)rdti | , (2.372) t t t where constant Cn < to depends on n and T — t (n = 1, 2,...).

Note that due to the above assumptions, the function Rpip2 (t1 , t2) is continuous in the open domains of integration of integrals on the right-hand side of (2.372) and it is bounded at the boundaries of these domains.

Let us estimate the first integral on the right-hand side of (2.372)

T t2

0 < J J (Rpip2 (tl,t2)^ dti dt2 = I I + ' 1 (R (t t ))2n tt

N—i i

< EE

(Rpip2 (ti ,t2))2n dtidt2 <

max (Rpip2(ti,t2))2n ATiATj + MSTe <

(ti ,t2)e[ri ,ri+i ]x[Tj ,Tj+i ]

¿=0 j =0

N-i i

<EE (RpiP2(Ti,Tj))2n ATiATj + i=0 j =0

N- i i

+ E E (RP.P2 (i<PiP2),tjnP2)^ 2" - (Rpip2 (T„ Tj ))

i=0 j =0 N- i

2n

ATiATj + MSpe < 1 "

N-i i £ / 1 \ < E E (RP^T3)fn Ar, Ar, + -L(T-i-3s)2 1 + - (2.373)

„'—n „■—n V /

i=0 j=0

where

De = {(ti,t2): t2 G [t + 2e,T — e], ti G [t + e,t2 — e]}, r = D\De,

D = {(ti,t2) : t2 G [t,T], ti G [t,t2]},

e is a sufficiently small positive number, Sr£ is the area of Te, M > 0 is a positive constant bounding the function (Rpip2(ti,t2))2n , (t(pip2), tjpip2)) is a point of maximum of this function when (ti,t2) G [ri,ri+i] x [t,, Tj+i],

Ti = t + 2£ + iA (i = 0,1,..., N), TN = T - £, A =

T-t-Se N :

A < e, £i > 0 is any sufficiently small positive number.

Getting (2.373), we used the well known properties of Riemann integrals, the first and the second Weierstrass Theorems for the function of two variables as well as the continuity (which means uniform continuity) of the function

(Gpip2(ti5t2))2n in the domain De, i.e. V > 0 3 > 0, which does not

depend on t\, to, pi, po and if y/2A < ¿(ei), then the following inequality takes place

RPiP2 (t( , ¿j )) - (RpiP2 (r«5T:?')) < ei.

Considering (2.346), let us write

lim lim (RPlP2(tl5i2))2n = 0 when (i1;i2) G De

and perform the iterated passages to the limit lim lim lim , lim lim lim (we use the property lim < lim in the second case; here lim means lim inf)

pi—yoo Pi^co Pl^oo Pl^CC

in the inequality (2.373). Then, according to arbitrariness of > 0, we have

T ¿2 T ¿2

\2n 7, 7, T-— T-- I I / r> / # # n

lim lim / / (Rpip2(ti, to)yn dt\dto = lim lim / / (RPlP2(ti, to)yn dt\dto =

p^oo'pi^co j j - pl->00 p2-ïoo j j

t t t t

T t2

= lim lim I I {Rprpr>{ti,to)fn dtidto = 0. (2.374)

Pi^-œp2^œ J J tt

Similarly to arguments given above we obtain

T ti

,2n

lim lim / / (Rp1p2(ti,t2)) dt2dti = 0, (2.375)

P1^œ p2^œ J J tt

T

lim Tim ( {RPiI»{ti,ti)?ndti = t). (2.376)

P1^œ p2^œ J t

From (2372), (2374)-(l2:3761) we get

lim lim M

J[Rp1P2]T2t 2 r =0, n G N. (2.377)

Note that (2.377) can be obtained by a more simple way. We have

t t2 t ti

J y\Rpip2(ti, t2))2ndtidt2 + ^ J (Rpip2(ti, t2))2n dt2dti = t t t t T t2 T T

= J J (Rpip2 (ti,t2))2n dtidt2 + ^ J (Rpip2 (ti,t2))2n dti dt2 = t t t t2

2n

(2.378)

[t,T ]2

Combining (2.372) and (2.378), we obtain

M

J [Rpip2 ]rt

2n

<

T

< Cn

>2n

'/,R...........^.....W —„

\[t,T ]2 t

\

/

(2.379)

where constant Cn < to depends on n and T — t (n = 1, 2,...).

From the one hand, we can use the above reasoning to the integrals on the right-hand side of (2.379) instead of integrals on the right-hand side of (2.372). However, we can get the desired result even easier.

Since the integrals on the right-hand side of (2.379) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim (Rpip2(ti,t2))2n = 0

pi—>-to p2—to

holds for all (ti,t2) e (t,T)2 and |RPiP2(ti,t2)| < to for all (ti,t2) e [t,T]2. According to (2.369), we have

pi

Rpip2 (tl,t2) = ( K*(ti,t2) - £ Cji (t2) j (ti) ) +

ji=0

Pi

P2

+ E Cji(t2) - E Cj2jijfa) j(ti) .

(2.380)

ji=0

j2=0

Then, applying two times (we mean here an iterated passage to the limit lim lim ) the Lebesgue's Dominated Convergence Theorem and taking into

account (2.344), (2.345), and (2.380), we obtain

lim lim / (Rpip2(ti,t2))2ndtidt2 = 0, (2.381)

[t,T ]2

T

lim lim I (Rpm(ti,ti))2ndti = 0. (2.382)

p1—to p2—TO J t

From (2.379), (2.381), and (2.382) we get (2.377). Recall that (2.382) for 2n = 1 has also been proved in Sect. 2.1.3.

Let us consider the case k = 3. Using (2.411) (see below) and (1.19), we have w. p. 1

N-i N-i N-i

J[RP1P2P3]T3t = l.i.m. ££ E RP1P2P3 (Til, Ti2, Ti3)Aw(;;) Aw(;22) Aw(;33) =

¿3=0 ¿2=0 ¿1=0 1 2 3

N-i l3-i ¿2 i /

= Li.rn.EEE RP1P2P3(Til,Ti2,Ti3)AwT;;)Aw(;22)Aw(;33) +

¿3=0 ¿2=0 ¿1=A 1 2 3

+RP1P2P3 (Til, T3 ,T72 )AwT;ii)AwT;;)AwT;23)+

+Rpip2p3 (n2,71!, 713 jAwij) Aw^Aw^ +Rpip2p3 (T12, T13, Til )AwT;i) Aw^Aw^

+RP1P2P3 (Ti3, Ti2, Til )AwTi3l) Aw^Aw^

+RP1P2P3 (Ti3, Til, Ti2 )AwT;3i)AwT;i2) AwT;23^ +

N-1 ¿3-1 /

+l.i.m. EE RP1P2P3(Ti2,Ti2,Ti3)Aw(;2l)Aw(;22)Aw(;33) +

¿3=0 ¿2=0 \ 2 2 3

=0 ¿2=0

+RPlP2P3 (Ti2, Ti3, Ti2 )Aw(i2l) Aw(;2)Aw(;3) +

+RPlP2P3 (Ti3, Ti2, Ti2 )Aw(;3l)Aw(i22) Aw^ J +

+i.i.m. EE RP1P2P3(t/1 ,t/3,t/3)AwT;;)AwT;32)AwT;33)+

/3=0 /1=0 \ 1 3 3

+RP1P2P3 (t/3 , t/1 , t/3 )AwT;31) Aw^Aw^

+RP1P2P3 (t/3 , t/3 , t/1 ^w^Aw^ Aw^ ) +

N-1

T;3) =

/ , ""'Pi P2P.3V '¿3 5 '¿3 5 'h/^'-''vTl „

N

+l.i.m. E RP1P2P3 (t/3, T/3, T/3)AwTi3) Aw^Aw^;

333

/3=0

T t3 t2

= II IRP1P2P3(^1 't2,t3)dwt(;1)dwi;2)dwt(33)+ ttt

T t3 t2

+// jRP1P2P3 (t1,t3 ,i2)dwt( i1) dwt(;3)dwt(32)+ ttt

T t3 t2

+// jRP1P2P3 (t2,t1 ,t3)dwt i2) dwt(;1)dwt(33)+ ttt

T t3 t2

+// jRP1P2P3 (t2, t3 ,t1)dwt( ;3) dwt(;1)dwt(32)+ ttt

T t3 t2

+ // IRP1P2P3 (t3,t2,t1)dwt( ;3)dwt(;2)dwt(31) + ttt

T t3 t2

+//1RP1P2P3 (t3,t1 ,t2)dwt( ;2) dwt(;3)dwt(31)+ ttt

T t3

+ 1{i1=i2=0^ y"RP1P2P3 (t2,t2,t3)dt2dwi33) + tt

T t3

+ 1{n=;3=0}^ y"RP1P2P3 (t2,t3,t2)dt2dwi32) + tt

T t3

+ !{i2=i3=0^y y"RPiP2P3 (t3,t2,t2)dt2dwt3l) + tt T t3

+i(.2=.3=o^/Rpip2p3 (ii

tt T t3

+ I{ii=i3=0^y y"RPiP2P3 (i3 ,ii,i3)dwt( ^3 + tt T t3

+ !(«// Rpip2p3 (i3,i3,ii)dw( (2.383)

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 (2.370) and (2.371), we obtain from (2.383)

M

T t3 t2

J [RPlP2P3 ]T,t

2n>| ^ <

~ J J I (RP1P2P3 (ti, t2,i3)^ n + (RP1P2P3 (ii,i3,i2)^ +

ttt

+ (Rp1p2p3 (i2, ^ ^))2n + (Rp1p2p3 (i2, i3,ii))2n + (Rp1p2p3 (i3, ^ ii))2n + + (Rp1p2p3(i3,ii,i2))2n^ diidi2di3 +

+ ff ( 1{:i=:2=0M (RP1P2P3 (i2, i2, i3))2" + (RP1P2P3 (t3, i3, h))2J + tt

+ l{:i=:3=0^(RP1P2P3 fe ^ ^^ + (RP1P2P3 (i3, i2, i3))2^ + +!{:2=:3=0} ^(Rp1P2P3(i3,i2,i2))2n + (Rp1p2p3^ di2di3^ , Cn <

(2.384)

Due to (2.366) and Theorem 2.11 the function RPlP2P3(ii, i2, i3) is continuous in the open domains of integration of iterated integrals on the right-hand side

of (2.384) and it is bounded at the boundaries of these domains. Moreover, everywhere in (t, T)3 the following formula takes place

lim lim lim RPiP2P3(ti, t2, t3) = 0. (2.385)

pi—p2—^TO P3—

Further, similarly to the estimate (2.373) (see 2-dimensional case) we perform the iterated passage to the limit lim lim lim under the integral signs

Pi—TO P2 — TO p3—

on the right-hand side of (2.384) and we get

2n

lim lim lim M

J[Rp1p2p3]T3t =0, n G N. (2.386)

From the other hand

T ¿3 ¿2 /

J J J ( (RPiP2P3 (ti,t2,t3))2n + (RPiP2P3 (ti,t3,t2))2n + (RPiP2P3 (t2,ti,t3))2n + ¿¿¿

+ (RPiP2P3 (t2? t3; ti))2n + (RPiP2P3 (t3,t2,ti))2n + (RPiP2P3 (t3,ti,t2))2^ dtidt2dt3 =

= J (RPiP2P3(ti,t2,t3))2ndtidt2dt3, (2.387)

[¿,T]3

T ¿3

l l ((RPiP2P3 (t2' t2' t3))2n + (RPiP2P3 (t3,t3,t2))2^ dt2dt3 = ¿¿

T ¿3 T T

= J J (RpiP2P3 (t2, t2, t3))2n dt2dt3 + J J (RPiP2P3 (t2,t2,t3))2n dt2dt3 =

¿ ¿ ¿ ¿3

= J (RpiP2P3(t2,t2,t3))2ndt2dt3, (2.388)

[¿,T ]2

T ¿3

I I ((RPiP2P3 (t2' t3' t2))2n + (Rpip2p3 (t3,t2,t3))2^ dt2dt3 = ¿¿

T ¿3 T T

= J j (Rpip2p3 (t2, t3, t2))2n dt2dt3 + J j (Rpip2p3 (t2,t3,t2))2n dt2dt3 =

¿ ¿ ¿ ¿3

= y (RPlP2P3 (t2 ,t3,t2))2n dt2dt3, (2.389)

[t,T ]2

T t3

I J ((RPlP2P3 (t3,t2,t2))2n + (RPlP2P3 (t2,t3,t3))2n^ dt2dt3 = t t

T t3 TT

= J J (RPlP2P3 (¿3, ¿2, t2))2n dt2dt3 + J J (RPlP2P3 (¿3,^2,t2))2n dt2dt3 = t t t t3

= J (RPlP2P3(¿3,t2,t2))2ndt2dt3. (2.390)

[t,T ]2

Combining (2.384) and (2.387)-(2.390), we have

/ „

M

J [RPlP2P3

(3) 2n

< Cn

y (Rplp2p3 (t1, ¿2, ¿3))2n dt1dt2dt3 + \t,T ]3

+ 1{il=i2=0} y (RPlP2P3 (t2,t2,t3))2n dt2dt3 + [t,T ]2

(2.391)

+ 1{il=i3=0} y (Rplp2p3 (t2, ¿3, t2)^ n dt2dt3 +

[t,T ]2

\

+ !{i2=i3=0} J (Rplp2p3 (t3,t2,t2))2n dt2dt3 • [t,T]2 /

Since the integrals on the right-hand side of (2.391) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim lim RPlP2P3(t1, t2, t3) = 0

Pl^TO

holds for all (t1,t2,t3) G (t,T)3 and |RPlP2P3(t1, t2, t3)| < to, (t1 ,t2,t3) G [t,T]3. According to the proof of Theorem 2.11 and (2.366) for k = 3, we have

Rplp2p3 (t1,t2,t3 ) = K *(t1,t2,t3) — £Cjl (¿2,t3) j fa) +

V jl=0 /

f Pi f P2 \ \

+ E Cji(t2,t3) - E Cj2ji(t3)0j2M j(tiH + \ji=0 V j2=Q / /

/ Pi P2 / P3 \ \

+ E E Cj2ji(t3) - E Cjj(t3) j(t2)0ji (tiH , (2.392)

\ji=Q j2=Q V j3=Q / /

where

T

Cji (t2,t3)^ K*(ti,t2,t3)0ji (ti)dti, ¿

Cj2ji (t3)= J K*(ti,t2,t3)0ji (ti)j (t2)dtidt2. [¿,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 P3—TO

lim lim lim / (RPiP2P3(ti, t2, t3))2n dtidt2dt3 = 0, (2.393)

Pi—TO P2 — TO P3—TO I

[¿,T]3

lim lim lim / (RPiP2P3(t2, t2, t3))2n dt2dt3 = 0, (2.394)

Pi—-TO p2—TO p3 — TO I

[¿,T ]2

lim lim lim / (RPiP2P3(t2, t3, t2))2n dt2dt3 = 0, (2.395)

Pi — TO P2—TO P3 — TO I

[¿,T ]2

lim lim lim / (RPiP2P3(t3, t2, t2))2n dt2dt3 = 0. (2.396)

Pi — TO P2—TO P3 — TO I

[¿,T ]2

From (2.391) and (2.393)-(2.396) we get (2.386).

Let us consider the case k = 4. Using (2.412) (see below) and (1.19), we have w. p. 1

J [RPiP2P3P4 ]T^ =

N-i N-i N-i N-i

= l.i.m. EEE E RpiP2P3P4 (Tii, Ti2, Ti3, Ti4 )Aw(;;) Aw[;22)Aw(;33) Aw[;44) =

N—to i 2 3 4

N —TO 14=Q 13=Q 12=Q 1i=Q

N-i 14-i 13-i 12-i ,

= H.m. ^^EE E (RPiP2P3P4 (T1i ,Tl2 ,Tl3 ,Tl4 )x

N—TO l4=Q l3=Q l2=Q 1i=Q (1i,l2,l3,l4) ^

xAw^Aw^ Aw^Aw^M +

''1 ''2 ''3 ''4 /

N — 1 ¿4 — 1 ¿3 — 1

+1NLm.£££ £ rpiP2P^P4(T2,T¿2,T¿3,т¿4)Aw(;!)Aw(;22)Aw(;зз)Aw(;44M+

N^ ¿4=0 ¿3=0 ¿2=0 (¿2,¿2,¿3,¿4)V J

N — 1 ¿4 — 1 ¿3 — 1 / \

+ljLm.£££ E RplP2P3P4(Til^^)Aw;*!)AW;;3)AW!;3)AW;;;M+

¿4=0 ¿3=0 ¿l=0 (¿l,¿3,¿3,¿4J J

N —1 ¿4 — 1 ¿2 — 1 , \

+l^.m. E E E E RP.P2P3P4(1.. ■T<2, T<4.T,4)AW(;!»AW«;;)AW«;43)AW(;4M+

^ ¿4=0 ¿2=0 ¿l=0 (¿1,¿2,¿4,¿4Г

+l.i.m. EE E RP1P2P3P4(■¿3,■¿3,■¿3,■¿4)Aw(;3l)Aw(;32)Aw(;33)Aw(;44M +

¿4=0 ¿3=0 (¿3,¿3,¿3,¿4^ ^^ j

N — 1 ¿4 — 1 , \

+lJLm. EE E RP1P2P^P4(■¿2,T¿2,T¿4^¿4^w^w^w^w^ +

N ^ ¿4=0 ¿2=0 (¿2,¿2 ,¿4,¿4^ j

N—1 ¿4 —1

N

+l-i.m^^E E RP1P2P^P4 (■¿1 ,■¿4 ,■¿4,■¿4)AwT;!)AWT;42)AWT;43)AWT;44M +

¿4=0 ¿1=0 (¿l^^^) N1

+l.i.m. £ RP1P2P3P4(■¿4,■¿4,■¿4,■¿4)Aw(;4l)Aw(;42)AwT;43)AwT;44) =

N^TO ,—' 4 4 4 4

¿4=0

T t4 t3 t2

//// E (RP1P2P3P4 (¿1, ¿2, ¿3 ,t4)dwt(;l)dwt(22)dwt(33)dwt(44^ +

t t t t (tl ,t2 ,t3 ,t4) T t4 t3

+1{;i=;2=0}/// E (^RPlP2P3P4fa,¿1,¿3,¿4)1 )dwt(44)) +

t t t (tl,t3,t4) T t4 t2

+1{;i=;3=0}/// E (RP1P2P3P4fa,¿2,^ 1,¿4)1 )dwt44)) +

t t t (tl,t2,t4) T t3 t2

+ 1{il = i4=0^^^y E ^RP1P2P3P4 fa ^^¿O^M^^ dwi33^ +

t t t (tl,t2,t3)

T ¿4 ¿2

+ l{i2=i3=Q^^^y E (RPiP2P3P4 (ti ,t2,t2,t4)dwiii)dt2 +

¿ ¿ ¿ ^,¿2^

T ¿3 ¿2

+ 1{»2=»4=Q^^^y E (^RPiP2P3P4 (ti ,t2,t3,t2)dwiii)dt2 dwi33^ + ¿ ¿ ¿ (¿1,¿2,¿3)

T ¿3 ¿2

+ 1{i3=i4=Q}/// E (RPiP2P3P4 (ti ,t2,t3,t3)dwiii)dwii2)dt^ + ¿ ¿ ¿ (^¿i ,¿2 ,¿3)

/ T ¿4

+ 1{ii=i2=Q}1{i3=i4=Q} ( J y"RPiP2P3P4 (t2,t2,t4,t4)dt2dt4 +

¿¿

T ¿4 \

+ J J Rpip2p3p4(t4, t4, t2, t2)dt2dt4 j + ¿¿

/ T ¿4

+ 1{ii=i3=Q}1{i2=i4=Q} ( J y"RPiP2P3P4 (t2,t4,t2,t4)dt2dt4 +

¿¿

T ¿4 \

+ J J Rpip2p3p4^ ^ ^)dt2dt4 j + ¿¿

/ T ¿4

+ 1{ii=i4=Q}1{i2=i3=Q} ( J y"RPiP2P3P4 (t2,t4,t4,t2)dt2dt4 +

¿¿

T ¿4 \

+ J J Rpip2p3p4 (t4, t2, t2, t4)dt2dt^ , (2.397)

¿¿

where the expression

E

(ai,...,afc)

means the sum with respect to all possible permutations (ai,..., ak). Moreover, we used in (2.397) the same notations as in the proof of Theorem 1.1 (see Sect. 1.1.3). Note that the analogue of (2.397) will be obtained in Sect. 2.7 (also see [10]-[15], [34]) with using the another approach.

By analogy with (2.391) we obtain

M

J [RPlP2P3P4 ]T,t

(4) 2n"

< Cn

I

/ (Rpip2p3p4 (^b ^ ^ ¿4^ dtldt2dt3dt4 + \t,T ]4

+ 1{il=i2=0} J (RP.P2P3P4^¿¿3^4 + [t,T ]3

+ 1{il=i3=0} J (RP.P2P3P4 ¿4^ " ^2^3^-4 +

[t,T ]3

+ 1{il=i4=0} J (RP.P2P3P4 ¿2^ " ^2^3^-4 +

[t,T ]3

+ 1{i2=i3=0} J (rp.p2p3p4))2n ¿^¿¿3^4 + [t,T ]3

+ 1{;2=;4=0} J (rp.p2p3p4t4, ¿2))2n ^2^3^-4 + [t,T ]3

+ 1{;3=;4=0} J (rp.p2p3p4^ ^ ¿2))2" ^2^3^-4 + [t,T ]3

+ 1{il=i2=0}1{i3=i4=0^ (RPlP2P3P4 (t2,t2,t4,t4))2n ¿¿2^4 +

[t,T ]2

+ 1{il=i3=0}1{i2=i4=0} J (RPlP2P3P4 (t2,t4,t2,t4))2n ¿¿2^4 +

[t,T ]2

\

+ 1{il = i4=0}1 {i2=i3=0W (RPlP2P3P4 (^2, ¿4, ¿4, ¿2)) ¿¿2^4 , Cn <

[t,T]2 /

(2.398)

Since the integrals on the right-hand side of (2.398) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover,

lim lim lim lim Rplp2p3p4(¿1, ¿2, ¿3, ¿4) = 0, (¿1, ¿2, ¿3, ¿4) G (¿,T)4

P.^to

and |RpiP2P3P4 (t1, t2,t3,t4)| < TO, (ti, t2, t3, t4) E [t,T] .

According to the proof of Theorem 2.11 and (2.366) for k = 4, we have

p1

Rpip2p3p4 (ti, t2,t3,t4) = K*(ti, t2,t3,t4) - E^'i (t2,t3,t4) j (ti) +

V ji=Q

Pi P2

+ E Cji (t2,t3,t4) - E Cj2ji (t3,t4)0j-2 M j (tiH +

\j'i=Q \ j2=Q / /

/ Pi P2 / P3 \

+ I EE ( Cj2ji (t3,t4) - E Cj3j2ji (t4)0j3 (t3H j (t2)0ji (t0 ) +

\ji=Q j2 =Q V j3=Q /

/ Pi P2 P3 / P4 \

+ I E E E ( Cj3j2ji (t4) - E Cj4j3j2ji j Ml j (t3)0j2 (t2)0ji (ti) Vji =Q j2=Q j3=Q V j4=Q /

where

T

Cji (i2,t3,t4)^y K*(ti,t2,t3,t4)j (ti)dti,

¿

Cj2ji (t3,t4)= J K*(ti,t2,t3,t4)0ji (ti)0j2 (t2)dtidt2,

[¿,T ]2

Cj3j2ji (t4) = J K*(ti ,t2,t3,t4) j (ti) j Mj (t3)dtidt2dt3.

[¿,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-

Pi—TO P2 — TO P3—TO P4 — TO

tain

lim lim lim lim / (RPiP2P3P4(ti,t2,t3,t4))2ndtidt2dt3dt4 = 0, (2.399)

Pi — TO P2—TO P3 — TO P4—TO /

[¿,T]4

lim lim lim lim / (RPiP2P3P4(t2,t2,t3,t4))2n dt2dt3dt4 = 0, (2.400)

Pi —TO P2—TO P3 — TO P4—TO /

[¿,T ]3

lim lim lim lim / (RPiP2P3P4(t2,t3,t2,t4))2n dt2dt3dt4 = 0, (2.401)

Pi —TO P2—TO P3 — TO P4—TO /

[¿,T ]3

lim lim lim lim / (RPlP2P3P4(¿2, ¿3, ¿4,¿2))2n di2di3di4 = 0, (2.402)

Pl —>-TO P2—TO P3 — TO P4—>-TO /

[t,T ]3

lim lim lim lim / (RPlP2P3P4(¿3, ¿2, ¿2,¿4))2n di2di3di4 = 0, (2.403)

Pl —TO P2—TO P3 — TO P4—TO /

[t,T ]3

lim lim lim lim / (RPlP2P3P4(¿3, ¿2, ¿4,¿2))2n di2di3di4 = 0, (2.404)

Pl —TO P2—TO P3 — TO P4—TO I

[t,T ]3

lim lim lim lim / (RPlP2P3P4(¿3, ¿4, ¿2,¿2))2n di2di3di4 = 0, (2.405)

l —TO P 2 —TO P 3 —TO P 4 —TO l 2 3 4

Pl —TO

[t,T ]3

lim lim lim lim / (RPlP2P3P4(i2,i2,i4,i4))2ndi2di4 = 0, (2.406)

l —TO P 2 —TO P 3 —TO P 4 —TO l 2 3 4

Pl —TO

[t,T ]2

lim lim lim lim / (RPlP2P3P4(i2,i4,i2,i4))2ndi2di4 = 0, (2.407)

l —TO P 2 —TO P 3 —TO P 4 —TO l 2 3 4

Pl —TO

[t,T ]2

lim lim lim lim / (RPlP2P3P4(i2,i4,i4,i2))2ndi2di4 = 0. (2.408)

l —TO P 2 —TO P 3 —TO P 4 —TO l 2 3 4

Pl —TO

[t,T ]2

Combaining (2.398) with (E399M2L308!), we get

2n>|

lim lim lim lim M

Pl — TO P2—TO P3 — TO P4—TO

J [RPlP2P3P4 ]T4t

= 0, n G N.

Theorem 2.13 is proved for k = 4.

Let us consider the case of arbitrary k, k £ 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-i j2 i S!v)(a) = £ . . . £ £ a(ji,..Jfc),

jfc=° j1=0 (jlv-jfc)

(k),

Csr • • • CsiSN (a) = N —1 jsr +2 — 1 jsr + 1—1 jSl+2 —1 jSl + l—1 j2 —1

e- E E ••• E E ••• E E

j'fc =0 jsr + l=0 jsr-l=0 Jsi+l=0 JS1-l=° jl=0 At • A ' = 1

1 1 nijSi,js;+l Ul.-jfc)

l=lll

a r ,

n ijS'.js. + 1 (jl,-,jfc)

where

r

j ,js i + i (ji , . . . , ) = Ijsr ,j'Sr + i . . . IjSi ,jSi + i (ji , . . . , ) ,

1=i

Q

Cso... CsisNk)(a) = 4k)(a), n j^+1 j..., j) = j... j),

1=i

j ,ji+i(jqi, . . . , jq2, , jq3,. . . , JV-2, j1, j'qfc-i, . . . , JV) = = (jqi, . . . , jq2 , j1+i , jq3 , . . . , JV-2 , j1+i , JV-i, . . . , j3fc ),

where l = qi,..., q2, q3,..., qk-2, qk-i,..., , l E N, aj;...j ) is a scalar value,

si,..., sr = 1,..., k - 1, sr > ... > si, qi,..., = 1,..., k, the expression

£

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 £ Csr... C., sNk) (a), (2.409)

jk=Q ji=Q r=Q sr,...,si=i

Sr >...>s i

where k = 2, 3, . . .

Hereinafter in this section, we will identify the following records

a(ji,...,jk) = a(ji...jk) = aj'i...jfc . In particular, from (2.409) for k = 2,3,4 we get the following formulas

N-i N-i

E E a(ji ,j2) = (a) + (a) =

j2=Q ji=Q

N-1 j2 — 1 N-1 N-1 j2 — 1

y] y] y] a(j1 j2) + ^^ a(j2j2) = ^^ 5^(aj1j2 + aj2j1 j2 =0 j1=0(j1,j2) j2=0 j2=0 j1=0

N — 1

+ E aj2j2, (2.410)

j2=0

N—1 N—1 N—1

£ £ £ a(jl,j2,j3) = sN3)(a) + C1SN3)(a) + C243)(a) + C2C^V) = j3 =0 j2=0 jl=0

N—1 j3 —1 j2 1 N—1 j3 —1

= + a(j2 j2 j3 ) +

j3 =0 j2=0 jl=0 (jl ,j2 J3) j3 =0 j2=0 (j2 ,j2,j3)

N —1 j3 — 1 N — 1

+ ^3 53 a(jlj3j3) + ^3 a(j3j3j3) =

j3=0 jl =0(jl,j3,j3) j3=0

N — 1 j3 —1 j2 1

= y ^ ^3 y ^ (ajlj2j3 + ajlj3j2 + aj2jlj3 + aj2j3jl + aj3j2jl + aj3jlj2) +

j3=0 j2=0 jl=0 N—1 j3 — 1 N —1 j3 —1

+ ^ ^ (aj2 j2 j3 + aj2 j3 j2 + aj3 j2 j2) + ^ ^ ^ ^ (ajlj3j3 + aj3jlj3 + aj3j3jl ) +

j3=0 j2 =0 j3=0 jl=0

N—1

+ £ aj3j3j3, (2.411)

j3=0

N—1 N —1 N —1 N —1

£ £ £ £ a(jlJ2J3J4) = ^IvV) + C1SN4)(a) + C2S^)(a) +

j4=0 j3=0 j2=0 jl=0

+C3S(4)(a) + C2C1S(4)(a) + C3C1S4V) + C3C2 (a) + C3C2C1S4V) =

N—1 j4 1 j3 —1 j2 1 N —1 j4 1 j3 —1

=

a(jlj2j3 j4) + a(j2j2j3 j4)

j4=0 j3=0 j2=0 jl=0 (jl,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

+

a(jl j3j3j4 ) a(jlj2 j4 j4 ) +

j4=0 j3=0 jl=0 (jl,j3,j3,j4) j4=0 j2=0 jl=0 (jl,j2,j4,j4)

N —1 j4 1 N — 1 j4 1

+

a(j3j3j3j4)

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(jlj4j4j4) + 53 aj4j4j4j4 =

j4=0 jl =0 (jl,j4,j4,j4) j4=0

N —1 j4 1 j3 —1 j2 — 1

= y ^ ^3 53 53 (ajlj2j3j4 + ajlj2j4j3 + ajlj3j2j4 + ajlj3j4j2 +

j4=0 j3=0 j2=0 jl =0

+ aji j4j3j2 + ajij4j2j3 + aj2jij3j4 + aj2jij4j3 + aj2j4jij3 + aj2j4j3ji + aj2j3ji j4 +

+ aj2 j3j4ji + aj3jij2j4 + aj3jij4j2 + aj3 j2jij4 + aj3j2j4ji + aj3j4jij2 + aj3j4j2 ji +

+ aj4 jij2j3 + aj4jij3j2 + aj4j2jij3 + aj4 j2j3ji + aj4j3jij2 + aj4j3j2ji ) + N-i j4 i j3-i

+ ^ ^ Ev (aj2j2j3j4 + aj2j2j4j3 + aj2j3j2j4 + aj2j4j2j3 + aj2j3j4j2 + aj2j4j3j2 +

j4=Q j3=Q j2=Q

+ aj3j2j2j4 + aj4j2j2j3 + aj3j2j4j2 + aj4j2j3j2 + aj4j3j2j2 + aj3j4j2j2) + N-i j4 i j3-i

+ ^ ^ Ev (aj3j3jij4 + aj3j3j4ji + aj3jij3j4 + aj3j4j3ji + aj3j4jij3 + aj3jij4j3 +

j4=Q j3=Q ji=Q

+ ajij3j3j4 + aj4j3j3ji + aj4j3jij3 +ajij3j4j3 + ajij4j3j3 + aj4jij3j3) + N-i j4 i j2 i

+ ^ ^ Ev (aj4j4jij2 + aj4j4j2ji + aj4jij4j2 + aj4j2j4ji + aj4j2jij4 + jjij2j4 +

j4=Q j2=Q ji=Q

+ ajij4j4j2 + aj2j4j4ji + aj2j4jij4 + ajij4j2j4 + ajij2j4j4 + aj2jij4j4) + N-i j4 i

+ ^ ^ 53 (aj3j3j3j4 + aj3j3j4j3 + aj3j4j3j3 + aj4j3j3j3) + j4=Q j3=Q

N-i j4 i

+ ^ ^ 53 (aj2j2j4j4 + aj2j4j2j4 + aj2j4j4j2 + aj4j2j2j4 + aj4j2j4j2 + aj4j4j2j2) +

j4=Q j2=Q

N-i j4 i

+ ^ ^ 53 (ajij4j4j4 + aj4jij4j4 + aj4j4jij4 + aj4j4j4ji ) +

j4=Q ji=Q

N-i

aj4j4j4j4 . (2.412)

j4=Q

Perhaps, the formula (2.409) for any k (k E N) was found by the author for the first time [67] (1997).

Assume that

/=1

where $ (ii,... ,ik) is a nonrandom function of k variables. Then from (1.16) and (2.409) we have

[k/2]

J rnS = E Ex

r=0 (sr ,...,Sl)GÄfc,r

N —1 jsr +2 — 1 jsr + 1 —1 jSl+2—1 jSl + l—1 j2 1

',i-E••• E E ••• E E ••■E E

x l.i.m

N

X

jk =0 jsr + 1=0 jsr-1=0 jSl+l=0 jSl-l=0 jl=0

,n IjS'.js' + 1 CilvJfc )

l=l

X

^ ( ■j'l , • • • , ■j's.-I , ■ S. + I , ■ S. + I , , • • • , ■jsr-1 , ■jsr + 1 , ■jsr + 1 , ■jsr+2 , • • • , ^'fc ) X

jsr-1' jsr + 1' jsr + 1' 'jsr +2-

xAwj • • • Aw^^^Awj) Aw^'Aw^^

• • • Aw(;sr-l)Aw(;sr) Aw(;sr+l) Aw(;sr+2) • • • Aw(ik)

Tjsr -1 Tjsr +1 Tjsr +1 Tjsr +2

jfc

[k/2]

£ £ I[*]Tk]Sl'-'Sr w. p. 1,

r=0 (sr,...,Sl)GÄfc,r

(2.413)

where

T tsr +3 tsr +2 tsr t«l+3 tSl + 2 t«l t2

I [*]

(k)sl,...,sr

T,t

t t t

t t t t

E

r

n its',««' + 1 (tl,...,tk )

X

^ ¿1, • • • , ¿s. —b ¿s^b ¿s^b ^+2 , • • • , ¿sr — b ¿sr¿sr+ b ¿sr +2, • • • , ¿k x

(il) J,T,.(;sl-l^ 7 (iSl ) , (iSl + l^„(iSl+2)

xdwt(;l) • • • dwt sl dwt sl dwt sl dwt sl

tl ts l — 1 ts l + 1 ts l + 1 t.

1+2

• • dwt(isr-l)dwt(isr) dwt(isr+l)dwt(isr+2) • • • dwt(ik)

tsr-1 tsr + 1 tsr + 1 tsr +2 tk

(2.414)

where k > 2, the set Ak,r is defined by the relation (12.351). We suppose that the right-hand side of (2.414) exists as the Ito stochastic integral.

Remark 2.3. The summands on the right-hand side of (2.414) should be understood as follows: for each permutation from the set

r

J^s i ^+i (tb ... ,tk) =

1=1

= |^ti, . . . , tSi-i, tSi+i, tSi+i, tSi+2, . . . , t.r-i, tSr+i, tSr+i, tSr+2, . . . , tkj

it is necessary to perform replacement on the right-hand side of (2.414) of all pairs (their number is equal to r) of differentials ^¿^dwj) with similar lower indices by the values 1{i=j=Q}dtp.

Note that the term in (2.413) for r = 0 should be understood as follows

T ¿2

/--./£ (*(ti.....tk)dw<;;'>...dw«;ik>),

¿ ¿ ^iv-^kO

where notations are the same as in (1.24).

Using (2.370), (2.371), (2.413), and (2.414), we get

M

J ^Tkt

2n

<

[k/2] f < E m\\i [$]Tk

r=0 (sr,...,S1)GÄfc,r

](k)s1,...,sr \t

2n

(2.415)

where

M

I №

(k)s1,...,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 niisi,tSI+1 (t1,...,tfc)

i=1

X$2n i ¿1. . . . . ts1 —1, ts1+1, ts1+1, ts1+2, . . . , tsr — 1, tsr+1 .tsr+1, tsr+2, • • • . tk ) X

X dt1... dts1—1dts1+1dts1+2... dtsr—1dtsr+1^+2... dtk, (2.416)

where Cnk and Ck" ^ are constants and permutations when summing are performed in (2.416) only in the values

^2n ( t1, . . . , tS1 —1, tS1+1, tS1+1, tS1+2, . . . , tsr — 1, tsr+1, tsr+1, tsr+2, . . . , tk ) .

Consider (23150 and (£33£l) for $(t1,...,tk) = RP1...Pk(tx,... ,tk)

where

M

(k)

J [RP1...Pk ]T.

2n

<

[k/2] . < E m||i [Rp1..pk ]

r=0 (sr,...,s1)eAfc,r

] (k)s1 ,...,sr ]T,t

2n

M

I [R ](k)s1 '...'Sr 1 [RP1...Pfc ]T,t

2n

<

(2.417)

T tSr+3 tsr + 2 tsr ts1+3 ts1+2 ts1 t2

s1...sr < Cnk

E

X

t t t t t t t nitSl,tS;+1 (t1,...,tk)

xRPr...p^ , . . . , ts1-1, ts1+1, ts1+1, ts1+2, . . . , tsr-b tsr+i , tsr^+2, . . . , x

x dti... dtSi-idtSi+idtSi+2... dt.r-idt.r+idt.r+2... dtk, (2.418)

where Cnk and Ck".^ are constants and permutations when summing are performed in (2.418) only in the values

R

2n

t1 5 ... 5 ts1 —1 5 tS1 + 1 5 tS1 + 1 5 tS1+2 5 ... 5 tSr — 1) tSr + 1) tSr + 1 5 tSr +2 5 - - - 5 tk ) .

From the other hand, we can consider the generalization of the formulas (2.379), (2.391), (2.398) for the case of arbitrary k (k E N). In order to do this, let us consider the sum with respect to all possible partitions defined by (1.52)

E

a

g1 g2,...,g2r-1g2r,q!...qfc-2r •

({{31>32}>--->{32r-1>32r }}>{91 >--->9fc-2r }) {31,32,---,32r-1>32r>91 >--->9fc-2r } = {1 >2 >--->k}

Now we can generalize the formulas (2.379), (2.391), (2.398) for the case of arbitrary k (k £ N)

/

2n) J 2n

(ti,... ,tk)) ndti.. .dtk+

M

J [RPl...Pk ]T.

(k) t

\jt,T ]k

[k/2] +E

E

1{iSl =ig2 =0} ••• 1{iS2r-l =;S2r =0}

X

r=1 ({{sl,32 },•••, {32r-l>32r }},{9lv.9k-2r }) {3l >S2v>S2r-l>S2r>9lv>9k-2r }={l,2v>k}

X

R

-pl...pfc

¿1, • • •, ¿k

2n

X

[t,T]

k-r

tgl =tg2 '...'t52r-l =tg2r.

x ( di1 • • • dik

(dtfll dtg2 )^dtfll ,...,(dtS2r-l dtg2r 2r-l

/

(2.419)

where Cnk is a constant,

¿1, • • •, ¿k

tgl =tg2 '...'t52r-l =tg2r

means the ordered set (ti,... ,tk), where we put tg = tg , Moreover,

• , ¿g2r-l ¿g2r '

di1• • • dik

(dtfll dtg2 )^dtfll ,...,(dtS2r-l dtg2r )^dtS2r-

means the product dti... dtk, where we replace all pairs by dtg1, ..., dtg2r correspondingly.

Note that the estimate like (2.419), where all indicators 1^} must be replaced with 1, can be obtained from the estimates (2.417), (2.418).

The comparison of (2.419) with the formula (1.53) (see Theorem 1.2) shows their similar structure.

Let us consider the particular case of (2.419) for k = 4

l

M

J [RP1P2P3P4 ]Tt

2n

< Cn4

(Rp1p2p3p4 (t1 5 12 5 13 5 t4))2n dt1 dt2dt3dt4 +

\jt,T ]4

+

E

1

({31,32},{91,92}) {31,32 ,91,92} = {1,2,3,4}

{ig1 =ig2 =0} J I JtP1P2P3P4 [t,T ]3

RP1P2P3P4 I t15 t2 5 13 5 14

2n

X

t31 =t32 .

x dt1dt2 dt3dt4

+

(dtfl1 dt32 )^dtS1

+ E 1{i31 =i32 =0}1{i33 =i34 =0}X

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

X J I Rp1p2p3p^ t15 t2513514 [t,T ]2

2n

X

t31 =t32 ,t33 =t34 ,

X ( dt1dt2 dt3dt4

(dt31 dt32 ) ^dt31 > (dt33 dt34 ) ^dt33

(2.420)

/

It is not difficult to notice that (2.420) is consistent with (2.398). According to (2.342) and (2.366), we have the following expression

RPi...Pk (t1, . . . , tk) = k / k-1 k-1 1 k-1 r k-1 ^

= Y[Mti) + 53 n^^+i} II Mti<ti+j

, 1=1 r= 1 sr,...,si = 1 1 = 1 i=1 /

y Sr>...>S1 i=S1,...,Sr /

1=1

P1 Pk k

n^j; (ti). (2.421)

¿1=0 jk=0 1=1

Due to (12.421) the function RP1...Pk(t1,..., tk) is continuous in the open domains of integration of integrals on the right-hand side of (2.418) and it is

bounded at the boundaries of these domains (let us remind that the iterated series

Pi Pk k

Urn ... Km^ ... £ Cjfc...ji J! j (ti)

P^TO Pk —»TO z-' z-' -1--1-

ji=0 jfc=0 l=1

converges at the boundary of the hypercube [t,T]k).

Let us perform the iterated passage to the limit lim lim ... lim under

pi — TO p2 — TO Pk — TO

the integral signs in the estimates (2.417), (2.418) (it was similarly performed for the 2-dimensional case (see above)). Then, taking into account (2.367), we get the required result.

From the other hand, we can perform the iterated passage to the limit lim . . . lim under the integral signs on the right-hand side of the estimate

p1—TO pk — TO

(2.419) (it was similarly performed for the 2-dimensional, 3-dimentional, and 4-dimensional cases (see above)). Then, taking into account (2.367), we obtain the required result. More precisely, since the integrals on the right-hand side of (2.419) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover,

lim ... lim Rpi...pk (ti,... ,tk ) = 0, (ti,..., tk) e (t,T )k

p1—TO pk —TO

and |Rpi...pk(ti,...,tk)| < to, (ti,...,tk) e [t,T]k.

According to the proof of Theorem 2.11 and (2.366), we have

Rpi...pk(ti,..., tk) = ( K*(ti,..., tk) - £ Cji(t2,..., tk) j(tiH +

V ji=0 J

Pi / P2

+ Cji (t2, . . . ,tk) - £ Cj2ji (t3, . . . ,tk ) j (t2) ) j (ti) ) +

ji=0 \ j2=0

/ pi Pk-i / Pk \

+ E • • • E ( Cjk-i...ji ) - E Cjk...jij (tk ) I 0jfc-i (tk-l) . . . j (t0 ) ,

\ji=o jk-i=o V jk=o /

where

T

Cji (t2,...,tk ) = ^ K *(ti,...,tk )j (ti)dti, t

Cj2ji(t3, . . . ,tk) = J K*(ti, . . . ,tk) j(ti) j(t2)dtidt2,

[¿,T ]2

» k-1 (tk) = K(ti,...,tk )JJ 0ji (ti )dti ...dtk-i.

7=1

[¿,T]k-i 7=1

Then, applying k times (we mean here an iterated passage to the limit lim . . . lim ) the Lebesgue's Dominated Convergence Theorem to the integrals

P,—to pk—TO

on the right-hand side of (2.419), we obtain

lim lim ... lim M

pi—œ p2—œ pk —œ

J [Rpi...pfc

(k) 2n~

= 0, n G N.

Theorems 2.13 and 2.10 are proved.

It easy to notice that if we expand the function K*(t1,..., tk) into the generalized Fourier series at the interval (t,T) at first with respect to the variable tk, after that with respect to the variable tk-1, etc., then we will have the expansion

Pk Pi k

K*(tl5...,tk )= lim ... lim E •••EC* ..¿ill ^ (ti ) (2-422)

'4, • • • 5 "k ! — ■ ■ ■ / v • • • / , ^¿fc...ji

pk—œ p1—t^oo z—' z—'

jfc=0 ji=0 1=1

instead of the expansion (2.343).

Let us prove the expansion (2.422). Similarly with (2.347) we have the following equality

oo T

<7,(7/,) (l.i, .. ,,;. + ^1.,, ,,;.) E / n-(li,)ojl(li,)<lli,ojl(l,,). (2.423)

jk=0 tk-i

which is satisfied pointwise at the interval (t, T), besides the series on the right-hand side of (2.423) converges when t1 = t,T.

Let us introduce the induction assumption

TT

to to „ „ k

E^E^M^ Wt3)j (t3)... J ^k (tk)j (tk)dtk ...dt^ 0ji (ti) =

jk =Q j3=Q ¿2 ¿k-1 7=3

k k-i 1

= + • (2-424) 7=2 7=2 ^ '

Then

T T

to to to „ „ k

E^EE^) / ^2) j fa) ... ^k (tk ) j (tk )dtk ...dt^ ^ fa)

jk=0 j3=0 j2 =0 I tk-i Z=2

TO to / 1 \

= • • • ( l{i!<i2} + j X jk=0 j3=0 ^ '

tt k

X J ^3(t3) j (t3) . . .J ^k (tk ) j (tk)dtk ... dt3 n j (t) =

t2 tk-i ^=3

1 \ TO TO

(1 \ TO TO

iifKfa} + 2 :1{*i=i2> ) 53 • • • 53 x

/ —n ;__n

2

jk=0 j3=0

TT k

XJ ^3(t3)0j3 (t3) ...J ^k (tk )0jk (tk)dtk ...dt^ (t1)

t2 tk-i ^=3 1 k k-l 1

^i(ii) (i{ii<i3} + |i{il=f2>) n^fa) n i1

\ / /—o o V

2~xh=h} j H'VW) 11 I Hti<ti+1} + ' /=2 /=2 ^

k k—i , 1 \

= Wl + 21™ • (2-425)

/=i /=i ^ '

From the other hand, the left-hand side of (2.425) can be represented in the following form

TO TO k

E-Ec-k ...jill ^ (t/) jk=0 ji=0 /=i

by expanding the function

T T

^i(ti^ ^2fa)j (t2) ...J ^k (tk ) j (tk )dtk . . .dt2

ti tk-i

into the generalized Fourier series at the interval (t,T) using the variable ti. Here we applied the following replacement of integration order

T T T

J ^i(ti) j (ti) y ^2fa)j (t2) ...J ^k (tk ) j (tk)dtk . . .dt2dti =

t ti tk-i

T is t2

= J ^k (tk (tk ) . . . J ^2(t2)0j2 (t2^ ^l(tl)0ji (tl)dtidt2 . . . dtk = t t t

= C

= Cjk•••ji.

The expansion (2.422) is proved. So, we can formulate the following theorem.

Theorem 2.14 [10] (2013) (also see [11]-[15], [32]). Suppose that the conditions of Theorem 2.10 are fulfilled. Then

œ œ k

J*[^(k)]T,i = E ...£Cj,.j,n<f, (2.426)

j =0 j1=0 1=1

where notations are the same as in Theorem 2.10. Note that (2.426) means the following

2n '

Pk Pi k

lim lim ...ÜM|( r[^]T,t - E • • • E (n Cj;0 ) H

pk—c» pk-i—œ p^^œ I \ z—' z—' A.A. jÎ i |

[ \ jk=0 ji=0 1=1

where n G N.

2.5.3 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,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

{( p p \ 2 |

№2)]T,f — ££ a,,, ¿j > = V ¿1=0 ¿2 =0 / J

{/ p p \ 2 I

p q

* ^ (2M \(~ Z E ^cji-'cf) \ +

ji =0 j2=0

\ 2

p q p p 2

-(ii)Ai2) V^ V^ n. . Aii) z(i2)

j2

ji=0 j2=0 ji=0 j2=0

2

+2" £ £ jiC'C' -EE jc

2

p q 2

2Ü-SM<(£ E j.crcf

ji =0 j2=P+1

p p q q

2j™ J™EE E E j, jiM{cj:',c|')}M{cj.;)cr}

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

j2 ji

p—TO q—TO

ji=0 j2 =p+1

p q p p

2 p—TO q—TO ( ^^ Cj 2j i 53 Cj 2j i \ji =0 j2 =0 ji =0 j2 =0

(2.427)

p q p p

2I pl—toEE ji- p—toEE j = <2-428)

ji=0 j2=0 ji=0 j2 =0 /

= J K2(ti,t2)dtidt2 — J K2(ti,t2)dtidt2 = 0, (2.429)

[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 (2.427) to (2.428) is based on the theorem on reducing the limit to the iterated one. Moreover, the transition from (2.428) to (2.429) is based on the Parseval equality.

Thus, we obtain the following Theorem.

Theorem 2.15 [14], [15], [32]. 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 (2.336) of multiplicity 2

*t *t2

J*[^(2)]T,t = / ^2) / ^l(tl)dwt(;i)dwi;2) (ii,i2 = 0, 1,...,m)

the following expansion

p

j * [V'(2)]t,(^ £ Cwi jj

j1j2=0

that converges in the mean-square sense is valid, where

T t2

Cj2ji = ^2(^2) j M / ^l(tl)0ji (tl)dtldt2

(2.430)

is the Fourier coefficient and

T

j = j (s)dw

jj t

(i) s

are independent standard Gaussian random variables for various i or j (in the case when i = 0), w[;) = f^ are independent standard Wiener processes (i = 1,..., m) and w[Q) = 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

«si ist, • • ■ "a. M<E-E II4*" - J* r'lr.

pi—œ p2—œ pk—œ

< lim lim ... lim

pi—œ p2—œ pk—œ

ji=0 jk=0 pi pk

1=l

k

<

m{£ ..■ECwII 4"1 - J '

ji=0 jk=0

T,t

< lim lim . . . lim M

pi—œ p2—œ pk—œ

1=l

pi pk

k

J '^It,, -E ..^Cjk^n Z

ji=0 jk=0

(il ) ijl 1=l

<

<

\ 2^ \1/2

pi pk k x 1 x

nön)

ji=0 jk=0 1=1

* J™ Jte,■ ■ ■ J™,Im{(r'fo-£••■£nc|") = o.

(2.431)

From the other hand,

/pi pk f k lim IE ... IE £ ... £ Cj^M nd"' - M {j'lAh,}

•00 p2—>-00 pk —TO \ z-' z-' J

ji=0 jk=0 ^ 1=1

pi pk ( k

lim lim ... lim E • • • E Ch~II ^ ~ M \r'.c'■'■'} '

p^TO p2—>-00 pk —TO z-' z-' _L_L J

ji=0 jk=0 u=1 J

(2.432)

Combining (2.431) and (2.432), we obtain

Pi Pk ( k

Mj-rr'' /,} lim Tim ... Tim E """ E M II " t2"433)

^ J pi—^oo p2—>-00 pk—TO z-' z-' -1--1-

ji=0 jk=0 I 1=1

The relation (2.433) with k = 2 implies the following

T

pi p2

lim !'•„ VV Cy.M {<f<H , (2.434)

-i—>-oo p2—to z——' L •yi 72 J

pi — TO p2—TO

ji=0 j2=0

where 1A is the indicator of the set A. Since

M {Cj(ii)C(22^ = 1{ii=i2=0}1{ji=j2}:

then from (2.434) we obtain

pi p2

l:— l:— \ \ n -I

-{ji=j2}1{ii=«2=0}

pi p2

m{j*[0(2)]tA = lim Tim V V CJ0J11 j ,, = ; 14

I J pi—TO p2—TO z—' z—'

ji=0 j2=0

min{pi,p2} to

l{ii=i2^o} lim lim E ^jiji = 1{h=«2^o}EC'^i' (2-435)

pi—TO p2—TO z-' z-'

j1=0 j1=0

where Cj is defined by (2.340) for k = 2 and j = j2, i.e.

T ¿2

Cjiji = J ^2(t2)0ji (t2^ ^i(ti)0ji (ti)dtidt2. ¿¿

From (2.434) and (2.435) we obtain the following relation

T

to 1

E < £ f Us)Us)ds. (2.436)

j1 2

ji=0 t

Combining (1.45) and (2.436), we have

p1 p2

J[*(2)]T,t = pl:p2mœ E E j ()11)()22) - 1{.,=.2=0}1{j1=,2}

j1=0 j2=0 V

P1 P2 œ

= pli^œ E E Cj2j1 Cjr)Cj22) - 1{i1=i2=0^ C

{ii=i2=0} / v Cjiji ji=0 j2=0 ji=0

T

pi p2

=„¡¿E^ E E cr1 - jif.^^oi / (2.437)

ji=0 j2=0 t

Since

T

J*^2^ = + j r,(.s-)r,(.s-)d.s- W. p. 1, (2.438)

t

then from (2.437) we finally get the following expansion

p1 p2

j * [V>(2']t,Î = l.i.m^ E^ cj:;1,c|;2).

p1;p2—œ z—* z—* 71 72

j1=0 j2=0

Thus, Theorem 2.2 is proved. We have

T

J*[<A(2)]?f = JlfXP + j>Pi(s)Us)ds =

t

pi P2 / \ 1 T

EEci;i}cj:2) - i{*i=^o}io-1=J2} + /

ji=0 j2=0 V J {

min{pi,p2}

pi p2 /1 T min{pi,p2} \

E E + i{i1=i2^o} 2 / Ms}us)ds ~ E ^ ji=0 j2=0 \ { ji=0 /

(2.439) where

pi p2

j(2>]!Tf2 =£ £ cj2ji cji-'cj:2»—i{!,=,2=0}i{ji=j2}

ji=0 j2=0 \

is the approximation of iterated Ito stochastic integral (2.337) (k = 2) based on Theorem 1.1 (see (1.45)).

Moreover, from (1.72), (1.73), and (2.8) we obtain

M { J*[v (2)]T,t — J*[v (2)]?f)2n} = M {(J [V (2)]T,t — J [V (2)]?tp02n} — 0

(2.440)

if pi,p2 — to (n e N). Further,

> 2n

^ (j * [v (2)]T,f —ee j, j

j =0 j2=0

= Mi 1 J*[^ (2)]T,t - J*(2)]Tf +

1 T min{p i,p2}

+ 1{*1=*2^0} ( 2 I ■Ms}ip2{s)ds - E

j =0

< K„ I M ^ I J*[^(2)]T,t - J*[^(2)]T1tf2

1 T min{p i,p2}

2 t j =0

where constant Kn < to depends on n.

Taking into account (2.436), (2.440), (2.441), we get

i / pi p2 \ 2n

„«¡2« M - £ £ j j j) ¡> = 0. (2.442)

^ V ji=0 j2=0

Thus, we obtain the following theorem.

Theorem 2.16. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, ^ (t), (t) are continuously differentiate nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral (2.336) of multiplicity 2

*t *t2

T* L/,(2)l _ I „1. f+ \ / ,/. \,7„,(:1^T,(:2)

Jfc = j ^2) j dw^ (ii,i2 = 0, 1,... ,m)

i i the following expansion

j *[^,2)]t,( = E j zj

-(«i)Z,:2 » j2

ji j2=0

that converges in the mean of degree 2n, n G N (see (2.442)) is valid, where the Fourier coefficient Cjj is defined by (2.430) and

T

j = J j (s)dw<!>

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 [107]. Suppose that

J 1/(X1,. . . )| (log+1/(x1, ... )|)k log+log+|/(X1,. .. )|x

[0,2n]k

x dxi . ..dxk < oo. (2.443)

Then, for the square partial sums

p p k

j 1=0 jfc=0 1=1 of the multiple trigonometric Fourier series we have

p p k lim E ...j i n^ji (xi) = f (xi,---,xk)

p—>■ oo z—» z—»

ji =0 jfc=0 1=1

almost everywhere in [0, 2n]k, where {0j(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) (xi )dxi.. .dxk

[0,2n]k 1=1

is the Fourier coefficient of the function f (xl5..., 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 (2.419) 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 (2.57)) we deduced that

n 1 n2 1

lim = -Mhyuti) = K^tuh), ti G (i,T),

n 1;n2—o z—' z—' 2

j 1=0 j2=0

(2.444)

where Cj2j1 is defined by (2.430).

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 (2.419), since Proposition 2.3 and (2.444) 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 (2.445)

p—to ' ' ji=0 j2 =0

to the function K*(t1 ,t2) (the proof of the series (2.445) boundedness is omitted). 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 (2.47) and Theorem 2.10. The case ii = i2 follows from (2.439) and (2.440). Consider the case ii = i2 = 0. We have

K *(ti,t2) + K *(t2,ti) = K'(ti,t2), (2.446)

where the functions K'(ti,t2) and K*(ti,t2) are defined by (2.47) and (2.77) correspondingly. Note that the function K'(ti, t2) is symmetric, i.e. K'(ti, t2) = K '(t2,ti).

By analogy with (2.368) we get w. p. 1

i n-i n-i J[K'/2]M = -l.i.m. E -

1

2 n-to ^^ v1 2/ tl1 "2 /2=0 1i=0

'N-1 /2-1 n-1 zi-r

= (EE+EE)A-v.,ri2)Af<;"Af««+

J2=0 Z1=0 Z1=0 Z2=0, N1

1 1 2

2 N—to f—^ V 11 ■

1N

= öLLm- E E + K'{Th,Th)) Af£>Af£> +

2 N-to /2=0 /1=0 1 2

1 N-1 2 +-l.i.m. =

2 N-to f—' v 11 /

/1=0

N-1 /2-1 1 N-1 2

n-to /2=0 /1=0 1 2 2 n-to /1=0 v 1

T t2 T

/1=0

N-1 /2-1

i/jo(to) I Mt^df^df^ + ^ I Uti)Uti)dti

* T * ¿2

= / ^i(ii}dft(il)dft(;i) =f J*[^(2)]T,t, (2.447)

t t

where we used the same notations as in (2.368).

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 (2.56))

K'{t iM) = 2

T T

1 Pi P2 n. n.

= 9 lim EE/ / =

2 J {

P1 P2 / T ¿2

= \ lim V V / ^2(i2)^3(i2) I MhWn{ti)dtidU+ ji=0 j2=0 \ t t T T \

+ ^ (t2)0j2 (t2^ ^2 (tl)0ji (tl)diH dt20ji (tl)0j2 fa) = t t2

1 Pi P2

= o lim + (2-448)

2 Pi,P2 — ^ z—' z—'

ji=0 j2=0

where the series (2.448) 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 (2.448) we replaced the order of integration in the second iterated integral.

Using (2.447), (2.448), and the scheme of the proof of Theorem 2.10 (k = 2), we can obtain the following relation (the proof of the series (2.448) boundedness on the boundary of the square [t, T]2 is omitted)

Pi P2 x

1

j1=0 j2=0

(2-449)

Let us rewrite the sum on the left-hand side of (2.449) as two sums. Let us replace j with j2, j2 with ji, pi with p2, and p2 with pi in the second sum.

Thus, we get

v 2n

P1 P2 x

lisro^(J*[^(2)iT,f-EECj2j1 w]

j1=0 j2=0

Theorem 2.16 is proved.

Let us consider another approach. The following fact is well known [104

Proposition 2.4. Let {xni,

: ... n=i be

a multi-index sequence and let

there exists the limit

lim xninfc < to.

ni,... , nfc—TO

Moreover, let there exists the limit

lim Xni,...,nfc = yni,...,nfc-i < to for any ni,... ,n*_i.

nfc —to

Then there exists the iterated limit

lim lim x

and moreover,

n!,...,nfc

n1,...,nfc_1—TO nfc—TO

lim lim Xn1,...,nfc = lim x„1)...int.

n1,...,nfc_1—TO nfc — TO n1,...,nfc — TO

Denote

Cjs...ji (ts+i,... ,tk )= i K (ti,...,tk) n j (ti)dti ...dts,

7=1

[t,T]s 7=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

pi p2

lim £ £ Cj2ji(ts,..., t*) j (ti) j (t2), (2.450)

Pi,P2—TO z-' z-'

ji=0 j2=0

P1 P1 P3

lim^ £ 13 Cj (t4,..., tk ) j (t1)0j2 M j (ts), (2.451)

j1=0 j1=0 j3=0

Pi Pfc-i

lim E"- E Cjfc-i...ji (tk )j (ti) --j (tk-i), (2.452)

ji=0 jfc-i=0

Pi Pk

lim E - - - E Cjk...ji j(ti)... j (tk), (2.453)

ji=0 jk=0

where t1, - - -, tk £ [t,T], (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 (I2.45UI)-(I2.453I) even for the case p1 = - - - = pk and trigonometric Fourier series. Obviously, at least for the case k = 2 and ^1(r), ^2(r) = 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 (r2.45Q)-('2.45,3) 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 Cj2ji(t3, - - -, tk) j(ti) j(t2) = (2.454)

ji=0 j2=0

Pi P2

= lim E ECj'2ji (t3,---,tk ) j (ti)0j2 (t2) = ji=0 j2=0

P1 P2 TO

lim E E E Cj3j2ji (t4, . . . , tk) j (tl)0j2 (t2)0j3 (ta) =

'Ho—^rsr, * * * * * *

P1,P2—TO

ji=0 j2=0 j3=0

Pi P2 P3

li m EE ECj-i (t4,...,tk ) j (ti)0j2 fa) j (ta) = (2.455)

P1,P2,P3^TO „ . „ . „

ji=0 j2=0 j3=0

TOTOTO

E E E Cj3j2ji(t4,..., tk)0ji(ti)jfa) j(ta) = (2.456)

ji=0 j2=0 j3=0

Pi P2 P3 TO

lim E EE ECj4...ji (t5,...,tk)0ji (ti) j (t4) = (2.457) ip

Pi,P2,P3^TO z

ji=0 j2=0 j3=0 j4=0

Pi Pk

= lim £ ...^Cjk ...ji j (ti) ...j (tk). (2.458)

ji=0 jk=0

Note that the transition from (2.455) to (2.456) is based on (2.454) and the proof of Theorem 2.11. The transition from (2.456) to (2.457) is based on (2.455) and the proof of Theorem 2.11.

Using (2.458), we could get the version of Theorem 2.10 with multiple series instead of iterated ones (see Hypothesis 2.3, Sect. 2.6).

2.5.4 Refinement of Theorems 2.10 and 2.14 for Iterated Stratonovich Stochastic Integrals of Multiplicities 2 and 3 (ii,i2,is = 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 [32]. Suppose that every (t) (/ = 1, 2,3) is a continuously differentiate function at the interval [t,T] and (x)}TO=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)]T,t and J*[^(3)]T,t (ii,i2,is = 1,...,m) defined by (2.336) are expanded into the converging in the mean-square sense iterated series

lim lim M

P1—TO P2 — TO

J *[^(2)]

T,t

P1 P2

EEc

j1=0 j2=0

Z(:1)Z(:2) j2j1 Zj1 Zj2

= 0, (2.459)

lim lim M

P2—TO P1 — TO

J *[^(2)]

P2 P1

C (:1) C (:2)

j2=0 j1=0

j2

= 0, (2.460)

lim lim lim M

p1—TO p2 — TO p3—TO

lim lim lim M

P3—TO P2 — TO P1—TO

P1 P2 P3

C (:1) C (:2)C (:3) T,t Z^ Z^ / ,CJ3j2j1 Cj1 Cj2 Cj3

j1=0 j2=0 j3=0

P3 P2 P1

c (:1) c (:2)C (:3) T,t Z^Z^ / ,CJ3j2j1 Cj1 Cj2 Cj3

j3=0 j2=0 j1=0

= 0,

(2.461)

= 0,

(2.462)

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 (2.340) and (2.338).

Proof. We will prove the equalities (2.459) and (2.461) (the equalities (2.460) and (2.462) can be proved similarly using the expansion (2.422) instead of the expansion (2.343)).

From (2.368) we have w. p. 1

Pi P2

J* [V'(2)]T,f-EE j jj = J [RpiP2 ]T,t =

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

+ l{ii=i2^ Rpip2(ti,ti)dti, (2.463)

t

where we used the same notations as in (2.368). Uning (2.463), we obtain

T t2 T ti

m{ (j [Rpip2 ]T2t)2} = J y RL (ti ,Î2)dMt2 + J y (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, ¿2)dtidt2 + J y R22iP2 (ti,t2)dtidt2 + t t t t2 T t2

+ 1{ii=i2} | y y Rpip2 (ti,t2)Rpip2 (t2,ti)dti dt2 + tt

TT \ / T

+ J j Rp1p2 (t1,t2)Rp^ (t2, t1)dt2dtM + 1{n=:2} i y*Rp1p2 (ti, t1)dt1 t t1 /

2

- J RPiP2(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 )dtidt^ J + 1{i1=i2^ J' RP1P2 (ti,ti)dti

t t2 / V t

= J (ti,t2)dtidt2+

[t,T ]2

/ / T \

+ 1{i1=i2} J Rp1p2 (ti,t2)Rp1p2 (t2, ti )dtidt2 + | J Rp1p2 (ti,ti)dti

V'T]2 Vt ! )

(2.464)

Since the integrals on the right-hand side of (2.464) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim RPiP2(ti,t2) = 0

Pi—TO P2 — TO

holds for all (ti,t2) G (t,T)2 and |RPiP2(ti,t2)| < to for all (ti,t2) G [t,T]2 (see

(E3B3)).

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—TO

account (2.344), (2.345), and (2.380), we obtain

2

[t,T ]2

lim lim / R:.p2(tb t2)dMt2 = 0, (2.465)

P1 — TO p2—TO / i'^2

lim lim / RP1P2(t1;t2)RP1P2(t2,t1)dt1dt2 = 0, (2.466)

p1 — TO P2—TO J

[t,T ]2

2

T

lim lim / (ti,ti)dti = 0. (2.467)

pi—œp2—œ J t

The relations (2.464) (2.467) imply the following equality

M<! (J[R^IS]2

Pi—œ P2—œ

m '—m M{ J [RP1P2 ]T201=°-

The relation (2.459) is proved.

Let us prove the relation (2.461). After replacement of the integration order in the iterated Ito stochastic integrals from (2.383) [1]-[15], [68], [105], [106] (see Chapter 3) we get w. p. 1

Pi P2 P3

J *[^(3)]T,t - £ £ £ Cj3j2 ji j jj = J [RPiP2P3 ]T,t = ji=0 j2 =0 j3=0

T t3 t2

= JJ I RP1P2P3(tl't2't3)dfiii)dfi2i2)df(3i3) + i i i T t3 t2

+//1RP1P2P3(ti't3,t2)dfi;i)dfi;3)dft(3i2)+ i i t T t3 t2

+//1RP1P2P3(t2'ti,t3)dfi;2)dft(;i)dft(3i3)+ t t t T t3 t2

+ 1 I I RP1P2P3 (t2't3,tl)dft(;3)dft(;i)dft(3i2) + i i i T t3 t2

+//IRP1P2P3(t3't2,ti)dft(;3)dfi;2)dft(3ii)+ i t t T t3 t2

+ // / RP1P2P3 (t3 ,tl,t2)dfi(;2)dfi(2i3)dfi(3i1 ) +

i i i

T / T

+ I{ii=i2^ y I J RP1P2P3 (t2,t2,t3)dt^ dft33 +

i \i

}(} \ (n)

+ 1{i2=i3W I / RPiP2P3 (ti , t2, t2)dt^ I +

T / T \

+ l{ii=i3^ J RPiP2P3(t3,t2,t3)dtJ dft(2i2)- (2.468)

(3)

Let us calculate the second moment of J[RPiP2P3]T t using (2.468). We have

M j fJ[RPiP2P3]T3]) | =

T t3 t2

2

= ///( S (ti,t2,t3H dtidt2dt3+ (2.469)

t t t \(ti, t2 ,t3) /

/ T t3 t2

+2 ( 1{ii=i2} J J J GPP1P2P3 (tb ^ t3)dt1dt2dt3 + ttt T t3 t2

+ 1{ii=i3^^y J GP2P2P3 (tb ^ t3)dt1dt2dt3 + ttt T t3 t2

+ 1{i2=i3^^y J GP3P2P3 (tb ^ t3)dt1dt2dt3 + ttt

T t3 t2 \

+ 1{ii=i2=i3^^y J GP1P3 (tb ^ J + ttt

+ I (1{ii=i2}RPiP2P3 (t1,t1,t3)RPiP2P3 (t2,t2,t3) + [t,T ]3

+ 1{i2 = i3}RPiP2P3 fe ^ ^№^3 (t3, t2, t2) +

+ 1{ii=i3}RPiP2P3 (t1, t3, t1)RPiP2P3 (t2, t3, t2) +

+2 • 1{ii=«2=i3^RPiP2P3 (tb ^ ^№^3 (t3, t2, t2) +

+ RPiP2P3 (tb ^ ^^P^Pij (t2, t3, t2) +

+Rp1p2p3(ta, ti, ti)^PiP2P3(t2, ta, t2) J j dtidt2dta, (2.470)

where permutation (ti,t2,ta) when summing in (2.469) are performed only in the value RpiP2P3(ti,t2,ta) and the functions Gp1P2P3(ti,t2,ta) (i = 1,..., 4) are defined by the following relations

G^i1p2p3 (ti,t2,ta) = RP1P2P3 (ti,t2,ta)RpiP2P3 (t2,ti,ta) + +Rp1p2p3 (ti, ta, t2)Rp1p2p3 (ta, ti, t2) +

+Rp1P2P3 (t2, ta, ti) Rp1p2p3 (ta, t2, ti),

GP2P2p3 (ti,t2,ta) = RP1P2P3 (ti,t2,ta)RpiP2P3 (ta,t2,ti) + +RP1P2P3 (tb ^ ^^^p^ (t2, ta, ti) + +RP1P2P3 (t2, ^ ^^p^^ (ta, tl, t2),

Gp3p2p3 (ti,t2,ta) = RP1P2P3 (ti,t2,ta)RpiP2P3 (ti,ta,t2) + +Rp1p2p3 (t2, ti, ta)Rp1p2p3 (t2, ta, ti) +

+RP1P2P3 (ta, t2, ti)Rp1p2p3 tl, t2),

GJP4P2P3 (ti,t2,ta) = RP1P2P3 (ti,t2,ta)RpiP2P3 (t2,ta,ti) + +RP1P2P3 (tb ^ ^^^p^ (ta, tl, t2) + +Rp1p2p3 (ti, ta, t2)Rp1p2p3 (t2, ti, ta) +

+RP1P2P3 (tb ^ ^^^p^ (ta, t2, ti) +

+RP1P2P3 ^ ^^p^p;? (ta, t2, ti) +

+RP1P2P3 (t2, ta, ti)Rp1p2p3 tl, t2).

Electronic Journal. http://diffjournal.spbu.ru/ A.380

Further (see (1.38)),

T ts t2

J J J i £ RpiP2Ps(ti,t2,tsM dtidt2dts

t t t \ (ti,t2,ts)

= J R2ip2ps(ti,t2,ts)dtidt2dts. (2.471)

[t,T ]s

We will say that the function $(ti,t2,ts) is symmetric if

$(ti,t2,ts) = $(ti,ts,t2) = $(t2,ti ,ts) = $(t2,ts ,ti) = = $(ts,ti,t2) = $(ts,t2 ,ti).

For the symmetric function $(ti,t2,ts), we have

T ts t2 / \

/// S ^(ti,t2,ts) I dtidt2dts =

t t t \(ti,t2,tS) /

T ts t2

= 6///^(ti,t2,ts)dtidt2 dts =

ttt

= J $(ti,t2,ts)dtidt2dts. (2.472)

[t,T ]s

The relation (2.472) implies that

T ts t2

J J j §{tiMM)dtidtodU = i J <S>{tht2,h)dtidt2dh. (2.473)

t t t [t,T]s

It is easy to check that the functions GpiP2Ps(ti,t2,ts) (i = 1,...,4) are symmetric. Using this property as well as (2.470), (2.471), and (2.473), we obtain

m{ J[Rpip2ps]TSt)2} = J RL^(ti,t2,ts)dtidt2dts+

[t,T ]s

+ 2 J ^{i1=i2}G^P2P3{tiA-2A-3)dtidt2dts+

[t,T ]3

+1{n=i3}GP2P2P3 (ti, t2, ta)dtidt2dta+ +1{i2=i3}G]piiP2P3 (ti, t2, ta)dtidt2dta+

+1{i1=i2=i3}G]p4P2P3 (ti ,t2,ta)dtidt2dt^j dtidt2 dta+

+ I (1{ii=i2}RPiP2P3(ti,ti,ta)RPiP2P3(t2,t2,ta) + [t,T ]3

+1{i2=i3}Rp1p2p3 (ta, ti, ti)Rp1p2p3 (ta, t2, t2) + +l{i1=i3}Rp1p2p3 (ti, ^ ^Äp^^ (t2, ta, t2) +

+ 2 • 1{ii = «2=i3^ RP1P2P3 (tl ,tl,ta)RpiP2P3 (ta ,t2,t2) +

+Rp1p2p3 (ti, ti, ta)Rp1p2p3 (t2, ta, t2) +

+Rp1P2P3 (ta, ^ ^^p^ (t2, ta, t2^ ^ dti dt2dta. (2.474)

Since the integrals on the right-hand side of (2.474) exist as Riemann integrals, then they are equal to the corresponding Lebesgue integrals. Moreover, the following equality

lim lim lim RPiP2P3(tl,t2,ta) = 0

Pi—TO p2—TO p3 — TO

holds for all (tl,t2,ta) G (t,T)a and |RPiP2P3(tl,t2,ta)| < to for all (tl,t2,ta) G [t,T]a (see (23671)).

Using (2.392) and applying three times (we mean here an iterated passage to the limit lim lim lim ) the Lebesgue's Dominated Convergence Theorem

Pi—>-TO P2—yTO P3—>-TO

in the equality (2.474), we obtain

lim lim lim M { (j[R«™} = 0.

Pi—TO P2 — TO P3—TO [V J

The relation (2.461) 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.

2.6 The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k (k E N) Based on Theorem 1.1

2.6.1 Formulation of Hypotheses 2.1—2.3

In this section, on the base of the presented theorems (see Sect. 1.1.3, 2.1-2.5) 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 (11.43) 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]-[15], [37]. Assume that [^ (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

I,*£5r,f = / ..•/ 'iwi:'1 ...<iw«-» (2.475)

t t

the following expansion

p k

CÄt = £ ...„n 4:'1 (2.476)

j'i,...jfc=0 1=1

that converges in the mean-square sense is valid, where

t t2

Cj* ...j = J j fa).. .J j (ti)dti... t t

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, il5..., = 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^ = f^ 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

I(t H*jT,t = E IlCf, (2.477)

j i,--.jfe=0 /=i

where

P™ M { (it '^.t - it "X*^ =0-

The integrals (2.475) will be used in the Taylor-Stratonovich expansion (see Chapter 4). It means that the approximations (2.477) may be very useful for the construction of high-order strong numerical methods for Ito SDEs (see Chapter 4 for detail).

The expansion (2.476) 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], [15], [37]. Assume that (x)}°=0 is 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 multiplicity k

*t *t2

J*[^(k)ht = ^k (tk )... (ti)dwt(;i)... )

the following expansion

p k

J* №<*>]„ = l.i.m. £ Zl") (2.478)

/1 H / ji,...jfc =0 1=1

that converges in the mean-square sense is valid, where

t t2

cjfc •••ji = / ^k (tk ) j (tk)... (ti)0ji (ti)dti

/1

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^ = f^ for i = 1,..., m and wT0) = t.

Hypothesis 2.2 allows to approximate the iterated Stratonovich stochastic integral J*[^(k)]T,t by the sum

p k

J *№(k) IT,( = £ CVjill j (2.479)

ji,-jfc=0 1=1

where

Hm M | J*[^(k)]T,t - J*[^(k)]T,t 1^=0.

Let us consider the more general hypothesis than Hypotheses 2.1 and 2.2.

Hypothesis 2.3 [14], [15], [37]. Assume that (x)}°=0 is 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 multiplicity k

*t *t2

tfc

J*№(k)kt = ^ (tk )... 1 )... dw

the following expansion

P 1 Pk k

J *[^(k)]r,( = l.i.m. E nCj^ (2.480)

p Pi.—' ' ' * J;

j i =0 jk =0 1=1

that converges in the mean-square sense is valid, where

T t2

Cjfc ...j 1 =

j

t

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, ii,..., ik = 0,1,..., m,

T

j = J j (s)dw<*>

t

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. Let us consider the idea of the proof of Hypotheses 2.1-2.3. According to (1.10), we have

p i pk k

l.i.m E i nCjg' ' = J №(k)]T,,+

j l =0 jk =0

p l pk k

+ J-^E -ECk- i y-m; E IK(Tig)Aw[;g) w.p.l,

(2.481)

^ ^ ^ ... ~ -m- H j (Tig1/xw(ig)

j i =0 jk=0 (1 i,...,1k)GGk g=1

where notations are the same as in (1.10).

From (2.481) and Theorem 2.12 it follows that

Pi Pk k

J *W'№)]r,i = l.i.m. E ..^Cj,...,^ j) (2.482)

ji=0 jk=0 g=l

if

[k/2] .

r=1 (sr ,...,si)eAk,r

Pi Pk k

= l.i.m. £ ..^Cjk...,i l.i.m. Y, II j K )AwT;g) w.p. 1,

where notations are the same as in Theorems 1.1 and 2.12.

Note that from Theorem 1.1 for pairwise different ii,..., ik (ii,..., = 0,1,... ,m) we obtain (2.482) (compare (1.43) and (2.482)).

In the case pi = ... = pk = p and ^ (s) = 1 (/ = 1,..., k) we obtain from (2.482) the statement of Hypothesis 2.1 (see (2.476)).

If pi = ... = pk = p and every ^(s) (/ = 1,..., k) is an enough smooth nonrandom function on [t,T], then we obtain from (2.482) the statement of Hypothesis 2.2 (see (2.478)).

In the case when every ^(s) (/ = 1,..., k) is an enough smooth nonrandom function on [t,T] we obtain from (2.482) the statement of Hypothesis 2.3 (see (2.480)).

2.6.2 Hypotheses 2.1—2.3 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 E [0,T]. Let fSi)p, p E N be some approximation of fs(i), i = 1,... ,m. Suppose that fs(i)p converges to fs(i), i = 1,...,m ifp —y oo 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. [64], [65], 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 iterated Ito stochastic integrals and solutions of Ito SDEs. The piecewise linear approximation as well as the regularization by convolution [64]-[66] 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 E [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 [108], [109] (also see Sect. 6.1 of this book for detail)

oo T T

p p

f«" - ff» = Y j(s)ds Cj", j = / j(s)dfj°, (2.483)

j=0

where t E [t, T], t > 0, (x)}°=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 (2.483) converges for any t E [t, T] in the mean-square sense.

Let fTi)p — ft(i)p be the mean-square approximation of the process fT^ — ft(i), which has the following form

p t

^ c

f(.»p - f(.)p = £ jj(s)ds j». (2.484)

_n "

- (i)

I (s)ds z

j=0

From (2.484) we obtain

f )p = £ j(T)Cfdr. (2.485)

j=0

Consider the following iterated Riemann-Stieltjes integral

T t2

^k (tk)... ^i(ti)dwt(; 1 )p 1... dwt(;k)pk, (2.486)

where p1,... ,pk E N, i1,..., ik = 0,1,..., m,

dw(i)p =

dfTi)p for i = 1,..., m

dr

p e N,

for i = 0

(2.487)

and dfri)p in defined by the relation

Let us substitute (2.487) into (2.486)

T

t2

^k (tk )

Pi

Pk

Vi (ti)dw';i)P1 ... dw<;k»Pk = £ ... £ Cjk..ji n j(2.488)

ji=0 jk=0

1=1

where p1,... ,pk e N,

T

Zj" = J j (s)dw

t

(i)

are independent standard Gaussian random variables for various i or j (in the

case when i = 0), wSv = fSv for i = 1,..., m and wS0) = s,

T t2

Cjk ...ji = ^k (tk )jjk (tk) .. ^i (ti)jji (ti)dti ...dtk

is the Fourier coefficient.

To best of our knowledge [64]-[66] the approximations of the Wiener process in the Wong-Zakai approximation must satisfy fairly strong restrictions [66 (see Definition 7.1, pp. 480-481). Moreover, approximations of the Wiener process that are similar to (2.484) were not considered in

(also see

66], Theorems 7.1, 7.2). Therefore, the proof of analogs of Theorems 7.1 and 7.2 [66] for approximations of the Wiener process based on its series expansion (2.483) (also see (6.16)) should be carried out separately. Thus, the mean-square convergence of the right-hand side of (2.488) to the iterated Stratonovich stochastic integral (2.6) does not follow from the results of the papers (also see [66], Theorems 7.1, 7.2) even for the case pi = ... = pk = p.

From the other hand, Theorems 1.1, 2.1-2.9 from this monograph can be considered as the proof of the Wong-Zakai approximation based on the iterated Riemann-Stieltjes integrals (12.486) of multiplicities 1 to 5 and the Wiener

process approximation (2.484) on the base of its series expansion. At that,

k

s

the mentioned Riemann-Stieltjes integrals converge (according to Theorems 2.1-2.9) to the appropriate Stratonovich stochastic integrals (12.6). Recall that

(x)}=0 (see (2.483), (2.484), and Theorems 2.1-2.9) is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space ¿2([i,T ]).

To illustrate the above reasoning, consider two examples for the case k = 2, ^i(s), ^2(s) = 1; ii,i2 = 1,... ,m.

The first example relates to the piecewise linear approximation of the multidimensional Wiener process (these approximations were considered in [64]-[66]).

Let b^(t), t G [0,T] be the piecewise linear approximation of the ith component ft(i) of the multidimensional standard Wiener process ft, t G [0,T] with independent components ft(i), i = 1,..., m, i.e.

/Om-fW I t~kAAfW

A \l) — k,A + ^ /cA>

where Af« = f^ — f« , t E [kA, (k + 1)A), k = 0,1,...,N — 1. Note that w. p. 1

db(i) Af(i)

= t G [kA,(k + l)A), k = 0,1,..., TV - 1. (2.489)

Consider the following iterated Riemann-Stieltjes integral

T s

Jy'dbJ;i)(T)dbJ;2)(s), ii,i2 = 1,...,m. 00

Using (12.489) and additive property of Riemann-Stieltjes integrals, we can write w. p. 1

T s T s (^) (^)

J J^(r)d^(s) = j /^M^Vi-0 0 0 0

E / E /

1=0 /A \q=0 à ^ y

Af <i2) A

N-1 1-1 . N-1 (1+i)A s

E E ^AAfS' + ¿5 E AfiA'AfS' J J drds =

1=0 q=0 1=0

1A 1A

N-1 1-1 i N-1

= EE^AAfS' + ^E^'^'- <2'49°)

1=0 q=0 1=0

Using (2.490), it is not difficult to show (see Lemma 1.1, Remark 1.2, and (2.8)) that

T s T s T

LLm. J J dbll] (T)db{^(s) = J J di^di^ + l-l[n=n} J d.s =

T s T s T

1

2 ^{n=«2}

0 0 0 0 0

*T *s

= J J df^df^, (2.491)

00

where A ^ 0 if N ^ to (NA = T).

Obviously, (2.491) agrees with Theorem 7.1 (see [66], p. 486).

The next example relates to the approximation (2.484) of the Wiener process based on its series expansion (12.483), where t = 0 and (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 „,.2 = ,...,., ^

00

where df()p is defined by the relation

Let us substitute (2.485) into (2.492)

T s

f p

J J df^df^ = £ Cj jj, (2.493)

0 0 j'lj2=° where

T s

Cj2j1 = j (S) / j (T)dTdS

is the Fourier coefficient; another notations are the same as in (2.488).

As we noted above, approximations of the Wiener process that are similar to (2.484) were not considered in [64], [65] (also see Theorems 7.1, 7.2 in [66]). Furthermore, the extension of the results of Theorems 7.1 and 7.2 [66] 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, 2.15, we obtain from (2.493) the desired result

T s p

l.i.m. / i df(il)pdfii2 )p = l.i.m. E Cj j0 j2)

p—to j j P—to z—' Ji J2

0 o j2=0

*T *s

= J J dfTil}dfs(i2). (2.494)

0 0

From the other hand, by Theorem 1.1 (see (1.45)) for the case k = 2 we

obtain from (2.493) the following relation

T s p

l.i.m. / i dfiil)pdfs(i2)p = l.i.m. E jjj =

p—TO j j P—TO z—' J1 J2

0 0 jiJ2=°

p to

= p—mTO S Cj2ji (C^Cj^ - 1{;i=;2|1{ji=j2^ + 1{;i=;2} E Cjiji = ji ,j2=0 ji =0

T s

T s TO

J J dtff) + 1{;i=;2} E Cjiji . (2.495)

0 0 ji=0

Since

E^m = ¿E (J Mr) A = \ (T[Mr)dr\ = \ Ids,

j!=0 / \o J o

then from (2.8) and (2.495) we obtain (2.494).

2.6.3 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 [64]-[66] (also see

110]-[117]). 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 [64]-[66] (also see [110]-[117]) choose a piecewise linear approxima-

tion 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 [75], [76] (see Chapter 4). It should be noted that the authors of the works [74] (pp. 438-439), [75] (Sect. 5.8, pp. 202-204), 76] (pp. 82-84), [84] (pp. 263-264) mention the Wong-Zakai approximation

64]-[66] 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], [15]) can be considered as the proof of the Wong-Zakai approximation for iterated Stratonovich stochastic integrals of multiplicities 1 to 5 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 2.2-2.10, 2.14, 2.17 and Hypotheses 2.12.3 of this monograph as statements on the convergence of the iterated Riemann-Stiltjes integrals (12.486) to the iterated Stratonovich stochastic integrals (2.336).

Theorem 2.18 [37] (reformulation of Theorem 2.2). Suppose that the following conditions are fulfilled:

1. Every ^¿(t) (l = 1, 2) is a continuously differentiate function at the interval [t,T].

2. (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

J*[#2)]T,t = J ^2(t2)y ^i(ti)dwJ1udw^ (ii,i2 = 0,1,...,m)

t t

the following formula

T ¿2

J*[^(2)ht = l.i.m. i ^fa) / ^i(ti)dwt(;i)pidwt(i2)p2

J J 12

i i

is valid.

Theorem 2.19 [37] (reformulation of Theorems 2.3 and 2.5). 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 * t3 * ¿2

J J J <>dft(2,2)dft(3,3) (ii,i2,i3 = 1,...,m)

i i i

the following formula

*T * ¿3 * ¿2 T t3 t2

JJJ dft(;l)dft(2i2)dft(зiз) = p H.m.^ JJ J fii)pif2i2)p2f;3)p3 t t t t t t

is valid.

Theorem 2.20 [37] (reformulation of Theorem 2.4). 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 * t; * Î2

ffi' =f (t - W (t - ¿s)'2/ (t - ¿i)'1

the following formula

T is t2

ffi) = l-i.m. /(i - t3)ls / (t - (t - ti)''dff)pi dff)p2dfi(s's)ps,

where ii, i2, i3 = 1,..., m, is valid for each of the following cases

1. ii = ¿2, ¿2 = ¿3, ii = ¿3 and li, /2,/3 = 0,1, 2,...

2. ii = ¿2 = i3 and /1 = /2 = /3 and /1, /2, /3 = 0,1, 2,...

3. ii = ¿2 = i3 and /i = /2 = /3 and /i, /2, /3 = 0,1, 2,...

4. ii,i2,i3 = 1,..., m; /i = /2 = /3 = / and / = 0,1, 2,...

Theorem 2.21 [37] (reformulation of Theorem 2.6). 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

where i1, i2, i3 = 1,..., m, the following formula

T is t2

J* [^<3)]t,( = 1^ / ^3) / ^m/ mfff*

is valid for each of the following cases:

1. ¿i = ¿2, ¿2 = ¿3, ¿i = ¿3,

2. ¿i = ¿2 = ¿3 and ^i(t) = ^2(t),

3. ii = ¿2 = ¿3 and ^2(t) = ^3(t),

4. ii,i2,i3 = 1,..., m and (t) = ^2(t) = ^3(t).

Theorem 2.22 [37] (reformulation of Theorem 2.7). Let (x)}=0 be a

complete orthonormal system of Legendre polynomials or trigonomertic functions in the space L2([i,T]). Furthermore, let the function ^2(t) is continuously differentiate at the interval [i,T] and the functions ^1(t), ^3(t) are twice continuously differentiate at the interval [t, T]. Then, for the iterated Stratonovich

stochastic integral of third multiplicity

* T * ¿3 * ¿2

J*[#3)]T,t = / ^3(taW «t2)/ ^i(ti)dft(;i)dft(2i2)dft(;i

where ii, i2, i3 = 1, . . . , m, the following formula

T t3 t2

J*['<3>]r,( = l.i.m. /\/>3(i3) /'2(t*2) /"^i(ii)df':i)pdft;2)pdf;C3)p

is valid.

Theorem 2.23 [37] (reformulation of Theorem 2.8). Let {j(x) j0 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 fourth multiplicity

>jc T ^ ¿4 ^ ¿3 ^ ¿2

i*ti:2i3:4) = J J J J dw<:i)dwi:2»dw<33)dwi:4»,

t t t t

where ii, i2, i3, i4 = 0,1,..., m, the following formula

T ¿4 ¿3 ¿2

i*f2:3:4)=l.i-m. ///J dwt(:i)p dwt(:2)pdwt(:3)pdwt(:4)p

tttt

is valid, where w[:) = f^ (i = 1,..., m) are independent standard Wiener processes and wT0) = t.

Theorem 2.24 [37] (reformulation of Theorem 2.9). 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 fifth multiplicity

>jc T ^ ¿5 ^ ¿4 ^ ¿3 ^ ¿2

/*;i:2,3:4,5)=J J J J J dw<:i»dw<:2»dwi:3>dw<44»dw«:5>,

t t t t t

where ii,..., i5 = 0,1,..., m, the following formula

T ¿5 ¿4 is ¿2

= LLm. J JJ J J dwi;i)pdwi;2)pdwiSs)pdwt(44)pdwt(55)p t t t t t

is valid, where wT) = f(i) (i = 1,..., m) are independent standard Wiener processes and w[0) = t.

Theorem 2.25 [37] (reformulation of Theorems 2.10 and 2.14). Suppose that every ^(t) (/ = 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)ht = J (tk).. J ^i(ti)dwi;i)... dw(ik), (2.496)

t t

where ii,..., ik = 0,1,..., m, the following formulas

2n'

lim lim ... lim M J*Wk)]T,t ~ Г[ф[к)]р^рк I }> = 0,

P1—œ p2—yœ pk—œ

2n

lim lim ... lim M { ( Г[ф{к)}т^ ~ Г[ф{к\уРк ) } = 0 pk pk-1 p1

are valid, where

J*[^ YTf^ = I ^ (tk )... I (ti)dwJl^1... dw™, (2.497)

T t2

n G N, and lim means lim sup.

Theorem 2.26 (reformulation of Theorem 2.17). Suppose that every ^(т) (/ = 1,2,3) is a continuously differentiate function at the interval [t,T] and {ф(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*[^(2)]T,t and J*[^(3)]Tji (ii,i2,= 1,...,m) defined by (2.496) the following formulais

lim lim M i f J*[^(2)]tî - J*[^(2)]ТГ = 0,

p1—œ p2—' 1

lim lim M < ( J*[

P2—^œ pi—œ

lim lim lim M

pi—œ p2—œ p3—œ

lim lim lim M

P3—œ p2—œ pi—œ

]T,t

- J *

= 0,

T,t

J

T,t

J

= 0,

=0

are valid, where J*['(2)]T/2 and J*['(3)]Tf2'P3 are defined by (24971).

Let us reformulate Hypotheses 2.1-2.3 in terms of the convergence of iterated Riemann-Stiltjes integrals to iterated Stratonovich stochastic integrals.

Hypothesis 2.4 [37] (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

st

t2

dw(i 1 )... dw

(ifc) tfc

(ii,... ,ik = 0,1,... ,m)

the following formula

* Î2

dw

(ii)

dwtik ) =

T

t2

l.i.m.

p—œ

dw

(ii)p

. dw

(¿fc )p ifc

is valid, where l.i.m. is a limit in the mean-square sense, w[:) = fT:) are independent standard Wiener processes (i = 1,..., m) and w[0) = t.

Hypothesis 2.5 [37] (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

*t2

J *

T,t = ^k (tk )

^i(ti)dwt(;i)

. . dw

(ifc) tfc

(ii,... ,ik = 0,1,... ,m)

*

*

i

i

the following formula

t t2

+ IT

tl tfc

J* = l.i.m. / ^ (tk)... ^i(ti)dwt(;i)p ... dwfk)p

t t

is valid, where l.i.m. is a limit in the mean-square sense, w^ = f^ are independent standard Wiener processes (i = 1,..., m) and w[0) = t.

Hypothesis 2.6 [37] (reformulation of Hypothesis 2.3). Let (x)}°=0 be a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Moreover, every ^(t) (/ = 1,...,k) is an enough smooth nonrandom function on [t,T]. Then, for the iterated Stratonovich stochastic integral of kth multiplicity

*t *t2

J*[^(k)]T,t = j ^k (tk) ..J ^i(ti)dwt;i) ...dw£k) (ii,...,ik = 0,1,..., m) tt

the following formula

T t2

J*[^(k)]T,t = l.i.m. [ ^ (tk)... i ^i(ti)dwt(;i)pi... dwtik )pk

is valid, where l.i.m. is a limit in the mean-square sense, w[i) = f(i) are independent standard Wiener processes (i = 1,..., m) and w[0) = t.

Note that Hypothesis 2.6 is valid under weaker conditions if i1,..., ik are pairwise different and i1,..., ik = 0,1,... ,m (see (1.43)).

2.7 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 2.7 and 2.8

2.7.1 Introduction

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 generalized iterated 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.

2.7.2 Another Proof of Theorem 2.7

Let us consider (2.365) for k = 3, p1 = p2 = p3 = p, and ii, i2, = 1,...,m

p p p

J*W'(3)]T,f = £ Cj jjj + J [R^plT3! w. p. 1, (2.498)

ji=0 j2=0 j3=0

where

N-1 N-1 N-1

J [RpppjT3! = l.i.m. £ £ £ RWt, ,rfe ,t,3 f Af^f,

^ /3=0 /2=0 li=0

p p p ji=0 j2=0 j3 =0

3

K*{ti,t2,u) = Y[^i{ti) (i{tl<t2]i{t2<t3} + ii{il=i2}i 1=1

11

+ 21{il<i2}1{i2=i3} + ^1{il=i2}1{i2=i3} 1 •

Using (2.383), 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

R(1jppp = JJ J Rppp(t1 ,t2,t3)dfi1il)dfi2i2)dfi3i3) + t t t T t3 t2

+//I Rppp(t1,t3,t2)dfi;i)dft(;3)dft(;2)+ ttt

T is t2

+// / Rppp(t2,ti,t3)dfi;2)dfi(;i)dfi(;s)+ t t t T ts t2

+// / ^^^^(t2 'tз,tl)dfi;s)dfi(;l)dfi(;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'ti't2)dfi(;2)dfi(2is)dfi(sil), t t t

T ts

R(2)ppp _ 1{.i=.2=0} J J Rppp(t2, ¿2, t3)dt2dftsis) +

t t

T ts

+ 1 {¿i=is=0W / Rppp(t2, ¿3, ¿2)dt2dftsi2 + tt

T ts

+ 1{i2=is=0W / Rppp(t3, ¿2, ¿2)dt2dftsi +

tt

T ts

+ 1{i2=is=0} J J ^ppp^b ¿3, ¿3)dft1ii)^3 + tt

T ts

+ 1{ii=is=0} J J ¿i, ¿3)dfti2)^3 + t t T ts

+ 1{ii=i2=0}^ J Rppp(t3,t3,t1)dfiГ)dt3. t t

We have M <!

J [RppJj) 2j < 2M { (RUT) 2j + 2M { (42)p

(2.499)

Now, using standard estimates for moments of stochastic integrals [91], we obtain the following inequality

M^ (RlM21 <

J

T is t2

~ 6 j J J I (Rpip2ps (t15 ¿25 ^3))2 + (Rpip2ps (t15 ^35 ^2))2 + (Rpip2ps (t25 ¿15 ^3))2 + t t t

222 + (Rpip2ps ^^¿l)) + (Rpip2ps (t35 ^25 )) + (Rpip2ps (t35 ¿15 t2)) J dtidt2dt3 =

6 ^ (Rppp (t15 ¿25 ¿3)) dt1dt2 dt3.

[t,T ]s

We have

J (Rppp(t15 ¿2 5 ¿3))2 dt1 dt2dt3 = [t,T ]s

// p p p \ 2

K*(i15 ¿25 ¿3) - E E E Cjsj2ji j j C^j (i3M ^3 [tTp V ji=0j2=0js=0 /

2

[t,T ]

where

// p p p . K5 ¿25 ¿3) - EE E^jij (¿1 ) j (^2) j (^3) I ^2^3,

s V ji=0 j2 =0 js=0 '

^1(^1)^2(¿2)^3(^3)5 ¿1 <¿2 <¿3 K (¿15*2 5^3)H 5 ¿1 5 ¿2 5 ¿3 G .

05 otherwise

So, we get

lim Mi i R

(1)ppp\ 2 T,t ^

/, p p p > K (¿1,^3) - YY1 Ejji j ^ j (isn "¿1 dt2dt

\ ; _n ;__n ;__n /

1) j (¿2) j (¿3) "¿1«¿2«t3 =

[t,T ]3

ji=0 j2=0 j3=0

= 0,

(2.500)

where K(ti, t2, t3) G L2([t,T]3).

After replacement of the integration order in the iterated Ito stochastic

integrals from R2]^ [1]-[15], [68], [105], [106] (see Chapter 3) we obtain w. p. 1

R

(2)ppp T,t

1

T t3 T t3

{ii=i2=0} I J J Rppp(t2, ¿2, ¿3)dt2dft33)+^ J Rppp(t3,Î3,Îl)dft(1i3)dÎ^ + i i i T t3 T t3

+1{i2=i3=0} I y J Rppp(Î3,Î2,Î2)dÎ2"fi(3ii) +J J Rppp(ti,t3,i3)dfi(1ii)di^ + .i Î Î i T t3 T t3

^2) , /

+ 1 ii =i

{¿1=i3=0} I y J Rppp(t2,t3,t2)dt2dft32 +y J Rppp(t3,t1,t3)dfti2 "¿3

i i i i T ti TT

1{i1=i2=0} ( J J Rppp(^ ^ ^"¿2dfii3)+y J ^pp^^"^^) ) + t t t t1 T ti TT

+ 1{i2=i3=0} I / / Rppp(^ ^ + / / Rppp(^1, | +

tt T t1

t t1 T T

+ 1{i1=i3=0} I / / Rppp, ¿2)"¿2dft1i2) + / / Rppp¿1, ¿2^^"f"^2''

,t t t t1 T / T

= 1{i1=i2=0} J I J Rppp(^2,^2, ) dft(3i3) + tt T / T

+ 1{i2=i3=0^y I J Rppp(^ ^ ¿2)d^2 ) dft\4U + tt

2

+ l(ii=is=0M I / Rppp^5^5^^ I dft22 tt

tt ppp

E E E Cjsj2ji j (¿2 )0j-2 (¿2)0j-s (¿3)) ^¿2 dft(;s)+

ji=0 j2 =0 js=0 T T

+1{*2=^o} ^l{ii<i2} + \^-{ti=t2} VkihYhihYkih)

v2 4

ppp

tt

E E E Cjsj2ji j (¿1)0j-2 (¿2)0j-s (¿2) di2dft(;i)+

ji=0 ¿2=0 js=0 /

TT ,

+ l{n=^0} I J {^{t2=t3}Mts)Mt2)Mts)-ppp

E E E Cjsj2jij (¿3)^2 Mj(¿3^ di3dft22) =

ji=0 ¿2=0 js=0

^ts

[Mt2)ut2)dt2-^^cnjljl(f)j3(t3)) <i3)+

{ ¿i=0 js=0

p p

-EE Cjsjsji j (¿0) dft(;i)+

ti ji=0 ¿3=0

T

p p

t ji=0 ¿2=0 T ts

iin^o} ( ^ I Uh) I Mt2)Mt2)dt2df^ -EEC^mC]:3) ) +

t t ji=0 ¿s=0

T T

p p

+ 1{*2=^0} [ 2 I Mtl) I ^2(i2)^3(i2)^2^r^EE^1C];i)

t ti ¿i=0 ¿s=0

(-l) E E Cjij2ji j (¿2)dft(:

1 /> /> p p

pp

j3

j1=0 j3=0

1{i1=i3=0} E E Cj1j3j1 Cj

From the proof of Theorem 2.7 we obtain

T is

m ; l d(2~)ppp^2' I .. I / 1

IVI S \riTi t , _ - , - , , 2

tt

p p 2 p p 2 EECj3j,j,zj33) +^.^m E ECj1j3j1 j" K

j1=0 j3 =0 / J [ Vj1=0 j3=0 /

T T

+i{i2=^o}M<; (i Î Mh) /'02(i2)'03(i2)«fi;i)-

t t1 p p 2

ÉÉCj-, C^M) ^ 0 (2.501)

j1=0 j3=0 /

if p ^ œ. From (I2.498l)-(l2.501l) we obtain the expansion (12.231). Theorem 2.7 is proved.

2.7.3 Another Proof of Theorem 2.8

Let us consider (2.365) for k = 4, p1 = ... = p4 = p, and ^1(s),..., ^4(s) = 1

% T ^ ¿4 ^ ts * ¿2

t t t t pppp

dw™ rfw™ dw™ =

=EEEE^j c,c,c)c)+jw.p. i, (2.502)

j1=0 j2=0 j3=0 j:=0

where

J [Rpppp]j4,t =

N-1 N-1 N-1 N-1

= E R™>(T<1, . T'3, T,4 )Aw«:;)Aw(;2»Aw(;3»Aw(;4:),

Nl:=0 l3=0 l2=0 l1=0

Rpppp^5 ¿25 ¿35 ¿4) — K5 ¿25 ¿35 ¿4)

pppp

)0j2 (^2)0js (^3)^¿4 (^4) 5 (2-503)

ji=0 ¿2=0 ¿3=0 ¿4=0

K*(¿15 ¿25 ¿35 ¿4)—— n (l

1=1 V

1 1

— I{tl<t2<i3<i4} + 2 1{tl=t2<i3<i4} + 2 1 {i 1 <^2=t3 <t4} +

1 1 1

+ 41{il=i2=i3<M + 2 1{il<t2<i3=i4} + ^1{il=i2<i3=i4} +

1 1

+ ^l{tl<t2=i3=i4} + g1{il=i2=t3=i4}-

We have

7

|(4) _ r>(i)pppp

J[Rpppp]T,t = £ R^,fpp w. p. 1 (2.504)

¿=0

where

N-1 14-1 1s-1 12-1

RT»tpppp =i i m. 53 53 53 53 (Rpppp(T1i5 T125 T1s5 T14) x

N

14=0 1s =0 12=0 1i=0 (1i,12,1s,14)

xAw^Aw^Aw^Aw^ ,

''1 ''2 ''3 ''4 1

where permutations (l1, /2, /3, /4) when summing are performed only in the expression, which is enclosed in parentheses,

N-1

RT1tpppp = 1{ii=i2=0}l.i.m. £ Rpppp(T1i 5T1i 5 T1s 5T14 )AT1i Aw(;ss) Aw^

N^to— 3 4

i4,is,ii = 0

'i='s>'i = '4>'s='4

N-1

RT2tpppp = 1{;i=;s=0}l.i.m. E Rpppp(T1i 5T12 5T1i ^4 )At1i AwT;22) AwT;44)5

N^to— 2 4

«4.'2.'i = 0 11 =¿2 'l i=^4 =l4

N-i

431P№P = 1{ii=i4=0}l.i.m. £ Rpppp(Tii ,Ti2 ,Tis ,Tii )At/i Aw^Aw^

__, ---- 'i' '2' 's' ''i/" ''i' ,i2 ,is

N^to 2 s

li=l2,li = ls,«2=ls

N-i

= 1{i2=is=0}l.i.m. £ Rpppp(Tii ,Ti2 ,Ti2 ,T/4 )Aw(;;)ATi2 Aw^,

' '2 ' ' '2 ' ' '4,

N ^TO

l4,l2,li = 0 'i='2>'i = '4>'2='4

N-i

RT.W = 1{i2=i4=0}i.i.m. £ Rpppp(Tii ,Ti2, T/s ,Ti2 )Aw(;;)ATi2 Aw^,

N^to— i s

ls,«2,«i = 0 li=i2,li = ls,l2=«s

N-i

4?PPP = 1{is=i4=0}l.i.m. y Rpppp(T/i ,T/2 ,T/s ,T/s ^w^Aw^ATs,

ls,l2,li = 0 i 2

li=i2,li = ls,l2=«s

N-i

N

Rt}PPPP = 1{n=i2=0}1{is = i4=0} lim Yl RPPPP(Ti2 > Ti2 , Ti4 , Ti4 )ATi2 At'4 +

i4,i2=0 l2 = l4

Ni

+1{ii=is=0}1{^2=i4=0} lim ^ ^ Rpppp (t/2 5 Ti4 ,Ti2 ,Ti4 )ATi2 ATi4 +

...... '2 ' ' '4 ' ' '2 ' ' '4 7" ' '2'

n^to ; ; 0

l4,l2 = 0 l2=l4

N-i

+ 1{ii = i4=0}1{i2 = is=0} lim Rpppp(T'2 ,T'4 ,T'4 ,T'2 )AT'2 At'4 .

i2 i4 i4 i2 i2

N ^TO

l4,l2 = 0 l2 = l4

From (2.502) and (2.504) it follows that Theorem 2.8 will be proved if

- M (RT)' = 0. i = 0,1,....7.

We have (see (1.19), (1.2

T ¿4 is ¿2

RTÜPPPP =//// E | Rpppp(ii, ¿2, ¿3, i4)dw||i)dw(.i2)dw<ss)dw«:4»

t t t t (ti,t2,is,t4)

where permutations (t1,t2, 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 t3 t2

RT0tpppp = E J J //Rpppp(i1,i2,i3,i4)dw<;i)dwi;2)dwii3)dw<i4),

(t1,t2,t3,t4) t t t t

where permutations (t1, t2, t3, t4) when summing are performed only in the values dw^W^dw^dw^. 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,t2,t3,t4), then ir swapped with in the permutation (i1, i2, i3, i4).

So, we obtain

T t4 t3 t2

M{ (4?^)'} < 24 £ J J J y,(RPppp(i1,i2,i3,t4))2 dt1dt2dt3dt4 =

(t1,t2,t3,t4) t t t t

= 24 J (Rpppp(t1, t2, t3, t4)))2 dt1dt2dt3dt4 ^ 0

[t,T ]4

if p ^ to, K*(ibt2,t3,t4) G L2([t,T]4) (see (2.503)).

Let us consider RTt

(1 )pppp

N1

RT1tpppp = 1{;i=;2=0}l.i.m. £ Rpppp (T1i 5T1i 5T1s 5 ^ )At1i Aw^ AwT;44) =

N^to— 3 4

¿4 >'s>'i = 0 li=l3>li = l4>ls=l4

N-1

= 1{;i=i2=0}l.i.m. V Rpppp(T1i 5T1i 5T1s 5T14 )AT1i =

N^to ^ 3 4

¿4 '¿s i=0

¿3 = l4

N-1 f 1

l3 = l4

1 1 1

+ + -^-{Th<Th=Th} + g1{7i1=7i3=7i4} —

p

£ ^'¿^¿3¿^¿i(T1i)<fe (T1i№¿3(T1s^¿4(T14) AT1i AwT;ss)AwT;44) =

¿4 ¿sj2ji=0 /

N-i '1

l{il=i2^}Ujn. y I -l{Tli<Tl3<Tk}

14 ,is i=0

ls=l4

£ C**Wi j (Tii ) j (t'i ) j (T's) j (Ti. ) AT, i AwTis Awr;4

j4jsj2ji=0 '

N-i N-i N-i /1

= l{n=^o}l-i-m. y S

N^TO ,4=0 is=0 ii=A 2 p \ E Cj4jsj2jij(t/i) j (t/i) j(Tis) j(t/4) At^ Aw(;s) Aw^-

j4,js ,j2,ji=0 /

N-i N-i /

-1{ii = i2=0}1{is=i4=0}l.i.m. 2J 0-

NlTO /4=0 /i=0 \

Yl Cj4 js j2 ji j (t'i ) j (t'i ) j (t'4 ) j K ) ) At'i At,4 j4 jsj2 ji=0

T ¡4 ts p \

iff diidw«;s>,iw«44» - £ Cjc]ss)cj:4M +

t t t j4jsJi =0 )

1

20

p

+ 1{ii=i2=0}1{is=i4=0} E Cj4j4jiji W. P. 1.

j4 ji=0

When proving Theorem 2.8 we have proved that

P 1 T t2

J™ Y, = 7

piTO ^ 4

j4,ji=0 t t

T t4 ts

LLm. £ C-,,1C]«C]r) = |/ //'iii'iwi:3,rfw«r» +

j4 ,js,ji=0 t t t

T t2

J J dtidt2 w. p. 1.

tt

Then

lim M* (RT1tppp^2 I =0.

p—>-to

T,t

Let us consider RT t

(2) pppp

N1

RT2tpppp = 1{;i=;s=0}l.i.m. £ Gpppp(T1i 5T12 5T1i 5T14 )AT1i AwT;22)AwT;44)

N^to— 2 4

¿4 '¿2 'l i =0 li=l2'li = l4'l2=l4

N-1

= 1{;i=;s=0}l.i.m. £ Gpppp(T1i 5T12 5T1i 5T14 )AT1i AwT;!)AwT;44) =

N^to— 2 4

¿4 '¿2'! i =0

l2=l4

1 . N-1 ( 1 1

—TO '4''2''i = 0 V

l2=l4

p

- £ C^i ^ (T1i (T12 ^ (T1i ^ (T14) A^ Aw(;2) AwT;44) =

¿4 ¿3 ¿2 ¿1=0 /

N-1 N-1 N-1 p

= 1{;i=;s=0}l^^' £££(-1) £ ¿¿¿ix

14=0 12=0 1i=0 ¿4, ¿3, ¿2 ,¿1=0

x&i (T1i )<fe (T12 ^ (T1i ^¿4 (T14 )AT1i Aw^Aw^-

iiy Y^V — 'i^' T2 T4 N-1 N-1 p

1{ii = i3=0}1{i2=i4=0}l.i.m. £ £(-1) £ ¿¿¿i x

N—to f—' ;—' . .

14=0 1i =0 ¿4 ,¿3, ¿2, ¿i =0

x ^¿1 (T1i ) ^¿2 K ) ^¿3 (T1i ) ^¿4 (T14 ) AT1i At14 =

p

1 V^ C Z (;2)Z (i4) I

1{;i = ;s=0} / v ^¿'¿^¿'i ^¿2 ^¿4 +

¿4,¿2,¿1 =0 p

+ 1{i1 = i3=0}1{i2=i4=0} £ ¿¿'i w. p. 1

¿4,¿'l =0

When proving Theorem 2.8 we have proved that

p

1,——TO. £ ¿fc C^'C^' = 0 w. p. 1,

¿4 ,¿2,¿1=0

lim y jij4j'i = 0.

p^to z—»

j4,jl=0

Then

i™ m{ n -a

Let us consider RTt

(3)pppp

N-1

431PPPP = 1{ii=i4=0|l-i-m. £ Gpppp(ni ,ri2 ,Ti3 ,T/i }Arii Aw^Aw^ =

N^to— 2 3

l3,l2,li=0

N-1

1{ii=i4=oil-i-m- V Gpppp(r/i,r/2,r/3,r/i}AtIiAw(;22)Aw(;33) -N^to— 2 3

l3,l2,li=0

l2=l3

N-1 ^

^^ i3,i2,ii = 0 \

l2=l3

E Cj4j3j2ji j (Tli } j (Tl2 } j (Tl3 } j (Tii } Ar/i Awi;2)Aw(i3)

j4 ,j3 ,j2 ,ji=0 /

N-1 N-1 N-1 p

/3=0 /2=0 /i=0 j4,j3,j2,ji=0

X j (T/i } j (t/2 } j (T/3 } j (T/i }AT/I Aw^Aw^

¿3/rj4V' ii/ — 'ii"' T2 T3

N-1 N-1 p

1{ii = i4=0|1{i2=i3=0}l-i-m- >J /J(-1} /J Cj4j3j2ji X

/3=0 /i =0 j4,j3j2ji =0

X j (T/i } j (t/3 } j (t/3 } j (T/i } AT/i At/3 -

/i /i

p

1 V^ c Z(i2) z(i3) I

1{»i = »4=0} / v Cj4j3j2j4 Zj Zj3 +

j4j3j2 =0 p

+ 1{ii=i4=0}1{i2 = i3=0} E Cj4j2 j2j4 W p 1

j4 ,j2=0

When proving Theorem 2.8 we have proved that

LLm. £ ¿f = 0 w. p. 1

p—TO

¿4 ,¿3 ,¿2=0

lim ¿¿2 ¿'2J4 =

p—VTO ' *

Then

p—TO z—

¿4 ,¿2=0

lim M^ (Ri3!ppp^ % =0.

p—TO

T,t

Let us consider RTtpppp

N-1

RT4tpppp = 1{;2=;s=0}l.i.m. £ Gpppp(T1i 5T12 5T12 5T14 )Aw(;i)Ar12 Aw^

N—to , , , 0 1 4

¿4 ''2 ''1= 0 ¿1 = ¿2' ¿1 = ¿4 '¿2 = ¿4

N-1

1{i2=;s=0}l.i.m. £ Gpppp(T1i 5T12 5T12 5T14 )AwT;i) At12 AwT;44) =

N—TO ; ^ 11 14

«± = ¿4

N1

.m. z ^ppppv'i^ '12 5 '12 5 '¿4,

¿ ¿ ¿ =0

N-1 1 N2

¿4d2 '¿1=0 \

11 = ¿4

1 1 1

+ 41{Tii=Ti2<T'4} + ~l*-{Th<Th=Th} + ~^-{Th=Th=Th}

p

53 fe (T1i^¿2 (T12^(T12^¿4fa) Aw^Ar^AwT;44) =

¿4 ,¿3,¿2,¿1=0 /

, • N-1 (1

N2

¿4 '¿2' ¿1 = 0 \

11 = ¿4

p

£ C^i^¿1 (T1i(T12^(T12^(T14) Aw^A^AwT;44) =

¿4 ¿3 ¿2 ¿1 =0 /

N-1 N-1 N-1 /1

= l{i2=^o}lfm. £ £ £ öMn^KnA

N—TO 14=0 12=0 1i=A 2

p

p

E Cj j j (T/i } j (t/2 } j (t/2 } j (t/4 } Aw(;;'At/2 Aw^ -

j4 ,j3 ,j2,j'i=0 /

N-1 N-1 p

-1{i2 = i3=0}1{ii=i4=0}l-i.m. >J /J(-1} /J Cj4j3j2ji X

N—to f—' ;—' . .

/4=0 /2=0 j4,j3,j2,ji=0

X j (t/4 } j (t/2 } j (t/2 } j (t/4 } At/2 At/4 -

p

i<«=**>> I!// jdw^>dtMf - V c^.cfcf ] +

t t t j4,j2,ji =0 /

2 1 1 1 tl'I'ltl, W-L --> K/A.A.A.A.i • 1

0

p

+ 1{i2=i3=0}1{ii=i4=0} E Cj4j2j2j4 W- P- 1-

j4 ,j2=0

When proving Theorem 2.8 we have proved that

p

L Cj4j2j2j4 — 0, j4 ,j2 =0

p T t4 t2

Li.m. ^ = I///dwii'Wwi:41 w. p. 1.

j4j2ji=0 t t t

Then

p—to m{ K^)2} - 0

Let us consider RT^t^

N-1

RT,tpppp - 1{i2=i4=0}l-i-m. E Gpppp(T/i,T/2,T/3,T/2}AwT;i)AT/2Aw^

N^to— i 3

I3 i =0 Ii=l2,li = l3,l2=l3

N-1

- 1{;2=;4=0}l-i-m. E GPPPP(T/I ,T/2 ,T/3 ,T/2 }A wTi i) A ^^/2 -

N^to— i 3

l3,l2,li=0 li=l3

1 . N-1 ( 1 1

li=l3

p

53 fe (T1i ^¿2 (T12 ^¿s (T1s ^¿4 (T12 ) ) Aw(;i)Ar12 AwT;ss)

¿4 ¿3 ,¿2,¿1=0

N-1 p

N

1{;2=;4=0}l.i.m. 53 (-1) 53 ¿¿ix

¿3 '¿2 '¿1 = 0 ¿4 ,¿3 ,¿2 ,¿1=0

x^i (T1i (T12 ^ (T1s (T12 )AwT;i) AT12 Aw£s)

p

= -1{;2=;4=0} 53 ^¿^¿s¿^¿'i Cj; C

p

= (;1)Z (;3) '¿3

¿4 ,¿S,¿1=0

N-1 N-1 p

1{i2 = i4=0}1{i1=i3=0}l.i.m. 53 53(-1) 53 C

N—to t—' ,—' . .

1s =0 12 =0 ¿4,¿S,¿2,¿1 =0

x ^¿1 (T1s ) ^¿2 (T12 ) ^¿3 (T1s ) ^¿4 (T12 ) At12 At1S =

12 12

p

1 V^ C Z (;1)Z (;s) +

1{;2 = ;4=0} / v ^¿^¿'l Z^ ^¿3 +

¿4,¿S,¿1 =0 p

+ 1{;2=;4=0}1{;i=;s=0} 53 ¿'i¿'¿'i w. p. 1.

¿4 ¿1=0

When proving Theorem 2.8 we have proved that

p

1,——to. £ ¿¿,1 Z^Cf = 0 w. p. 1,

¿4 ¿s^0

lim 53 C¿4¿l¿^¿i =

p—TO ' *

¿4,¿1=0

Then

lim M^ f RT5tpppp) > =0.

p—TO

Let us consider RT,t

(6)pppp

N1

RT6tpppp = 1{;s=;4=0}l.i.m. 5] Gpppp(T1i 5T12 5T1s 5T1s ^w^Aw^ AT1s

N—TO ^ • - - - -1 .¿2

¿3 '¿2 '¿1=0 ¿l^' ¿1 = ^ '¿2 =¿3

p

N-1

= 1{;3=;4=0}l.i.m. 53 Gpppp(r1i5T125T1s5T1s)AwT;i)AwT;22)Ar1s = N^to— 1 2

¿3 ¿2 ¿1=0 !l = !2

(1

N2

¿3 ¿2 ¿1=0 11 = 12

1 1 1

p

53 C¿4¿s¿2¿lfe (T1i^¿2 (T12^¿s(T1s^¿4(T1s ) j Aw(;i) Aw(;22) AT1s

¿4 ¿3 ,¿2,¿1=0 /

i-i (1

N2

¿3 ¿2 ¿1=0 ¿1 = !2

p

53 ¿¿¿l ^¿1 (T1l ^^ ^¿2 (T12 ^¿3 (T1s ^¿4 (T1s ) ] A w(-;i ) A w(-;2 ) AT1s

¿4 ¿3 ¿2 ¿1 =0 /

p

1(^,^0)1 dw'^rfw^rfts - v c3№J2J1C«;;')C]:2)

¿4 ,¿2,¿1=0

N-1 N-1 p

-1{;s=;4=0}1{;i=i2=0}l.i.m. £53(-1) 53 CJ

N—TO T—■ ,—' . .

1s =0 1l =0 ¿4,¿S,¿2,¿1=0

x fe (T1l ) ^¿2 (T1l ) ^¿3 (T1s ) ^¿4 (T1s ) AT1l At1s =

^p

¿4,¿2,¿i=0

p

C¿4¿4¿1¿1

+ 1{;i=i2=0}1{;s=;4=0} 53 CJ

¿4 ¿1=0

^T ts

dw^dw^dt3 + -l{n=i2^0} / / dtidts-

tt

(;lV(;2)

EC z (;i)Z (;2M + ¿¿'l Z¿i '¿2 J +

¿4 ,¿2 ,¿1=0

p

/ p T t3

+1{*i=*2^0}l{i3=M^0} ( E (J dtldti 1 W' P'

\j4,ji=0 t t

When proving Theorem 2.8 we have proved that

p 1 T t3

E = 4 / /

j4Ji=° "t "t

p T t3 t2

_ 1 / / /

^jA]A]2jIS

j4,j2,ji=0 t t t

T t3

i.Lm. v -III

J J dtidt3 w. p. 1.

tt

Then

I—TO 4 (46^) 1 - 0

Finally, let us consider RJtpppp

N-1

,(7)pppp

RT,tpppp - 1{ii = i2=0}1{i3=i4=0}l-i-m- E Gpppp(T/2 ,T/2 ,T/4 ,T/4 }At/2 At/4 +

N

14 ,l2=0 l2=l4

N-1

+ 1{;i = ;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{;i=;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{;i=;2=0}1{;3=;4=0}l-i.m. E E^ppppK,t/2,t/4,t/4}At/2At/4 +

N—TO /4=0 /2=0

N-1 N-1

+ 1{ii = i3=0}1{i2=i4=0}l-i-m- E EGpppp(T/2 ,T/4 ,T/2 ,T/4 }At/2 At/4 +

N—TO /4=0 /2=0

N-1N-1

+ 1{;i=;4=0}1{;2=;s=0}l.i.m. 53 53 Gpppp(t12 ^ ^ ^)At12 AU

12 5 '14 5 '14 5 "12/" '12" '14

N—TO 14=0 12=0 N-1 N-1 1 1

14=0 12=0

53 ¿¿i¿i ^¿i (T12 ) fe (T12 ) ^¿3 (T14 ) ^¿4 K ) ) At12 At14 +

¿4,¿S,¿2,¿1 =0

N-1 N-1 1

+l{il=i3^0}l{i2=i4^0}^m- 53 53 ( g - 53 chj3j231x

TO 14=0 12=0 \ ¿4,¿3,¿2,¿1=0

x ^¿1 (T12 ) ^¿2 (T14 ) ^¿3 (T12 ) ^¿4 (T14 ^ At12 At14 +

N-1 N-1 1 p

53 53 ( g ~ 53 CJ4nnnx

14=0 12=0 \ ¿4 ,¿3 ,¿2 ,¿1=0

x ^¿1 (T12 ) fe K ) ^¿3 (T14 ) ^¿4 (T12 ) AT12 AT14 =

1

40

T t4 p \

J J dt'2dt4 - 53 ¿¿'i I

t t ¿4 ,¿1=° /

p

1{;i=;s=0}1{;2=;4=0} 53 C¿

¿4 ¿1=0 p

1{;i=;4=0}1{;2=;s=0} 53 C¿4¿2¿^ •

¿4 ,¿2=0

When proving Theorem 2.8 we have proved that

p 1 T t4

¿is, 53 = 4 / /dt2dt^

¿^¿i^ i i p

lim 53 C¿4¿i¿4¿1 = 05

p—TO z-'

¿4,¿1=0

p

p

Urn y j j2j4 — 0. p—f

Then

p^œ z—

j4,j2=0

lim RT7tPPPP — 0.

Theorem 2.8 is proved.

2.8 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 and Wong—Zakai Type Theorem

Let us prove the following theorem.

Theorem 2.27 [31]. 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— y ^(*2)y ^(ti)dfi;i)dfi;2) (M2 — i,...,m)

the following expansion

Pl P2

J*[^<2>]s,( = l.i.m. £ (sjj (2.505)

jl =0 j2=0

that converges in the mean-square sense is valid, where s G (t,T] (s is fixed),

S t2

Cj2ji(s) = J Mh(t2)J (ti)dtidt2, (2.506)

t t

and

T

cf=/ (t )df«

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 = JbPi2)]s,t + I UhyUt^dh, (2.507)

t

where s G (t,T) (s is fixed), 1A is the indicator of the set A. From the other side according to (1.242), we have

Pi P2

J ^ = Ä £ £ C'2jl (s) ( jj - 1(!1=,2}1tj1=j2}

jl=0 j2=0

Pl P2 min{pi,p2}

^ E ECj2ji(s)C];i)Cii2) - 1{ii=i2} lim E Cjiji(s). (2.508)

1 Po—Vno f ■« ' * Ji J2 L J p p —1.,-v-, ^ *

= l.i.m. Z^j (s)Zj1,1)Cj2,2) - i{i1=i2> ^

—^to ^—y z—' 71 72 P1,P2—TO

¿1=0 ¿2=0 ¿1=0

From (2.507) and (2.508) it follows that Theorem 2.27 will be proved if

1 s TO

- / UtiyU^^Y,0^)- (2-509)

2

t ji=0

Note that the existence of the limit on the right-hand side of (2.509) for s = T is proved in Sect. 2.1.2 (Lemma 2.2). The case s G (t,T) can be considered by analogy with the proof of Lemma 2.2 (see (2.621), (2.622) for details).

To prove (2.509), we multiply the equality (2.12) by the function

1{t2<s} + 21{i2=s}> t-2 € [t;T],

where s G (t,T) (s is fixed). So we have

2~ {t2 = SM jiy I {t2 <S} I 2 {t2 = S}

ji=0

(2.510)

where ti = t, T,

K*(tut2) = Mti)h(t2) (i{il<i2} + ii{il=f2}), ti,t2 G [t,T],

t2

Cj(t2)= ^2(t2^ (ti)0j! (ti)dti. t

The function

has the same structure as the function K*(ti;t2). Then, by analogy with (2.12), we get

Cjdh) (l[t2<s} + il{i2=a}) (/■>). (2.511)

V 2 J j2=0

where t2 = t,T and the Fourier coefficient Cj2j1 (s) is defined by (2.506). Let us substitute (2.511) into (2.510)

(1 \ to TO

' j1=0 j2=0

where (ti,t2) E (t,T)2.

Note that the series on the right-hand side of (2.512) converges at the boundary of the square [t,T]2.

It is easy to see that substituting ti = t2 in (2.512), we obtain 1 / 1 \ to to

^ ' j1=0 j2=0

where ti = t, T. Denote

Rpip2{tl,t2,s) = K*{ti,t2) ( 1 {t2<s} + o1^^}

2

Pi P2

5353C;-2ji fa) j (t2), (2-514)

ji=0 j2=0

where p1,p2 < to. Then

RPlP2{ti,ti,s) = i'0l(il)'02(il) i l{ii<S} + ^l{ii=S}

Note that

pi p2

E E Cj2j1 (s) j (t1)0j2 (t1)-j1=0 j2=0

T s

j* RPlP2{ti,ti,s)dti = i y —

t t

T

p1 p2 ji=0 j2=0 t

s

1 r, P1 P2

= 2 / ('S'!1:./ ./Vi =

t j1=0 j2=0

t j1=0 Using (2.515), we obtain

t 1 s to

lim lim / = - / -y^C'jmis). (2.516)

P1^TO P2 ^TO / 2 J Z-'

t t j1 =0

The equality (2.516) means that Theorem 2.27 will be proved if

T

lim lim / RP1P2(ti,ti,s)dti = 0. t

Since the integral

T

Rp1p2 (t1 5 ^ S)dt1

t

exists as Riemann integral, then it is equal to the corresponding Lebesgue integral. Moreover, |RPlP2 (t1,t1,s)| < to (t1 E [t,T]) and the following equality

lim lim RPlP2 (t1 ,t1,s) = 0 when t1 E [t,T]

Pl—TO p2—^TO

holds with accuracy up to sets of measure zero (see (2.513)). We have

Rpip2{tl,t2,s) = ^K*{ti,t2) ^l{i2<s} + -

-i^GniU) (l{t2<s} + ^2=s}) +

+E Cj (t2W 1{t 2 <4 + ) ~~ E ^hjii^^hfa) j (2.517)

j1=0 V ^ ' j2=0 J

Let us substitute ti = t2 into (2.517)

+E (mho} + "E^^^i)) (2-518)

j1=0 V ^ ' j2=0 J

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 (2.510), (2.511), and (2.514) for the case ti = t2, we obtain

T

lim lim / RP1P2(ti,ti,s)dt1 = 0.

p1 — TO p2—TO J t

Theorem 2.27 is proved.

Let us reformulate Theorem 2.27 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 (2.488) for k = 2, ii ,i2 = 1,...,m, and s G (t,T] (s is fixed) we obtain

S ^ p1 p2 J Mt2)j ^i(ti)dft(1l1)p1 dft(2l2)p2 = EE j (s)Cj1)C]:2), (2.519) t t j=0 j2=0

where pi,p2 E N and dfT:)p is defined by (12.4851); another notations are the same as in Theorem 2.27.

The iterated Riemann-Stiltjes integrals

S t2 S

Ys(t1!2)p'p2 = /m*)iV'i(ti)<!1)wff)K, xSf1 = /^i(ii)df«;')p'

are the solution of the following system of ODEs

dYS(,r)P1P2 = ^2(s)xS;ti)P1 f:2)p2, Yt,;1i2)p1p2 = 0 dxS;tL)P1 = ^i(s)f1)p1, xt(,i1)p1 = 0

From the other hand, the iterated Stratonovich stochastic integrals * s * t2 * s Yir2) = / ^ / ^i(ii)df«:;',dft;2), xs:;> = / wtof;1'

t t t

are the solution of the following system of Stratonovich SDEs

',iY<r>=fcwxij) * dfi*2), Yt!;i,2) = o

iX™ = ^(s) * ifi'l), X™ = 0

where * ifs(i), i = 1,... ,m is the Stratonovich differential.

Then from Theorem 2.27 and (1.241) we obtain the following theorem.

Theorem 2.28 [31]. 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(t) are continuously differentiate functions on [t, T]. Then for any fixed s (s E (t, T])

l.i.m. Y(t1i2)pip2 = Y(t1i2), Xi^1 = l.i.m. X^.

P1;P2—TO ' ' ' p1—TO '

2.9 Modification of Theorem 2.7 for the Case of Integration Interval [t, s] (s G (t, T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 and Wong—Zakai Type Theorem

Let us prove the following theorem.

Theorem 2.29 [31]. 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 (t), ^3(t) are twice continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* S * t3 * t2

J*[^(3)]s,t = J ^3(t3)| ^2(t2)J ^(tifdft(2l2)dft(3l3) (ii,i2,i3 = 1,...,m) t t t the following expansion

p

J*[^(3)]s.t = E jj wcfcfzj33)

j1 j2 j3=0

that converges in the mean-square sense is valid, where s G (t,T] (s is fixed),

S t3 t2

Cj3j2j1 (s) = J ^3(t3)jto) J lh(t2)j(t2^ ^i(ti)j(ti)dtidt2dt3

ttt

and

T

j = J to(tf

t

are independent standard Gaussian random variables for various i or j.

Proof. The case s = T is considered in Theorem 2.7. Below we consider the case s G (t,T). First, let us consider the case of Legendre polynomials. From (1.243) for the case pi = p2 = p3 = p and standard relations between Ito and Stratonovich stochastic integrals we conclude that Theorem 2.29 will be proved if w. p. 1

p P 1ST

= 2 J J MsMiMdsrfM, (2.520)

j1=0 j3=0 t t

p p 1

£ E = 2 / / (2.521)

p TO j1=0 j3=0 "t {

p p

'¿s- EECj.j»j1 (s)zf = °- (2'522)

j1=0 j3=0

The proof of the formulas (2.520), (2.522) is absolutely similar to the proof of the formulas (2.232), (2.234). It is only necessary to replace the interval of integration [t,T] by [t, s] in the proof of the formulas (2.232), (2.234) and use Theorem 1.11 instead of Theorem 1.1. Also in the case (2.522) it is necessary to use the estimate (1.201).

Let us prove (2.521). Using the Ito formula, we have

ST s s

i JMt)Mt) j Usi)di^dr=l- Jmsi) j Ur)Ur)drdiM w.p. 1.

t t t S1

Moreover, using Theorem 1.11 for k = 1 (also see (1.241)), we obtain w. p. 1

s s p

\ fusi) J Ur)Ur)drdf^ = ¿qwcf,

t S1 j1=0

where

s s

Cj (s) = J ^i(si)0j1 (si)/^s(t )^2(t )dTdsi. (2.523)

t s1

We have

| / p p 1 p x 2

km = m E E ^.wcj;1' -, E

( Vj1=0 j3=0 j1=0

/ \ \ 2

p p 2

j1 =0 \j3=0

2

p p 1

j1=0 V3=0

i

s e t

Cj3j3ji(s) = (0) / ^2(t)0j3(t) / (si)dsidTd0 =

t t t s s s

= J Wsi)j(si)y ^MjMy (0)d0dTdsi. (2.525)

t S1 T

From (2523)-(2525) we obtain

p i s s / p

Ep(s) = £ /^i(si) j (si) / E«t) j(t)x

j1=0 \ t S1 Vj3=0

s \ \ 2

X j UOWjMdOI drdsi I . (2.526)

We will prove the following equality for all t G (t, s)

to S 1

Y^'UtWjAt) / U0m0)d0 = -'h(r)Ur). (2.527)

j3=0 T

Let us denote

^i(ii5i25s) = /M(i1,i2,.s) + il{il=i2<sr02(i1)^3(ii), (2.528)

where

Ki(ti , t2,s) = ^2(ti)^3(t2) l{t1<t2<s};

ti,t2 G [t,T], s G (t,T) (s is fixed).

Let us expand the function Ki*(ti,t2,s) using the variable t2, when ti is fixed, into the Fourier-Legendre series at the interval (t, T)

to s

K*(ti,t2,s) = E ^2(tiW ^3(t2)toj3(t2)dt2 • j(t2) (t2 = t, s, T). (2.529)

j3=0 t1

The equality (2.529) is fulfilled in each point of the intervals (t,s), (s,T) with respect to the variable t2, when ti G [t, T] is fixed, due to piecewise smoothness of the function K*(ti,t2,s) with respect to the variable t2 G [t,T] (ti is fixed).

Obtaining (2.529), we also used the fact that the right-hand side of (2.529) converges when ti = t2 < s (ti is the point of a finite discontinuity of the function K1*(ti, t2, s)) to the value

i (Ki(ti,ti — 0, s) + Ki(t\,ti + 0, s)) = ^ih{ti)h{h) = I<;(ti,ti,s), where t i < s.

Let us substitute t i = t2 into (2.529). Then we have From (2.526) and (2.527) we get

E(s) = El / (si(si) / E ^(t) j(T) / «0) j(^)dödrds i )

ji=0

si

j3=p+ 1

p / N - 1 s ^ s ^

E lim E M)/ E «t(t)/ (^)d^drAu/

. N^to

ji=0 y /=0

j3=p+1

)

pi N-1 to s s ^

E lim e Wu?) j(u?) e / «t) j(t) /j(0)d0drAu/

. N^TO

ji=0 y /=0

j3=p+1,

7

(2.530)

where t = u0 < ui < ... < uN = s, Au/ = u/+i — u, uf is a point of minimum of the function (1 — (z(t))2)-a (0 < a < 1) at the interval [u/, u/+i], l = 0,1,...,N — 1,

max Au/ —> 0 when N —>

oo.

0/N1

The last step in

is correct due to uniform convergence of the Fourier-

Legendre series of the piecewise smooth function Kf (t, t, s) at the interval [u1 + £, s — e] (with respect to t) for any £ > 0 (the function Kf(T, t, s) is continuous at the interval [uf,s]).

Using the inequality of Cauchy-Bunyakovsky and the estimates (11.202), (2.117), we obtain

^2(T)0j3 (tW «0) j (^T

<

2

2

2

s

s

< (^ (T ) j (t ))2 dT

^(0)j(0)d0 dT <

C

<T2

1

+

1

3S J V (1 - (z(s))2)1/2 (1 - (z(T))2)1/2

+ Ci dT <

<

C2

" jeH (1 - (z(s))2)1/2

+ C3 ,

where constants C, Ci, C2, C3 do not depend on j3.

We assume that s G (t,T) (z(s) = ±1) since the case s = T has already been considered in Theorem 2.7. Then

^(t) j (t) / «0) j (0)d0dT

-J32'

(2.531)

where constant C4 is independent of j3.

Let us estimate the right-hand side of (2.530) using (2.531)

2

n-1

oo

1

lim EE ~2<

. . N^TO ji=0 \ 1=0

j3=P+1

L1

p

n- 1

1

r j=0 £0 (1 - (z(ui))2)1/4

AuJ <

p2 z

1 ji=0

E | lim i

dT

(1 - (z(T))2)1/4

¿1 sr^ I f d,T

p2 fa [J (i - (z(t)T-)1'4

L2p L2

< -Ar = — 0 p2 p

<

(2.532)

2

s

ss

s

1

s

s

2

2

2

s

if p —y to, where constants L,Li,L2 do not depend on p and we used (2.25) and (2.117) in (2.532). The relation (2.521) is proved. Theorem 2.29 is proved for the case of Legendre polynomials.

For the trigonometric case, we can use the following estimates

oo

si

£ ^(T(t)J ^ (0)0 (0)i0i

T

<

Ci p

(2.533)

to (0)0 (0)i0

si

<y OVO),

(2.534)

where constant Ci is independent of p and constant C2 does not depend on j; l = 1 or l = 3; t < si < s < T (see the proof of Theorem 2.7 for details).

The estimate (2.534) is obvious and the estimate (2.533) can be obtained similarly to the estimate (2.33). Theorem 2.29 is proved for the trigonometric case. Theorem 2.29 is proved.

Let us reformulate Theorem 2.29 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 (2.488) for the case k = 3, pi = p2 = p3 = p, ii, i2, = 1,..., m, and s E (t, T] (s is fixed) we obtain

s ¿3 t2 p

J to (t3) / v te) / v (tiff f" = £ Cj (s)cj:;i,cj.;2 'cjj3',

t t t j1'j2 j3=0

(2.535)

where p E N and dfT:)p is defined by (2.485); another notations are the same as in Theorem 2.29.

The iterated Riemann-Stiltjes integrals

is

t2

Z

(ili2«s)p

s,t

^3(iaW «to / ^i(ii)df((;i)pdfi(2!2)pdf((;s )p,

t2

y;(;i;2)p = / to(t2w to (ti

(;i)p if (;2)P i

s

s

s

s

s

s

xsiti)p = / ^1(i1)dft(;i)p

t

are the solution of the following system of ODEs

' dZ^2 i3)p = ^3 (s) Ys(tii2)p dfs(i3)p, Zt(;i;2;3)p = 0

< dYs(ti;2)p = ^2(s)xs(;^)pdfi;2)p, yt;;ii2)p = o.

k dxs(-)p = ^1(s)dfs(;i)p, Xt(;i)p = 0

From the other hand, the iterated Stratonovich stochastic integrals

* s * t3 * t2

zsfi3) = / Mt3)j Mt2)j ^1(t1 K'X^f«,

t t t

* s >jc t2

y(;i*2) = / v^)/ MffK

tt

>Jc s

X™ = / ^1(i1)df«;;i)

t

are the solution of the following system of Stratonovich SDEs

'dz(;fi3) = ^3(s)Ys(r2) * dfs(i3), Zt(;ii2i3) = 0

< dYs(r2) = ^2(s)Xs(;t) * dfs(i2), Yt!r2) = 0 , ^ dXj^ = ^1(s) * dfs(ii), X^ = 0

where * dfs(i), i = 1,..., m is the Stratonovich differential.

Then from Theorems 2.28 and 2.29 we obtain the following theorem.

Theorem 2.30 [31]. 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 (t), ) are twice continuously differentiable nonrandom functions on [t, T]. Then for any fixed s (s E (t, T])

l.i.m. Zs(it1i2i3)p = Zs(it1i2i3), l.i.m. Ys(;ii2)p = Ys(;ii2),

Xf)p = l i.m. xSit>.

2.10 Modification of Theorem 2.8 for the Case of Integration Interval [t,s] (s G (t,T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 4 and Wong—Zakai Type Theorem

Let us prove the following theorem.

Theorem 2.31 [31]. Suppose that {0-(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

= /// / dw^dw^dw^dw™ (ii,i2,i3,i4 = 0,1,...,m) t t t t

the following expansion

p

J^'(4,]-"< ^ £ j (s)cj;',c«22)c«33)cj44)

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)^/ M/ 03 M/ j M/ j (s1)ds1ds2ds3ds4 tttt

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), wTl) = f(l) for i = 1,..., m and wT0) = t.

Proof. The case s = T is considered in Theorem 2.8. Below we consider the case s G (t, T). The relation (1.244) (in the case when pi = ... = p4 = p ^ to) implies that

p

Um. £ Cj (s)C':')C«22)C«33)Cjr) = J №(4)]s,t+

p—>-TO

ji,j2,j3,j4=0

, 1 , /l(:2:4)

+ 1{:i=:2=0}A1 3'" (s) + 1{:i=:3=0}A2 2"' (s) + 1{:,=:4=0} A^2^ («) +1{;2=:3=0} A^ ^ (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{:;=:3=0}1{:2=:4=0}B2(s) - 1{:i=:4=0}1{:2=:3=0}B3(s), (2.536)

where J[^(4)]s,t has the form (1.225) for ^1(t),... ,^4(t) = 1 and ii5.. 0, 1, . . . , m,

. ,i4 =

A1:3i4)(s

A2:2:4)(s

A3:2:3)(s

A4:i:4)(s

A5:i:3)(s

^ E Cj(s)ci3:3)ci;i4),

j4j3ji=° p

^ E c jwcf^',

j4,j3,j2=0 p

= lim. E j.wcfcf.

p—TO z

j4j3j2=0

= lim. E Cjwcfcf,

p—TO z

j4j3,ji=0

= l^ E Cj(s)cj:;')cj33).

p—TO

j4j3,ji=0 p

A«:i:2)(s) = l.i.m^ Cjj (s)Cji;')C«:2),

p—TO ' * J i J2

j3j2ji=0

pp

B1(s)= Um E Cj4j4jiji (s) B2(s)= lim E Cj3j4j3j4 p—TO ' * p—TO '

(s),

ji j4=0

j4,j3=0

p

p

B3(s) = lim E jj3j4 (s)-

34,33=0

Using the integration order replacement in Riemann integrals, Theorem 1.11 for k = 2 (see (1.242)) and (2.509), Parseval's equality and the integration order replacement technique for Ito stochastic integrals (see Chapter 3) [1]-[15],

[68]

(2.263))

or Ito's formula, we obtain (see the derivation of the formula

A(r\s) = -

1

s T Si

ds2 dw(i3)dwTi4) +

t t t

1

+ ¿l{<3=i4^0} I (51 - t)ds 1 - Ap4)(.s) w. p. 1, (2.537)

where

Ap4)(s) = l.i.m. V 3 (s)C(i3)C(i4),

1 V > ? / j 3433 v > 3 3 '

33,34=0

<fj4j3(s) = \ JhsM E ( j <t>h(s2)ds2\ dsidr. (2.538)

ji=p+1 \ t

Let us consider A2i2i4)(s) (see the derivation of the formula (12.265))

4i2i4)(s) = -A2i2i4)(s) + Aii2i4)(s) + A3i2i4)(s) w. p. 1, (2.539)

where

A2i2i4)(s) = l.i.m. V , (s)C(i2)Cii4),

2 V ' p / V 3432 V ' 32 3 '

(«2^(«4)

A3i2i4)(s) = l.i.m. y 3 (s)Cf2)tf4),

3 V y p / J 3432 V y 32 3 '

34,32=0 p

(«2^(«4)

34,32=0 T

= IJ^jAt) E {JfasMdsA J (f)j2{.Si)dsidT,

33=p+1 \ t

^ I <t>h(r) I folM E I I hsMdsi I rfsadT.

t 33=p+1

oo

V.s3

s

p

p

2

s

2

s

Let us consider A^l1l3)(s) (see the derivation of the formula (2.268)))

4l1l3)(s) = -A4l1l3)(s) + A5l1l3)(s) + Ag 1l3)(s) w. p. 1, (2.540)

where

A4:i:3)(s) = l.i.m. V dp. (s)C((:i)C(:3),

4 v > p )TO / J j3ji V ^ji . '

j3 ,ji=0 p

A5:ii3)(s) = i^- E euCs)c.iii)cj335,

j3 ,ji=0

p

A6:i:3)(*) = ^ E .i (s)cj;l)cj33),

j3,j'i=0

\ J J2 (/ MT)dr) J 0;Jt)(IT(IS-,.

2

s / s \ s

1 n TO ' X

dP; ; (s) = -

t j4=p+i

S S OO / T \ 2

^ J hiM J <f>3s(T) J2 / drd.s3,

t S3 j4=p+i \S3 /

s s / s \ 2

./£.;» ^ J J <f>j3M E (y <fe4(si)dsij ds2ds3 =

t S3 j4=p+i \S2 /

S / S \ 2 S2

= \J <f>j3M E y J (f>jl{s3)ds3ds2.

t j4=p+i \s2 / t

Moreover (see the derivation of the formula (2.272)),

(l2l3)/0\ _ OA(l2l3)/„\ /l^^)/^ _ A^^)/^ A(l2l3)/„\ i /\ (l2l3)

A3(s) = 2Ag^ (s) - A5(s) = A4(s) - A5 2i3J (s) + Ag2i3; (s) w. p. 1.

(2.541)

Let us consider A4l1l4)(s) (see the derivation of the formula ((2.273))

s S2 S1

AfH\s) = \j [ [ dw^dsidw^ - A^4)(s) w. p. 1. (2.542)

2

ttt

Let us consider Ag1i2)(s) (see the derivation of the formula (12.2741))

S si S2

4ili3)(s) = UII dw^dw^dSl + t t t

+ jl{i1=i2*)} J(S - s2)ds2 - A^2)(.s) w. p. 1. (2.543)

t

Let us consider Bi(s), B2(s), B3(s) (see the derivation of the formulas (2.275), (2.276))

s „

1p

1

Bi(s) = - / (.Si -t)dSl - lim (2.544)

{ P ^ 34=0

pp

B2(s) = lim £ ap333(s) + li^ CP333(s) - lim £ 6P333(s). (2.545)

p^œ ' ■* J3J3 p^œ ' ■* j3J3 p^œ ' * J^3

33=0 33=0 33=0

Moreover (see the derivation of the formula (2.277)),

p

B2(s) + B3(s) = 2pim£ fj (s).

p

j4=0

Therefore (see the derivation of the formula (2.278)),

p p p p

B$(s) = 2 lim V j (s) - lim V ap (s) - lim V cp . (s) + lim y bp . (s).

p_^^^ ^^^^^ ^3 ^3 p_^^^ ^^^^^ J3 J3 p_^^^ ^^^^^ J3 J3 p_^^^ ^^^^^ ^3 ^3

j3=0 j3=0 j3=0 j3=0

(2.546)

After substituting the relations (2.537), (2.539)-(2.546) into (2.536), we obtain

p

u™. £ C34333231 (s)ci;i)c322)c333)c!4

31,32,33,34 =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

p

1

T si

dS2dS1 + R(s) = J *[^(4)]s,t +

tt

+ R(s) w. p. 1,

(2.547)

where

R(s) = -^^Ap^s) + 1{:i=:3=0} ("A*"'« + A^4^) + A^4^)) +

+ 1{:i=:4=0} (A4:i:3)(s) - A'*"'^) + A^W) - 1{:'2=:3=0|A3:;i:4)(s) +

+ 1{::=:1=0} ("A^s) + A^^) + A'™^)) - X^^ A^'(s) -

p

p

p

1{:i=:3=0}1{:2=:4=0^ ^ E a.p3j3 (s) + ^ E Cp3j3 (s) - ^ E .. (s)

s)

.3=0

.3=0

.3=0

pp

pp (s) - lim ap . (s) -

p TO

.3=0 .3=0

p

1{:i=:4=0}1{:2=:3=0^ 2 ^m E /.3.3(s) - Um E .3 (s)

Sjto E (s)+plij^E bp3.3(s))+

p TO

.3=0

.3=0

+ 1{:i=:2=0}1{:3=:4=0} U™ E «La (s)

(2.548)

.3=0

Fro

and

it follows that Theore 2.31 will be proved if

Ak:.)(s) = 0 w. p. 1, (2.549)

p p p p lim y ap . (s) = li m 7 bp . (s) = lim Cp . (s) = li m /p. (s) = 0,

-j_^ TOO ^^^^^^ J3J3 p_^TOO ^^^^^^ J3J3 p_^TOQ J3J3 p_^TOQ J3J3

.3=0 .3=0

_ (s)= _

p—TO ^—' •/3J3 p—TO

.3=0 .3=0

(2.550)

where k = 1, 2,..., 6, i, j = 0,1,..., m.

Consider the case of Legendre polynomials. Let us prove that

Ail3l4)(s) = 0 w. p. 1.

(2.551)

p

p

We have

, (T-tUMTmn±T)x

j4j3 V / 32

z(s) y to /

x / Pj4 (y) / Pj3(yi) £ (2ji + 1) i Pji ^fe) dyidy

-i -i ji=p+i V-i

_ (r-*)y(2j4 + l)(2/3 + lL

— -X.

32

z (s) z (s) x / £ 2T^{Pjl+l{yi)~Pjl-l{yi))2 I P^y)dydyi =

-i j1=p+i i yi

_ (T-^y^TT^ 32V2J4 + 1

x / Pj3(yi)((Pj4+i(z(s)) - Pj4-i(z(s))) - (Pj4+i(yi) - Pj4 i(yi))) X -i

TO

ji=p+i

if j4 = 0 and

El o

+ 1 №1+1(2/1) "

j4j3 V / 32

z(s)

1

E qry (^1+1(2/1) -

-i

ji=p+i

if j4 = 0, where z(s) is defined by (2.20).

We assume that s E (t,T) (z(s) = ±1) since the case s = T has already been considered in Theorem 2.8. Now the further proof of the equality (2.551) is completely analogous to the proof of the equality (2.290).

It is not difficult to see that the formulas

A2i2i4)(s) = 0, A4i1i3)(s) = 0, A6i1i3)(s) = 0 w. p. 1 (2.552)

can be proved similarly with the proof of (2.551).

Moreover, the relations

p p p

limV ap ■ (s) = 0, lim V № . (s) = 0, lim V . (s) = 0 (2.553)

p—>■ œ ' ^ .3.73 p— œ / j J3J3 p— œ / j .3.3

j3=0 .3=0 .3=0

can also be proved analogously with (2.292), (2.293). Let us consider A3'2'4)(s) and prove that

A3'2'4)(s) = 0 w. p. 1. (2.554)

We have

A3i2i4)(s) = Aii2i4)(s) + Ag2i4)(s) - A7i2i4)(s) =

= -A7i2i4)(s) w. p. 1, (2.555)

where

p

A72'4)(s) = i^. E (s)ci22)ci44).

j2j4=0

oo ' s

gj4j2 (s) = J j (T) J j (si) E ( I j (s2)ds2 / j (s2)ds2 ) t t j'l=p+1

VSl

Note that (see (2.296

/ s s

9iM(s) = jr \ ¡1 <f>j4{r) I ^{s2)ds2dr ) . (2.556)

j1=p+i \t T

The proof of (2.554) for the case i2 = i4 = 0 differs from the proof of the equality

A3l2l4) = 0 w. p. 1

for the case i2 = i4 = 0 (see the proof of Theorem 2.8). In our case we will use Parseval's equality instead of the orthogonality property of the Legendre polynomials.

s

s

Using the Parseval equality, we obtain

p

E^L (s) =

.4=0

p to 1 i s s

= E E 2 I feM / ) =

.4=0 Ji=p+1 \{ T

= E E feW ^y" fe(S2 )d«2 - y (t>ji(s2)ds2 J rfr J <

< El / fe (t )dn E I / fe (s2)dsj +

+ e e y fe (t^ . (s2)ds2dT j =

.4=0 .i=p+^ t t /

= e / 1{T<s}^.4 (t)dT e / fe (s2)dsH +

.4=0 \ { J j1=P+A t J

( t T \ 2

+ E E / V^fe (t ) / fe (s2)ds2dT <

.4=° \ {{ J

< e / V<s}fe (t)dT e / fe (s2)dsn +

.4=0 \ t ) j1=P+A t )

/ T T > 2

TOTO

+ E E I V<s}fe (t) I fe (s2)ds2dT

.^p+l-.4=0 \ t t ,

T oo / s \ 2

= (1{T<s^2 dT E / fe (s2)ds2 +

t jl=P+1 \ t J

TO T 2 V

+ E / (1{T<s^ M / fe (s2)ds2 I dT =

^p*1 tVt

s x 2

oo ' s

= (s - t) £ i j (s2)dsj +

ii=p+A i )

+ £ / ( / j(s2)ds^ dT. (2.557)

ji=p+i * V * /

We assume that s E (t,T) (z(s) = ±1) since the case s = T has already been considered in Theorem 2.8. Then from (2.557) and (2.118) we obtain

p c

j4=° P

where constant C is independent of p.

By analogy with (2.306), we can derive the following estimate

p c

j2,j4=°

for the case s E (t,T) or z(s) E (—1,1) (the case s = T has already been considered in Theorem 2.8), where constant Ci does not depend on p. For this we have to use that

T t

gp4j2 (s)^/ 1{T<s|^j4 (T ) / j (si)Fp(T. si,s)dsidT =

= J Kp(T,si,s)j(t)j(si)dsidT

[t,T ]2

is a coefficient of the double Fourier-Legendre series of the function

Kp(T, si,s) = 1{T<s}1{si<T<s}Fp(T, si,s),

where

s s

00 ,, ,,

def

£ / j(s2)dsW j(s2)ds2 = Fp(r,si,s). (2.560)

p\

ji=p+i si

Moreover, we have to use the Parseval equality

p1 n

phi5TO s j(s))2 = J (kp(t,si,s))2dsidT =

j4j2=0 [t,T]2

sT

p

tt

(fp(t, s1, s))2 ds1 dT.

Then from analogue of (2.329) for s G (t, T) (s is fixed), (2.558), and (2.559) we have

m(it guww)}<

[ Vj2 ,j4=0 / )

p / p \2

{l2 = l4=0} 1 ^ ^ gj4j4 (s) I < j2,j4 =0 \j4=0 '

C2 p2

if p —to, where constant C2 does not depend on p. The equality (2.554) is proved.

Let us consider A^1l3) (s)

A5l1l3) (s) = A4l1l3) (s) + Agl1l3) (s) - A8l1l3) (s) w. p. 1,

where

p

_ 1 : ™ V^ up ( A(l1^(l3)

A™ = li^ E "In(s).'.3

.3.1 =0

hp3.1 (s)= / fe (s3) . (t)Fp(s3,T,s)dTds3,

.3 .1

t s3

where Fp(s3,T, s) is defined by (2.560).

Analogously to (2.554), we obtain that A^l1l3)(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)

s

s

and the relations (see (2.556

hp33i(s)= J Kp(s3,T,s)0i(s3)033(T)dTds3,

[t,T ]2

s S

œ i

h'j j (sî E 2 j j ^^dsidr

j4=p+1 V t T /

Let us prove that

2.

p

•UsE 3, (s) = o. (2.561)

33=0

We have

Moreover,

3,(S) = (S) + dp333(S) - g3333(S)- (2.562)

pp fP

Hm£ /jp3j-3(s) = 0, dp3j3(s) = 0, (2.563)

j3=° j3=°

where the first equality in (2.563) has been proved earlier. Analogously, we can prove the second equality in (2.563).

From (2.558) we obtain

p

p—mTO ' gj3j3 ( ) j3=°

So, (2.561) is proved. The relations (2.549), (2.550) are proved for the polynomial case. Theorem 2.31 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 estimate

03(0)d0

<j (JVO),

where constant C is independent of p, t < t < s < T, and {j(x)}°=0 is a complete orthonormal system of trigonometric functions in the space L2([t,T]) (see the proof of Theorem 2.8 for details). Theorem 2.31 is proved.

s

Let us reformulate Theorem 2.31 in terms on the convergence of the solution of system of ODEs to the solution of system of Stratonovich SDEs.

By analogy with (2.488) for the case k = 4, p1 = ... = p4 = p, ii,..., i4 = 0,1,..., m, and s G (t, T] (s is fixed) we obtain

S ¿4 ¿3 ¿2

HI/dwi:i,pdw<22)pdw<ss)pdw<44»p = £ Cj(s)ii:;,,4,2)cjss)zj44),

t t t t

j'l j2 J3J4 =0

where p £ N and is defined by

as in Theorem 2.31.

The iterated Riemann-Stiltjes integrals

I; another notations are the same

s ¿4 ¿S ¿2

y (¿1Î2Î3Î4)P

dwi:i)p dwi;2)p dwi;s)p dwi;4)p,

i i i i

S ¿3 ¿2

Z

(:i:2:3)p

s^

dwi:i)pdwi:2)pdwi;3)p,

i i i

S ¿2

Y

(:i:2,p

s,i

dwi:i)pdwi:2)p,

i i

(2.564)

(2.565)

(2.566)

S

Xj)p — j <K

i

(:i)p

(2.567)

are the solution of the following system of ODEs

'dVr(:i:2:3:4)P — Z (:i:2:3)pdw(:4)p V (:i:2:3:4)p

s,i

— Z

s,i

dw(:4)p, y

t,t

— 0

dZ

(:i:2:3)P — y^i^P dw(:3)P

s,i

dYs(ii:2)p — xf^dwp,

dX^)p — 1 • dw(:i)p,

Z(:i:2:3)P — 0

Y(f:i:2)p — o

)p — 0

From the other hand, the iterated Stratonovich stochastic integrals

s t4 t3 t2

Vr(ti

s,t

(H«2«3«4)

dw<;i)dw<;2)dwii3)dw<44),

t t t t

s t3 t2

Z

(¿i«2«3)

s,t

dwt(;i) dwt(;2)dwt(;3),

t t t

s t2

Y(iii2) _ Ys,t _

dwi;'»dw<;2),

tt

s

x_ I dw

(ii)

s,t ti

t

(2.568)

(2.569)

(2.570)

(2.571)

are the solution of the following system of Stratonovich SDEs

' dVr(iii2i3i4) _ Zdw(i4) V(iii2i3i4)

s,t

_ Z

s,t

* dwS, Vt

dZ

s,t

dYs(tii2) _ X^ * dw(i2),

dX^0 _1 * dw(ii),

t,t

_ 0

_ Ys(t * dw(i3), Z

(¿i«2«3)

t,t

_0

Yt(;ii2) _ o

_0

where * dwSi), i _ 0,1,... ,m is the Stratonovich differential, * dwS0) _ ds.

Then from Theorems 2.28, 2.30, and 2.31 we obtain the following theorem.

Theorem 2.32 [31]. Suppose that {03- (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then for any fixed s (s G (t,T])

l.i.m. V

(ii«2«3M)p

p^œ

s,t

_ V

(H«2«3«4)

s,t

l.i.m. Zs°fi3)p _ Zs°fi3),

l.i.m. Ys(tii2)p _ Ys(tii2),

X^)p _ l.i.m. X

p^œ

(ii ) s,t .

2.11 Modification of Theorem 2.9 for the Case of Integration Interval [t,s] (s E (t, T]) of Iterated Stratonovich Stochastic Integrals of Multiplicity 5 and Wong—Zakai Type Theorem

Let us formulate the following theorem.

Theorem 2.33 [31]. 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 fifth multiplicity

j*[V'(5)w = //// / dwi;1'dw<22)dwi33)dw<44)dwi55)

t t t t t the following expansion

p

J*[*(5)]«.' =E j jwc^Wcj44)c]55)

j1 j2 J3J4 j5=0

that converges in the mean-square sense is valid, where s E (t,T] (s is fixed), ii,..., i5 = 0,1,..., m,

s S5 S4 S3 S2

Cj5j4j3j2ji (s) = J j J j (s4^ j (s3) J j M J j (si )dsids2ds3ds4ds5, ttttt

T

Cf = / h (T)dw«

t

are independent standard Gaussian random variables for various i or j (in the case when i = 0), wTi) = fj(i) for i = 1,..., m and w[0) = t.

The proof of Theorem 2.33 can be carried out on the basis of proof of Theorem 2.9 (the case s = T) and the method of proof of Theorem 2.31.

Let us reformulate Theorem 2.33 in terms on the convergence of the solution of system of ODEs to the solution of system of Stratonovich SDEs.

By analogy with (2.488) for the case k = 5, pi = ... = p5 = p, ii,..., i5 = 0,1,..., m, and s E (t, T] (s is fixed) we obtain

p

tt (ili2i3i4i5)p \ ' C (s)Z (i2VZ (i4V(is)

Us,t — Cj5 j4j3j2j1 Zj2 Zj3 Zj4 Zj5 ' j1 j2 j3j4 ,j5=0

where

S ¿5 ¿4 is ¿2

ti *2 t3 *4 s,t '

ttttt

p E N, and dwT)p is defined by (12.487); another notations are the same as in Theorem 2.33.

(ii)p Y (ii«2)p Z (iii2«3)p V (¿ii2i3«4)p

The iterated Riemann-Stiltjes integrals Xst, Ys

s,t

s,t

, Vs

s,t

(see (I2564D-(I2W1)): and U^2*3^ are the solution of the following system of ODEs

'dU(ili2isi4«5)p _ у(ili2«S«4)pdwSi5)P Ц(¿1 ^¿SM^P _ Q

dV

s,t

_ ZS72 Wdw(

< dZS(i^i2is)p _ Ys(r)pdw(is)p,

dYS

(i1i2)P _ X(i1)Pdw(i2)P

s,t

- s,t

dXS(ï)P _ 1 • dw.

(i1)P

V (n

(¿1¿2«S«4)p

_Q

Z

(¿1i2«s)p

_Q

Y(t1i2)p _ Q

Xj)p _ Q

From the other hand, the iterated Stratonovich stochastic integrals X^1 ),

Ys((i1i2), Z<f4 Vs((i1i2isi4) (see (2568)-(25711)), and U^3^ are the solution of the following system of Stratonovich SDEs

'dU(*1i2isi4«5) _ V(«1«2is»4^ ^ dw(i5) Ц(¿1г2«3«4«5) _ Q

dV

(¿^¿ЗЧ) _ Z(¿1*2«з) dw(i4)

S,t

S,t

dw (

< dZ (¿Ji2is) _ Ys(i1i2) * dw (is),

dYS(i1i2) _ X * dw <4

S(t

dXSiJ1) _ 1 * dw sn;,

(¿1)

^(¿згзч) _ q

Z

(¿1«2«з)

t(t

_Q

Yj(J1i2) _ Q

_ Q

s

where * i = 0,1,... ,m is the Stratonovich differential, * = ds.

Then from Theorems 2.28, 2.30, 2.32, and 2.33 we obtain the following theorem.

Theorem 2.34 [31]. Suppose that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). Then for any fixed s (s e (t,T])

l i m U(i1*2^4^ = U(i1*2^5) l i m V= V(i1*2*3*4)

s,t s

t , 1.1.HI. — vst ,

p—»O ' ' p—>• OO ' '

l . i . m . Zs(ri3)p = Zs(ri3), l . i . m . Ys(r)p = Ys(;ii2)

p—>• œ ' ' p—>• œ ' '

Xt)p = i.i.m. xS;,1 >.

' p—œ '

2.12 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 4 in Theorems 2.2, 2.7, and 2.8

2.12.1 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 2.35 [31]. 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

;T

<Î2

J *

T,t = ih(h)

(ii,¿2 = 1,... ,m)

the following estimate

M

J *[^(2) ]

T,t

Er Âh)Âi2) <

j1j2=0

C

p

(2.572)

p

is valid, where constant C is independent of p,

T S2

Cj2ji = ^2(s2)0j2 (s2) ^i(si)0ji (si)dsids2,

and

T

c«!)=j to (t )df«

t

are independent standard Gaussian random variables for various i or j.

Proof. From (2.8) we obtain

p2 I j*W'(2)]T,f - £ Cj2j,cji-'cj:2'

V ji,j2=0

m{ ( J^lK + ^ii^,,}/to(iiW'2(ii)<iii- v C^cfcf

t ji,j2=0

= m{ J[to2>]T,( - £ Cj2ji( cj:i,cj22) - i{.,=!2}i{ji=j,})+

I V ji,j2=0

T x 2-

1 i p x 2

+ 21{h=»2} / - l{i1=i2} E Qu!

t ji=0

M j J[^(2)]T,t - £ Cj2j-YCj:i)Cj:2) - 1{:i=:2} 1{ji=j2^ \ +

{ V ji ,j2 =0 V

h t p

{ii = i2W ^i(ti)^2(ti)dti - 1{ii = i2} Cjiji t ji=0

M^ (j [^(2)]T,t - J «t H +

+ l{i1=i2} ( \ ¡Mtiyut^dh 1 • (2.573)

t ji=0

From Remark 1.7 (see (1.215)) we have

J- J

M

where constant C1 is independent of p. From Theorem 2.2 (see (2

<

Ci p

(2.574)

we get

T

oo

- / ißiihyißoit^dh -J2CJU1 i ji=0

Cj1 j1 '

ji=p+i

(2.575)

Let us consider the case of Legendre polynomials. The estimate (2.65) implies that

oo

Cj1j1

j1=p+1

oo

E ¿1.

p j1=p+iji

(2.576)

where constant C2 does not depend on p. From (2.25) and (2.576) we have

Sp —

El Cj1j1 j1=p+i

<

Cs p

(2.577)

where constant C3 is independent of p.

Applying the ideas that we used to obtain the relations (12.671). (I2.71l)-(I27731). we can prove the following estimates for the trigonometric case

S2p =

El Cj1j1 j1=2p+i

<

Kl p

S2p-i =

El Cj1j1 j1=2p

< Sop h—, p

(2.578)

(2.579)

where constants Ki, K2 do not depend on p.

From (2.578) and (2.579) we get the estimate (2.577) for the trigonometric case. Combining (2.573)-(2.575), (2.577), we obtain (2.572). Theorem 2.35 is proved.

2

p

2.12.2 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 2.36 [31]. 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 ^(t), ^3(t) are twice continuously differentiable nonrandom functions on [t,T]. Then, for the iterated Stratonovich stochastic integral of third multiplicity

* T * is * ¿2

J*[^(3)]T,i = / (is)/ «t2)/ ^i(ii)dfi(;i)dfi;2)fs) (ii,i2,i3 = 1,...,m)

i i i

the following estimate

Mjfj'I^lr,,- t, Q3«,Cj:',Ci,Cj:3)) 1 < ^ (2.580)

[ V jl ,j2,js=0 /J ^

is valid, where constant C is independent of p,

T ss S2

Cjsj2ji = / ^3(^3)j(S3)/ V^Mj(s2^ ^i(si)j (si)dsids2ds3,

i i i

and

T

j=/ to (t )df«"

t

are independent standard Gaussian random variables for various i or j. Proof. We have (see (2.362))

m^ (j№>]T,t - x zfzj:2)zjss)X

j1,j2,j's=0

T is

M<! ( J[0(3)]T,. + ii{il=i2} /^3(i3) /'02(i2)'0i(i2)«fi;3)+

# t

T is p

juh)uh) [Mti)dt]dh- E c>

jij2,jS=0

= m j (j [^(3)]T,i - j [^(3)e'p+

T ts

p

+l{il=i2} ( - / ^3) / ^(^ife)«^ - E 1 +

t t jU3=0

1 T r p

+i{i3=i3} ( ö / / E ^C(n)

2 / r3v3/r^v3/ / r^"^—ti —3 / j jsjsjij

t t ji ,js=0

pp

i(ii=.yEECjijsjiejs2') (2.581)

ji =0 js=0

where (see (1.46))

p

J[#>]-p = £ Cjsj2j4 zjr'zfzjss)-

j'i,j2,j's=0

_1 1 Z (;s ) _ 1 1 Z _ 1 1 z (;2)

1{;i = ;2}1{ji =j2 } Z js 1{;2 = ;S}1{j2=jS}Zji 1{;i=;S}1{ji=jS} Zj2

From (2.581) and the elementary inequality

(a + b + c + d)2 < 4 (a2 + b2 + c2

we obtain

m J (j*[^(3)]t., - E ¿jzj;;i)zj:2)zjss^ <

^ V ji,j2,jS =0

< 4^ M { (J [^(3)]s,t - J [^(3)]?,p'P) ^ + 1{;i =;2}EP ) + 1{;2=;s}Ep ) +

+1{;i=;s}Ef), (2.582)

2

where

T ¿3 \ 2

= juh) J Ut2)Mt2)dhd^ - X CJ3J1J1

, t t ji ,js=0

T is p

t t ji ,js=0

if p p

E E Cjijsji cjs

t(i2) js

ji=0 js=0

From Remark 1.7 (see (1.215)) we have

M | - J[^3)]pTf J| < j, (2.583)

where constant C1 is independent of p.

Moreover, from (2.256) and (2.260) we have the following estimate

M3> < ^ (2.584)

1 p

for the polynomial and trigonometric cases, where constant C2 does not depend on p.

Using Theorem 1.1 for k = 1 (also see (1.44)), we obtain w. p. 1

T s p

\J Us) f Usi)Msi)dSldf^ = ±CJ5C^,

t t P ^ j3=0

where

T s

Cjs = J tojs(s)^3(s) / ^2(si)^i(si)dsids. tt Applying the Ito formula, we have

Ts TT

J Ms)Ms)J ^i(si)dfS;i)ds = J USl)j ^3(s)^2(s)dsdfS;i) w. p. 1. t t t s1

Using Theorem 1.1 for k = 1 (also see (1.44)), we have w. p. 1

1 p

^ I Ms) I h(si)Usi)dSldi^ = -LLm.

t s P ^ ji=0

where

T T

Cji = I ^i(s)j(SW )^2(si)dsids.

Further, we get

ji

t

E^1) < 2G^1) + 2Gp2), (2.585)

E^2) < 2Hp1) + 2Hp2), (2.586)

where

gP] = M {\ (j hit2)^{t2)dt2di^ -

4 t t js=0

p p 2

js=0 ji ,js=0

T ts p

4 t t ji=0 p p 2

ji=0 ji,js=0

From Remark 1.7 (see (1.215)) we have

G«1» < H^ < (2.587)

where constant C2 is independent of p. The estimates

Gi2) < —, #i>2) < — (2.588)

p p p p

are proved in Sect. 2.2.5 (see the proof of Theorem 2.7) for the polynomial and trigonometric cases; constant C3 does not depend on p.

Combining the estimates (l'2.582l)-(2.588l). we obtain the inequality (12.580). Theorem 2.36 is proved.

2.12.3 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.

Theorem 2.37 [31]. 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

J*[^(4)]T,t = II J I dwt(;i)dwt(;2)dwt(33)dwt(44) (ii,i2, i3, i4 = 0,1,..., m) t t t t

the following estimate

M j (r^h, - £ ^.cfcjrW') I * c~ <2'589)

^ V j1j2j3j4 =0 / J

is valid, where constant C is independent of p,

T S4 S3 S2

Cj4j3j2j1 = ^ j (s4) / j My j (s2^ j (s1 )ds1ds2ds3ds4 , tttt

and

T

Cf = J h (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. First, let us prove that Theorem 2.7 is valid for the case i1, i2, i3 = 0,1,..., m. The case i1, i2, i3 = 1,..., m has been proved in Theorem 2.7. From (1.46) and the standard relation (2.362) between Stratonovich and Ito stochastic integrals of third multiplicity we have that Theorem 2.7 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, (2.590)

12, i3 = 1,..., m, i1 = 0, (2.591) ii, i3 = 1,...,m, i2 = 0. (2.592)

The relations (1.46) and (2.362) imply that for the case (2.590) we need to prove the following equality

E / ^(ti) / j (Î2)^2(Î2W j (ti)^i(ti)dtidt2 dt3 =

ji=° t t t

T t3

= ^ y^3(i3) J Mti)Mti)dhdt3. (2.593)

tt

Using the relation (2.10), we get

t t3 t2

» ft rt /»

E / ^(ti) / j (Î2)^2(Î2W j (ii)^i(ii)dMÎ2 dt3 =

j1=0 t t t

T T T

œ

— E / j (ti)^i(tiW j (¿2)^2) / (i3)dt3dt2dti — j1=0 t ,1 ,2

^ T T

œ

— E / j (ti)^i(tiW j (¿2)^2)^2 dti = j1=0 t ,1

œ T ^

= E / j (^2)^2(^2)/ j(ti)^i(ti)dtidt2 = j1=0 t t T

= ^ i MtoJh(to)dto, (2.594)

where

T

^(¿2) = ^2) / ^(¿3)^3. (2.595)

t2

From (2.594) and (2.595) we obtain

t t3 t2

OO ft rt ft

X / / j (t2)^2(t2W j (ti)^i(ti)dtidt2 dt3 =

ji=0 t t t T T

= 5 j Mh)h{to) j h{h)dhdto = t t2 T t3

= ^ y^3(i3) J 'Mh}Hh)dUdt3. (2.596)

tt

From (1.46) and (2.362) it follows that for the case (2.591) we need to prove the following equality

O ft 3 2

X / (¿3)^3) / j(¿2)^2) / ^i(ii)dtidt2dt3 = j2=0 t t t T t3

= ^ J h(h)Hh) J i/Ji(ti)dtidh. (2.597)

tt

Using the relation (2.10), we have

O T ^ y

X / (¿3)^3(i3W j (¿2)^2(i2W Wti)dMi2 dt3 =

j2=0 t t t

o T ^

= X I j (t3)Mt3)l j (¿2)^/^2(¿2)dt2dt3 = j2=0 t t T

where

t2

^(¿2 )= ^(¿2)^ ^i(ti)dti. (2.598)

t

The relation (2.597) is proved.

The relations (1.46) and (2.362) imply that for the case (2.592) we need to prove the following equality

E / j(¿3)^3(£3) / ^(¿2) / j(ii)^i(ii)dtid^dtj = 0. (2.599)

ji=01 t t

We have

O « 3 2

E / j(tOWtj) / W*2) / j(ii)^i(ii)d£id£2d£3 = ji=01 t t

T t3 t3

00

= E J jfa^hfa) I j(ti)^i(tiW ^2(t2)dt2dtidt3 = ji=0 t t ti w T ts / T T \

E / j (t3)^3(t3^ j (ti)^i(tiW J ^2 (t2 )dt2 - J ^2(t2)dtJ dti dt3

ji=0 t t ti ts

T ts T

w ft ft ft

= E / j(t3)^3(t3W j(ti)^i(tiW ^2(t2)dt2dtidt3-

ji=0 t t ti

w T fc T

/ j(t3)^3(t3W j(ti)^i(tiW ^2(t2)dt2dtidt3 = ji=0 t t ts w T t3

= E / j(t3)^3(t3W j(ti)^Ai(ti)dtidt3-ji=0tt

00 T t2

E J j(t3Yh(t3) J j(ti)^i(ti)dtidt3 ji=0 t t

T T

= \ J h{ti)Mti)dti - ^ j h{ti)iJi{ti)dti = tt T T T T

= \ j h{h)Mti) J hitojdtidti - ^ j ik{h)h{h) J h{t2)dt2dti = 0, t ti t ti

where

T

Wti ) = ^2(t2)dt2, (2.600)

ti

T

) = ^(¿3) ^ ^2(t2)dt2. (2.601)

t3

The relation (2.599) is proved. Theorem 2.7 is proved for the case ii, i2, i3 = 0,1,..., m.

Using (2.362) and (2.363), we obtain

(j*[^4)]T,t - E Cjzf zfzfzj x

p

(«iM^M^Mm)

j4

jU2 j3j4=0

i/ T s Si

( JbP^kt + J J Jdsodw^dw^ +

t t t

T S2 si T si S2

J J J dw^dsidw^ + J J J dw^dw^ds^ t t t t t t

T si p ^ 2

+ 41{n=^o}l{i3=^o} J J dsod.si - y CjMjiCj[i)Cjf Cjf Cjf

t t jl,j2,j3,j4=0

*T *s *si

= M { ( J[^,(4)]T>i + il{il=i2^} III ds2dw^dw^

ttt

T si *T *s2 *si

|1{ii=Mo}l{i3=i4^o}f I ds2dsi + il{i2=i3^0} I I I dw^dsidw^A

t t t t t

*T *si *S2 T si

+il{i3=i4^0} J J J dw^dw^ds! - il{il=i2^0}l{z3=i4^0} J J ds2dsi + t t t t t T si

+ 41{H=*2^0}l{i3=M^0} J J ds2dsi - E Cnnj2jiCj[1\jl2\jl3)Cjl4)

t t ji,j2,js,j4=0

*T *s *si

M { ( J[^(4)]T,t + il{il=i2^0} III ds2dwM(faM +

ttt

*T *S2 *si *T *si *S2

III dw^dSldw^ + il{,3=^0} III dw^dw^dSl

t t t t t t

T si x 2

j J ds2dsi - E CjdshhC3\ Cn (js CJ4

t t ji,j2,js,j4=0

= m i j[#4j]T,t - j[#4

*T *s *si

+51«.=«*» [iff - S^" ) +

ttt *T *s2 *si

[iff (to^ds^ - sf ^ I +

ttt

*T *si *s2

+ ^{<3=^0} ( [ [ [ dw^dwifdSl ~ Sf^

ttt

1 T si

-l{n=i2^0}l{i3=M^0} I 4 / /

p1

tt T s

j4=0 2 t t

2

where S(i3i4)p, s2hm)p, S3i1i2)p are the approximations of the iterated Stratonovich stochastic integrals

*T *s *si *T *S2 *si *T *si *S2

///JJJ ^W^, JJJ ^Wi^i.

t t t t t t t t t

respectively (these approximations are obtained by the version of Theorem 2.7 for the case ii,i2,i3 = 0,1,...,m); J[^(4)]yp'p'p is the approximation of the iterated Ito stochastic integral J[^(4)]Tjt obtained by Theorem 1.1 (see (1.47))

7To/i(4)l P'P'P'P — V^ C Z («i)Z («2)Z (i3)Z («4)

J [/ ]T,t - Cj4j3j2ji Zji Zj2 Zj3 Zj

ji J2,j3,j4=0

-t /l(i3i4)P n /l(i2i4)P n /l(i2i3)P n /l(iii4)P

-1{ii = «2=0}A1 - 1{ii=i3=0}A2 - 1{ii = «4=0}A3 - 1{«2 = i3=0} A4

1 /l(iii3)P T /l(iii2)P^1 1 DP I

-1{i2 = «4=0}A5 - 1{i3=«4=0}A6 + 1{ii = i2=0}1{i3=i4=0}B1 +

+ 1{ii=i3=0}1{i2 = «4=0}B2 + 1{ii = i4=0}1{i2=i3=0}B3 ,

where

PP

A (i3i4)P — C Z («3)Z («4) A (i2i4)P — C Z («2) Z (i4)

A1 — / v Cj4j3jiji Zj3 Zj4 , A2 — / v Cj4j3j2j3 Zj2 Zj4 ,

j4 ,j3,j'i=0 j4j3j2=0

PP

i («2«3)P — V"^ c Z (i2) z («3) a (iii4)P — V"^ c Z («i) Z («4)

L3 — Cj4j3j2j4 Zj2 Z j3 ' A4 — Cj4j3 j3ji Zji j ,

j4 ,j3,j2=0 j4,j3ji=0

PP A (ii«3)P — V"^ c Z(«i)Z(«3) A (iii2)P — V"^ c Z(«i)Z(i

A5 — Cj4j3j4ji Zji Z j3 , A6 — Cj3j3 j2 ji Zji Zj2

j4 ,j3,j'i=0 j3,j2,j'i=0

PP

BP — V^ C- ■ ■ ■ BP — V^ C

B1 — / v Cj4j4jiji , B2 — / v Cj3j4j3j4, ji j4=0 j4 ,j3 =0

P

BP — V^ C- ■ ■ ■ •

B3 — / v Cj4j3 j3j4 • j4 ,j3=0

is the expression on the right-hand side of (2.280) before passing to the limits, i.e.

Rp = -1{n=i2=0}^i + 1{ii=i3=0} + Ai + A3 ) +

+ 1{ii=i4=0} ^A4 - A5 + A6 ) - 1{i2=i3=0}A3 +

+ 1{i2=i4=0} ( -A4 +A5 +A6 j- !{i3=i4=0}A6 -

p p p

■ ' — ' "J3J3

"1{i1=i3=0}1{i2=i4=0} ( E ap3j3 + E cp3j3 - E j

V?3=0 j3=0 j3=0

p p p p

1{i1 = i4=0}1{i2=i3=0^ 2 E fp3j3 - E ap3j3 - E cp3j3 + E bp3j3 +

j3=0 j3=0 j3=0 j3 =0

p

where

+ 1{i1 = i2=0}1{i3=i4=0} E app3j3 ,

j3=0

pp A (i3i4)p = V"^ ap Z(i3)ZM A (i2i4)p = V"^ IÜ z(i2) Z(i4)

^i = aj4j3 Zj3 Zj4 ' ^2 = °j4 j2 Zj2 Zj4 '

j3,j4 =0 j4,j2=0

pp

A (i2i4)p = ^p z(i2) Z(i4) A (i1i3)p = dp z(i1) Z(i3)

^3 = Cj4j2 Zj2 Zj4 ' = 2-^ dj3j1 Zj1 Zj3 '

j4,j2 =0 j3,j1=0

pp

A (i1i3)p = p Z(i1)Z(i3) A (i1i3)p = fp Z(i1)Z(i3)

j3,j1=0 j3,j1=0

where

ap ■ bp ■ cp . dp ■ ep ■ fp .

j4j3' j4j2' j4j2' j3j1' j3j1'

are defined by the relations (2.264), (2.266), (2.267), (2.269)-From (2.602) and the elementary inequality

(ai + ... + a-6)2 < 6 (ai + ... +

we get

2

p2

M{ (J*[#>]«- E CjCi:i)zi22)ziss)zj44M <

ji,j2,js,j4 =0 / J

< 6 (qPi) + QP2) + QP3) + Q4) + QP5) + Q6)) , (2.603)

where

QPi) = m J[^(4)]t,; - J[^(4)]Tf'p

I / *T *s *s ttt

*T *s *si

ds2dwiis)dwi;4) - s(;s;4)p

I / *T *s2 *s

Qf = ¿i{.3=M0iM /'/'/'

ttt

*T *s2 *si

dwi;i)dSidwi;4) - s2;i;4)p

I / *T *si *s;

Q(p] = ///

ttt

*T *si *s2

dw(;i)dw(^2)dsi - S3;i;2)p

QP5) = 1{;i=;2=0}1{;s=;4=0}x

jp ^{;i T p T s

1 T , , A1

42

t j4=0 t t

Q6) = M{ (Rp)2}

From Remark 1.7 (see (1.215)) we have

Q{P1] < —, (2.604)

pp

where constant Ci is independent of p.

2

2

2

2

2

Let us prove the version of Theorem 2.36 for the case ii, i2, i3 = 0,1,..., m. The case ii, i2, i3 = 1,..., m has been proved in Theorem 2.36. It is easy to see that, in addition to the proof of Theorem 2.36, we need to prove the following inequalities

T

t3

I I hih) I Mh)Mh)dtidt3-

P T t3 t2

/ /3(*3)/ j (t2)/2(t2^ j (t1)/1(t1)dt1dt2dt3 ji=0 t t t

C

< -, (2.605) p

T

t3

/3(t3)/2(t3) / /i(ii)di1dt3-

T

t3

t2

E I j (t3)/3(t3W j (t2)/3(t2W /1(t1)dMMt3

j3=0 t t t

C

< —,

p

(2.606)

1

2

T

t3 t2

I j (t3)/3(t3W /2^2) / j (t1)/1(t1)dt1dt2dt3

ji=0

C

< -, (2.607) p

where constant C is independent of p.

The inequalities (2.605) and (2.606) are equivalent to the following inequalities (see the proof of the cases (2.590), (2.591))

t p t t2

/1(t2)/Mi2)dt2 I j (t2)Mt2) I j (t1)/1(t1)dt1dt2

ji=0 t t

<

C

t p t t3

<

p

(2.608)

c p

(2.609)

where ?/2(t2), V>2(£2) are defined by (2.595) and (2.598), respectively. The inequalities (EOT), (EM) follow from (23751), (2577)-(2579).

\ J J (pj-M)hih) J (pJ3{toyß2{u)dudh

t j3 =0 t t

Let us prove (2.607). By analogy with the proof of (2.599) we have

p T t3 t2

E J j(£3)^3) y W^y j(£i)^i(£i)d£id£2d£3 = ji=01 t t

p T t3

= E i j(tOWtO / j(£i)^^i(£i)d£id£3 ji=01 t

p T t3

E i jfaYhfa) ( j(ii)^i(ii)d£id£3 ji=0 { t

00 T t2

E J j(t3)^3(t3W j(ti)^^i(ti)dtidt3

ji=0 t t

w T ^

-E / j^3)^3) / j(ti)^i(ti)dtidt3-

ji=0tt

w T ^

E / j(t3)^3(t3M j(ti)^^i(ti)dtidt3+

ji=p+i t t

w T ^

+ E / j^3)^3) / j(ti)^i(ti)dtidt3 =

ji=p+i t t

w T ^

= - E / j(t3)^3(t3M j(ti)^^i(ti)dtidt3+

ji=p+i t t

t ts

ws

+ E / j^3)^3) / j(ti)^i(ti)dtidt3, (2.610)

ji=p+i t t

where ^(tO, ^3(t3) are defined by (2.6001). (12.6011). respectively.

Now the estimate (2607) follows from (26TQ and (2577) (2579). Theorem 2.36 is proved for the case ii, i2, i3 = 0,1,..., m.

Using the version of Theorem 2.36 for the case ii, i2, i3 = 0,1,..., m, we obtain the following estimates

C

C2

q? < Q? < Qi4)<

p

P

p

P

c* p

where constant C2 does not depend on p. From Theorem 2.2 (see (2.36)) we get

1

2

T T

T p T

(S1 - t)ds1 - y I j(s) / j(S1)(s1 - t)ds1ds

j4=0 t t

(2.611)

oo

T

Y / j (sW j (s1)(s1 - t)ds1ds-

j4=P+1 { {

(2.612)

Let us consider the case of Legendre polynomials. From (2.577) and (2.612) we have

T s

I j (s) / j (si)(si- t)dsids

j4=P+i t t

<

9i p

(2.613)

where constant C3 is independent of p.

By analogy with (2.578) and (2.579) we have the estimate (2.613) for the trigonometric case. Then

C4 p2

(2.614)

where constant C4 does not depend on p.

Analyzing the proof of Theorem 2.8, we conclude that

qP6) <

p

(2.615)

for the polynomial and trigonometric cases; constant C5 is independent of p.

Combining (2.603), (2.604), (2.611), (2.614), (2.615), we get (2.589). Theorem 2.37 is proved.

s

s

2.13 Rate of the Mean-Square Convergence of Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 2 to 4 in Modifications of Theorems 2.27, 2.29, and 2.31 for the Case of Integration Interval [t,s] (s G (t,T])

Let us prove the following theorem.

Theorem 2.38 [31]. 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 ^i(ti)dft(;i)dfi2i2) (M2 = 1,..., m) t t

the following estimate

M I - ± J ^ (2.616)

is valid, where s G (t,T] (s is fixed), constant C(s) is independent of p,

S t2

Cj2j1 (s) = J ^2(Î2)0j2 ^l(tl)0ji (tl)dtidt2, tt

and

T

cj,:)=j h (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 2.35. Below we consider the case s G (t,T). By analogy with (2.573) we obtain

( J*W'(2)kt - £ j(s)cj1°zj X

p

= (n)Z (i2) j2

jl,j2=0

\ 2'

m ji^(2)].,,t - JW'(2)iP:P +

+ l{il=i2} I \ f uhyut^dh ¿r^i,) I , (2.617)

2

t ji=0

where (see (1.242))

p

j['<A,2)]P:P = £ Cj2ji(«) zjij - i{.,=,2=0}i{ji=j2}

ji,j2=0 \

From Remark 1.12 (see (1.239)) we have

m i - ^ < ^

(2.618)

where constant Ci(s) is independent of p.

From (2.509) we obtain (the existence of the limit on the right-hand side of (2.509) will be proved further in this section)

s

1 P o

2 ]Mti)№i)dti-Y,Cjiji{s)= E CJin(s). (2.619)

t ji=0 ji=P+i

Consider the case of Legendre polynomials. By analogy with (2.64) we get for n > m (n, m £ N)

s 9

n n „ „

e Cjiji(«)= e (0)J ^i(r)j(t)drd^ =

ji=m+i ji=m+i t t

z(s)

T - £ f

= —/ (Ma;)) (Pn+1(x)Pn{x) - Pm+i{x)Pm {x)) dx--i

n z(s)

ji=m+i J _i

X ( (Pji + i(z(«)) - Pji-i(z(«))) - (Pi+i(y) - Pji-i(y)) «%))-

z (s)

[ (Pjl+i(x)-Pjl.1(x))i/2(h(x))dx\dy, (2.620)

where

, / \ T -t T +1 ( T +1

Hy) = —y + —, = s -

2

2

2

T-t

and , ^2 are derivatives of the functions ), ^2(r) with respect to the variable h(y) (see (2.54)).

Applying the estimate (2.61) and taking into account the boundedness of the functions ^1(r), ^2(r) and their derivatives, we finally obtain

E Cjiji (s)

ji=m+1

z (s)

<C1(1- + ^ r dx

+

n m

1

n / z(s)

If

(1 - x2)1/2 z (s)

+c2 E 4 [

ji=m+1 ^M^

+

dy

(1 - y2)1/2 (1 - z2(s))1/4J (1 - y2)1/4

z(s)

z(s)

+

dx

-1 \

1

(1 - y2)1/4J (1 - x2)1/4

dy

(2.621)

y

where constants C1, C2 do not depend on n and m.

We assume that s G (t,T) (z(s) = ±1) since the case s = T has already been considered in Theorem 2.35. Then

E Cjiji(s)

ji=m+1

< c-M ( i+- + è 4

» n m ^^ j2

ji =m+1

Ji

(2.622)

where constant C3(s) does not depend on n and m. Thus, the limit

Y Cjiji(s)

p

ji=0

(2.623)

exists for the polynomial case. For the trigonometric case, the existence of the limit (2.623) can be proved by analogy with the proof of Lemma 2.2 (Sect. 2.1.2).

The relations (2.622) and (2.25) imply that

E Cjiji(s) ji=p+1

p ji=+1 j27 p

(2.624)

n

1

1

n

p

where constant C4(s) is independent of p.

For the trigonometric case, the analog of the inequality (2.624) can be obtained by analogy with (2.578) and (2.579) (see the proof of Lemma 2.2).

Combining (I2T6T71)-(I2T6T91), (127624), we obtain the estimate (I26T6). Theorem 2.38 is proved.

The arguments given earlier in Chapters 1 and 2 of this book allow us to formulate the following two theorems.

Theorem 2.39 [31]. Suppose that (x)}°=0 is a complete orthonormal sys-

tem 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)]s,i = / «is)/ W*2)/ ^i(ii)dwi;i)dwii2)dwiss), t t t

where ii, i2, i3 = 0,1,... ,m, the following estimate

M ((V>W - t c^wwe')'} < S®

[ V jU2 ,js=0 /J 1

is valid, where s £ (t,T] (s is fixed), constant C(s) is independent of p,

S is t2

Cjsj2ji (s) = / ^3(t3)0js ^ / ^2(¿2)0j2 (^2^ / ^i (ti)^ji (¿i)dMMt3, ttt

and

T

j = J to(tf

t

are independent standard Gaussian random variables for various i or j.

Theorem 2.40 [31]. 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

* s * t4 * tS * t2

J*[^(4)]s,t = /// / dwt;i)dwt;2)dwtss)dwt44) (ii,i2,i3,i4 = 0,1,...,m) t t t t

the following estimate

Er (Q\An)Ai2)Ai-i)Ai4)\ \ ^ C(s)

1 ./:./:■,/:./ S/V A'3 S/: ( ..

M < I J'[f'],< - £ ^«.(»iT

j'lj2 jsj4=0

is valid, where s G (t, T] (s is fixed), constant C(s) is independent of p,

S ¿4 is t2

Cj4jSj2j1 (s)^/ j (t4^ j ^ 0j2 (t2^ j (t1 )dt1dt2dt3dt4, 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.

2.14 Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity k. The Case ¿1 = ... = ik = 0 and Different Weight Functions ^1(r),..., (t)

In this section, we generalize the approach considered in Sect. 2.1.2 to the case ¿1 = ... = ik = 0 and different weight functions ^i(t),... (t) (k > 2).

Let us formulate the following theorem.

Theorem 2.41 [32]. 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

* T * ¿2

= J 1(tk). J 1 (ti)dfi;i}..f (¿1 = 1,...,m) t t the following equality

is valid, where n £ N,

T t2

Cjfc -ji = / (tk ) j (tk)... ^i (ti)toji (ti)dti ...dtk

is the Fourier coefficient and

T

j) = / h (t(ii = 1,..., m)

■(ii)

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'{ti,t2,h) = \ { 6

tl)^2(t2)^3 (t 3

tl)^2(t3 )^3 (t2

t2)^2(tl)^3 (t3 t2)^2(t3)^3(tl t3)^2(t2)^3(tl

^l(Î3)^2(tl)^3(Î2), t3 < tl < t2

, ti < t2 < t3

, ti < t3 < t2

, t2 < ti < t3

, t2 < t3 < ti

, t3 < t2 < ti

ti,t2,t3 £ [t,T].

Using Lemma 1.1, Remark 1.1 (see Sect. 1.1.3), and (2.362), we obtain w. p. 1

N-i N-i N-i

J[K']<3t = l.i.m. £ £ £ K'(Tii, t,2, TisfAf^f =

/3=0 /2=0 1i=0 N-1 /3-1 /2-1

lNi-m- EEE^i^2,T/3)f f f +

V/3=0 /2=0 /i=0 N-1 /3-1 /i-l

+E E E K '(T/, , T/2 , T,3 f ff >+

/3=0 /,=0 /2=0

N-1 /2-1 1l-1

+ £ £ £ K'(t/1 . t/2 , t/3fff^

/2=0 /1=0 /3=0

N-1 /2-1 /3-1

+ £ £ £ K'(t/1 . t/2 , t/3 ) AfTi;11) AfTi;21) AfT,;31) +

/2=0 /3=0 /1=0

N-1 /1-1 /2-1

+ £ £ £ K(t/1 , t/2 , t/3ff

/1=0 /2=0 /3=0

N-1 /1-1 /3-1

+ £ £ £ k (t/1 , t/2 , t/3 fff^

/1=0 /3=0 /2=0

N-1 /2-1 2

+ ££K'(t/1 -t/2,t/1 )(Af;;>) f >+

/2=0 /1=0

N-1 /3-1 2

+ ££K(t/1 ,t/3,t/3)(Af£>) AfT;1)+

/3=0 /1=0

N-1 /1-1 2

+ ££ K'(T/1 .t/2 ,T/2 ^f) f +

/1=0 /2=0

N-1 /3-1 2

K'(T/3 .T/2 . T/3 )(f>)

/3=0 /2=0 3 2

N-1 /3-1 2

K (T/2 .T/2 . T/3 ^f) f +

/3 = 0 /2=0

N-1 /2-1 2

K' (T/2 .T/2 .T/3 }(f) f

/2=0 /3 = 0

/ T t3 t2

= UI Uh) J Mh) J Mt^df^dtb ttt T t2 t1

+ / Mt2)J i1(t3)df£1)dft(1;1)df(;1) +

ttt

T t2 t3

+ j ^3(t2^ ^2(t3)/^ 1 (11 )dft(;1 )dft(i 1 )dft(2i 1 ) + t t t T t3 t,

+J Mt3)J «t^y ^df^d^f;i)+ ttt T ti t2

+ j Mh) J Ut2)f ^1(t3)dft(3;i)dft(2;i)dft(i;i) + ttt T ti t3

+ /^3(ti^y «t^ ^1(t2)dft(2;i)dft(3;i)dft(i;i) + ttt T t2 T ti

+ J ^3(t2^ ^2(t1)^1(t1)dt1dft(2;i) + J «t1)/^2(t2)^1(t2)dt2dft(1ii) + t t t t T t3 T t3

+ / ^J ^2(t1)^1(t1)dt1dft(3;i) + J Mt3)Mt3) j Äfc^W t t t t T t3 T t2

+ J Mt3)Mt3)J ^1(t2)dft(2;i)dt3 + J Mt2)Mt2)J ^1(t3)dft(3;i)dt2 t t t t T t3 t2

= j Ut3)f Mt2)J ^1(t1)dft(i;i)dft(2;i)dft(3;i) + ttt T t3 T t3

y^3(i3) JihitiïMtiïdtidf^ + ^ j Uh)Uh) J'Mhïdf^dt^ t t t t * T * t3 * t2

= J U^J Mt2)J ^1(t1)dft(i;i)dft(2;i)dft(3;i) =f J*[^(3)]T,t, (2.625) t t t

where the multiple stochastic integral J[K']T3t is defined by (11.161) and {t^}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 Fourier-Legendre expansion (2.56) for the function K'(ti, t3), we obtain

p p p 1 / n .;„— n;—n V

ji=0 j2=0 j3=0

g \ ~ hhh ^ './ '../ ./: + ^ './: ./ ./:■. +

+ Cj2jSj1 + Cj1j2 jS + Cj1jSj2 J j (t0 j (t2)tojs (t3 ) , (2.626)

where the multiple Fourier series (2.626) converges to the function K'(t1, t2, t3) in (t,T)3. Moreover, the series (2.626) is bounded on [t,T]3; (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 (2.56) to the three-dimensional case (see (2.626)) is omitted as well as the proof of the boundedness of the Fourier-Legendre series (12.626) on the boundary of the cube [t,T]3.

Denote

p p p 1

p p p 1 / Kppiti'^'k) = K'ihMM"EEEë c

ji=0 j2 =0 j3=0

g \ - J-îJUl + ^ './:■../ ./: + ^ './: ./ ./:■.

+ Cj2j3j1 + Cj1j2 j3 + Cj1j3j2j j (t0 j (t2)0j3 (t3 )-

Using Lemma 1.3 and (2.625), we get w. p. 1

p p p i / ji=0 j2=0 j3=0 ^

■ ■ + c- ■ ■ + c . ■ V^V(n) + fi(3) =

+ Cj2j3j1 + Cj1j2j3 + Cj1j3j2 J j Zj2 zj3 + J LRpppJT,t = ppp

= Z(i1)z(i1)z(i1 ) + Tri(3)

= / , Cj3j2j1 Zj1 zj2 Zj3 + J [RpppJT,t-

j1=0 j2=0 j3 =0

Then

M

J J

M

p p p

z (n)z (ii)z (ii)

/ , / , / , Cj3j2j1 Sji j Zj3 ji=0 j2=0 j3=0

2n

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 (2.419) for k = 3 and R(ti,t2, t3) instead of RPiP2Ps(ti, t2, t3), we obtain

pirn M{ (j[RPpp]T3r ^

= 0.

Theorems 2.41 is proved for the case k = 3.

To prove Theorem 2.41 for the case k > 3, consider the auxiliary function

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

) ...^ (tgk), tgi < ... < tgk, ii5...5ik e [t,TL

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

(2.627)

where {gi,..., gk} = {1,..., k} and we take into account all possible permutations (gi,..., gk) on the right-hand side of the formula (2.627).

Further, we have w. p. 1

[k/2]

r=1 (sr ,...,si)GÄfc,r

J

sr v,S 1

T,t ,

(2.628)

where the function K'(ti,...,tk) is defined by (2.627); another notations are the same as in (2.350) and Theorem 2.12 (ii = ... = ik = 0 in (2.350)).

From (2.628) and Theorem 2.12 we obtain w. p. 1

J*[^(k)]T,t = J [K']Tkt. (2.629)

Generalizing the above reasoning to the case k > 3 and taking into account (2.629), we get w. p. 1

p p i

j-n, = E • • ■ E h I E ) <2* • ■ ■ + W..J8 =

ji=0 jk=0 \(ji,-,jfc)

pp

EV^ c Z(ii) Z(ii) + T[ R ](k) . . . Z^ Cjk...ji j . . . j + J [Rp...p]T,t,

ji=0 jk =0

where

R' (ti,... ,tk) = K '(ti,...,tk)-

p p i

ji=0 jk=^ \(ji,-.,jk)

the expression

E

(ji,-,jk)

means the sum with respect to all possible permutations (j,..., jk). Further,

2n ^ ^ p p N 2n

m j J[R j£!) f = ^ (J*[^(k)]T,f - E. ■ ■ E cj;'»...

ji=0 jk=0

where n £ N.

Applying (we mean here the passage to the limit lim ) the Lebesgue's Dom-

piM

inated Convergence Theorem to the integrals on the right-hand side of (2.419) for Rp...p(ti,... ,tk) instead of RPi...;Pfc(ti,... ,tk), we obtain

hm M j (J [R...p]

Theorems 2.41 is proved.

2.15 Comparison of Theorems 2.2 and 2.6 with the Representations of Iterated Stratonovich Stochastic Integrals With Respect to the Scalar Standard Wiener Process

Note that the correctness of the formulas (2.34) and (2.220) 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 (2.34) and (2.220) the well known equalities (see Sect. 6.7)

t2 / T

T

m) I ^(ii)«0 = ^

i

2!

t )df

tt T t3 t2

T

i

T )df

w. p. 1, where ^(t) is a continuously differentiable nonrandom function at the interval [t,T].

From (2.34) (under the above assumptions and p1 = p2 = p) we have (see (2.410) and (1.58))

p

J-l^'k, = l.i,m. £ C,2j1 Cji'd

p—TO ' * J1 J

(;)z (;) ^¿1 Zj2

P ¿1-1

= l.i.m.

p—to

¿1j2=0

p j1-1 / \ P

S S ( Cj2j1 + Cj1j2 j Cj(;)Ci;) + E Cj1j1 (Zj( ¿1=0 ¿2=0 7 ¿1=0

(; )

¿1

¿1=0 ¿2=0 p ¿1-1

(p ¿1-1 p

EE^Ä + ^ici ¿1=0 ¿2=0 ¿1=0

2 /7(;) ¿1

= l.i.m.

p—>-to

2

\ j1j2 = 0 \ j1=j2

p

p 1 '

1

p

¿1=0

l

2!

¿1=0 T

/

t )df

(;)

(2.630)

3

2

2

2

2

w. p. 1. Note that the last step in (2.630) is performed by analogy with (1.54).

From (2.220) (under the above assumptions) we obtain (see (2.411) and (1.59)—(1.61))

p

(i) (i) (i)

^ E Cjicjj

ji,j2,j3 =0

p ji i j2 i / \ ^ E^ ( Cj3j2ji + Cj3jij2 + Cj2jij3 + Cj2j3ji + Cjij2 j3 + Cjij3j2 j X

P 00 ji=0 j2=0 j3=0 '

>(;U(;U(;) Xj j j +

ji j2 j

p ji i /

+ El E! ( Cj3jij3 + Cjij3j3 + Cj3j3ji ) (j^ Zj(;) + ji =0 j3=0V 7

p ji i / \ o p Q

' (i)W(i) , /7(;)x3

j'ij'ij'i Vcji

ji=0 j3=0 v 7 ji=0

+E E (Cj3jiji + cjijij3 + cjij3ji J (Zj;) j + E c

/ p ji-i j2 i ^ E EE Cji Cj2 Cj3 j cj

\ji=0 j2=0 j3=0

p j'i-i 2 p j'i-i 2 ji=0 j3=0 ji=0 j3=0

1 p 3

ji=0 1p

=(J E c,c,c33cj:'c<:'c«+

ji ,j2 j3 = 0 ji =j2 ,j2 =j3 ji =j3

p ji i 2 p ji i 2 (ci;,)"cj;) + ¿EE^ (cj;')"^

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

1 p q

ji=0

=(\ E w«

\ ji,j2,j3=°

1 ( p ¿1-1 2 p ¿1-1 2

\ 3 E E <%ch (cj:') - Cj;'+3 E E ^ (cj:')'

\ ¿1=0 ¿3=0 ¿1=0 ¿3=0

- ¿3i +

p

+ E c3 (ci

¿1=0

p ¿1-1 2 p ¿1-1 2

4 E E (<jj>) - cj;1 + \ E E

¿1=0 ¿3=0 ¿1=0 ¿3=0

1 p qN ¿1=0

¿ 3 / T

p \ 3 / T

w. p. 1. Note that the last step in (2.631) is performed by analogy with (1.57).

2.16 One Result on the Expansion of Multiple Stratonovich Stochastic Integrals of Multiplicity k. The

Case i1 = ... = ik = 1,..., m

Let us consider the multiple stochastic integral (1.16)

n-1

l.i.m. £ «(t*^Aw<j)=i J[*]&"*(2.632)

JivJfc =0 1=1

where we assume

that «(t1,..., tk): [t,T]k ^ R1 is a continuous nonrandom function on [t,T]k. Moreover, {t*}n= _0 is a partition of [t, T], which satisfies the condition (1.9) and i1,..., ik = 0,1,..., m.

The stochastic integral with respect to the scalar standard Wiener process (i1 = ... = ik = 0) and similar to (2.632) (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 [118]). The integral (2.632) 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[$]&■■■«' = J... Jw. p. 1,

k

where $(ti,..., t*) = Vi(ti)... V*(t*) and

T

JV® = i V(s)dw(i1) (l = 1-----k).

s

t

It is not difficult to see that for the case i1 = ... = = 0 we have w. p. 1

N-1

l.i.m. V $ (t?1 ,..., T7k) Aw^... AwTn) =

31,-3=0

= l.i.m. v if v $(tJ1,...)rjj|AwP...AwP,

i.e.

where

J[$]Tit1-i1) = J[cj?]^1-'1) w. p. 1, (2.633)

\ \(ii,...,ifc)

is the symmetrization of the function $ (t1,..., ); the expression

E

(ai,...,afc)

means the sum with respect to all possible permutations (ai5..., ak).

Due to (2.633) the condition of symmetry of the function $(t1,..., ) need not be required in the case i1 = ... = = 0.

Definition 2.1 [119]. Let $(tb ... ) e 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 $(t1' . . . )031 (t1 )031 (t2) . . . j (t2q-1)03q (t2q)X

3'l,...,3q=0 32q+1,...,3fc =0 [t T]fc

X 032q+1 (t2q+1) . . . 03k (tk)dt1 . . . • 032q+1 (t2q+1) . . . j (tk) (2-634)

k

k

converges in L2([t, T]k-2q) if p ^ œ 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 ..., ) exists, which by definition is the limit of the sum (2.634) and is denoted as Tr9Moreover,

—def -x-

Tr $ = $.

Consider the following Theorem using our notations.

Theorem 2.42 [119]. Let <&(t1,..., ) G L2([t,T]k) is a symmetric non-random function. Furthermore, let all limiting traces for (t1,..., ) (see Definition 2.1) exist. Then the following expansion

Jo i^Tr11 = LLm. è Cf ..j

j1 ,...jfc=0

that converges in the mean-square sense is valid, where

k

j o[$]T:'-:i 1

is the multiple Stratonovich stochastic integral defined as in [120] (1993) (also see [119], pp. 910-911),

C = •••ji =

/k

$(t!,...,tA )JJ j (t/)dt1 ...dtk

[t,T ]k l=i

is the Fourier coefficient, l.i.m. is a limit in the mean-square sense, ii = 1,..., m,

T

j1 = J to(s)df«;'»

t

are independent standard Gaussian random variables for various j.

In addition to the conditions of Theorem 2.42, we assume that the function $(ti,... ,tk) is continuous on [t,T]k. Then [118]

J°[$]T;t1-; 1 ) = J[$]T;r-;i ) w. p. 1, (2.635)

where the multiple Stratonovich stochastic integral

j pfer1 '

k

k

k

is defined by (2.632). As a result, we get the following expansion

j[$]T;r» = 1^. E Cjk...jizjl1»■••cjkl)- (2-636)

ji,---jk=o

It should be noted that the expansion (2.636) is valid provided that for the function ... ) there exist all limiting traces that do not depend on the choice of the complete orthonormal system of functions (x)}°=0 in the space L2([t,T]). The last condition is essential for the proof of the equality (2.635) (this proof follows from Theorem 1.5.3, Remark 1.5.7, and Propositions 2.2.3, 2.2.5, 4.1.2 [118]). More precisely, in [118], to prove Proposition 4.1.2 (p. 65)

a special basis (x)}°=0 was used. This means that the existence of a limit of the sum (12.634) for the function $(t1,... ) in the case when (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 (2.629))

kk J*[^k)]T:r': 11 = J[K]TV': 1 » w. p. 1, (2.637)

where

k

j 1}

is the iterated Stratonovich stochastic integral (2.336) (¿i = ... = = 0),

k

j [k 'iTP1'

is the multiple Stratonovich stochastic integral (2.632) (i1 = ... = = 0) for the continuous function K'(t1,... ) defined by (2.627).

If we assume that the limiting traces from Theorem 2.42 exist, then we can write (see (2.637))

k

j'iTr11 = u£- y, Ckjcf...j1, (2.638)

jlvj'fc =o

where

t t2

Cjk j = j ^ (tk ) j (tk ) ..J Wti) j (ti)dti.. .dtk (2.639)

t t

is the Fourier coefficient; another notations are the same as in Theorem 2.42.

The equality (2.638) agrees with Hypothesis 2.2 (see Sect. 2.6) for the particular case i1 = ... = ik = 0.

From the other hand, the following expansion (see (1.43))

pi pk

j*№(k)]Tr1=p1l-^ £...j1'...j1 (2.640)

ji=0 jfc=0

is valid, where Cjk...j1 has the form (12.639), 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 (2.640) corresponds to Hypothesis 2.3 (see Sect. 2.6) for the pairwise different numbers i1,..., ik = 0,1,..., m.

2.17 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, 118]-[120]) the conditions of theorems related to multiple stochastic integrals (see Theorem 2.42 in Sect. 2.16) 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 [118]. The concepts of traces

considered in [118]-[120] are close to some expressions that we used in this chapter. For example, the following integral (see (2.10))

T

T

\ iMh)Mti)dti = f K^tiMdti

is an example of a trace introduced in [118] (Definition 2.2.1). However, the function K*(t1,t2) defined by (2.77) is not symmetric compared with [118] (De-

finition 2.2.1). In addition, the expression (see (2.10))

TO OO p

E^rn = E J K(ti,t2(ti)j(t2)dtidt2

j!=0 j!=0 ]2

is an example of a limiting trace (see Definition 2.1) for the function K(t1,t2), which is not symmetric (see (2.78)).

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. 2.6 (Hypotheses 2.1 and 2.2 are special cases of Hypothesis 2.3) in the form of Theorem 2.43 (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 2.43. Assume that continuously differentiate nonrandom functions (t ),... (t ) : [t,T] ^ R and complete orthonormal system of functions {fy (x)}°=0 in the space L2([t,T]) (each function fy (x) of which for j < to satisfies the condition (*) (see Sect. 1.1.7)) such that the following equality is fulfilled

[k/2] 1

Ei E -w{k)]irsi

r=1 (sr ,...,si)eAfc,r

Pi Pk k

= l.i.m. ]T ..^Cjk..J-i l.i.m. Y, II J K 'AwT;g' (2.641)

w. p. 1, where

i

is, ,...,si def

j№<k>]?,r'si sn1

{isp =*sp+1=0}

P=1

X

T ts,+3 is, + 2

X J ^k(tk) . . . J ^s,+2(ts, +2) J ^ s, (ts,+1 )^s,+1(ts,+1) x t t t ts, + 1 ts1+3 ts1 + 2

^s,-1(ts,-1) ... J ^s1+2(ts1+2^ « (ts1+1)^s1+1(ts1+1)X t t t

ïs1+I

t2

X / ^si-l(isi-l)

.. dw|isi_il)dtsi+idw-

(«si+2) tsl + 2

t

... dw^rfis,+i(iwi:;;2)...dw<:k),

T t2

Cjfc ...ji = J 1 (tk ) j (tk) ...J 1 (tl)0ji (ti)dti t t is the Fourier coefficient,

Gk = Hk\Lk, Hk = {(/i,...,/k) : li,... ,1k = 0, 1,..., N - 1},

Lk = {(/i,...,/k): li = 0, 1,. .. , N -1; lg = lr (g = r); g,r = 1,...,k},

Ak,i = {(si ,...,si): si > sz_i + 1,...,s2 > si + 1, si,..., si = 1,...,k - 1},

(sz,..., si) G Ak,z, l = 1,..., [k/2] , ii,...,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++i _ 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 *t2

J*[^(k)]T,t = / ^k (tk )... i ^ (ii)iwt;i)... iwir) (2.642)

the following expansion

Pi Pfc k

= pi,1:^ j' (2.643)

ji=0 jfc=0 1=1

that converges in the mean-square sense is valid, where

T

j = J to (s)dw<°

t

are independent standard Gaussian random variables for various i or j (in the

,(0 = fW for i = 1 m w(0)

case when i = 0), w( = f(i) for i = 1,..., m and w( ) = t.

The proof of Theorem 2.43 follows from Theorems 1.1 and 2.12 (see (1.10) and (2.352)).

Note that a significant part of Chapter 2 is devoted to the proof of Theorem 2.43 (see (12.641)) for various special cases (Theorems 2.1-2.9). More precisely, in Theorems 2.1-2.9 we assume that (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t, T]). It should be noted that these two 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 (2.643) for the case k = 2. At that ^2(t) is a continuously differentiable nonrandom function on [t, T] and ^i(t) is twice continuously differentiable nonrandom function on [t, T] (Theorem 2.1). In Theorem 2.2, the functions ^(t) and (t) are assumed to be continuously differentiable only one time on [t,T].

Theorems 2.3-2.7 prove the case k = 3 of (2.64,3). Moreover, in Theorems 2.3 and 2.5, the case (t), ^2(t), ^3(t) = 1 (p1,p2,p3 ^ to) is considered. Theorem 2.7 proves the expansion (2.643) for the case when ^2(t) is a continuously differentiable nonrandom function on [t, T] and ^1(t), ^3(t) are twice continuously differentiable nonrandom functions on [t, T] (p1 = p2 = p3 = p ^ to). In Theorems 2.4 and 2.6, we consider narrower particular cases of the functions ^1(t ),^2(t ),^(t ).

The cases k = 4 and k = 5 of (2.643) are considered in Theorems 2.8 and 2.9. Moreover, we choose ^1(t),..., ^5(t) = 1 and (p1 = ... = p5 = p ^ to) in these theorems.

Remark 2.4. The equalities (1.10) and (1.53) imply that the condition (2.641) is equivalent to the following condition

[k/2] Pi Pk [k/2]

E1 E J^r" = -£-££<-irx

r=1 (sr ,...,si)eAfc,r ji=0 jfc=0 r=1

r k—2r

x E n 1{ig2s-i = ig2s=0}1{jg2s-i = jg2s} n c

({{3i>32 }>--->{32r-i>32r }}>{9i>--->9fc-2r}) s=1 1 = 1

{3i,32 ,---,S2r-i>S2r>9i>'">9fc-2r }={i>2>'">k}

(2.644)

w. p. 1, where notations are the same as in Theorems 1.2 and 2.12.

Remark 2.5. Using (1.256), we reformulate the equality (2.644) as follows

[k/2]

Pi

Pk

2 r 53

r=l (sr ,...,si)GÄk

J

sr ?...,s 1

T,t

= l.i.m. y ...ji

x

ji=0 jfc=0

X

c

(ii) A'i-k)

ji

j) - II

1=1

V

1=1 ^ (j)) , if = 0

s=1 v ' 7

l{m; =0} + l{m;>0} <

V

n (00

s=1 v

(0)

if ii = 0

/

/

w. p. 1, where notations are the same as in Theorems 1.14 and 2.12 (Hn(x) is the Hermite polynomial

Remark 2.6. Recently, in [121], an approach to the proof of expansion similar to (2.478) was proposed. In particular, this approach uses the representation of the multiple Stratonovich stochastic integral as a finite sum of multiple Wiener stochastic integrals. Note that a similar representation in a different form can be obtained using the formula (1.247). Namely, using the equality (1.247) without passing to the limit l.i.m. (pi = ... = pk = p), we have for

$(ti,...,tk) = Kp(ti,...,tk) w. p. 1

[k/2]

j [kjTT ) = j' [KJrfc) _E (_1)r x

r=1

X

£

n^»*,- = i»,=0}J I*?

gi...g2r,qi ...qfc-2r l(iqi .. iqk —2r )

JT,t

({{Si >32}>'">{32r-i>32r}}>{9i>--->9fc-2r }) S=1 {Si ,S2v>S2r-i>S2r>9iv>9fc-2r }={i>2>--->k}

(2.645)

where J [Kp]

(ii ...ife) •

PJT.t

is the multiple Stratonovich stochastic integral (2.632) and

J' [*.] tV^ is the multipte Wiener stochastic integml (1.245),

Cjfc ...ji n^ji (ti),

ji v-Jfc =0

1=1

KS,i...g2r,9i...9fc-2r I

(tqi, • • • , tqfc-2r ) = ^ ^ Cj'fc . ji 1

k- 2r

{j

S2s-1 »2

jS2, }

n j (tqi )•

ji V",j'fc =0

s=1

1=1

r

k

Thus, the multiple Stratonovich stochastic integral of multiplicity k is expressed by the formula (2.645) in terms of the multiple Wiener stochastic integral of multiplicity k and multiple Stratonovich stochastic integrals of multiplicities k — 2, k — 4, ..., k — 2[k/2]. As a result, by iteratively applying the formula (2.645), we can obtain a 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.

It should be noted that an expansion similar to (2.478) was considered in 121] for arbitrary k. The system of basis functions (x)}°=0 in the space L2([t,T]) can also be arbitrary. However, in [121], the condition on the convergence of traces is used as a sufficient condition for the validity of expansion similar to (2.478) (see [121] for details). Note that the verification of the above condition is a separate problem.

In Theorems 2.35-2.40, the rate of mean-square convergence of expansions of iterated Stratonovich stochastic integrals is found. Determining the rate of mean-square convergence in approach [121] is an open problem.

2.18 Invariance of Expansions of Iterated Ito and Stratonovich Stochastic Integrals from Theorems 1.1 and 2.43

In this section, we consider the invariance of expansions of iterated Ito and Stratonovich stochastic integrals from Theorems 1.1 and 2.43.

Consider the multiple Wiener stochastic integral J' [j ... j defined

by (11.23) ($(ti,... ) = j(ti)... j(tk)), where {0»(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).

Taking into account the equalities (1.35), (1.39), and (1.42), we can write the expansion (1.10) in the following form

pi pk

j'[*]&•'"' = Ujm £• •.êc»'"»J'[j..-jtr"' w.p.1, (2.646)

ji=0 » =o

where J'[K]^'''^ is the multiple Wiener stochastic integral defined by (11.2,3) ($(t1,... ,tk) = K(t1,... ,tk)), the function K(t1,... ,tk) has the form (1.6).

On the other hand, the expansion (2.643) can be written as follows (see

Lemma 1.3 and (2.364))

pi pk

J [K *|,r,k) = l.i.m. £ J [j ...j jfe"" ) w.p.1, (2.647)

ji=0 jk=0

where J[K^T;1,''^ and J[ j... jj^1,'"ifc) are multiple Stratonovich stochastic integrals defined by (26321). the function K*(ti,..., tk) has the form (127342).

It is not difficult to see that w. p. 1

j [Kj<ir") = J' [k iT1,''").

Therefore, the expansions (2.646) and (2.647) have the same form. At that the expansion (2.646) is formulated using multiple Wiener stochastic integrals and the expansion (2.647) is formulated using multiple Stratonovich stochastic integrals.

The expansions (2.646) and (2.647) can be written in a slightly different way. Using (1.39), (1.42), and (1.83), we obtain

.1 pk

J[^<k)j<tru) = l.i.m. £ ... £ X

p1,''',pk^^ z—' z—'

j1=0 jk=0

T t2

X E j j (tk ) ..J j (ti)dw,(;1)... dw,:k ) w. p. 1, (2.648)

(j1vJk) t t

where J[^( ^jy1,''':k) is the iterated Itô stochastic integral (11.5),

E

means the sum with respect to all possible permutations (j,..., jk). At the same time if jr swapped with in the permutation ..., 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 * ,2

i j(tk).../ j(ti)dw<:1)...dw<-k)

satisfy the following equality

* T * ¿2

Cj::,) .■■Cj;k) = E / j (tk) .■■/ j (t1)dw!;i) ...dw<;k) w.p. 1, (2.649)

(jiv",jk) t t

where

T

j = / 0j(s)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

(j'iv-Jfc )

has the same meaning as in (2.648).

For the case i1 = ... = ik = 0 we obtain from (2.649) the following well known formula from the classical integral calculus (see (1.

T t2

J j (t1)... j (tk)dt1... dtk = E J j (tk). . J j (t1)dt1... dtk =

[t,T]k (jivJfc) t t

T t2

= E /..^^ji (t1) ...^jk (tk)dt1 ...dtk, (2.650)

(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 (2.650)) 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 (2.649) for the cases k = 2 and k = 3. Using (1.45), (2.361), and (2.648) (k = 2), we have

* T * ¿2

E/ j(¿2)/ j(ii)dwi;'W<22)

*T * ¿2 *T * ¿2

J j(¿2)J j(ii)dwt(;i)dwi;2) + y j(t2)J j(ti)dwi;2)dw^ t t t t T t2 T t2

= / j (¿2 )/ j (ti)dw<;'W<;2) + / j <h)j j (ti)dw<:2)dw<;')+

t t t t

T

+ 1{;i=;2=0W j(ti)0j2(ti)dt

i=

- Cj(:1)Cj(22) - 1{:1=:2=0}1{j1=j2} + 1{:1=:2=0}1{j1=j2} —

= c(:1)C(:2) w. p. 1. j1 j2

Applying (1.46), (2.362), (2.648) (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 * ,3 * ,2

E f j(*>)/ j(t2)/ j (ii)dw<;1)dwi:2)dw<33) —

/ ... \ U \J \J

(j1'j2,j3) , , ,

T ts t2

E / j (ta)/ j fe)/ j (ti)dw<;iW<;2)dwiss)+

(jU2js) t t t

/ T ts

+1{;i=;2=0} J j toj j (ti)0ji (ti)dtidwt(Ss) + tt T ti

+ f j (ti) j (ti)^ j(ta)dwt(ss)dt^ + tt

/ T t2

+ I{n=i3=0} I J jJ j (ti)0j3(tl)dtldWt(i2) + \t t T ti \

+ / j (ti) j (ti)/ j (i2)dw(i2)dtj + tt / T ti

+ !{i2=i3=0} J j (t0/ j (t3)0j3 (t3)dt3dwt(ii) + tt T t3

+ /j(t3)0j2(*)/ j(ti)dwt(ii)dt3 tt T t3 t2

E f jte) /jfe) /j(ii)dw<;i)dwi;2)dw<33>+

(ji,j2,j3) t t t

/ T t3

+!{ii=i2=oH J jtoj j (ti)0ji (ti)dtidwt(33) + tt TT \

+ J j(*)/ j (ti)0ji(ti)dtidwt(33M +

t t3

/ T t2

+ l{ii=i3=oH J j ^ J j (t0 j (ti)dtidwt(J2) + tt TT \

+J j fe)/ j (ti)0j3(ti)dtidwi;2M +

t t2

/ T ti

+ 1{i2=i3=oU J j (t0/ j (t3)0j3 (t3)dt3dwt(ii) + tt T T

+ f j (ti)/ 02 (t3)0j3 (t3)dt3dwt(ii) t ti

T t3 t2

= E / j ^ j M/ j (t1 )dw(;i)dwt(;2)dwt(;3)+

(ji ,j2 ,j3) t t t

T T

+ 1{:i=:2=0}/j ^ j (t1)0ji (t1)dt1 dwt(33) + tt T T

+ 1{:i=:3=0}/j (t2^ j (t0 j (t1)dt1 ^t^ tt T T

+ 1{:2=i3=0} / j (t1 ^ j (t3)0j3 (t3)dt3dwt(;i) = tt T t3 t2

= £ / j(*)/ j j (t1 )dwi;i)dw<;2)dwii3» +

(ji,j2,j3) t t t

+ 1{:i=:2=0}1{ji=j2}Cj(33) + 1{:i=:3=0}1{ji=j3}Cj(2:2) + 1{:2=i3=0} 1{j2=j3} =

= Z(;i)Z(:2) z(;3)_

j Z j2 Zj3

— 1 1 Z(;3^ 1 1 Z(:2) _ 1 1 Z(;i) +

1{:i=:2=0}1{ji=j2}Sj3 1{:i=:3=0}1{ji=j3} j 1{:2=i3=0} 1{j2 =j3}Sji +

+ 1{:i=:2=0}1{ji=j2}Zj33) + 1{:i=:3=0}1{ji=j3}Zj:2) + 1{:2=i3=0} 1{j2=j3} Cj^ =

= Z (;i)Z (:2 )Z (;3)

j Z j2 Zj3 .

Using (2.649), we can write the expansion (2.643) as follows

Pi Pk

•n^'iTr*1 = i-i-m Ex

pi,.. . ,pfc^to ^^

ji=0 jfc =0

* T * t2

x e / j(tk).../ j(t1)dwt(;i)...dwt:k) w.p. 1, (2.651) (ji,...,jk) t t

where J*[^(k)]T1t":k) is the iterated Stratonovich stochastic integral (26421): another notations are the same as in (2.648).

Obviously, the expansions (2.648) and (2.651) have the same form. At that the expansion (2.648) is formulated using iterated ItO stochastic integrals and the expansion (2.651) is formulated using iterated Stratonovich stochastic integrals.

Taking into account the expansions (2.648) and (2.651), we can reformulate the condition (2.641) as follows

pi pk

J» - J= l.i.m. £ . ■ ■ E C.-A x

PivvPfc ^^ z-' z-'

ji=0 j=0

* ¿2

£ j (tk ) ...f j (i1)dw<;1' ...dw<kk) -

(j1,• • • Jfc ) \ t t

T t2

- /j(tk)... [ j(ti)dwi;i)...dwi:k)

w. p. 1.

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 arise 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) == 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 (ti) ...J fa (tk) J Фт dwTk+1)dwt(k)... dw^,

t t t

where ф(т, u) == фт, фт G S2([t,T]), every ^(т) (/ = 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), (^i,...,^k) d=f ^(k),^(1) =f .

We will call the stochastic integral J^,^(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[^ikt = J dw^dw^

t t

the formula on integration order replacement, which is similar to (3.2), is valid, then we will have

T s T T

y ^i(s)y dw^dw^ = J j ^i(s)dwi1}dwT2). (3.3)

t t t T

If, in addition w(1), w(2) = fs (s G [t, T]) in (13.3), then the stochastic process

T

nT = 0T y ^(s)dw(1)

T

does not belong to the class M2([t,T]), and, consequently, for the Ito stochastic integral

T

t

'VT\—ry

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 T

J (/ - /t)ds = y J d/ds = y J dsd/ w. p. 1.

t t t t T

Then

T

T

T t

T T

T T

J (s - t)dfs + J (fs - ft)ds = J J dsd/V + J J dsd/V = 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 (13.31) for the case w(1), w(2) = fs (s £ [t, T]) as well as its analogue corresponding to the iterated Ito stochastic integral J[0, 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

_(N)

_(N)

t = TO" ' < Tl"'' < . . . < T

(N) = T N = T

max

0<j<N-1

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

0 if N oo.

(3.5)

In [100] 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. ^ 0Tj (fTj+1 j =0

T

=f / 0T d/r 0T,

(3.6)

where 0t, ^ £ 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 £ S2([t,T]) and #T 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,^(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 0Tdf g(T) = J 0Tg(T)d/r w. p. 1,

tt

where g(T) is a continuous nonrandom function at the interval [t,T],

T T T

f + №) dfT* = af ^dfT* + p / ^dfT* w. p. 1,

t t t T T T

/<MfTMt + №) = a/f + p/f w. p. 1,

t t t where a, p G R1.

At that, we suppose that the stochastic processes 0T, #T, and 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 /[^(k)]T,s, k > 1 of the form

T T T

%(k)]T,s = J (tk )dw^| ^k—1 (tk—1)dwt(k--11) ..J fa (t1)dwt(11)

s tfc t2

in accordance with the definition (3.6) by the following recurrence relation

N—1

0 [^(k)]T,t = l.i.m. £ fa(t)AwTk)1 [^(k—1)]t,ti+1 , (3.7)

1=0

where k > 1, I[^(0)]T,s = 1, [s,T] C [t,T], here and further AwT? = wTj+1 — wTl), i = 1,.. .,k + 1, l = 0,1,. ..,N — 1.

Then, we will define the iterated stochastic integral J[0,^(k)]Tjt, k > 1

T

J[0,^(k)]T,t = i 0sdw(k+1)I[^(k)]T,S

similarly in accordance with the definition (3.6)

N-1

j[^.<k>]T,( d= l.i.m. £ Aw<f+1>/[^>]T:

._. . - - Jl+1-

Let us formulate the theorem on integration order replacement for iterated Ito stochastic integrals.

Theorem 3.1 [105] (1997), (also see [1]-[15], [68], [106]). Suppose that G

S2([t,T]) and every fa(t) (l = 1,..., k) is a continuous nonrandom function at the interval [t,T]. Then, the stochastic integral J[0, fa(k)]T,t, k > 1 exists and

J= J[fa#feV>t 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[0,fai]T,t = l.i.m. V fai(Ti)AwT1W 0tdwT2) = N^TO f—' ' J

t

N— 1 1 — 1 Tj+1

l.i.m. V fa^Arn^V / fadwT2), (3.8)

N^œ 1=0 j=o 7

N- 1 T

J[fafafat =f l.i.m. V Aw^ / fafa)^ N^œ ^ 3 J

j=0 T

7 Tj+1

N — 1 N — 1 Tl+1

l.i.m. E ^Tj Aw(2^ / fa(s)dw« =

j=0 1=j+1 Tl

N —1 Tl+1 1 — 1

l.i.m. £ / fa(s)dw«E^Aw(2(3.9) N^œ f-' J ^ 33

1=° 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 ^ to in the mean-square sense, then the stochastic integral J[0, exists and

J= Jw. p. 1. The difference £N can be represented in the form £N = £N + £N , where

£N =

N -1 1-1 Tj+X

E>(t)Aw(1 ^ / - dwT2); 1=0 j=0 i

£ N =

N-1 Tl+1 1-1

E / (^1(ti) - ^1(s)) dwi1^ AwTJ)

1=0 T j=0

We will demonstrate that w. p. 1

l.i.m. £N = 0.

N ^TO

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 [91]

M

T

£t dfT

T

M{ |£t |2} dT,

M

T

£t dT

T

< (T - t) / M{ |£t|2} dT,

(3.10)

where G 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

2

ity) of the process on the interval [t, T], we obtain

N-1 k-1 Tj+1

m{ |£n\2} = £ ^2(Tk)AT^ / m{ |0T - I2}

7. n ;_n J

dT <

k=0

j=0

n-1 k-1

2 (T-f) fc--—

2

< C2e E An E ArJ < C k=0 j =0

i.e. M j\£n\2} ^ 0 when N ^ to. Here At, < %), j = 0,1,... ,N - 1

(£(£) > 0 exists for any £ > 0 and it does not depend on t), \^1(t)\ < C.

Let us consider Case 2. Using the Minkowski inequality, uniform mean-square continuity of the process 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 y (fa - )dT

k=0 V=0 Tj )

\ <

N-1

< E ^2(Tk)ATk k=0

k-1

E

j=0

/ f / rl+1 \ 2 \

M ( fa - 0Tj )dT >

V J Vi J / /

/

\ 2

/ /

<

N1

k1

2 (T-f);

< y Ark y Atj < c

k=0 \j=0 J

i.e. M {|£n|2} ^ 0 when N ^ to. Here At} < ¿(¿), j = 0,1,...,N - 1

(^(¿) > 0 exists for any £ > 0 and it does not depend on t), |^1(r)| < C.

For Case 3 using the Minkowski inequality, standard moment properties for the It o stochastic integral as well as uniform mean-square continuity of the process , we find

2

M

{\£n\2}

/ N-1 / (k-1 Ti+1 ^ 2 \ 1/2 \

< E\^(Tk)\ ATk M E / - )d/T >

k=0 V \ j j J / / /

2

I k-1

N-1 k-1 'jt1

(Tk)|ATk E / M{I^T - |2} dr

ivk=0 v=0 j

\ i/2

/

<

/

N1

k1

1/2

<C2£ ^ At* £ ATj

'k i / k=0 \j =0

<6 £ -

i.e. m{|£N|2} ^ 0 when N ^ to. Here Arj < J(e), j = 0,1,...,N - 1 (£(e) > 0 exists for any £ > 0 and it does not depend on r), |fa(r)| < C.

Finally, for Case 4 using the Minkowski inequality, uniform mean-square continuity of the process 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

< EE^i(rk )iArk M < / (0T - )dr >

k=0 j=0 \ V U / / / /

<

N 1 k 1

< C2£ E ArkE Arn <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), |fa(r)| < 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. £N = 0 w. p. 1.

N

Theorem 3.1 is proved for the case k = 1.

2

2

2

Remark 3.1. Proving Theorem 3.1, we used the fact that if the stochastic process 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

{|0ti - I2} < £

is fulfilled (here 5(£) does not depend on t1 and t2).

Proof. Suppose that the stochastic process 0 is mean-square continuous at the interval [t,T], but not uniformly mean-square continuous at this interval. Then for some £ > 0 and V 5(£) > 0 3 t1,t2 G [t,T] such that |t1 — t2| < 5(£), but

m{ — |2} > £.

Consequently, for 5 = 5n = 1/n (n G N) 3 t1n), t2n) G [t, T] such that

,(n) _ ,(n) 62

<

n

but

M

0,(n) -

(n)

> e.

The sequence t1n) (n G N) is bounded, consequently, according to the Bolza-no-Weierstrass Theorem, we can choose from it the subsequence t1kn) (n G N) that converges to a certain number t (it is simple to demonstrate that t G [t,T]). Similarly to it and in virtue of the inequality

,(n) _ ,(n) 61 62

<

n

we have t2kn) ^ £ when n ^ to.

According to the mean-square continuity of the process at the moment t and the elementary inequality (a + b)2 < 2(a2 + b2), we obtain

0 < M

0,(fc„) — 0,(fc„)

<

<2 M

when n —to. Then

+ M

) —

¿2

lim M

n—7>00

0,(fc„) — 0,(fc„)

= 0.

0

1

2

2

1

2

2

2

2

It is impossible by virtue of the fact that

M

0,(fc„) — 0,

kn) — 0. (fcn) r2

2

> £ > 0.

The obtained contradiction proves the required statement.

3.4 Proof of Theorem 3.1 for the Case of Iterated Ito 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= J (ti).. ■ / (WiMwtr'+T'. ■ ■ dwtq',

s s

0 tr tr + 1

J [0, ^ks = J faq (ti) ...J faq+r(ir+0 ^ fa dw^dw^ . . . dwtf,

s s s

n—1 jq-1 jq+r-1-1 r+q

cfa^'u= E E ... £ w[fak•

jq=mjq+i=m jq+r=m l=q

«.,,..., faq+r) = fa'r+1), fa',,

.....far+1) =f fa1r+1), fa1r+1)=f fa(r+1).

Note that according to notations introduced above, we have

s

I [fa,]s,0 = i fa (T )dwTl).

To prove Theorem 3.1 for k > 1 it is enough to show that

J[0,fa(k)]T,t = l.i.m. S[fafa(k)fa = J[0,fa(k)]T,t w. p. 1, (3.11)

N ^TO

where

j'fc-1

S[0, fa V =

1=0

where AwTk+1) = wTk++1) - w(k+1).

At first, let us prove the right equality in (3.11). We have

N — 1

J [^(k)]T,t = l.i.m. £ Aw[k+1)/ [^(k)]r,Ti+1. (3.12) n^to 1=0

On the basis of the inductive hypothesis we obtain that

I[^(kWi+i = 1[^W w. p. 1, (3.13) where /[^(k)]T,s is defined in accordance with (3.7) and

T tfc-2 tfc-1

(k) (k—1) (1)

i[^(k)]T,s = ^1(t1)... ^k-1(tk-1W ^(tk)dwt^...dwt;

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 — 1 Tj'1+1 ^

I[^(k)]T,Ti+1 = £ / ^fa) / ^2(t2)1 [^3k—2)]t2,ti+1 dWfdw« = ¿1=1+1

Ti+1

N1

TJ1 + 1 / 1 TJ2 + 1 tA

wto E +

■■■ j2=i+1 7 7 /

\ '72 j1 /

¿1=1 + 1 ,

'71

^2(t2)1 [^3k 2)]t2,Ti+1 dwt(22)dwt(1)

= ... = G[^(k)] N,1+1 + H)]n,1+1 w. p. 1, (3.14)

where

N-1 Tj'1+1 s H[^(k)]N,1+1 = £ i ^(s) / ^2(t)1 [^3k-2)]T,Ti+1 dw(2)dw(1) +

j1=1+17 71

k-2 jr_1-1 Tj'r-+1 s

+ £ G[^(r-1)]N,1+^ / ^r (s) / ^r+1(T)1 [^ik+-2r-1)]T,Ti+1 dw(r+1)dw(r) +

r=2 jr =1+1 / /

jk_2 1

+ 22>]n,w E / [V&] T7k_1+1,T7k_1. (3.15)

jfc_1 = 1+1

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 fa, AwTf+1) + H. (3.16)

Since

N-1 ji-1 jfc-i-1 N-1 N-1 N-1

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 = E ■■• £ IP№k+i.Tj,• (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^G^^m = 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)ki+1 = 0. (3.21)

Nl

1=0

Analyzing the second moment of the prelimit expression on the left-hand side of (1.3/21) and taking into account (1.3.15), the independence of , Aw(f+1), and Has 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[фк(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, fa^ in (13.22) the same arguments,

which resulted to the relation (3.14) for the integral I[fa(k)]T,T;+1. After that let us substitute the expression obtained for the integral J[0, fa^-1^t into (3221). Further, using the Minkowski inequality and standard estimates for second moments of stochastic integrals it is easy to obtain that

J[0,0(k)]Tt = l.i.m. R[0,0(k)]N w. p. 1, (3.23)

N

where

N —1 jW Tl+1

ji=0 1=0 J

We will demonstrate that

l.i.m. R[0,0(feV = l.i.m. S[0,#fe)]N w. p. 1. (3.24)

N^TO N^TO

It is easy to see that

R[0,0(k)k = U[0,0(k)k + V[0,0(k)k + S[0,0(k)k w. p. 1, (3.25)

where

N—1 jk —1

U [0,0(% = £ «j ^wj^f-1°ко£ I [Дф]т+1,т , ji=o 1=0

N—1 jfc — 1

V= £ I[Д^1]тл+„тлС^Г'коЕ Фт,Д№<к+1»,

ji=o 1=0

I^J^ij = J (01 (j) — 01 (т))dw(1),

тл

I[A0Wi = J (fa — )dw(k+1).

Using the Minkowski inequality, standard estimates for second moments of stochastic integrals, the condition that the process belongs to the class S2([t, T]) as well as continuity (which means uniform continuity) of the function ^1(t), we obtain that

l.i.m. V[0,^(k)]N = l.i.m. U[0,^(k)]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[0, ^(k)]T,t, which exists in 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 Dk = {(t1,... ,tk) : t < t1 < ... < tk < T} and the following conditions are fulfilled:

AI. £T g S2([t,T]).

AII. $(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

T T T

•-%, = /itkdwi-> ...J dwg2'/$(t1,t2,...,tk_1)dwi:1» =

t is t2

N—1 T T T

d=f Nim. ? & AwT:k) J dwt:--1) ...J dw? j $(t1,t2 ,...,tk—1)dwt:1)

^=0 Ti+1 is t2

for k > 3 and

T T

J K. $iT2! = / it, dwi22)| $(t1)dwt;1) =

t t2

N-1 T

= l.i.m. £ £„Aw<;2> / $(ti)dw«;')

1=0 T

for k = 2. Here wT:) = fT:) for i = 1,..., m and wT0) = t, fr:) (i = 1,..., m) are FT-measurable for all t G [0,T] independent standard Wiener processes, 0 < t < T, i1,..., ik = 0,1,.. .,m.

Let us denote

T tfc-1

J[i, <4k] = / ...J ^(t1-----tk—1)itkdwt(:k) ...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 TT

1) = | dwfc'1 dwg2'/ $(ti,t2,...,tk—i)dwi

t ¿3 ¿2

T TT

N —1

= l.i.m A (;' ')

/ v ' i

> 7_n

Tl + 1 ¿3 ¿2

N —1 T 11 =fU.m. £ Awi;k-1^ dwl;--2) ..J dw^] j $(t1,t2,...,tk—1)dwt(;i),

T ifc-2

j/[^]Tk—1) = / -J $(t1,... ,t*—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 J[$]Tk—1) exists and

J'[^TV1' = ./^TV1' 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 [105] (1997) (also see [1]-[15], [68], [106]). Suppose that the

conditions AI, AII of this section are fulfilled. Then, the stochastic integral J[£, exists and for k > 2

jk, = jk, w.p. i.

(k)

Let us consider the following stochastic integrals

T T T ¿2

i = i dfi:2) i j = , ,

T T t2

t t2 If we consider

T

^i(ti,t2)dft(

(ii) i

t2

as the integrand of I and

t2

^2(ii,i2)dft(;i)

as the integrand of J, then, due to independence of these integrands we may mistakenly think that M{iJ} = 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 ¿1 T ¿2

/ = / J «Mti, i2)f2)f;i) = / |$1(i2,ii)dfi(if.

i i i i

So, using the standard properties of the Ito stochastic integral [91], we get

T t 2

M{1J} = l^} / I $l(t2,ti)$2(tl,t2)dtidt2,

i i

where 1A is the indicator of the set A.

Let us consider the following statement.

Theorem 3.3 [105] (1997) (also see [1]-[15], [68], [106]). 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( + )i[0( )]t,t w. p. 1, (3.28)

i i

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

/ h(r)dw( + )]t,T =

N-1

N->oo

l.i.m. £ Ah(r1)Aw(.f+1)/[^(k)] T,T; + 1 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

jk -i

l.i.m. G[^(k)jN,o£ fatAh(T)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 ^ to. Theorem 3.3 is proved.

Let us consider one corollary of Theorem 3.1.

Theorem 3.4 [105] (1997) (also see [1]-[15], [68], [106]). In the conditions

of Theorem 3.3 the following equality

T ti

i h(t1) / fa dwTk + 2)dWt(k+1)/

T,ti —

T T

= J fadw!k+2) y h(ti)dw(k+1)/[^(k)]T,i1 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 fadw^ / h(t1)dwt(f+1)/

T tfc-i tk

= J ^i(ti).. .J fa (tk ) J Pt dwTk+1)dwtk)... dwi(11) = t t t TT T

: J Pt dwTk+1) y fa (tk )dwt(fck) ..J fai(ti)dwt11) w. p. 1,

t T t2

where

T

Pt = h(t) / 0sdwik+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 in 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 ¿2 T

J J f dt2 = J (T - ti)dft1, ¿ ¿ ¿

T ¿2 T

i cos(t2 - T) / dftidt2 = / sin(T - ti)dfti,

i

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-^) dftl, a ^ 0, t t t T ¿2 T

J(t2-Trj dftldt2 = -^1 J(ti-T)a+1dftl, a^-1, t t t

T T

fai00)T,t =2 J(T ~ tl fdftn J(oio)t,t = J[t\ - t)(T - ti)dftl, tt T ¿2

J(iio)T,t = J(T - t2) J dftldft2, (3.31)

tt T t2 T t3 t2

J(101)T,t = J j(t2 - ii )dftidft2 , J(1011)T,t = J J J(t2 - ii )dftidft2dft3 , t t t t t

T t3 t2

J(1101)T,t = J j(h - t2) J dftidft2dft3, t t t T t3 t2 T t2

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 t2

J(ioio)T,t = J(T - t2) y (t2 - ti)dftidft2, (3.32)

tt

T t2 T t2

^(iooi)T,t =2 y Jto~ ti)2dftldft2, J(ouo)T,t = J(T - t2) J [t\ ~ ^)dftidft2, t t t t T t2

J(oioi)T,t = y J(h - t1)(t1 - t)dftidft2, tt

T T

J{ooio)T,t =2 J(T — t\)(ti - t)2dftl, J(oioo)T,i =2 J(T ~ ~~ t)dftr.

t t

T

J{iooo)T,t = ^ J(T — ti)3dftl, t

T

k-1 t

t t2

^n^o^o^t = ^ _ 2)! f(T ~ t2^k~2 J dftidfhi

k-2 t t T

0)T,t = J(T - J^^ti,td/ti,

k-1 t k-2 T t2

^i^o^TO = ^ _ 2)! J J^2 ~ ¿i)*-2^!^

k-2 t t T ts t2

J(i0 i...i )T,t = J -J J (t2 - ti)dfti dft2 • • • d/tfc-i, k-2 t t t

T tfc-1 tfc-2 t2

Ji^ 0i)T,t = J J (tk-i - ik-2^ y -J dfti •••d/tfc-3 d/tfc-2 d/tfe-i, k-2 t t t t

J(10)T,t + J(01)T,t = (T — t) J(i)T,t5 J(ii0)T,t + J(101)T,t + J(0ii)T,t = (T — t) J(ii)T,t5

^(ooi)T,i + ^(oio)T,t + ^(ioo)T,t = -—2—

J(ii00)T,t + J(i0i0)T,t + J(i00i)T,t + J(0ii0)T,t + + ^(0101)T,i + ^(0011)T,i = --^—~

/ + 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 — t) J(111)T,t,

k 1 k-l_ £ -^^O^I^O^)^ = ^ _ - 1

1=1 i-1 k-i k

£ 0— (T - t)

1 = 1 i-1 k-i k-1

_^ (T _ t)k-m

¡1+...+ i k=m ¡¿£{0, 1}, ¿=1,...,fc

(A: — m)!

where

T t2

J(1i...1k)T,t = J ...J ^i1 • • • dwt(k), tt

/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 t3 t2 T T T

J(iio)T,t = J J J dftidft2dt3 = J dfti y d/t^ y dta = t t t t ti t2

T T T T

= y d/t^ d/t2 (T - t2) = y d/t^(T - t2)d/t2 =

t ti t ti

T t2

= J(T - t2)J dftid/t2 w. p. 1. tt

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 dfa j dt2 J f J dt4 =

t t t t t t1 t2 t3 T ^n ^p ^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^ d/, = 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 £ [0, T] such that

M

s

^ - MtY | F^ = M < J PtdT

{(Ms - Mt)2 | Ft}

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 ^2ptdt| < oo. For any partition r?(N), j = 0,1,..., N of the interval [0, T] such that

0 = T<N> < Tf > < ... < TNN> = T, max

u 1 N 0<j<N-1

we will define the sequence of step functions

T(N) _ T(N)

— 0 if N —to (3.33)

^(N)(t,w) = V) w. p. 1 for t G

.(N )

T(N) T(N)

where j = 0,1,..., N - 1, N =1, 2,...

Let us define the stochastic integral with respect to martingale for ^(t, w) £ H2(p, [0, T]) as the following mean-square limit [91]

N-1

j=0

T

=f / dMT

where is any step function from the class H2(p, [0,T]), which con-

verges to the function in the following sense

T

lim M

N—TO J 0

^(N)(t,w) - } Ptdt = 0

It is well known [91] that the stochastic integral

T

J dMT 0

exists and it does not depend on the selection of sequence ^>(N^(t,^).

Let iH2(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 ifc-i tk

S[fa,fa( )]T,t = J fai(ti).. .J fa (tk) J fa dMik+1)dMt(kk)... dM^, (3.34) t t t

T tk-i

S [fa(k)]T,t = J fai(ti).. .J fa (tk )dMf ... dM(11). (3.35)

tt

Here faT G HH2(p, [t,T]) and fa(T),... , fa(t) are continuous nonrandom functions at the interval [t,T], M^ = MT or M^ = t if t g [t,T], l = 1,... ,k + 1, MT is the martingale defined above.

Let us define the iterated stochastic integral S[fa(k)]T,S, 0 < t < s < T, k > 1 with respect to martingale

T T

S [fa(k)]T,s = J fa (tk )dMt(kk) ..J fai(ti)dMt(1)

S t2

by the following recurrence relation

N-i

S[fa(k)]T,t d=f l.i.m. £ fa(Ti)AMTf)S[fa(k-i)]T,Ti+i, (3.36)

1=0

where k > 1, S[fa(0)]T,S d=f 1, [s,T] C [t,T], here and further amT? = mT^ -M(), i = 1,..., k + 1, l = 0,1,..., N - 1, {t1}N0 is the partition of the interval [t,T], which satisfies the condition similar to (3.33), other notations are the same as in (3.34), (3.35).

Further, let us define the iterated stochastic integral S[fa, fa(k)]T,t, k > 1 of the form

T

S[fa,fa(k)]T,t = J fasdMs(k+i)S[fa(k)]T,s t

by the equality

N-i

S[fa, fa<k)]T,( = l.i.m. £ fa, AM<k+i)S[fa<k»]T,t,+, ,

7=0

where the sense of notations included in (3.341) (13736) 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 [122] (1999) (also see [1]-[15], [106]). Let fa £ #2(0, [t,T]),

every fa(t) (/ = 1,..., k) is a continuous nonrandom function at the interval [t,T], and |pT| < K < 00 w. p. 1 for all t £ [t,T]. Then, the stochastic integral S[fa0(k)]T,t exists and

S[fafafe)fat = S[fafafeV,i 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 execution of the following additional conditions:

1. M{|pT|} < 00 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 [100].

The stochastic integral S[fafak)fat is also the stochastic integral with integrable process, which is not an FT-measurable stochastic process. However, in the conditions of Theorem 3.5

S[fa0(k)fat = S[fa0(k)fat w. p. 1,

where S[fafak)fat is 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[faand Stratonovich stochastic integral is solving as a standard question on connection between Stratonovich and Ito stochastic integrals [100 .

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. ^ G H2(p, [t,T]).

BII. $(ti,..., tk-i) is a continuous nonrandom function in the closed domain Dk-i (recall that we use the same symbol Dk-i to denote the open and closed domains corresponding to the domain Dk-i).

Let us define the following stochastic integrals with respect to martingale

T T T

S K, = / itk dM«k> ..J dMf | $(ii, t2,..., tk-i)dM('i» =f

t t3 t2

N-i T T T

=f l.i.m. £ ^ A<> i d<"1}... / dM<2) /

N^TO -1=0 J J J

for k > 3 and

T' + i is ¿2

T T

SK, = / 4dMf J $(ii)dMii1) =f t ¿2

T

N-1 »

=f l.i.m. £ i„AM<,2W *(ti)d<>

T + 1

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

SK, ^]Tkt = / ...J ^(ti,...,tk-i)^tkdMf ...dMt(i), 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

S'[$]Tk(-i) = / dMi£;i) ... / dM2> / $(ii, t2,..., tk-i)dM«:» =f

t t3 t2

def

N-1 T

l.i.m. V AM?-1) / dMt 1=0 1 /

T

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).. .dM^, 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^lTfa^ 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 [122] (1999) (also see [1]-[15], [106]). 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 S[£, exists and for k > 2

s[«, =sK, w.p. i.

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 [122] (1999) (also see [1]-[15], [106]). 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[0(k)]t,t = J fah(T)dM(k+1)S[0(k)]T,r w. p. 1, (3.38) t t

(k)

where the stochastic integrals in (3.38) exist.

Theorem 3.8 [122] (1999) (also see [1]-[15 Theorem 3.5

). In the conditions of

T

hfao / fadM(k+2)dMi(ik+1)S[0(k)]

T,ti =

t

i

T T

= J far dM^y h(ti)dMt(f+i)S [fa(k)]T,t 1 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

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 = xo + J a(xT, t)dr + J B(xT, t)dfT, xo = 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) [91]. The second integral on the right-hand side of (4.1) is interpreted as an Ito stochastic integral. Let x0 be an n-dimensional

, ^ -v

random variable, which is F0-measurable and M{|x0| ) < to. Also we assume that x0 and ft — f0 are independent when t > 0.

It is well known [75], [76], [84], [123], [124] (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 [75], [76], [84], [123], [124] (also see [13]).

Numerical integration of Ito SDEs based on the strong convergence criterion of approximations [75] 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

125]-[129].

stability and bifurcations analysis [73], [75], [76], [83], [84], [108],

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 [73], [75], [76], [83], [84] to the numerical integration of Ito SDEs based on the stochastic analogues of the Taylor formula (Taylor-Ito and Taylor-Stratonovich expansions) [130], [131] (also see [132]-[

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., [73]-[76], [83], [84]) 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-Ito and Taylor-Stratonovich expansions [132], [134] (also see [50], [68]) which makes it possible (in contrast

with its classical analogues [75], [130]) 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 [132] with

using of the slightly different approach (which is taken from [134]) in comparison with the approach from [132]. Moreover, we obtain another (second) version

of the unified Taylor-Ito expansion [72], [135]. In addition we construct two

new forms of the Taylor-Stratonovich expansion (the so-called unified Taylor-Stratonovich expansions [134]).

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 [73], [75], [76], [83] (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:

T t2

J = fa (tk)... fai(*i)dw^ ... dw^, (4.2)

i i *T *t2

J [fa(fe fc = y fak (tk) ...J fai(ti)dwt(;i)... dw(ik), (4.3)

t t

where every fa/(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 [19], [38 .

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 Ito 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 [130], [131] (also see [73], [75]) 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] ^ R1 is a sufficiently smooth nonrandom function.

The first result in this direction called the Ito-Taylor expansion has been obtained in [131] (also see [130]). 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) tfc

(4.4)

as a multiplier factor, where i1,..., ik = 0,1,..., m. Obviously, the iterated Ito stochastic integral (4.4) is a particular case of (4.2) for 01(r),... , fa(t) = 1.

In [130] 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) tfc

(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 [132] the Ito-Taylor expansion [130] 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)

(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 72], [135] (also see [1]-[15]). Terms of the latter expansion contain iterated Ito

stochastic integrals of the form

(t - tk )1k... I (t - ti)11 ..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) [136] (also see [134]) 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)1k(t-1^1 dfi(;i)..f(4.8)

t t : S * ¿2

(s - tk )1k ..J (s - ti y1 f;1) ...dftf), (4.9)

t

where /1,..., 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) ([4791), 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 ^ 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) = : [0,T] x Q ^ 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

M {|bs — br|4} < C|s — t|y

for all s,t G [0,T] and for some C, 7 G (0, to).

The second integral on the right-hand side of (4.10) is the Ito stochastic integral.

Consider a function F(x, t) : R1 x [0, T] ^ R1 for fixed t from the class C2(—to, to) consisting of twice continuously differentiable in x functions on the interval (—to, to) such that the first two derivatives are bounded.

Let us recall 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 [100] (also see [75])

;T t T

F(rjT,r)dfT= f F(VT,r)dfT + ^ i ^(i]T,r)bTdr w. p. 1. (4.11)

If the Wiener processes in (4.10) and (4.11) are independent, then

* t T

J F(nT,r)d/T = J F(nT,r)d/T w. p. 1. (4.12)

t t

Let us remind that a possible variant of conditions providing the correctness of the formulas (4.11) and (4.12) consists of the following conditions

nr G Q4([t,T]), F(nT,T) G M2([t,T]), and F(x,t) G C2(-to, to).

Let us apply Theorem 3.1 (see Sect. 3.2) to derive one property for Ito stochastic integrals.

Recall 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).

Lemma 4.1 [14], [15], [50]. Let h(r), g(r), 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

J g(t)J h(0) y W = J(G(s) - dfUi)df<j}

t t t t t

w. p. 1, where i, j = 1, 2 and fT1^, 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

0

J g (t ) J h(0) y & ff w = J & y > y g (t )dr =

t t t tu 0

s s s s

= G(s) y h(ö)df0j) - y y G(0)h(0)f) =

t u t u

s 0 s 0

= G(s) y ) - j ^ff) =

t t t t s 0

= j(G(s) - G(0))h(0^ ) w. p. 1.

tt

s

s

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 [134] (also see [1]-[5], [12]-[15], [50]). Let h(r), g(r), 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 Q4([t,s]) and

T T

= f audu + J 6„dfU°, l = 1, 2.

t t

Then

S * T ^ 0 >jc s ^ 0

J g(t)J fc(0)y ^df«df0j)dT = J (G(s) -^df«dfj (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 G [t, s].

Proof. Under the conditions of Lemma 4.2, we can apply the equalities (4.11) and (4.12) with F(x,0) = xh(0) and

* 0

no = y ^df«,

t

since the following obvious inclusions hold: n0 G Q4([t, s]), xh(0) G C(-to, to), and n0h(0) G M2([t,s]). Thus, we have the equalities

t ^ 0 T * 0 T

y h(6)J tfdfMdfP = I h(9)y ei^f^df^ + il^} y /i(0)^d0, (4.14) t t t t t

*0 o o

y ei/^f^ = y ^dfW + ±l{i=i} y budu (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

sT

e

g(T) I He) I Âf W =

u

t t t

s s s

= J ^f0 | h(ö)df0j) | g(T)dT+

tu

s s s

IT, r ......1

I Kdu I h(0)d4j) / g{r)dr + -!{,=?} / /

2 — {o—i - «— i v / —-0 i j \ j — ■ 2

t u t

s

u

s s

+il{/=i} f budu f h(0)df®

2

tu

s

CiPdf^ I G(6)h(e)df^ + ±i{i=j} I G{ß)h№fdß+

2

s s

1

+-l{/=i} / budu /

2 _ 1 o—» u

t 0

= G(S) J h(9) J ^df^df^ + il{i=j} J h(0)^d0+

u 'dig' + 2

tt

s 0

J h{0) J bududï{0j)

tt 0 s

J ^dï^dï^ + ±l{i=j} J G(0)h(0)^l)d0+

tt s 0

[ h{6)G{6) [ bududi{ej) ) (4.16)

s

s

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—Itô Expansion

In this section, we use the Taylor-Ito expansion [130] 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] ^ 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

dR

i=1

d x(i)

j=1 1,i=1

d2R <9xW<9x«

n (x,^),

(4.17)

G^x, t) = £ bUi) (x> y (x, i), i = 1,..., m,

dR

j=1

d x(j)

(4.18)

where x(j) is the jth component of x, a(j)(x,t) is the jth component of a(x,t), and B(ij)(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 + E / Gf)i)R(xT,t)dfW (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

H=1,..,ik=1

m1,..., mk > 1,

(4.20)

n

n

(k+i)^ l (i)b(k) =

mi m;

¿1 = 1 ;; = 1

m; + i ... m;+fc

¿;+i=1,---,i;+fc=1

for k > 1

mi ml

^ ... Y A(ii...i;)B(ii...¿;)

^ ¿1=1 i;=1

mi ... mfc

for k = 0

(¿i)

= (k)Ak+1Dk Ak ...A2 D1A1R(x,t),

ii=1,...,ifc=1

. (4.21)

where and D^9^ are operators defined on the space L for p = 1,..., k + 1, q = 1,..., k, and = 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(n."i i— ii i+i...ifc)

mi ... m;-i m;+i ... mfc

¿i=1,...,i i—i=1,i;+i=1,...,ifc=1

(shortly, (k-1)A).

We also introduce the following notations

mAi ... mA;

=f >Qa, ...Qai R(x,i),

qA;i) ...q^m)

¿i = Ai,...,i; = A ;

(pk)J

(Afc...Ai)s,t

(ife...; i)

(Ak ...Ai)s,t

mAi ... mAk

ii—Ai,...,ifc=Ak

Mk = {(Ak,..., Ai) : A/ = 1 or A; = 0; l = 1,...,kl, k > 1,

S ¿2

J

(ifc ...ii) (Ak ...Ai)s,t

dwt(ik) ...dwj;^, k > 1,

(;i)

tfc

t t

where A; = 1 or A; = 0, qA;) = L and %/ = 0 for A; = 0, Q^ %l = 1,..., m for A/ = 1,

) and

pl = ^^ Aj for l = 1,..., r + 1, r E 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-Itô expansion [130

R(xs, s) = R(xt, t) + E E ^ •.. QaiR(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 =

= E ['-( / (Pr+l)QAr+i ...QAi R(xti ,t1) Ar-+1 dwtJ ... Al dwtr+i.

(Ar+i,...,Ai)GMr+i { \ t J

(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^ ... Q™R(x, t) G L for all (A/,..., Ax) G U Mg;

g=i

(ii) Q^... 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 mAk

R(x.,s)=R(x(,t)^ E E ... £QAf...QA':)R(xf,i)j(t;A1))S,(+

k=1 (Ak,...,Ai)eMk ii=Ai ik=Ak

+ (Dr+1)s,t W. p. L

Denote

Grk = |(Ak,..., Ax) : r + 1 < 2k - A1 - ... - Ak < 2r J,

Eqk = ^(Ak,..., A1) : 2k - A1 - ... - 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 mAfc

= Rx,t) + E E E-EQ: 1...eA!>(x<,t)J£:Ai,,,+

q,k=1 (Afc,...,Ai)GEqfc ii=Ai ¿fc =Afc

+ (Hr+i)s,t w. p. 1, (4.24)

where

r mAi mAk

(Hr+i)s, =e e E.. ^ qA::1 ... q^rK t)jAtAw+(Dr+i).,.

k=1 (Afc,...,Ai,GCrfc :i=Ai ¿fc=Afc

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 t2

= y(t-tk)1k... J(t-ti)01 dft(;i)...ff) for k>i

(4.25)

and

I

) = 1 for k = 0,

'l-.-'fcs.i '

where i1,..., = 1,..., m. Moreover, let

(k)1

h—hs,t

(ii...ifc)

ii,...,ifc—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(H ...G((ik )Lj R(x, t)

d=f (k)G/i... G/kLjR(x, t),

ii,...,ik=1

Lj R(x, t) = I

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

(s-ty ß

E Gi;i)...G((ik)LjR(xt,t)/;;xk;+

q=1 (k,j,/i ,...,/k )eAg ^ ii,...,ik = 1

+ (Dr+1)s,t ,

(4.27)

where (Dr+1)st is defined by (14.23). Proof. We claim that

(Pq )

E (Pq )QAq ...Qai R(xt,t)"q (Pq )J(Aq ■■■Ai)s,t

(Aq .■■■,Ai)GM,

E

(k,j,/i v^k ) G Aq

j!

Y g(;i)...Gii'>LjR(*,t)j££) (4.28)

(ii ■■■ik)

i1,...,ik=1

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+i) Qai . . . QAq+i R(xt,t) (Pq+i) ^■■■Aq+i )s,t

(Aq+i .■■■,Ai )GMq+i

E / E i(Pq+i)QAi ...QAq+iR(xt,t)

^q+ie{1, 0} { (Aq.■■■,Ai)GMq\

Pq

•q (PqJ

(Ai ■■■Aq)0,t

Aq+1

• dw^ =

E / E

Aq+i G{1, 0} t (k^/^^/k )GAq

j!

X

m

m

s

x I (^V1 >Gi,... G/kLjQa„+iR(xt,t)k Aq-+i dw9 =

(k,j,/l,...,/k )EAq \ t

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...iksi - (-1)j+1 ■ (k)//i...ik—i ik+j+1si for k> 0

(4.30)

w. p. 1. In addition (see (4.25)) we get

s

...¿kik+l) (A Q1 \

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, t) = - E(-it), c; = (4.32)

(this equality follows from (4.26)), we note that the obtained expression equals to j

(kj,il,...,ik)eAq+i

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

= R(X„t) + j: v fail- V 6t'»...6f»L^(x(li)/,(;'.:C.:+

+ (Hr+i)a>i w. p. 1, (4.33)

where

( _ ,)j m

E far1 E

(k,j,/i,...,/fc)GUr ii,...,ifc = 1

+ (Dr+1)s,t ,

k

Dq = < (k,j,/i,...,/k) : k + 2 1 j + > I = q; k, j, li,..., Ik = 0,1,...J>,

(k, j, li,..., Ik) : k + 2 + E ^

q _

\ \ p=i /

(4.34)

i k

Ur = < (k, j, li,... ,4) : k + j + E ^ r,

I p=I

k + 2^' + ^ > r + 1; k, j, li,..., Ik = 0,1,.. J, (4.35)

and (Dr+1)st is defined by (14.23). Note that the remainder term (Hr+1)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

= /(* - tk)/k - J(s - t1)1'f1... <W for k > 1

and

J

(n-.-ifc)

'l—'fcs , t

= 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£ct = £ • • ■ £ IK (t - s)'1^-*--" j; j) w. p. 1, (4.36)

k

r(;i"-;: )

"1=0 j:=0 g=l

where

ck =

l!

k!(1 - k)!

is the binomial coefficient. Thus, the Taylor-Itô expansion of the process n = R(xs,s), s G [0,T] can be constructed either using the iterated stochastic integrals 1 ' l^ similarly to the previous section or using the iterated stochastic

integrals Jf1 ''• ifc). This is the main subject of this section.

Denote

J

(;1 ;:)

'l • • • 'ks , t

=f (k)J

'l.• • 'ks,t '

;i, • • • ,;:=1 m

LjG^1) •••G((;:)R(x,t)

:

=f (k) Lj Gii ...Gifc R(x,t).

¿1,'",ifc = 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 ß

E lG((;i)•••G((;:)R(xt^Ji^

q=1 (k,j,/i,...,/fc )eAq 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)

£

j !

£ LjG«;1' • ••G^'«(xt.OJ^ =

;l,•--,«:=1

i

m

m

( _ j m

= E far- E t)j£;£> w.p.1 (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 E L'Gf^ ... Gf^Rfa,

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 (14.231): 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-Itô expansion (14.22) (without the remainder term (Dr+1)st) to the process R(xs,s) as r ^ to for all s,t G [0,T] such that s > t and T < to has been proved in [75] (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 [75] converge to the process R(xs,s) w. p. 1 as r ^ 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 [130] 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) G M2([0,T]) for i = 1,...,m and consider the Itô formula (4.19).

In addition, we assume that G0i)R( x, t) G C2(-œ, œ) for i = 1,..., m and R(xT,t) G Q4([0,T]). In this case, the relations (4.11) and (4.12) imply that

J = J G^R{xr,T)df^ j GfGfR{-xT,T)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 + E / G0;)R(xT, t)df(;) , ¿=1 .

w. p. 1, (4.40)

where

2

0 G0

¿=1

Introduce the following notations

mAi ... mA|

(Pfc) 7*

J(Afc •••Ai)s,t

=f (pi)Dai •••DaiR(x,t),

¿i=Ai ,•••,¿1 =A;

J

(A: •••Ai)s,t

mAi ••• mA:

¿i—Ai ,•••,«:=A:

(4.41)

Mk = ^•••faO : A' = 1 or A' = 0; l = 1,_,kL k > 1,

<Î2

J

= («: •••¿i) (A: •••Ai )s,t

dw£:) •••dwj;u, k > 1,

r(n)

t t

where A' = 1 or A' = 0, D^) = L and i = 0 for A' = 0, D^0 i = 1, • • •, m for A' = 1,

GO1 ) and

P' = ^^ Aj for I = 1, • • •, r + 1, r G N,

j=1

s

s

s

s

s

m

s

'

wT^ (i = 1,... ,m) are FT-measurable for all т G [0,T] independent standard Wiener processes and w[0) = т.

Applying the formula (4.40) to the process R(xs, s) repeatedly, we obtain the following Taylor-Stratonovich expansion [13

R(xs, s) = R(xt, t) + £ Y, ^... DaiR(xt, t) ? (pk' J(*Afc...Al)s , +

k=1 (Afc, Ai)eMfc

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

Q(i;}...QC;}R(x,t) - Qiil}... Qi;1 }R(y,t) < K|x - y|, (4.44)

Q™ ...QAi1i)R(x,t) < K (1 + |x|),

(4.45)

and

... QAii)R(x,t) - QAr • • • QArR(x, s) < K|t - sr(1 + |x|),

r»(ii)

where K < to is a constant, Q^ = L and iz = 0 for Az = 0, Q^ = G01) and iz = 1,..., m for Az = 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(,

k=1 (Afc,...,A 1 )eM: ii=Ai i: =A:

+ (Dr+i )s,t w- p. 1.

Denote

Grk = {(A*,..., Ai) : r + 1 < 2k - Ai - ... - A* < 2r J,

Eqk = j(A*,..., Ai) : 2k - Ai - ... - A* = 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(x(, t) + £ £ E---EDÏ)... DAii)R(xi, t) +

q,k=1 (A:,...,Ai)eEg: ii=Ai i:=A:

+ (Hr+iw. p. 1, (4.46)

where

r mAi mA:

(Hr+i ),, = E E E •. . E D<:)... Di)R(x„ t)+

k=i (A:,...,Ai)eGr: ii=Ai i: =A:

+ (Dr+i )s,t .

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

t2

I

c(i i —)

'i •••':s,t

= (t - tk)'

(t - t1)'idf^

• • dft(;:) for k > 1 (4.47)

and

where i1, • • • , ik = 1, Moreover, let

I *(«i •••¿: ) 'i—':s,t

= 1 for k = 0,

m^

(k) T

^ I

'i •••':s,t

s(«i-«:) 'i •••':s,t

i i j^^^ji:=1

= i (G^L - LG^i) , p=l,2,..., i = 1,..., m, (4.48)

s

m

where G0i) == G0i), i = 1,... ,m. The operators L and Gfa, i = 1,... ,m are determined by the equalities (4.17), (4.18), and (4.41).

Denote

(¿)

Aq = <j (k, j, ¿1, • • •, Ik) : k + j + E = q; k, j, l1, • • •, Ik = 0,1,

p=1

g ((;l) •••(G ((;: )l j R(x,t)

=f (k)G'i • ••(':LjR(x,t),

; i ,•••,;:=1

Lj R(x,t) =f 4

R(x,t)

for j = 0

Theorem 4.3 [134] (also see [1]-[15], [50], [136]). 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) =

( f\j ''

+ (Dr+i)a,t, (4.49)

where (Dr+1)st is defined by (14.4,3). Proof. We claim that

(A, ,...,Ai)GM,

( _ ,)j m

= E iiXL E (4-50)

(k,j,/l,...,/fc)GAq ' i1,...,ife = 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 (P,+ l)DAi . . . Da,+iR(xf, t) (Pl+'> J(-Ai...Ai+i)S,( =

(Aq+1 ,...,Al )GM,+ 1

E f E i(p'+l)D^i-Dw.R(xt,t)<»JV,,JA-

^q+iejl, 0} { (A,,...,Al)GMq\ /

A,+l ,

dw, =

E / E

Aq+lG{1, 0} t (k,j,ll,...,lfc)GAq

-n—x

J!

X [ (k+A,+l)GGil... GIkLDA,+lR(xt, t) k (k)/iUsJ Aq+l dw, =

E r^ • • • GhP+1R(^t)k j

(k,j,/l,...,/fc)gA, v t

: S

+ (i+1)<5;1... G\^G0Ä(x(, t) < f 1 1

J ß I • dfe | (4.51)

w. p. 1.

s

Using Lemma 4.2, we obtain

(.e-ty

■{k)I* d0 =

j!

t

j ! ti---tfce,t

(s - +1 for k = 0

(j + 1)!

(s - ^+1 • (k)li...ifcsi - (-l)j+1 • ik+j+isi for k > 0

(4.52)

w. p. 1. In addition (see (4.47)) we get

f (0 — j*(h...ik)1) _ j*(ii-ifcifc+i) ^ ^ 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

£ {-^(k)Glt...GlkL1Mk(k)IUv

j

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

= «<*.*)+£ £ far1 s

q=l (k,j,/l,...,/fc )GDq il,...,ifc = 1

+ (Hr+i)si 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)

i k

Ur = < (k, j, li,... ,1k) : k + j + E ^ r,

I p=I

k + 2^' + Z^ > r + 1; k,j,Zi,...,Zk = 0,1,.. j, (4.57)

and (Dr+1)st is defined by (14.43). Note that the remainder term (Hr+1)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'r:' = / (s - tk)'k... / (s - ti)''df<;;'»...ffor k > 1

t t

and

j;(il,-ifc) = 1 for k = 0,

where i1,..., ik = 1,..., m.

The additive property of stochastic integrals and the Newton binomial formula imply the following equality

C; = £ ■■.. £ II1Cg (t - «)"+• • •+'k-j;-• • jw. p. 1, (4.58)

j ; =0 j; =0 g=1

where

nk — 1 ~ k\(l-k)\

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 similarly to the previous section or using the iterated

stochastic integrals J*/^'^. This is the main subject of this section.

Denote

J

= (n...ifc)

/i.../ks,i

=f (k) J

/i.../ks,i ;

ii,...,ik=1

LjG(il).. .G((ik)R(x,t)

def

(k)Lj G ¿i.. .G ik R(x,t).

ii,...,ifc=i

Theorem 4.4 [134] (also see [1]-[15], [50], [136]). 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 ß

E gi;^...(Giik)R(xt,t)Ji

(ik)

r*(;i."ifc)

ll...lks,t

+

q=l (k,j,/l,...,/fc )eAq il,...,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!

E (G((;i)... (G((;k)R(xt, t) J*(iik.;k)

11 ...lk

(k,j,/i,...,/fc )eAq ;i,...,;k = 1

(s - t)j m

£

£ Giii)...gikk'-£jR(xt.t)/*«;'/-» w.p.1

(k,j',/i,...,/k)GAq

j!

V(ik) fj lk

(i1...ik)

ii,...,^=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

m

m

,t) + £ E

q=l (kj,/ i,...,1fc)GDq

R(xs, s) =

(s - t)j m

j !

E L G

(- 1 ) ...G(:k)fi(xi,i)j*(:'i;i': )+

i 1,...,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 1 ) /

G((ik )R(xt ,t)j/;(i-.ik )

11.../fcs,i

+

+ (Dr+i )

s,t '

The remainder term (Dr+i)st is defined by (14.431): 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 ^ to for all s,t G [0,T] such that s > t and T < to has been proved in [75] (Proposition 5.10.2). Since the expansions (14.49) and (14.591) 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 [75] converge to the process R(xs,s) w. p. 1 as r ^ to for all s, t G [0, T] such that s > t and T < to.

r

m

4.9 Comparison of the Unified Taylor—Ito and Taylor-Stratonovich Expansions with the Classical Taylor— Ito and Taylor-Stratonovich Expansions

Note that the truncated unified Taylor-Itô 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 [130 .

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 Ito and Stratonovich stochastic integrals, which are included in the sets (4.6) (479) 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 Ito stochastic integrals

I(il) I(¿1«2) I(il) I(¿li2«3)

J0M, J00M , JiSji, J000Sii •

The same truncated classical Taylor-Ito expansion (4.24) [75] contains 5 various iterated Ito stochastic integrals

T (il) T(ili2) T(i10) T(0i1) t (i1i2i3)

J(1)s,t' J(11)s,t' J(10)s,t' J(01)s,t' J(111)s,t*

For r = 4 we have 7 (rankD(4) = 7) stochastic integrals

I(ii) I(ili2) I(i1) I(i1i2 i3) I(i1i2) I(i1i2) I(i1i2 i3i4)

J0M, J00Sji , J1M5 J000Sji , J01Sji , J10M , J0000Sji

against 9 stochastic integrals

T (il) T ( i 1 i2 ) T ( i 10) T (0i l) r(ili2i3) j (il0i3) T (ili20) r(0ili2) T (ili2i3i4) 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 [134] (also see [1]-[15], [50], [136]) with their classical analogues 75], [130] (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 [73], [75] (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 [73], [75]. For example, rankA(4) = 15 while the total number of various iterated Ito stochastic integrals (included in the classical Taylor-Ito expansion [75] when r = 4) equals to 26.

Let us show that [3]-[15], [50]

rankA(r) = 2r - 1.

Let (Z1;..., Zk) be an ordered set such that Z1,..., Zk = 0,1,... and k =

def

1, 2,... Consider S(k) = Z1 + ... + Zk = p (p is a fixed natural number or zero). Let N(k,p) be a number of all ordered combinations (Z1,...,Zk) such that Z1,..., Zk = 0,1,..., k = 1, 2,..., and S(k) = p. First, let us show that

N (k,p) = Cp-k-i,

where

n !

(Jm _

n

m!(n - m)!

is a binomial coefficient.

It is not difficult to see that

N (1,p) = 1 = Cp+i_i, N (2,p) = p +1 = Cp+2-i,

Moreover,

N(k + 1,p) = £ N(k, l) = £ Ci+k- i = Cp+k,

1=0 1=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)) =

= eg + ( c0 + Ci ) + ( C2 + Ci + ci ) +...

... + (Cr_ i + Cr _ i + Cr_ i + ... + Cr_ 1 ) = = 20 + 2 1 + 22 + ... + 2r_ 1 = 2r — 1.

p

p

Let nM(r) be the total number of various iterated stochastic integrals included in the classical Taylor-Ito expansion (4.22) [75], 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 _ t)j

then

j !

nM (r) =

= (2i - 1) + (22 - 1) + (23 - 1) + ... + (2r - 1) =

= 2(1 + 2 + 22 + ... + 2r^) - r = 2(2r - 1) - r

It means that

lim -^¿M- = 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]-[15], [50]

rankD(r) =

fr-i (r-i)/2+[s/2]

E

s=0

E c/

for r = 1 , 3, 5,

/=s

=

(4.61)

r-i r/2-i+[(s+i)/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

n-M(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) [75] 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) [75]. 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 1=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) = «E(^)/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

(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

yP+i = y, + E E E W

q=l (k,j,h,...,lk)eDq il,...,ifc = 1

, 1f {Tp+1 -Tp)(,+1)/2 (r+1)/2

+ l{r=2d-l,dGN} l)/2)! ^ ^

r

where J^1is an approximation of iterated Ito stochastic integral

11 • • • lkTp+i,TP

if 1 • I¿k' of the form

11 • • • lkTp+1,rp

S ¿2

i^1 ii:! = fa - tk ),k ■■■[ (t - ti)11 fi1'■ '■

Note that we understand the equality (4.64) 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 [75] 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).

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^, G0i)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 + £ ß.14:p+„Tp + Aa + £ G«>2)ß,1 /<;;■> ,TP ■ (4.65)

¿1 = 1 ¿1 ,¿2 = 1

Scheme with strong order 1.5

m

yp+1 = yp + £ B,1 + Aa + £ G«'2>ß,1 +

¿1 = 1 ¿1,i2 = 1

+ V [c^'a (A/?1' + /f1' ) - LBi1 if1'

l 0 v u:p+1':p 1:p+1-:p / 1 1:p+1>:p_

+

I Y^ G(i3'G(i2 / (i3i2i1) I

+ G0 G0 B1 J000:p+1,:p +

¿1,i2,i3 = 1

m

m

m

m

m

A2 + ~

(4.66)

Scheme with strong order 2.0

yp +£«1 +Aa + z g^X «cL+

¿1=1

«1,«2 = 1

+ £ [G<'l)a (A/0

i1=1

(:1) + /(:1) ) - LB:1 /1(il) T:+l.T: 1r:+1>T:/ 1 1r:+l.T:_

+

A2

, \ rii't-V riin)-T) f{'<3'<2n; i — t ^ i

+ Z^ 0 0 £'ii'/000T + —

¿1 ,¿2 ,¿3 = 1

T:+l.T: 2

m

+ £ [g0'2)lb.1 (/

(¿2:1) 10

-I,

( ¿ 2 ¿ 1)

i1,i2=1

':+1>T:

01.

':+1>T:

TH^U T (¿2¿l) I

':+1>T:

+G0¿2)G0¿l)a (/Of^ + A/Or^

0 0 V 01 T:+l.T: 00T:+l> T:

+

_I_ v^ g^g^g^) R T (¿4¿з¿2¿l) + G0 G0 G0 «¿11 0000t:+1,T: '

i1,i2,i3,i4=1

(4.67)

Scheme with strong order 2.5

yP+1 —

yp + £ «¿1 TL, + Aa + £ Ä/TSsri+'l.,, +

i1=1

i1,i2=1

+ y ^a (a Tt +T

i1 =1

0n) +A(il) ) - ^«¿l) 0r:+l.T: 1r:+l.T:/ 1 1r:+l.T:

+

+ £ g^gC^bJ,

i1,i2,i3=1

A2

m

+ £ [G0'2)LB,1 (/

(i2i1) 10

- /,

( ¿ 2 ¿ 1)

i1,i2=1

>+l,T:

01

T:+l.T:

Tn^U T(¿2¿1) I

— LGo «¿1 i 10 +

T:+l.T:

1 гi(¿2)Гi(¿l)* ( T(¿2¿l) + G0 Go a I 1 01

0 0 V 01r:+l.T:

+ A /(1¿2¿l)

00

T:+l.T:

+

m

m

m

m

m

m

m

m

0000T +

+ Z^ G0 G0 G0 Bii h

¿1,i2,i3,«4 = 1

G^La (

+£

ii=i

+\llbul

1

(il)

JTp+i>T; A2

(n) +Af(ii) i +

______—LG^ali^

2 ^ 2Tp+i_Tp 0 ^ 2Tp+1,Tp

-r^(Ti)

2t; i 1 ,Tp 1 Tp+i?Tp

+ A/}

+

+ £

ii,i2,i3 = 1

r'(i3) T r'(i2) n ( f (i3i2ii)

G0 LG0 Bii |v/100Tp+i,Tp

f(i3i2ii) . I - h010 ) +

Tp+i>T;

I r1^) n(i2) T n ( f(i3i2ii) f (i3i2ii) \ I

+G0 G0 LBii l/010Tp+i,Tp - /001tp+i,tJ +

I A f(i3i2ii) I f(i3i2ii)

+G0 G0 G0 a I A/000T , _ T; + h001

000

Tp+i>T;

Tp+i>T;

^3)^2) D f(i3i2ii)

_ 7"/^(i3)^(i2) D f LG0 G0 Bii J100-

"p+i>Tp_

+

\ " g^g^g^g^) R f(ißi4i3i2ii) + + G0 G0 G0 G0 Rii h00000 +

ii,i2,i3,i4,i5 = 1

Tp+i>TP

A3

H--LLsl.

6

(4.68)

Scheme with strong order 3.0

yp+1 = yp + £ Bii4Tp+i,Tp + Aa + E G^Bii/00r;;+i,Tp +

ii=1

ii,i2 = 1

+ £ [G0ii)a (A/0

ii=1

(ii) + f1(ii) ) - LBnf1(ii)

T;+i>TP 1Tp+i>TP/ i 1T;+i,TP

+

+ E G0i3)G«i2)Bii /,

ii,i2,i3 = 1

A2

00%+bTP 2

m

+ £ [G«i2)LBii (/

(i2ii) 10

-I

( i 2 i i)

ii,i2 = 1

Tp+i>TP

01

Tp+i>TP

LG(i2)R f (i2ii) 1

— LG0 Rii h10 +

Tp+i>T;

m

m

m

m

m

m

m

where

+Go¿2^(¿^a (/Of^ + A JnO2 ¿l)

0 0 V 01 T:+l.T: 00T:+l.T:

+

+ ^^^«¿ltet+qp+1,p+rp+1,p, (4.69)

¿1,¿2,¿3,¿4 = 1

qp+1,p

E

¿1=1

# 1) , a2^ 1)

G^La I - ffl} + Air' + — VA*1' I +

0 1 2 2r::l.T: 1r::l.T: 2 0t::i,t: /

+-LLBJ^] -LG^'alPo

2 ¿1 2T::l.r: 0 V 2T::l.T:

^ ( T(¿l)

+A

^ 1)

1t::I,t:

+

+

¿1, ¿ 2^ 3=1

r'(¿3) T ^2)v ( T (¿3¿2¿1)

G0 LG0 R ¿ 1 ^1 100r::1.r:

T (¿3¿2¿1) . 1 - 010 +

T::l.T:

^rl(¿з)п(¿2)^U (T(¿з¿2¿l) T (^¿O ^ 1

+G0 Go l( 1 010r::l.r: - 1 001^1^ +

1 r^^^r^ / A T (¿3¿2¿1) I T ^¿O

+G0 G0 G0 a 1000r::l.r: + 1 001^:

( 3) ( 2) ( 3 2 1)

( 3) ( 2) LG0 G0 R l1100

T::l,T:i

+

and

I ¿5)^'(¿4)A~^(¿3)^f(¿2) ^ T

+ G0 G0 G0 G0 «¿l1 <

¿1^ 2, ¿3^ 4, ¿5=1

A3

H--LLa,

6

( ¿5¿4¿з¿2¿l) + 00000r::l,r: +

r

p+1,p

E

¿1^ 2=1

^(^Wü) T _ I 1 f(«2H) Go Go 2 0%+1,rP

+ A T ¿ 2¿l)

+ 2 0(Wp 1

_l rrn^u r^n) 1

2 0 1 T::l.T:

+G0¿ 2)LG0¿ l)af /if0

-I,

(¿2¿l)

>:l.T:

02

::l.T:

+ A (/{O2^

-I,

( ¿2¿l)

>:l.T:

01

>:l.T:

+

+LG0¿2)Lß¿ 1 (/1fl) 0 1 V T::l.T:

- /^02¿l) ) +

m

m

m

+G<r»LLB,1 ( „ + „ - '["A

1)

-LG^G^a A/<;?"» t. +

Jrp+1>TP -LJ-Tp+l>TP

+

+

»1,i2,i3,«4=l

GMG^W^W^ / A f(i4i3i2i1) I T(i4i3i2i1) \ ,

Go Go Go Go a 1 ooooT.+1iTp + 1 oooiT.+1iTp I +

_|_g(®4)g(®3) rG(®2) R ^ T («4i3ï241) _ T (24232221)

+Go Go LGo Rn ^1 oiooTp+1iT. 1 ooioTp+1iT.

_rG(i4)G(i3)G(i2)R T (i4i3i2i1) +

LGo Go Go Ri11 ioooT1iT. +

+ g(Ï4) LG^G^) R T (i42322^1) _ T (¿4«3«2Î1) \ +

+Go LGo Go Rn ^1 ioooT.+1iTp 1 oiooTp+1,Tl +

+ ^(24)^(23)^(22) LR ^ T (24232221) _ ^(24232221)

+Go Go Go ^1 ooioT.+1iTp 1 oooiT.+1iTp

+

I \ A /^(26^(25^(24^(23^(22^ T (262524232221)

+ Go Go Go Go Go R211 (

21,22,23,24,25,26=i

1)

o Go Go Go Go R21-f ooooooTp+1iT. •

It is well known [75] 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) [75] (also see [13])

M{( if 1 • ;• - v ik) ^ 1 < C Ar+1, (4.70)

| l ^ • •;fcrp+1,rp 11 • • • ;fcrp+i,rp I 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 (I4.65I)-(I4jB91). The mentioned material based on the results of Chapter 1.

m

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

(o _ t)(r+1)/2

i^TTTW ¿(-+1,/2fi(x(,i) + (Är+1),( w.p.l, (4.71)

-{r=2<i-l,<feN} ((r + 1)/2)! where

_ (s _ t)(r+!)/2

Consider the partition {tp}N=0 of the interval [0,T] such that

0 = t0 < Ti < ... < tn = T, AN = max |t?-+1 _ Tj1 .

0<j<N _i J J

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 / m

yP+i=y,+E E ^p1 E ■ ■ ■ Gir<vyP +

-Li (Tp+1 ~rp)('+1)/2 r(r+l)/2,r (A70\

where //f1^'^) is an approximation of iterated Stratonovich stochastic integral /;(il/"ik) of the form

* S ^ ¿2

(i1...ifc) _ / /, , \/fc / /, , \/^f(i1) Jf(ifc)

err = / (t - tk )" ... / (t - il)" da?;" ...df(t

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 [75] 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), Ggi)Bj(yp,Tp), ... correspondingly. Moreover, the operators L and g)^, i = 1,... ,m are determined by the equalities (I4.17I), (I4.18I), and (4.73).

Scheme with strong order 1.0

yp+1 = yp + £ Btl + Aa + £ G('2)B21 Î^ • (4.74)

21=i 21,22=i

m

m

Scheme with strong order 1.5

yp+i = yp + £ Bi1 + Aa + £ g?2^ z,;':.;1^. +

21=i

21,22=i

+ £ |G«21)a(A V1' +1

(21)

21=i

i(21) )- lB21T;(21)

+

m

m

m

i g^W^r f*(i3i2ii) i

+ G0 G0 BiiJ00°rp+i,rp +

¿i,«2,«3 = 1

A2

+ —La. (4.75)

Scheme with strong order 2.0

yp+i=yp +L ßü +Aa+ £ G0,2)B.i «c, +

ii = 1 ii,i2 = 1

m

ii=i

+

0 * 0rp+i.T: 1Tp+i,Tp/ ii 1rp+i.Tp_

m A2

+ £MoooTp+1,Tp + "2"^+

¿i,«2,«3 = 1

m

+ £ [g0,2)lBii (- /^l) - Lg02)b„+

i i,i2 = 1

+G0i2 )G0i i W /7lii2i i: + A i: 1

0 0 * 01tP+ i,Tp

+

I Y^ R rl(i4i3i2i i) (Aia\

+ 2^ G0 G0 G0 Bi i10000tp+ i,TP •

i i,«2,i3,i4 = 1

Scheme with strong order 2.5

m

yp+1 = yp + £ Bi, ^ ,TP + Aa + £ g0-2)b. i/010TP'+:,TP +

i =1 i ,i2=1

m

+ V [G i:a (A /1(i i: + i: ) - Lßi ^ i:

L 0 V 0Tp+ i-Tp 1Tp+ i i i 1Tp+ i ,Tpj

i =1

+

m A2 + £MoooTp+1,Tp + ~2 a+

i i,i2,i3 = 1

m

m

m

m