SDE-MATH: A software package for the implementation of strong high-order numerical methods for Ito SDEs with multidimensional non-commutative noise based on multiple Fourier—Legendre series

Mikhail D. Kuznetsov; Dmitriy F. Kuznetsov

г

DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N. 1, 2021 Electronic Journal, reg. N $C77-39410 at 15.04.2010 ISSN 1817-2172

http:// diffjournal.spbu.ru/ e-mail: [email protected]

Computer software for the investigation of differential equations, dynamical systems, and control processes

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

SDE-MATH: A software package for the implementation of strong high-order numerical methods for Ito SDEs with multidimensional non-commutative noise based on multiple Fourier—Legendre series

Mikhail D. Kuznetsov1, Dmitriy F. Kuznetsov2

1 Faculty of Computer Technologies and Informatics, St. Petersburg Electrotechnical University, Saint-Petersburg, Russia 2Institute of Applied Mathematics and Mechanics, Peter the Great St.

Petersburg Polytechnic University e-mail: [email protected]

Abstract. The article is devoted to the implementation of strong numerical methods with convergence orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito stochastic differential equations with multidimensional non-commutative noise based on multiple Fourier-Legendre series and unified Taylor-Ito and Taylor-Stratonovich expansions. Algorithms for the implementation of these methods are constructed and a package of programs in the Python programming language is presented. An important part of this software package concerning the

Differential Equations and Control Processes, N. 1, 2021

mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 with respect to components of the multidimensional Wiener process is based on the method of generalized multiple Fourier series. More precisely, we used multiple Fourier-Legendre series converging in the sense of norm in Hilbert space for the mean-square approximation of iterated Ito and Stratonovich stochastic integrals.

Key words: Software package, Python programming language, numerical method, strong convergence, Ito stochastic differential equation, multidimensional Wiener process, non-commutative noise, unified Taylor-Ito expansion, unified Taylor-Stratonovich expansion, Milstein scheme, high-order strong numerical scheme, iterated Ito stochastic integral, iterated Stratonovich stochastic integral, mean-square approximation, generalized multiple Fourier series, multiple Fourier-Legendre series, Legendre polynomial.

Contents

1 Introduction |94

2 Theoretical Results Underlying the SDE-MATH Software Package |98

2.1 Strong Numerical Methods with Convergence Orders 0.5, 1.0, 1.5, 2.0, 2.5, and

3.0 for Ito SDEs Based on the Unified Taylor-Ito Expansion...............|98

2.2 Strong Numerical Methods with Convergence Orders 1.0, 1.5, 2.0, 2.5, and 3.0

for Ito SDEs Based on the Unified Taylor-Stratonovich Expansion..........|106

2.3 Method of Expansion and Approximation of Iterated Ito and Stratonovich

Stochastic Integrals Based on Generalized Multiple Fourier Series........|113

2.4 Approximations of Iterated Ito Stochastic Integrals from the Numerical Schemes

(11)-(16) Using Legendre Polynomials ....................................1123

2.5 Optimization of Approximations of Iterated Ito Stochastic Integrals from the

Numerical Schemes (12)-(16) ............................................1132

2.6 Approximations of Iterated Stratonovich Stochastic Integrals from the Numerical

Schemes (24)-(28) Using Legendre Polynomials ............................1141

2.7 Numerical Algorithm for Linear Stationary Systems of Ito SDEs Based on Spectral Decomposition....................................................|147

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3 The Structure of the SDE-MATH Software Package |148

3.1 Development Tools.........................................................|148

3.2 Dependency Libraries......................................................|149

3.3 Architecture..............................................................|149

3.3.1 Integration with SymPy.............................................1151

3.3.2 Purpose of NumPy...................................................|151

3.3.3 Purpose of SQLite Database.........................................|152

3.3.4 Purpose of Matplotlib .............................................|153

3.4 Implementation Plan.......................................................1153

3.4.1 Calculation of the Fourier-Legendre Coefficients...................|153

3.4.2 Differential Operators L, L, G0\ i = 1,... ,m......................1154

3.4.3 Approximations of Iterated Stochastic Integrals....................|154

3.4.4 Strong Numerical Schemes for Ito SDEs .............................|154

3.4.5 Graphical User Interface...........................................1154

4 Software Package Graphical User Interface 154

4.1 Information Model of The Graphical User Interface.........................|154

4.1.1 Processing Screens.................................................|156

4.1.2 Greetings Dialog...................................................|156

4.1.3 Main Menu Dialog...................................................|156

4.1.4 Visualization Tool.................................................|156

4.1.5 Data Input Dialogs.................................................|156

4.2 The User Experience and Implementation Results............................1157

5 The Results Obtained Using the SDE-MATH Software Package |175

5.1 The Calculated Fourier-Legendre Coefficients .............................|175

5.2 Accuracy Settings.........................................................|176

5.3 Testing Example (Nonlinear System of Ito SDEs)............................1179

5.4 Visualization and Numerical Results for Nonlinear System of Ito SDEs Obtained

via the SDE-MATH Software Package.........................................|18()

5.5 Example of Linear System of Ito SDEs (Solar Activity).....................|211

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5.6 Visualization and Numerical Results for Solar Activity Model ..............|211

5.7 Example of Abstract Linear System of Ito SDEs..............................|214

5.8 Visualization and Numerical Results for Abstract Linear System of Ito SDEs

Obtained via the SDE-MATH Software Package.....................................|215

6 Source Codes of the SDE-MATH Software Package in the Python Programming Language 217

6.1 Source Codes of Graphical User Interface.................................|217

6.1.1 Source Codes of Main Menu.........................................|217

6.1.2 Source Codes of Charts Window.....................................|230

6.1.3 Source Codes of Input for Nonlinear Systems of Ito SDEs...........|240

6.1.4 Source Codes of Input for Linear Systems of Ito SDEs.............|255

6.2 Source Codes for Nonlinear Systems of Ito SDEs...........................|275

6.2.1 Source Codes for Calculation of the Fourier-Legendre Coefficients .... |275

6.2.2 Source Codes for Supplementary Differential Operators and Functions . . |299

6.2.3 Source Codes for Iterated Ito Stochastic Integrals Approximations Subprograms ..................................................... 303

6.2.4 Source Codes for Iterated Stratonovich Stochastic Integrals Approximations Subprograms.........................................................|329

6.2.5 Source Codes for Calculation of the Numbers q, qi,..., q\5 ........|351

6.2.6 Source Codes for Strong Taylor-Ito Numerical Schemes with Convergence

Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs...............|359

6.2.7 Source Codes for Strong Taylor-Stratonovich Numerical Schemes with

Convergence Orders 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs........|381

6.3 Source Codes for Linear Stationary Systems of Ito SDEs...................|400

6.4 Source Codes for Utilities and Initialization ...........................|407

7 Future Work |412

References 413

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1 Introduction

As known, Ito stochastic differential equations (SDEs) have appeared in the theory of random processes relatively recently [1] (1951). Nevertheless, to date, a large number of mathematical models for dynamical systems of different physical nature under the influence of random perturbations have been built on the basis of such equations [2]-[16]. Among them we note mathematical models in stochastic financial mathematics [5]-[7], [10]-[12], geophysics [2], [4], genetics 13], hydrology [2], epidemiology [9], chemical kinetics [2], [9], biology [8], [15], seismology [2], electrodynamics [16] and many other fields [2], [9], [14]. In addition, Ito SDEs arise when solving a number of mathematical problems, such as filtration [2], [3], [17]-[21] , stochastic control [2], [17], stochastic stability [2], parameter estimation of stochastic systems [2], [3], [22].

Exact solutions of Ito SDEs are known in rare cases. For this reason, it becomes necessary to construct numerical methods for Ito SDEs. Moreover, the problem of numerical solution of Ito SDEs often occurs even in cases when the exact solution of Ito SDE is known. This means that in some cases, knowing the exact solution of the Ito SDE does not allow us to simulate it numerically in a simple way.

This article is devoted to the implementation of high-order strong numerical methods for systems of Ito SDEs with multidimensional non-commutative noise. More precisely, we consider strong numerical methods with convergence orders 1.0, 1.5, 2.0, 2.5, and 3.0. The article also considers the Euler method, which under suitable conditions [2] has the order 0.5 of strong convergence. To construct the mentioned numerical methods in this article, we use the so-called

unified Taylor-Ito and Taylor-Stratonovich expansions [24], [25] (also see [26

Chapter 4). The important components of these expansions are the iterated Ito and Stratonovich stochastic integrals, which are functionals of a complex structure with respect to the components of a multidimensional Wiener process.

It should be noted that it is impossible to construct a numerical method for Ito SDE in a general case (multidimensional non-commutative noise) that includes only increments of the multidimensional Wiener processes, but has a higher order of convergence (in the mean-square sense) than the Euler method (simplest numerical method for Ito SDEs). This result is known as the ”Clark-Cameron paradox” [23] (1980) and well explains the need to use high-order numerical methods for Ito SDEs, since the accuracy of the Euler method is insufficient for solving a number of practical problems related to Ito SDEs [2].

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According to the ” Clark-Cameron paradox” [23], avoidance of the problem of mean-square approximation of the mentioned iterated stochastic integrals is impossible in the general case when constructing high-order strong numerical methods for Ito SDEs.

The problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of the numerical integration of Ito SDEs was considered in a number of works [2], [3], [7], [8], [27]-[39 .

It should be explained why the results of these works are insufficient for constructing effective procedures for the implementation of strong numerical methods of order 1.5 and higher for Ito SDEs.

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There exists an approach to the mean-square approximation of iterated stochastic integrals based on integral sums [27], [34], [35]. Note that one of the variants of this method is based on reducing the problem of mean-square approximation of iterated stochastic integrals to the numerical integration of systems of linear Ito SDEs by the Euler method [39]. However, this approach 27], [34], [35], [39] implies the partitioning of the interval of integration for iterated stochastic integrals. It should be noted that the length of this interval is an integration step for numerical methods for Ito SDEs, which is already a fairly small value even without additional partitioning. Computational experiments show that the numerical modeling of iterated stochastic integrals by the method of integral sums [27], [34], [35], [39] leads to unacceptably high computational

[42]

cost and accumulation of computation errors

More efficient approach of the mean-square approximation of iterated stochastic integrals is based on the expansion of the so-called Brownian bridge process into the trigonometric Fourier series with random terms (version of the so-called Karhunen-Loeve expansion) [2], [3], [7], [27], [28], [33], [34], [37], [38.

However, in [27], [33], [34], [38], this approach was used to approximate only iterated stochastic integrals of multiplicities 1 and 2, which makes it possible to implement numerical method with order 1.0 of strong convergence for Ito SDEs (Milstein method [27]). In papers [2], [3], [7], [28], the approximation of iterated stochastic integrals of multiplicities 1 to 3 was considered by the above approach, which makes it possible to implement numerical methods with orders

1.0 and 1.5 of strong convergence for Ito SDEs. However, formulas concerning integrals of multiplicity 3 turned out to be too complicated and did not find wide application in practice. Moreover, these formulas (for iterated stochastic integrals of multiplicity 3) were obtained without strict theoretical justification and exclude the possibility of effective estimation of the mean-square error of

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approximation (see discussion in [26] (Sections 2.6.2, 6.2) for details).

It should be noted that in papers [29], [30], a similar approach was used to approximate iterated stochastic integrals of multiplicities 1 to 3 based on the series expansion of the Wiener process using trigonometric functions and Haar functions. In [40] orthonormal expansions of functions in terms of Walsh series were used to represent the iterated stochastic integrals.

Note that the iterated stochastic integrals under consideration are the random variables with unknown density functions. The only exception is the iterated Ito stochastic integral with multiplicity 2 [31]. However, the knowledge of density function of the mentioned stochastic integral gives no simple way of its approximation [31 .

In this work, we use a more efficient method of the mean-square approximation of iterated Ito and Stratonovich stochastic integrals than the methods considered above. This method (the so-called method of generalized multiple Fourier series) is based on the theory constructed in Chapters 1, 2, and 5 of monograph [26] (also see bibliography therein). The method of generalized multiple Fourier series made it possible in this work to successfully implement the procedures for the mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6. In this case, we use multiple Fourier-Legendre series, that is, we have chosen Legendre polynomials as a basis system of functions for approximating iterated stochastic integrals. It is important to note that the Legendre polynomials were first applied in the context of this problem in [43] (1997), while in the works of other authors Legendre polynomials were not considered as a system of basis functions for approximating iterated stochastic integrals (an exception is work [36]). As shown in [44], the Legendre polynomials are an optimal system of basis functions for approximating iterated Ito and Stratonovich stochastic integrals.

In this article, to build the SDE-MATH software package in the Python programming language, we use a database with 270,000 exactly calculated Fourier-Legendre coefficients to approximate iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6. It should be noted that the procedures for the mean-square approximation of iterated stochastic integrals of multiplicities 4, 5, and 6 constructed in this work have no analogues in the literature. At the same time, we propose a much more convenient procedure for the mean-square approximation of iterated stochastic integrals of multiplicity 3 than in works [2], [3], [7], [28]. This procedure provides an accurate calculation of the mean-square error of approximation of the mentioned stochastic integrals.

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Another important feature of the presented software package is the use of unified Taylor-Ito and Taylor-Stratonovich expansions [24], [25] (also see [26], Chapter 4) for constructing strong numerical methods with convergence orders

1.5, 2.0, 2.5, and 3.0 for Ito SDEs. Unified Taylor-Ito and Taylor—Stratonovich expansions make it possible (in contrast with its classical analogues [2]) to use the minimal sets of iterated Ito and Stratonovich stochastic integrals. This property well explains the motive for using the mentioned unified expansions.

The results of this work on the approximation of iterated stochastic integrals can be used to numerically solve various types of SDEs. For example, for

semilinear SPDEs with multiplicative trace class noise [26] (Chapter 7), [45], 46]. This is due to the fact that iterated stochastic integrals are a universal

tool for constructing high-order strong numerical methods for various types of SDEs. In recent years, the mentioned numerical methods have been constructed for SDEs with jumps [7], SPDEs with multiplicative trace class noise 47]-[49], McKean SDEs [50], SDEs with switchings [51], mean-field SDEs [52], Ito-Volterra stochastic integral equations [49], etc.

There are many publications in which codes of programs in various programming languages are given for the numerical solution of SDEs [3], [9], [14], Among them, we note the software described in [3], [54], [56

61

60

Some of the mentioned works [3], [54], [56], [57], [60] are based on the results of monograph [2] on the approximation of iterated stochastic integrals (see above discusson on the disadvantages of approach [2]). Other publications [9], [14], 53], [55] do not use the modeling of iterated stochastic integrals for the case of

multidimensional non-commutative noise at all.

In this article, we develop software for the numerical integration of Ito SDEs based on theoretical results and MATLAB codes from monographs [58], 61] for modeling iterated stochastic integrals of multiplicities 1 to 6 (the case of multidimensional non-commutative noise). In addition, we provide software (as a part of the SDE-MATH software package) for the numerical integration of linear stationary systems of Ito SDEs based on the results of article [62] and MATLAB codes from monographs [58], [61 .

In Section 7 we discuss possible directions for the development of the SDE-MATH software package. In particular, the parallelization of computations, the implementation of methods of the Runge-Kutta type [2], [7], [42], [61] and multistep numerical methods for Ito SDEs [2], [7], [42], [61], the development of a part of the software package for solving filtering problem and stochastic optimal control problem [2], as well as improvement of the graphical user interface.

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2 Theoretical Results Underlying the SDE-MATH Software Package

2.1 Strong Numerical Methods with Convergence Orders 0.5, 1.0,

1.5, 2.0, 2.5, and 3.0 for Ito SDEs Based on the Unified Taylor— Ito Expansion

Let (Q, F, P) be a complete probability space and let {Ft, t £ [0,T]} be a nondecreasing right-continuous family of n-algebras of F. Let wt be a standard m-dimensional Wiener stochastic process with independent components wt(i) (i = 1,... ,m), which is Ft-measurable for any t £ [0,T]. Consider an Ito SDE in the integral form

Xt = Xo +

tt

/Ш

a(xT, т )dr + Bi (xT

i=l

, т)dw(i), Xo = x(0,w), (1)

where xt £ Rn is a solution of the Ito SDE (1), the nonrandom functions a(x,t) : Rn x [0,T] ^ Rn, B(x,t) : Rn x [0,T] ^ Knxm guarantee the existence and uniqueness up to stochastic equivalence of a solution of (1) [63], the second integral on the right-hand side of (1) is interpreted as an Ito stochastic integral, Bi(x,t) is the ith colomn of the matrix function B(x,t), x0 is an n-dimensional and F0-measurable random variable, M |x0| < to (M is an

expectation operator). We assume that x0 and wt — w0 are independent when t > 0.

It is well known that one of the effective approaches to the numerical integration of Ito SDEs is an approach based on the Taylor-Ito and Taylor-Stratonovich expansions [2], [7], [42]. The essential feature of such expansions are the so-called iterated Ito and Stratonovich stochastic integrals, which have the form

T t2

J [£k)b = J Щ (tk).. J ... dwtik >,

tt

*T *t2

J*[£k)]T,t = f (tk) ...jmodwC ...dwt

tt

where every (т) (l = 1,..., k) is a continuous nonrandom function on [t,T],

, ( ik)

'tk ,

(2)

(3)

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w(i) (i = 1,... , m) are independent standard Wiener processes and w<0) = t,

and

denote ItO and Stratonovich stochastic integrals, respectively; A,...,ik = 0,1,..., m.

Note that Ф1 (t ) = 1 (l = 1,... ,k) and i1,... ,ik = 0,1,..., m in the classical Taylor-Ito and Taylor-Stratonovich expansions [2]. At the same time Ф1 (t) = (t - t )qi (l = 1,...,k; qi ,...,qk = 0,1, 2,...) and ii,...,ik = 1,...,m in the

unified Taylor-Ito and Taylor-Stratonovich expansions [24], [25] (also see [26], Chapter 4).

Let C2,1(Rn x [0, T]) be the space of functions R(x, t) : Rn x [0, T] ^ R1 with the following property: these functions are twice continuously differentiable in x and have one continuous derivative in t. Let us consider the following differential operators on the space C2,1 (Rn x [0,T])

n m n 2

L = ^ + Y a<*)(x,t)+ 2 YY B(,j)<x’t)B(*i)(x’t)

j= l,i=1

i=1

dx(l)dx(i) ’

A = Y B t) ФФ), 1 = 1>..., m

d

j=1

d x<j)

(4)

(5)

where a(i) (x, t) is the ith component of the vector function a(x, t) and B<ij) (x, t) is the ijth component of the matrix function B(x,t).

Consider the following sequence of differential operators 1 (^

Tp-1L LGp-1

Gp? = 1 (Gp-L - LGP-1) , p =1,2,..., i = 1,...,m,

p

where L and g0^, i = 1,... ,m are defined by (4), (5).

For the further consideration, we need to introduce the following set of iterated Ito stochastic integrals

t2

I

<ii***ife)

<l1 •••lk )s,t

= I (t - tk )lk... /(t - гl)ll dw/1’... dw(/ ^,

(6)

t t

where 11,..., 1k = 0,1,... and i1,..., ik = 1,..., m.

s

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Assume that R(x,t), a(x,t), and B^x, t), i = 1,... , m are enough smooth functions with respect to the variables x and t. Then for all s,t £ [0,T] such that s > t we can write the following unified Taylor-Ito expansion [24] (also see 26], Chapter 4)

R(xs,s) =

R(xt,t) + E E

(s - t)j

q 1 (k,j,Z1 ,***,1fc )£Dq

j!

E G

(ii)

li

G((ik )Lj R(x*,t)

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T(i1---ik) i

J(1i ...Zfc )s,t +

i 1 ,...,ik =

1

+ (Hr+1 )s,t w. p. 1,

(7)

where

Lj R(x,t) =

L . „ L R(x, t)

j

<

for j > 1

5

R(x, t) for j = 0

Dq (k,j,/l,...,/k) • k + 2 j + ^ ^ q; k, j, l1, . . . , lk 0, 1, . . .

and (Hr+1)st is the remainder term in integral form [26].

Consider the partition {тр}^0 of the interval [0, T] such that

(8)

0 = то <-л < ... < tn = T, An = max |t,-+1 — Tj | . (9)

о<j<N—1 j+ j

Let yTj = yj, j = 0,1,...,N be a time discrete approximation of the process xt, t £ [0,T], which is a solution of the Ito SDE (1).

Definiton 1 [2]. We will say that a time discrete approximation yj, j = 0,1,..., N, corresponding to the maximal step of discretization AN, converges strongly with order 7 > 0 at time moment T to the process xt, t £ [0,T], if there exists a constant C > 0, which does not depend on AN, and a 5 > 0 such that M{|xT — yT|} < C(An)y for each AN £ (0, 5).

From (7) for s = тр+1 and t = тр we obtain the following representation of explicit one-step strong numerical scheme for the Ito SDE (1)

Ур+1 = Ур +

q 1 (k,j,Z1 ,...,Zk )£Dq

(тр+1 трУ

j!

E

G

(ii)

Z1

G(kk)LjУр 0

(ii...ik)

(Z1 ...Zk )Tp+1,Tp

i

,ik 1

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+ 1 (7>+1 - тр)(Г+1)/2 (r+1)/2y

+ 1{r=2d-i,d&n} ((r + i)/2)! L yp’

(10)

where -^(/11"/*fcfc))T +1 T is an approximation of the iterated Ito stochastic integral (6) and 1A is the indicator of the set A. Note that we understand the equality (10) componentwise with respect to the components yP of the column yp. Also for simplicity we put Tp = pA, A = T/N, p = 0,1,..., N.

Under the appropriate conditions [2] the numerical scheme (10) has strong order r/2 (r G N) of convergence.

Below we consider particular cases of the numerical scheme (10) for r = 1, 2,3,4, 5, and 6, i.e. explicit one-step strong numerical schemes with convergence orders 0.5,1.0,1.5, 2.0, 2.5, and 3.0 for the Ito SDE (1) [26], [64], [65]. At that for simplicity we will write a, La, B*, G0*)Bj etc. instead of a(yp,Tp), La(yp,Tp), B*(yp,Tp), G0*)Bj(yp,Tp) etc. correspondingly. Moreover, the operators L and G0*), i = 1,... ,m are defined by (4), (5).

Scheme with strong order 0.5 (Euler scheme)

yp+i=yp+E B*i Aw* + Aa.

*1=1

Scheme with strong order 1.0 (Milstein scheme)

(11)

yp+1 = yp + E B*10(0)bi,Tp + Aa + E G0 0B*2°(00)Li,r,. (12)

*1=1

*1,*2 = 1

Scheme with strong order 1.5

yp+1 = yp + E B*1 °0)lp+1,rp + Aa + E G0 0 B*2 feP+1 ,Tp +

*1=1

*1,*2 = 1

+ G0n)a (AL(i1) _ + 1 - LB*. o

*1=1

o(*1) ^

(0)tP+1,tP (1)tP+1,tP

(*1)

*1 1(1)G+bG

+

+ \ ' G(*1)G(*2) B О(*1*2*3)

+ G0 G0 B*31(000)rp+1,rp 1 2

A2

+ — La.

*1,*2,*3 = 1

(13)

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Scheme with strong order 2.0

Ур+I - + J2 ВчЩ]Тр+ъТр + Aa + G01 Bn 1{щ2;р+1 ,Tp+

*1=1

*1,*2 = 1

[Go

i1 =1

(il)a (AI™ + /IV ) - LB*1 №

(0)TP+i,TP (1)Tp+1,TW 41 (1)Tp+1,Tp

+

m , , A 2

I \ Л ГА(*1^А(*2) n I (*1*2*3) |____1„ I

+ / . G0 G0 B*3 ^(pnnW . ^ ^ La+

*1,*2,*3 = 1

■(000)Tp+1,Tp ~ 2

m

+ £ [g0,1)lb,2 (I

Г (*1*2) _ I (*1*2) A LG(*1)B I (*1*2) +

L(10)Tp+1,Tp J(01)Tp+1,TW 0 *2/(10)Tp+1,Tp +

*1 ,*2 = 1

_i_/^(*1 V',(*2 A / Г (*1*2)

+ G0 G0 a lJ(01)Tp+1,7

+ AI

(*1*2)

(00)Tp+1 ,Tp

+

\ G(*1)G(*2)G(*3) в Г(*1*2*3*4)

+ G0 G0 G0 B*4 J(0000)Tp+1,Tp •

*1 ,*2,*3,*4 = 1

Scheme with strong order 2.5

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yp+1=yp+^2 b*1 /(0)Tp+1,Tp+Aa + S G01 B*21(00)2Tp+1 ,Tp+

*1=1

*1,*2 = 1

+Y1 [G0

*1=1

(*1)

a AI

(*1)

(0)tp+1,t

+ /,(n)

(1)tp+1,

- LB. I

(*1)

+ \ ' G(*1)G(*2) в I (*1*2*3)

+ Z_^ G0 G0 B*31 (000)Tp+1,Tp 1 2

41 (1)tp+1,tp

A2

+—~ La+

+

*1,*2,*3 = 1

m

+ £ [Gf>LB!2 (/

I (*1*2)

[(10)Tp+1,7

-I.

(*1*2)

(01)tp+1,

*1,*2 = 1

-lgJVb*. i(i\\2} _ +

(i1) (i2) (i1i2)

+G041)G042)a (I

(01)tp+1 ,t

+AI

(i1i2)

(00)tp+1 ,Tp

0 *2 (10)tp+1,tp

+

(i1) (i2) (i3) (i1i2i3i4)

+ G0 G0 G0 Bi4I

'0 G0 G0 B*4 J(0000)Tp+1,Tp +

*1,*2,*3,*4 = 1

p

(14)

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G(ii)La ( ^I,

+£

*i=i

+—LLBi, I.

1

Кч)

+ ДI

(ii)

+Д lii ’ V

2 (2)TP+i,TP (1)tp+i,tp 2 (0)rp+i,T)

(ii)

(*iU ( I(ii)

2

i (2)tp+i ,Tp

- LG0n)a( I

(2)tp+i,tp

+ Д1

(ii)

(1)Tp+i,Tp

+

+ £

*i,*2,i3 = l

G!,*i)LGi*2]B„, 11

__ I(iii2i3)

0 Bi3 \ff (l00)Tp+i,Tp J(0l0)Tp+i,r?

(iii2i3)

+

+G<ii)G<i2)LB,, (I

_ I(iii2i3)

i3 V I(0l0)Tp+i,Tp J(00l)rp+i,r3

(iii2i3)

+

+G0ii)G0i2)G0i3)a ^I(iii2i3)

+1

(iii2i3)

'(000)rp+i,Tp (00l)rp+i,r3

_LG(ii)G(i2)B I(iii2i3)

+

0 0 i3 (l00)Tp+i,Tp

(ii^'»(i2^'f(i3^'f(i4) Г? I(iii2i3i4i5)

I \ л /^4(ii^4(i2^'f(i3^'f(i4) TD I = »=3'4'5/ I

+ G0 G0 G0 G0 Bi5 J(00000)Tp+i,Tp +

ii,*2,*3,i4,i5=l

Д3

+ — LLa. 6

(15)

Scheme with strong order 3.0

УР+1 - yP+^2 Bii-I(0)Tp+i,Tp+Да + G0i Bi2I(00)2rP+i,Tp+

ii=l

ii,i2=l

+Y1 [G0

ii=l

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(ii)

а ДТ,

(ii)

(0)tp+i,t

+ I(ii)

(1)tp+i>

- LBr I

(ii)

+ \ ' G(ii)G(i2) B I(iii2i3)

+ 2^ G0 G0 Bi3 J(000)rp+i,rp 1 2

ii (1)tp+i,tp

Д2

+—~ La+

+

ii,i2,i3 = l

m

+ £ [Gf>LB!2 (I

(iii2)

___ I(iii2) \ ____ LG(ii)B I(iii2) +

L(l0)Tp+i,Tp 1 (0l)Tp+i,Tp у 0 i2 J(l0)Tp+i,Tp +

ii,i2=l

(ii)^1(i2ff f I(iii2)

+G0fi)G0i2)a (I

(0l)Tp+i,7

+ Д/,

(iii2)

(00)Tp+i ,гг

+

\ Л /^4(ii)^4(i2)^'f(i3) D I(iii2i3i4)

+ G0 G0 G0 Bi4

ii ,i2 ,i3,i4 = l

(0000)rp+ilrp + qp+l,p + rp+l,p,

(16)

P

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where

qp+i,p

E

*1=1

G^La ( ^I

1

(*i)

r0

+ AI

(*i)

+ A1

(ii)

2 (2)rP+i,T^ (1)rp+i,r^ 2 (0)tp+i,tp

+

+1 LLB* 112;E- iG0-'^ в . + AI

*(*d. (Iff + A/dd

(2)Tp+i,Tp (1)Tp+i,Tp

+

+ E

*i,*2,*3 = 1

G^LGl*2>B*. 11

__ I(*i*2*3)

0 B*3 \ J(100)rp+i,Tp J(010)Tp+i,r?

(i1i2i3)

+

+G0‘i)G',‘2)LB„ (I

(i1i2i3)

r0 G° LB*3 l J(010)Tp+i,Tp I(001)Tp+i,Tp

+

+G0*i)G0*2)G0*3)a (A/^i*^ , _ + I

+ I ( 1 2 3)

(000)Tp+bTp + J(001)rp+i,Tp

and

r

_LG(*i)G(*2)B I(*i* 2*3)

+

0 0 *3 (100)Tp+i,Tp

(*d^d*2)^'f(*3)^'f(*4) Г) I(*i* 2* 3* 4* 5)

I \ л /=f(* ih*( *2ff'1(*3ff'1( *4) n I-i-2-3-4-5/

+ G0 G0 G0 G0 B * 5 1(00000)тр+ьтр +

*i, * 2,* 3, *4, *5 = 1

A3

+ LLa, 6 ’

’p+1,p

E

*i, *2 = 1

g0* i * G0* 2)La( 1I( *i,2)

, +AI(* ^2) +—T('' 1 +

2 (02)Tp+i,Tp + 1 (01)Tp+i,Tp + 2 1 (00)rp+i,rp

A I*i* 2)

1

+-LLG0* "B*,1

(*i) D I(*i*2)

2

0 * 2 (20)Tp+bT?

+

+G0*i)LG0*2)a (I(*^ 2)

( 1 2)

(11)Tp+i,Tp 1 (02)Tp+i,Tp

I(*i* 2) _ j(*i* 2)

d10)rp+i,rp 1 (01)tp+i,tp

_ J(*i* 2) + A ( I(*i*2) _ Ц,"=2/ | | +

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+LG0*i)LB*0 fI(*^ 2)

___ I(*i*2) 1 +

* 2 V1 (11)tp+i,tp 1 (20)tp+i,t^

+G(*i) LLB ( 1 I(*i* 2) +1 I(*i* 2) __ I(*i* 2)

0 * 21 21 (02)tp+i,tp + 21 (20)tp+i,tp 1 (11)rp+i,rp

- LG0*i) g0* 2)a (A I

i(* i* 2) I(10)rp+i,7

+ I

( 1 2)

(11)Tp+i,Tp

+

p

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Differential Equations and Control Processes, N. 1, 2021

+ L

*1,*2,*3,*4 — 1

(*l) (*2) (*3) (*4) T

txq Gq Gq Gq a /-Л.1,

T(*1*2*3*4)

i(QQQQ)tp+i,7

+ I

(*1*2*3*4)

(0001)Tp+i,Tf

+

+ G(*1)G(*2)LG(*3)B I(*1*2*3*4)

Q *4 (Q1QQ)Tp+1,Tp

1T

(*1*2*3*4)

(QQ1Q)Tp+1,Tp

_LG(*1)G(*2)G(*3)B I(*1*2*3*4) +

*4J (1QQQ)Tp+1,T

+ G(*1)LG(*2)G(*3)B f I(*1*2*3*4)

__ T(*1*2*3*4) 1

(1QQQ)Tp+1,Tp 1(Q1QQ)tp+1,tp ' +

+gQ‘1)gQ‘2)g!,!3)lb! i

'0 w0

0

*4

r(*1*2*3*4)

IT

(*1*2*3*4)

■(QQ1Q)Tp+1,Tp (QQQ1)Tp+1,Tp

+

L

(*1) (*2 ) (*3) (*4) (*5) I( * 1 * 2 * 3 * 4 * 5 * 6)

G G G G G 1,

r0 vjq vjq vjq

*6 (QQQQQQ)Tp+1 ,Tp

*1 ,*2,*3,*4,*5,*6 — 1

Under the suitable conditions [2] the numerical schemes (12)-(16) have strong orders 1.0, 1.5, 2.0, 2.5, and 3.0 of convergence correspondingly. Among these conditions we consider only the condition for approximations of iterated Itl stochastic integrals from (12)-(16) [2] (also see

M

I.

(*1...*fc)

IT

(*1 ...*k)

(11 ..Ik )Tp+1,Tp (11 ..Ik )Tp+1,Tp

< CД’’+1,

(17)

p

2

where constant C is independent of Д and r/2 are the strong convergence orders for the numerical schemes (12)-(16), i.e. r/2 = 1.0,1.5, 2.0, 2.5, and 3.0.

Note that the numerical schemes (12)-(16) are unrealizable in practice without procedures for the numerical simulation of iterated Itl stochastic integrals from (10). In Section 2.3 we give a brief overview of the effective method of the mean-square approximation of iterated Ito and Stratonovich stochastic integrals of arbitrary multiplicity k (k £ N).

2.2 Strong Numerical Methods with Convergence Orders 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs Based on the Unified Taylor—Stra-tonovich Expansion

Let us consider the following differential operator on the space C2,1(Rn x [0,T])

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Differential Equations and Control Processes, N. 1, 2021

т _ L _ _ \ л G(i)G(i)

L _ L 2 G° G° ’

(18)

i=1

where operators L and G°i), i _ 1,... ,m are defined by (4), (5). Define the following sequence of differential operators

£(,) _f _ fgW^L - L (3p-1) , p _1,2,..., i _ 1,... ,m,

■Г p P-1'

(19)

where G°i) _f G°i), i _ 1,... ,m. The operators L and G°i), i _ 1,... ,m are defined by (18) and (5) correspondingly.

For the further consideration, we need to introduce the following set of iterated Stratonovich stochastic integrals

l

*t2

*(il"'ifc) _ [ (t - tk )lk... [ (t - f1)/l dwt(il)... dw((ik),

'-(h...ik)s,t _ J (t tk) .. J (t t1) dwti ...dwtk , (20)

t t

where l1,..., lk _ 0,1,... and i1,... ,ik _ 1,...,m.

Assume that R(x,t), a(x,t), and Bi(x,t), i _ 1,... ,m are enough smooth functions with respect to the variables x and t. Then for all s,t £ [0,T] such that s > t we can write the following unified Taylor-Stratonovich expansion [25] (also see [26], Chapter 4)

R(xs,s) _

R(xt,t) + ^ ^

(s - t)j

q 1 (k,j,l1 ?***,lk )£Dq

j!

E G

i1,...,ik=1

(ii> ..G(ik>LjR(x(,t)

l1

+ (Hr+0 st w. p. 1,

(21)

where

Lj R(x,t) =

. ..L R(x, t) for j > 1

R(x, t)

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for j _ 0

the set Dq is defined by the equality (8) and (Hr+1) st is the remainder term i integral form [25] (also see [26], Chapter 4).

in

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Differential Equations and Control Processes, N. 1, 2021

Consider the partition (9) of the interval [0, T]. From (21) for s = rp+\ and t = Tp we obtain the following representation of explicit one-step strong numerical scheme for the Ito SDE (1)

Ур+1 = Ур +

q 1 ,...,1k )GDq

(tP+1 ТрУ

j!

E G

;...;*fc 1

(ii)

li

..G(ik)Ljyp ГГ1;\K) +

lk p (li ...Ik )Tp+1,Tp

r*(*1...*k )

+1

{r=2d-l,deN }

(tP+1 - Tp)(r+l)/2 L(r+l)/2y

((r + 1)/2)! L Ур’

(22)

where In ’1'1"/1, ^ is an approximation of the iterated Stratonovich stochastic

integral (20) and 1A is the indicator of the set A. Note that we understand the equality (22) componentwise with respect to the components yP^ of the column yp. Also for simplicity we put tP = pA, A = T/N, p = 0,1,..., N.

Under the appropriate conditions [2] the numerical scheme (22) has strong order r/2 (r G N) of convergence.

Denote

m

a(x,t) = a(x,t) - ^ E Goj)Bj(x,t),

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 (18))

L

d_

dt

+ E aM(x,t)

i=l

d

dx(i),

(23)

where a(i)(x,t) is the ith component of the vector function a(x,t).

Below we consider particular cases of the numerical scheme (22) for r = 2,3,4, 5, and 6, i.e. explicit one-step strong numerical schemes with convergence orders 1.0,1.5, 2.0, 2.5, and 3.0 for the Ito SDE (1) [26], [66], [67]. At that for simplicity we will write a, La, La, Bi, G0i)Bj etc. instead of a(yp, tP), La(yp, tP), La(yp,Tp), Bi(yp,Tp), G0i)Bj(yp,Tp) etc. correspondingly.

Scheme with strong order 1.0

yp+i yp

= yp + Bi11

i 1=l

n(i1)

'(0)Tp+1,Tp

+ Aa+ E G0*^ Bi21

n(i1i2)

(00)Tp+1,Tp

(24)

i 1 ,i 2 =

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Differential Equations and Control Processes, N. 1, 2021

yP+1

+

yP+1

+

m

+ E

*Ъ*2 =

УР+1

+

Scheme with strong order 1.5

yp + J2 B*1 I(0)rp+1,rp + Aa + G01 B*21^+1 ,Tp +

* 1=1

*1,*2=1

У2 |G0':,)a (a/(*(,1)

1=1

+ I

*(*1)

(0)tp+1,tp (1)Tp+1,Tp

ТЭ T*(i 1*2*з)

- L B' I

*(*1)

11 (1)Tp+1,Tp

+

+ E G;0‘ G0“2>В'з.

1, 2, з=1

A _

(000)Tp+1,Tp + 2 a‘

(25)

Scheme with strong order 2.0

yp + ^2 B“ 1 I(0)1Tp+1,Tp + Aa + ^2 G01 B*2 .(o0)Tp+1 ,Tp +

1 =1

1, 2=1

У2 |g0“‘’a (aI(*(,1)

1=1

+I

*(* 1)

(°)tp+1,tp (1)tp+1,t?

- L B* 11

*(* 1)

(1)tp+1,Tp

+

+ E gCg^B'31

1, 2, 3=1

A2

(* 1^(* 2) p, I *(* 1“ 2* 3) 1 A Ta I

(000)Tp+1,Tp +2 +

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__ I(i 1* 2) 5 T G(i 1) B I*(* 1“2) +

*2 VI(10)tp+1,tp I(01)tp+1,tW 0 *2I(10)tp+1,tp +

g0* 1)lb*. (I*(* 1i2)

_i_/^(* /" I(i 1 “2)

+ G0 G0 L lI(01)Tp+1,7

+ AI

*(*1*2)

(00)Tp+1 ,Tp

+

I \ л г~1(*1) r~1(*2) r~1 (*3) D I *(* 1 *2*3*4)

+ 2-^ G0 G0 G0 B*41|

*1 ,*2,*3,*4 = 1

0 ^0 ^0 ^*4J (0000)tp+1,tp-

(26)

p

Scheme with strong order 2.5

yp + ^2 B*1 I(*0)Tp+1,Tp + Aa + ^ G^ B*2 I(00)Tp+1 ,Tp +

i1 =1

*1,*2 = 1

riG'*11 a(Al;0',1) + I//

| 0 V (0)tp+1,tp (1)tp+1,tp

- L В*. I

*(*1)

41 (1)Tp+1,Tp

+

1=,

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+ £ g0,i)g<*j)b,3I

*1,*2,*3 = 1

A2

(*1)^'f(*2) Д. ;*(*1*2*3) | A Га |

(000)Tp+i,Tp +2 +

m

+ £ [g/Ab,, (/,

K*1*2)

=(*1*2)

'(10)rp+1,Tp (01)Tp+bTp

- T g0,1)b,2 i

;(*1*2)

(10)tp+1,t

|

*1,*2=1

+ G(*1)G(*2)T ( T*(i 1*2) + A f*(* 1*2)

0 0 V (01)tp+1,tp + 1(00)tp+1 ,TP

I \ л G'f(*1)G'f(*2)^'f(*3) TD f*(*1*2*3*4)

+ 2-^ G0 G0 G0 B*41

*1 ,*2 ,*3,*4 = 1

|

0 G0 G0 B*41 (0000)rp+1,rp +

+£

*1=1

G(n)La ( =-1,

1

*(*1)

0

2 (2)tP+1,TP

+ A I

*(*1)

+ A1

*(*1)

+1 LLB4- м0*1)^II*1) + AI(1)

2 1 (2)tP+1 ,TP 0 V (2)tP+1,tP (1)tP+1,tP

(1)tp+1,tp 2 (0)tp+1,tp

*(i1)

|

|£

i1,i2,i3=1

G(*1)LG(*2)B (I*(*1*2*3)

-I,

*(i1i2i3)

■(100)Tp+1,Tp (010)rp+1,r3

|

+Gi*1)Gi*2)LB^ (I

*(i1i2i3)

'0 G0 LB*3 l J(010)rp+1,Tp 1(001)Tp+1,Tp

|

(i1) (i2) (i3) *(i1i2i3)

+G041)G0t2)G0t3)a ( AI

+1

*(i1i2i3)

■(000)rp+1,Tp ~ (001)rp+1,r3

__ T r'i(*1)G4(*2) ]Э I *(* 1 *2*3)

LG0 G0 B*3 J(100)Tp+1,Tp

|

I \ л G4(*2) ^'f(*3) A'f (*4) TD I*(*1*2*3*4*5)

+ 2_^ G0 G0 G0 G0 B*5 J(

i1,i2,i3,i4,i5=1

0 G0 G0 G0 B*51 (00000)rp+1,rp +

A3

+ — LLa. 6

(27)

Scheme with strong order 3.0

yp+1 - yp + ^ B*11(0()1P+1,Tp + Aa + ^ G0 0 B*2 1

(*1) 0

*(*1*2) t(00)Tp+1 ,tj

|

*1=1

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*1,*2=1

+Y^ [G0

*1=1

(*1)

a AI

*(*1)

(0)7p+b7

+ I

*(*1)

(1)Tp+1,Tp

- L B*11

*(*1)

(1)Tp+1,Tp

|

p

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Differential Equations and Control Processes, N. 1, 2021

*1,*2,*3 = 1

A2

(*1)^'f(*2) Д. f*(*1*2*3) | A Га |

(000)Tp+i,Tp +2 +

*1,*2 = 1

(10)tp+1,tp (01)tp+1,tp

+ £ [g0,1)LB!2 (/‘А2’ _ - I*'*1*2’

+g0!1)g0‘2)a f Л'1!2 A + A/;(,1!2)

(i1) 70 f*(* 1 *2)

L G((1) Bn I

(01)Tp+1,7

(00)tP+1 ,Tp

0 *2 (10)Tp+1,Tp

|

(*1)G'f(*2)W'(*3) D f*(*1*2*3*4) + q + r

(0000)tp+1,tp + qP+1,P + rP+1,P,

_i_ \ л GO(*1 А^(*2 A(*3 A f

+ 2_^ G0 G0 G0 B*41|

*1 ,*2,*3,*4 = 1

where

qp+1,p —

*1=1

G0*1)La f 1 /Г(!1)

+ AI*(*1) +— f*-1' | +

2 (2)tp+1,tp + I(1)tp+1,tp + 2 I(0)tp+1,tW +

A f*(*1)

+~L L Bu I.

*(*1)

hda ( f*(*1)

2

1 (2)tP+1 ,Tp

- LG01) a( I

+ AI

*(*1)

(2)tp+1,tp (1)tp+1,tp

|

+ £

*1,*2,*3=1

G0!1)LGi!2>B*„ II,

0 ^*3

f*(*1*2*3) __ I*(*1*2*3) 1 +

■(100)Tp+1,Tp (010)tp+1,t3

+g0,1)g0,2)l b„ (I

__ f*(*1*2*3)

*3 V i(010)tp+1,tp I(001)tp+1,t)

Г*(*1*2*3)

|

(*1)п(*2)г<(*3)~ ( A f*(*1*2*3)

+G041)G0t2)G0t3) a(AI

+1

*(*1*2*3)

■(000)Tp+1,Tp ^ (001)tp+1,t3

__ T W*(*1)G'1(*2) TD f*(*1*2*3)

LG0 G0 B*31(100)Tp+1,Tp

|

and

r

I \ л G'1(*1^'1(*2A'f(*3A'f(*4) 70 f*(*1*2*3*4*5)

+ 2_^ G0 G0 G0 G0 B*5 J(

i1,i2,i3,i4,i5=1

0 G0 G0 G0 B*51 (00000)Tp+1,Tp +

A3--_ + L La,

6 ’

p+1,p

£

i1,i2=1

G^G^Lal

1

(i1i2)

00

+ AI

(i1i2)

+ A1

(i1i2)

2 (02)tp+1,tp (01)tp+1,tp 2 (00)tp+1,tp

I1 T T П(*1) R f*(* 1 *2) 1

+ 2 LLG0 B*21(20)Tp+1,Tp +

P

(28)

|

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Differential Equations and Control Processes, N. 1, 2021

_ ,*(*1*2) + Д J*(*1*2) _ J*(*1*2) +

*(*1*2)

(ll)Tp+i,Tp 1(02)Tp+i,Tp

(10)TP+b

(01)Tp+1,7

+LG0*1)LBin (/*(i1*2)

_ f*(*1*2) 1

*2 I(11)TP+1,TP I(20)TP+1,TP *

I G(*1) T T B 1 f*(*1*2) 1 1 f*(*1*2)

0 *2 2 J(02)rp+1,rp + 21(20)Tp+1,Tp I(11)rp+1,rp

-i;

=(*1*2)

_T g0*° G0*2)a(A/,

*(*1*2)

I(10)rP+b7

+;

(*1*2)

(11)Tp+1,Tp

I

+ z

*1 ,*2,*3,*4 — 1

G^^G^G^G^a f A/*(*1 *2*3*4) + ;*(*1*2*3*4) 1 +

■(0000)rp+1,7

■(0001)rp+1,Tp

+Gi,1)Gi,2)LG,(,!3)B! Z

00

___ f*(*1*2*3*4)

0 B*4 J(0100)rp+1,Tp 1 (0010)тр+1,тг

г *(*1*2*3*4)

_£G(*1)G(*2)G(*3)B ;*(*1*2*3*4) |

(1000)rp+1,7

+G(*1) t g(*2)g(*3) в (* 1*2 *3*4) _;*(*1*2 *3*4) +

+G0 LG0 G0 B*4 J(1000)rp+1,Tp J(0100)Tp+1,Tp 1 +

+g<*1)g<*2)g0*3)Zb,, (I,

r*(*1*2*3*4)

*4 *(0010)rp+1,rf

-I;

*(*1*2*3*4)

(0001)Tp+1,Tp

+

z

G(*1)G(*2)G(*3)G(*4)G(*s) B ;*(* 1 *2*3*4*5*6)

G0 G0 G0 G0 G0 B*61 (000000) Tp+1 ,Tp •

*1 ,*2,*3,*4,*5,*6 — 1

Under the suitable conditions [2] the numerical schemes (24)-(28) have strong orders 1.0, 1.5, 2.0, 2.5, and 3.0 of convergence correspondingly. Among these conditions we consider only the condition for approximations of iterated Stratonovich stochastic integrals from (24)-(28) [2] (also see [42])

M

1

*(*1...*fc)

-I,

*(*1...*fc)

(11 ..Ik )тр+1,тр (11 ..Ik )тр+1,тр

< C Ar+1,

(29)

P

2

where constant C is independent of A and r/2 are the strong convergence orders for the numerical schemes (24)-(28), i.e. r/2 = 1.0,1.5, 2.0, 2.5, and 3.0.

Note that the numerical schemes (24)-(28) are unrealizable in practice without procedures for the numerical simulation of iterated Stratonovich stochastic

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Differential Equations and Control Processes, N. 1, 2021

integrals from (22). The next section is devoted to the effective method of the mean-square approximation of iterated Ito and Stratonovich stochastic integrals of arbitrary multiplicity k (k £ N).

2.3 Method of Expansion and Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series

Let us consider the effective approach to expansion of iterated Ito stochastic

integrals [42] (2006) (also see [26], [44]-[46], [58], [61], [67], [68]). This method

is reffered to as the method of generalized multiple Fourier series.

Suppose that every ф1 (t) (l = 1,..., k) is a continuous nonrandom function on [t,T]. Define the following function on the hypercube [t,T]k

i^i(ti). ..фк (tk) for ti < ... <tk

, ti,... , tk £ [t, T],

0 otherwise

(30)

where k > 2 and K(t1) = ффФ) for t1 £ [t,T].

Suppose that (ф-(ж)}°=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) £ 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,...,pk

pi

pk

k ^...^k) -^2 ...^2 C-k ...ji П(ti)

j 1=0 jk=0

l=1

L2([t,T ]k)

where

k

c-k ..j = K (tl,..., tk) п(ti)dt 1.

[t,T]k l=1

is the Fourier coefficient and

. . dtk

(t \

f IL2([t,T]k) = f 2(t1,. . ,tk)dt1 . . . dtk

\t,T ]k /

0,

(31)

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Differential Equations and Control Processes, N. 1, 2021

Consider the partition [tj}N=0 of the interval [t,T] such that t = t0 < ... < tn = T, AN = max Arj ^ 0 if N ^ ж, Atj = Tj+1 — Tj.

0 <j<N—1

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

(32)

Theorem 1 [42] (2006) (also see [26],[44]-[46], [58], [61], [67], [68]). Suppose

that every Ф1 (t) (l = 1,...,k) is a continuous nonrandom function on [t,T] and [fj(x)}j=0 is a complete orthonormal system of continuous functions in the space L2([t,T]). Then

Pi Pk /к

J[

^ = c/pm» .. • Cjkj

ji=0 jk=0 1=1

(il)

jl

lim. T jС)Aw//...j(T‘k)Aw/k) ,

N^“ (ii. -.Ik)€Gk

(33)

where J[^(k)]T,t is defined by (2),

Gk = Hk\Lk, Hk = [ (li,..., Ik)• li,...,lk = 0,1,...,N — 1},

Lk = [(l1, . . . , lk) • l1, . . . , lk = 0, 1, . . . , N 1, lg = lr (g = r); 9,r = 1, . . . , k},

l.i.m. is a limit in the mean-square sense, i1,... ,ik = 0,1,... ,m,

T

j = / Фj(s)dw(i) (34)

are independent standard Gaussian random variables for various i or j (in the case when i = 0), Cjk...j is the Fourier coefficient (31), Aw(j = wj+1 — w(j (i = 0,1,... ,m), [tj}N=0 is a partition of the interval [t,T], which satisfies the condition (32).

Note that a number of modifications and generalizations of Theorem 1 can be found in [26], [67].

Consider transformed particular cases of (33) for k = 1,..., 6 [26], [58], [61],

[67], [68]

P1

J[^(1)]T,t = U.m. £ Cjij0,

P 1^Ж

(35)

ji =0

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Differential Equations and Control Processes, N. 1, 2021

p i p 2

T,t =

'is, E £«, C(:i,zi22) -1

Р1,Р2^ГО

{*1=*2=0}1{,1=,2} I ,

,1=0 ,2=0

(36)

Р1 Р2 Р3

Ji(3)b=Р,1:^, E £ E «-3,2,3 zjll)ci,2)c<:3)-

,1=0 ,2=0 ,3 =0

1{*1=*2=0}1{,i=,2}C73 — 1{*2 = *3 =0}1{,2 =,3}Cji — 1{*1=*3=0}1{,i=,3}C72 I , (37)

J ^(4) ]T,t = 1:1:Ш:

Р1,...,Р4

Р1

Р4

Li;S;, Е ...£C',4..,1 П с

•31

,1=0 ,4=0

./=1

__1 1 /^(*з£(*4) __ 1 1 Z(*2)Z(*4)__

1{*1=*2=0} 1{,1=,2}Z,3 Z,4 1{*1=*3=0} 1{,1=,3}Z,2 Z,4

__1 1 Z(*2)Z(*3) __ -I 1 Z(*1)Z(*4)__

1{*1=*4=0} 1{,1=,4}Z,2 Z,3 1{*2=*3=0} 1{,2=,3}Z,1 Z,4

1{*2=*4=0} 1{,2=,4}Z,1 Z,3 1{*3 = *4=0}1{,3=,4}Z,1 Z,2 1

|1{*1=*2=0} 1 {,1 =,2 } 1{*3=*4=0} 1{,3=,4} 1 1{*1 = *3=0}1{,1=,з} 1{*2=*4=0} 1{,2=,4}1

1 1{H=*4=0}1{,1=,4}1{*2 = *3=0}1{,2=,3} I ,

(38)

Р1

J [^(5) ]T,t = l:i:S: E

Р1,...,Р5 Z--'

,1 =0

__1 1 Z (i3)Z (i4) Z (i5) _

1{*1 = *2 =0}1{,1=,2}Z,3 Z,4 Z,5

1 1 Z(i2) Z(i3)Z(i5) _

1{*1 = *4 =0}1{,1=,4}Z,2 Z,3 Z,5

__-| -| Z(*1E(*4E(*5)

1{*2 = *3 =0}1{,2=,3}Z,1 Z,4 Z,5

__1 1 Z(i1)Z(i3)Z(i4) _____

1{*2 = *5 =0}1{,2=,5}S,1 Z,3 Z,4

__1 1 f Z(i1)Z(i2) Z(i4)

1{*3=*5=0}1{,3=,5}S,1 Z,2 Z,4

= (i5)

П Z

/=1

Ы

3i

1{*1=*3=0} 1{,1: 1{*1=*5=0}1{,1 1{*2=*4=0} 1{,2: 1{*3=*4=0} 1{,3: 1 {*4 = 75=0} 1 {,4:

Z(*2)Z(*4) Z(*5)____

=,3}Z,2 Z,4 Z,5

Z (*2)Z (*3) Z (*4)_

=,5}Z,2 Z,3 Z,4

Z (*1)Z (*3) Z (*5)_

=,4}Z,1 Z,3 Z,5

Z (*1)Z (*2) Z (*5)_

=,4}Z,1 Z,2 Z,5

Z (*1)Z (*2)Z (*3) I =,5}Z,1 Z,2 Z,3 +

E1{*1=*2=0}1{,1=,2}1{*3=*4=0}1{,3=,4} Z,5 1 1{*1 = *2=0}1{,1=,2}1{*3=*5=0}1{,3=,5}C,4 1

E1{*1=*2=0}1{,1=,2}1{*4=*5=0}1{,4=,5} C,3 1 1{*1 = *3=0}1{,1=,3}1{*2=*4=0}1{,2=,4}C,5 1

= (*5)

E1{*1=*3=0}1{,1=,3}1{*2 =*5=0}1{,2=,5} Z,4 1 1{*1 = *3=0}1{,1=,3}1{*4=*5=0}1{,4=,5}C,2 1

= (i2)

E1{*1=*4=0}1{,1=,4}1{*2 =*3=0}1{,2=,3} Z,5 1 1{*1 = *4=0}1{,1=,4}1{*2=*5=0}1{,2=,5}C,3 1

= (*3)

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Differential Equations and Control Processes, N. 1, 2021

+ 1{*i=*4=0}1{j’i=j4}1{*3=*5=0}1{j3=j5} Cj2 + 1{*i = *5=0}1{j’i=j5}1{*2=*3=0}1{j2=j3}Zj'4 +

+ 1{*i=*5=0}1{j’i=j5}1{*2 =*4=0}1{j2=j4} j + 1{*i = *5=0}1{j’i=j5}1{*3=*4=0}1{j3=j4}Zj'2 +

+ 1{*2=*3=0}1{j2=j3}1{*4=*5=0}1{j4=j5} j + 1 {*2 = *4=0} 1 {j2=j4} 1 {*3=*5=0} ^ j3=j5 } j +

+ 1 { *2 = *5 =0} 1 {j2 = j5 } 1 { *3=*4 =0 } 1 {j3 = j4 } C

(il)

ji

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

(39)

pi ре /6

J[^(6)b = ,1.!:^ £ £ C'e...ji (n cf-

ji=0 je=0 \/=1

__1 I # Z(*2) Z(*3)Z(*4) Z(*5) _____ -I -I # Z(*1)Z(*3) Z(*4) Z(*5)____

1{*i=*e=0} 1{j’i=j’e}Zj2 Zj3 Zj4 Zj5 i{*2=*e=0}1{j2=je}Zj'i j j j

__1 1 Z (*i)Z (*2) Z (*4) Z (*5) _ 1 1 Z (*i)Z (*2) Z (*3)Z (*5)_

1{*3=*е=0} 1{j3=je} Zj'i Zj2 Zj4 Zj5 1{*4=*e=0} 1{j4=je}Zj'i j Zj3 Zj5

__1 1 Z(*i)Z(*2) z(*3)z(*4) ______ I I z(*3)z(*4) z(*5)z(*e)_____

i{*5=*e=0}1{j5=je}Zj'i Zj2 Zj3 Zj4 1{*i=*2=0}1{j'i=j2}Zj3 Zj4 Zj5 Zj'e

__1 1 f Z(*2)Z(*4) Z(*5)Z(*е) ______ 1 1 f Z(*2)Z(*3) Z(*5)Z(*e)_____

1{*i=*3=0}1{j'i=j'3}Zj2 Zj4 Zj5 Zje 1{*i=*4=0}1{j'i=j4}Zj2 Zj3 Zj5 Zj'e

__1 1 z (*2)z (*3)Z (*4) z (*e) __ I n z (*i)z (*4) z (*5)z (*e)_

1{*i=*5=0}1{j'i=j5}Zj2 Zj3 Zj4 Zj6 1{*2=*3=0}1{j2=j3}Zj'i Zj4 Zj5 Zj'e

__1 1 z (*i)z (*3)z (*5)z (*e) ___ I I z (*i)z (*3) z (*4)z (*e)_

i{*2=*4=0}1{j2=j4}Zj'i Zj3 Zj5 Zj6 1{*2=*5=0}i{j2=j5}Zj'i Zj3 Zj4 Zj'e

__1 1 f z(*i)z(*2) z(*5)z(*e) ______ 1 1 f z(*i)z(*2) z(*4)z(*e)_____

i{*3=*4=0}1{j3=j4}Zj'i Zj2 zj5 Zje 1{*3=*5=0}i{j3=j5}Zj'i Zj2 Zj4 Zj'e

__1 , 1 , Z (*i )Z (*2)Z (*3)Z (*б) I

1{*4 = *5 =0}1{j4 =j5 }Zj'i Zj2 zj3 Zj6 +

+ 1{*i=*2=0}1{ji=j2}1{*3=*4=0}1{j3=j4}Cj^ ^ +

+ 1{*i=*2=0}1{j’i=j2}1{*3=*5=0}1{j3=j5}Cj'4 Cjg +

+ 1{*i=*2=0}1{j’i=j2}1{*4=*5=0}1{j4=j5}Cj'3 Cjg +

+ 1{*i=*3=0}1{ji=j3}1{*2=*4=0}1{j2=j4}Cj^ +

+ 1{*i=*3=0}1{j'i=j3}1{*2=*5=0}1{j2=j5}Zj'4 Zj'e +

+ 1{*i=*3=0}1{j'i=j3}1{*4=*5=0}1{j4=j5}Zj'2 Zj'g +

+ 1{*i=*4=0} 1{j'i=j'4} 1{*2=*3=0} 1{j'2=j<3} Zj5^ (je^ +

+ 1{*i=*4=0}1{j’i=j4}1{*2=*5=0}1{j2=j5}Zj'3 Zj'g +

+ 1{*i=*4=0}1{j'i=j4}1{*3=*5=0}1{j3=j5}Zj'2 Zj'g +

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Differential Equations and Control Processes, N. 1, 2021

+1{г1= + !{*! = + !{*! = + 1{*2 = + 1{*2 = + 1{*2 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 = + 1{*6 =

:*5=0}l{j'1=j5} 1{*2 = :*5=0}l{j'1=j5} 1{*2 =

=*5=0} 1{j1=j5} ±{*3=*4=0} ±{j3=.

h

:*3=0}1{j2:

=*4=0}1{j2: - 1 -f jo

:*3=0} 1{j2=j3} 1{*4 = :*4=0}1{j2=j4} h{*3 = :*5=0}1{j2=j5} h{*3 =

=*1=0} 1{j6=jl} h{*3=*4=0} 1{j3=.

h

=*5=0}1{j4: =*5=0}1{j3: =*4=0}1{j3: - 1 -T jo

:*1=0} 1{j6=j1} 1{*3 = :*1=0}1{j6=j'l} h{*2 = :*1=0} 1{j6=j1} 1{*2 =

=*1=0} 1{j6=j1} 1{*4=*5=0} 1{j4=.

h

=*5=0}1{j3:

=*5=0}1{j2:

=*4=0}1{j2: - l I j A

:*1=0} 1{j6=j1} 1{*2 = :*2=0} 1{j6=j2} 1{*3 = :*2=0} 1{j6=j2} 1{*4 =

=*2=0} 1{j6=j2} 1{*3=*4=0} 1{j3=.

h

=*3=0}1{j2: :*5=0}1{j3: =*5=0}1{j4: hj

:*2=0} 1{j6=j2} 1{*1 = :*2=0}1{j6=j2} h{*1 = :*2=0} 1{j6=j2} 1{*1 =

=*3=0} 1{j6=j3} 1{*2=*5=0} 1{j2=.

h

=*5=0}1{j1:

=*4=0}1{j1:

=*3=0}1{j1:

hj

:*3=0} 1{j6=j3} 1{*4 = :*3=0} 1{j6=j3} 1{*2 = :*3=0} 1{j6=j3} 1{*1 = :*3=0} 1{j6=j3} 1{*1 = :*3=0} 1{j6=j3} 1{*1 = :*4=0} 1{j6=j4} 1{*3 =

=*5=0}1{j4:

=*4=0}1{j2:

=*5=0}1{j1:

=*4=0}1{j1:

=*2=0}1{j1:

=*5=0}1{j3

Z (*4)z (*6) 1

=j3} Zj4 zj6 +

Z(*3)z(*6) I =j4}Zj3 Zj6 +

Z (*2)Z (*6) I =j4 } Zj2 Zj6 +

Z(*1)z(*6) i

=j5} Zj 1 j +

Z (*1)z (*6) _i_

=j5 } zj 1 Zj6 +

Z(*1)z(*6) I =j4 } Zj 1 Zj6 +

Z (*2)Z (*5) +

=j4 } Z j2 Zj5 +

Z (*2)Z (*4) I =j5} Zj2 Zj4 +

Z (*3)Z (*4) I

=j5} Zj3 Z j4 '

Z (*3)Z (*5) I

=j4} Zj3 Z j5 '

Z (*2)Z (*3)I

=j5} Zj2 Zj3 '

Z(*4) Z(*5) I

=j3} Zj4 Z j5 ^

Z (*1)Z (*4) I

=j5 } Zj 1 Zj4 ^

Z(*1)z(*3) I

=j5 } Zj 1 Zj3 ^

Z (*1)Z (*5) I

=j4 } Z j 1 Zj5 _l_

Z (*3)Z (*4) I

=j5} Zj3 Z j4 ^

Z (*3)Z (*5) I

=j4} Zj3 Z j5 '

Z(*4) Z(*5) I

=j3} Zj4 Z j5 ^

Z (*1)Z (*4) I

=j5 } Zj 1 Zj4 ^

Z (*1)Z (*2) I =j5 } Z j 1 Zj2 ^

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

Z (*1)Z (*5) I

=j4 } Z j 1 Zj5 ^

Z (*2)Z (*4) I =j5 } Z j2 Zj4 ^

Z (*2)Z (*5) I

=j4 } Z j2 Zj5 ^

Z(*4) Z(*5) I

=j2} Zj4 Z j5 ^

(*1) (*2)

=j5

Z (*1)Z (*2) I

'5}Zj1 Z j2 ^

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Differential Equations and Control Processes, N. 1, 2021

+ 1{*6=*4=0}1{j6=j4} 1{*2=*5=0}1{j2=j5}Cj1 Cj3 +

+ I{*6=*4=0}l{j6=j4} 1{*2=*3=0}1{j2=j3}Cj1 Cj'5 +

+ 1{*6=*4=0}1{j6=j4} 1{*1=*5=0}1{j1=j5}Cj2 Cj'3 +

+ 1{*6=*4=0}1{j6=j4} 1{*1=*3=0}1{j1=j3}Cj2 Cj'5 +

+ 1{*6=*4=0}1{j6=j4} 1{*1=*2=0}1{j1=j2}Cj3 Cj’5 +

+ 1{*6=*5=0}1{j6=j5} 1{*3=*4=0}1{j3=j4}Cj1 Cj'2 +

+ 1{*6=*5=0}1{j6=j5} 1{*2=*4=0}1{j2=j4}Cj1 Cj3 +

+ 1{*6=*5=0}1{j6=j5} 1{*2=*3=0}1{j2=j3}Cj1 Cj4 +

+ 1{*6=*5=0}1{j6=j5} 1{*1=*4=0}1{j1=j4}Cj2 Cj’3 +

+ 1{*6=*5=0}1{j6=j5} 1{*1=*3=0}1{j1=j3}Cj’2 Cj4 +

+ 1{i6=*5=0}1{j6=j5}1{*1 = *2=0}1{j1=j2}Cj3 Cj’4 —

h

{*6 = *1=0}1{j6=j'l} 1{*2=*5=0} ±{j2=j5} Х{*3=*4=0} ±{j3=j4} ' " 1{*6 = *1=0} 1{j6=j1} 1{*2=*4=0} 1{j2=j4} 1{*3=*5=0} 1{j3=j5} " "1{*6 = *1=0}1{j6=j’l}1{*2=*3=0}1{j2=j3}1{*4=*5=0}1{j4=j5}" '1{*6 = *2=0}1{j6=j2}1{*1=*5=0}1{j'l=j5}1{*3=*4=0}1{j3=j4}' '1{*6 = *2=0}1{j6=j2}1{*1=*4=0}1{j'l=j4}1{*3=*5=0}1{j3=j5}' "1{*6 = *2=0}1{j6=j2}1{*1=*3=0}1{j'l=j3}1{*4=*5=0}1{j4=j5}" '1{*6 = *3=0}1{j6=j3}1{*1=*5=0}1{j'l=j5}1{*2=*4=0}1{j2=j4}' '1{*6 = *3=0}1{j6=j3}1{*1=*4=0}1{j'l=j4}1{*2=*5=0}1{j2=j5}' "1{*3 = *6=0}1{j3=j6}1{*1=*2=0}1{j'l=j2}1{*4=*5=0}1{j4=j5}" 'I{*6 = *4=0}1{j6=j4}1{*1=*5=0}1{j'l=j5}1{*2=*3=0}1{j2=j3}' 'I{*6 = *4=0}1{j6=j4}1{*1=*3=0}1{j'l=j3}1{*2=*5=0}1{j2=j5}' "1{*6 = *4=0}1{j6=j4}1{*1=*2=0}1{j'l=j2}1{*3=*5=0}1{j3=j5}" ' 1{*6 = *5=0} 1{j6=j5} 1{*1=*4=0} 1{j'l=j4} 1{*2=*3=0} 1{j2=j3} ' '1{*6 = *5=0}1{j6=j5}1{*1=*2=0}1{j'l=j2}1{*3=*4=0}1{j3=j4}'

h

- 1 -f jo

"1{*6=*5=0}1{j6=j5}1{*1=*3=0}1{j'l=j3}1{*2 = *4=0}1{j2 =j4} I ,

(40)

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Differential Equations and Control Processes, N. 1, 2021

where 1A is the indicator of the set A.

A detailed discussion of advantages of the method based on Theorem 1 over the approximation methods from works [2], [3], [7], [8], [27]-[35], [37]-[39] can be found in [26] (Section 1.1.10) or in [67].

As it turned out, Theorem 1 can be adapted for the iterated Stratonovich stochastic integrals (3) of multiplicities 1 to 5 [26], [67]-[69] (also see bibliography therein). Let as collect some of these results in the following theorem.

Theorem 2 [26], [67]-[69]. Suppose that {ф (x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L2([t,T]). At the same time i2(s) is a continuously differentiable function on [t,T] and iff (s), 'фз(в) are twice continuously differentiable functions on [t,T]. Then

Pl P2

- (D/dk)

C j2jl Z

jl=0 j2=0 Pl P2 P3

Лф(3)Ь = hum. ^ ^ ^ Cjj 1 (ib i2, га = 0, 1,...,m),

Pl,P2,P3^^ 123

ji=0 j2=0 j3=0

(42)

^ Cj3j2ji Cj(il)Cj22)Cj(33) (гь г2, га = 1,...,m), (43)

J *

T,t =

LLm. ££Cj2nj1 j2 (ii,г2 = l,...,m), (41)

1 p 12

Pl ,P 2^^

J *

T,t = l.i.m.

P^-IXI

j1, j2 j3=0 P

J*[i(4)]t, t = Lhm. Cj4j3j2ji(гь ..., г4 = 0, 1,...,m),

j1,...,j4=0

(44)

P

J*0/,(5)l„ _ 1 \ ™ \ л ..piaffAffAffAffiA

T,t =

l.i.m. C

A(*im(*2m(*3m(*4m(*5)

j5j4j3j2j1 j j j j j (гl, . . . , г5 0 L . . . , m),

j1,...,j5=0

(45)

where J *[i(k)]T,t is defined by (3), and ф/ (s) = 1 (l = 1,..., 5) in (42),

(45); another notations are the same as in Theorem 1.

Consider the following Hypothesis on expansion of the iterated Stratonovich stochastic integrals (3) of arbitrary multiplicity k (k £ N).

Hypothesis 1 [26], [67], [68]. Assume that {ф(x)}°=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the

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space L2([t,T]). Moreover, every ф1 (t) (l = 1,..., k) is an enough smooth nonrandom function on [t,T]. Then, for the iterated Stratonovich stochastic integral

(3) of multiplicity k the following expansion

P k

J *№<‘>b = lpi;m. E C..j П A (46)

ji,...jk =0 1=1

that converges in the mean-square sense is valid, where notations are the same as in Theorem 1.

Hypothesis 1 allows to approximate the iterated Stratonovich stochastic integral J*[^(k)]T,t by the sum

J *[Г’]У =

E Cv..ji П4“’’

ji,...jk=0 1=1

(47)

where

lim M JЧф(к’]т,, - J

p^><Xi

*U ,(k)| P T,t

= 0.

Assume that J[^(k)]T t is the approximation of (2), which is the expression in (33) before passing to the limit for the case p1 = ... = pk = p, i.e.

J

p

t, t =

ji,...,jk =0 1=1

Ы.

ji

u.m. j (tii )a4;)... j (Tik)Aw(;

r(ik ’

N—>00

(l;,...,lk )eGk

(48)

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Let us denote

MJ

T,t

J

p

T,t

= EP,

IK

def

A([t,T ]k)

K 2(tb... ,tk) dt 1 ...dtk = Ik.

[t,T ]k

For the futher consideration, we need the following useful estimate [26], [67

EP <k! /k - E C?k„.ji

(49)

j1,...,jk =0

2

k

2

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where i1,... ,ik = 1,... ,m for T — t £ (0, to) and i1,..., ik = 0,1,... ,m for T — t £ (0,1); another notations are the same as in Theorem 1.

The value Ep can be calculated exactly.

Theorem 3 [26], [67], [70]. Suppose that the conditions of Theorem 1 are satisfied. Then

Ep = Ip — E x

ji,...,jh =0

{T t2

J[^(p)b E / j(tk)...f j(ti)dwt!l}...dwtlfc^, (50)

{ji,...,3k) t t

where ii, . . . , ik = 1, . . . , m; expression

E

(j 1 ,...,jk )

means the sum with respect to all possible permutations (jl,... ,jk). At the same time if jr swapped with jq in the permutation (jl,... ,jk), then ir swapped with iq in the permutation (il,..., ik); another notations are the same as in Theorem

1.

Note that

м|т[^(к)]т,^ j (tk) ...j Фп (ti)dw(il) ...dwtlk)| = Cjk ...ji.

Then from Theorem 3 we obtain

p

Ep = Ik — E Cj...j (i1, . . . ,ik are pairwise different), (51)

ji,.--,jk=0

Ep = Ik — Cjkj I Cjk...ji I (ii = ... = ik). (52)

j1,...,jk=0 (j1 ,.--,jk)

Consider some examples of the application of Theorem 3 (il,...,i5 =

1,... ,m):

pp

EP = I2 — E j — E C/2jiCjij (il = г2), (53)

jl,j2=0 j 1 ,j 2 =0

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P P

i2

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

E3 — h 53 Cj3j2ji 53 Cj3jij2Cj3j2ji — "2 — га),

PP 2

У j3j2j1

j3,j2 ,j'l=0 j3,j2,j1=0

El — ^3 Cj3j2jl E Cj2j3j 1 Cj3j2j1 (i1 — г2 — га),

PP 2

j3j2j1

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

El — 53 Cj3j2j1 53 Cj3j2j 1 Cj1 j2 j3 (i1 — га — г2),

P

EP — /4 - £ I E Cj4...j1

j1,--,j4=0 \(j1,j2)

(i1 — "2 — г4; га — г4^

EP — I4 - E Cj4...j1 I E Cj4...j1

j1,...,j4=0 \(j1,j3)

(i1 — "l — ^, г4; г2 — г4),

EP — /4 - E Cv.j I E Cj-4...j1

j1,...,j4=0 \(j2,j3)

(i2 — "l — i1, г4; г1 — "4^

E\ — /4 - E С,^ I E Cj4...j1

j1,...,j4=0 \(j1,j4)

(ii — "4 — ^,"a, "2 — "a),

rn — /4 - E c,4..,. I E I E с,4...,111 ("i — ч—"2—*a)i

j1,...,j4=0 \(j1,j4) \(j2,j3)

P

EP — /4 - E C,4...j1 I E C,4...j1 I ("1 — "2 — "a — "4),

j1,...,j4=0 \(j1,j2,j3)

j1

EP — /5 - E Cj5...j. I E Cj,...

j1,...,j5=0 \(j1,j2)

where "1 — "2 — "a, "4, "5 and "a, "4, "5 are pairwise different,

EP — /5 - E Cj5..j I E Cj5...

j1,...,j5=0 \(j2,j3)

j1

where "2 — "a — "1, "4, "5 and "1, "4, "5 are pairwise different,

(54)

(55)

(56)

(57)

(58)

(59)

(60) (61) (62)

(63)

(64)

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ji

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E — /5 - с*..л I Cj-,...

j1,--,j5=0 (j4,j5)

where 24 = i5 — i\,i2,h and 21,22,23 are pairwise different,

(65)

EP = /5 - Cj-5...j, I I Cj5...ji

j1,--,j5=0 (j2,j4) (j3,j5)

(21 = 22 — 24 = 23 — 25 = ii).

(66)

2.4 Approximations of Iterated Ito Stochastic Integrals from the Numerical Schemes (11)—(16) Using Legendre Polynomials

This section is devoted to approximation of the iterated Ito stochastic integrals (6) of multiplicities 1 to 6 based on Theorem 1. At that we will use multiple Fourier-Legendre series for approximation of the mentioned stochastic integrals.

The numerical schemes (11)-(16) contain the following set (see (6)) of iterated Ito stochastic integrals

/(*1) T(il) T(il) /(*1*2) /(*1*2)

1(0)T,t, 1(1)T,t, T (2)T,t, T(00)T,V T(10)T,t,

/(*1*2*30*5) / (*1*2) /(*1*2) / (*1*2)

T(00000)T,t , T (02)T,t, T (20)T,t, T(11 )T,0

/(*1*2*3*4) /(*1*2*3*4) /(*1*2*3*4)

/(0001)T,t, /(0010)T,t, /(0100)T,t,

/(*1*2) T(01)T,t, T(i1i2i3) , /(000)T,t, T(i1i2i3i4) T(0000)T,t

T(i1i2i3) , /(100)T,t, T(i1i2i3) , /(010)T,t, T(i1 i2 i3) T(001)T,t

/ (*1*2*3*4) / (*1*2*3*4*5*6)

T (1000)^0 T(000000)T,t .

(67)

(68) (69)

Let us consider the complete orthonormal system of Legendre polynomials in the space L2([t,T])

Фj (x)

x - T±l)

T -1 Pj 2 T -1

j — 0,1, 2,...,

(70)

where Pj (x) is the Legendre polynomial

Pj(x)

1 dj 2j j! dxj

4Y>

L

Using Theorem 1 and well known properties of the Legendre polynomials, we obtain the following formulas for numerical modeling of the stochastic integrals (67)-(69) [26], [42]-[46], [58], [61], [67], [68], [70]-[72]

T(*1)

T(0)T,t

(71)

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T(*1) __

1(1)T,t _

(T - t)3/^„(i.)

i

r(*1) \ __r(

Z0 + Z3Zl )'

r(.i) (t —1)5/2

L(2 )T,t

r(*i) _ ^ ~> i A*.) i v “zv*.) i - z(

^' _ ' ' Zo + " Zl + 2^5Z'2

^iM^ An АСА

V3 z (i.) 2

-(ii)

i

-(ii)

T

(ii*2)q

(00)T,t

T - ‘'zW + E 1

*=1

л/4 i2 — 1

2л/б

(*i)z(*2) Z(ii)Z(*2M _ i r. ^ 1

i—1Zi Zi Zi —1 / 1{*i=i2} I ’

(72)

(73)

qi

T

-(iib(i2b(i3)

4*з)

(iii2*з)qi \ л c000 | z(ii)Z(i2)z(*з) I i ^ z(*з)

(000)T,t _ / V Cj3j2ji l Zji Zj2 Zj3 1{ii=i2}±{j'i=j2}Zj3

Jbj2,j3=0

___1 1 z(ii) I i z(i2)

1 {i2 =*3} X{j2=j3}zji 1 {i 1 = *3} X{j’i=j3}Zj2

(74)

(75)

T

(iii2)q2

(10)T,t

q2

EC 10 I Z (ii)Z (i2) 1 1 |

Cj2ji I Zji Z j2 1{ii=i2} 1{ji=j2} ’

T

(iii2)q2

(01)T,t

J1,J2=0

q2

EC 01 I Z (ii)Z (i2) 1 1 |

Cj2ji I Zji Zj2 1{ii=i2} 1{ji=j2} ’

JbJ2=0

(76)

(77)

T

(*1*2*з*4^з

(0000)T,t

q3

EC0000 I Z (*i) Z (i2) Z (*3)z (i 4)_

Cj4j3j2j1 I Zji Zj2 Zj3 Zj4

Jbj2,j3,j4=0

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__1 1 f Z(i3)Z(i4) _______ 1 1 f Z(i2)Z(i4)______

1{ii=i2} 1{ji=j2}Zj3 Zj4 1{*1=*3} 1{ji=j3}Zj2 Zj4

—Is■ ■ i1r ■ ■ iZ(i2)Z(i3) — is■ ■ ■ ■ t,Z(ii)Z(i4) —

1{i1=i4} 1{ji=j4} Zj2 Z j3 1{*2=*3}1{?2=?3}Чп Zj4

— 1s ^ ^ \1r ■ ■ \Z(ii) Z(i3) — 1s ^ ^ i1r ■ ■ iZ (ii)Z (i2) +

1{*2=*4}1{?2=?4}Ул ■ 1 {*3 = *4 } 1 {^3=^4} Z■i ■ +

+ 1{*1=*2}1{31=32}1{*3=*4}1{33=34} + 1{*1 = *з} 1{?1=?’з} 1{i2=i4} 1{^2=^4} +

+ 1{i1 = i4} 1{ji=j4} 1 {*2 = *3} 1{j2=j3} ) ’

(78)

T

(*1*2*з*4*5^4

(00000)T,t

q4

C

00000 / ТТ z(i).

■5j4.73.72.711 X X ■

J1,J2,J3,J4,J5=0 Vl = 1

E

= (*зЕ(*4Е(*5) 1 1 Z(i2E(i4E(i5)

1 1 Z(*3)Z(i4) Z(i5) 1 1 Z(i2 ) Z(i4) Z

1{ii=i2} 1 {■ 1 = ■ 2} z ■з Eh Eh 1 {*1=*з} 1{Ji=j3} Eh E'4 Z.

■5

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+ l{i + l{i + l{i + l{i + l{i + l{i

+ 1 i

1

= *2}1{j1: = *2} 1{j1: = *3}1{j1: = *4}1{j1 =

1 Z (*2)z (*s) _ 1 r. 1 Z (i2 ) Z (i3)z (*4)_

{*1 = *4 }1{j1=j4}Zj2 Zj3 Zj5 i{*1=*5}i{j1=j5}Zj2 Zj3 Zj4

1 f Z (i1)Z (i4) Z (i5) _ 1 1 f Z (i1)Z (i3)Z (i5)___

{*2 = *3 }1{j2=j3}Zj1 Zj4 Zj5 i{*2=*4}i{j2=j4}Zj1 Zj3 Zj5

1 Z (i1)Z (i3)Z (i4) __ 1 1 z (i1)Z (i2) Z (i5 )_

{*2 = *5 }1{j2=j5}Zj1 Zj3 Zj4 1{i3=*4} 1{j3=j4}Zj1 Zj2 Zj5

1 f Z (i1)Z (i2) Z (i4) _ 1 1 f Z (i1)Z (i2)Z (i3) I

{*3=*5}i{j3=j5}Zj1 Zj2 Zj4 i{*4 = *5}i{j4=j5}Zj1 Zj2 Zj3 +

1 1 Z (i5) I 1 , .1. , 1 . . 1 , Z (i4) I

=j2} 1{i3=i4} 1{j3=j4} Zj5 + 1{i1 = i2} 1{j1=j2} 1{i3=i5} 1{j3=j5} Zj4 +

1 1 Z (i3) 1111 Z (i5) I

=j2} 1{i4=i5} 1{j4=j5} Zj3 + 1{i1 = i3} 1{j1=j3} 1{i2=i4} 1{j2=j4} Zj5 +

1 1 Z (i4) I 1. 1 , 1 . . 1 , Z (i2) I

=j3} 1{i2=*5} 1{j2=j5} Zj4 ^ 1{i1 = *3} 1{j1=j3} 1{i4=*5} 1{j4=j5} Zj2 ^

1 1 Z (i5) I 1 , 1 , 1 . -1- Z (i3) I

=j4} 1{i2=i3} 1{j2=j3} Zj5 ^ 1{i1 = i4} 1{j1=j4} 1{i2=i5} 1{j2=j5} Zj3 ^

= *4}1{j1=j4}1{i3=i5}1{j3=j5} Zj2 ^ 1{*1 = *5}1{j1=j5}1{i2=i3}1{j2=j3} Zj4 ^

= *5} 1{j1: = *3} 1 {j2:

1 1 Z (i3) I 1. 1 , 1 . . 1 , Z (i2) I

=j5} 1{i2=i4} 1{j2=j4} Zj3 ^ 1{i1 = i5} 1{j1=j5} 1{i3=i4} 1{j3=j4} Zj2 _l_

1 1 Z (i1) I 1 . , 1 , , 1 . -1- Z (i1) I

=j3} 1{*4=*5} i{j4=j5} Zj1 ^ 1{i2 = i4} 1{j2=j4} 1{i3=i5} 1{j3=j5} Zj1 ^

_I_ 1 { *2 =*5 } 1 {j2 = j5 } 1 { *3=*4 } 1 {j3 = j4 } Z

(i1)

j1

(79)

I

(*1*2)95

(20)T,t

95

V C20 I Z(n)Z(*2) — 1 r. . , 1 r. -ii

/ V Cj2j1 1 Zj1 Z j2 i{*1=*2}i{j1=j2} h

I

(*1*2)96

(11)T,t

j1,j2=0

96

EC 11 I z (*1)Z (*2) 1 1 1

Cj2j1 I Zj1 Z j2 i{*1=*2}i{j1=j2}

I(*1 *2)q7 \ л C02 I Z(*1)Z(

1 (02)T,t _ Cj2jM Zj1 Zi

j1,j2=0

97

02 Z(*1)Z(*2)

j2j1 I Zj1 Zj2

1{*1=*2}1{j1=j2} I ,

j1,j2=0

1

(*1*2*3)98

(001)T,t

98

E

j1,j2,j3 0

C001 I z(*1)Z(*2)z(*3)

Cj3j2j1 I Zj1 Zj2 Zj3

1 1 Z (*3)

1{*1 = *2 }i{j1=j2}Zj3

__1 1 Z(*1) 1 1 Z(*2)

1{*2 =*3} i{j2=j3}Zj1 1{*1 = *3} i{j1=j3}Zj2

(80)

(81)

(82)

(83)

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T(*1*2*3)99

T(010)T,t

99

E

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jbj2,j3=0

Coio I z(il)z(i2)г(*з)

Cj3j2j1 I zj 1 z j2 j

1 1 z(i3)

Х{*1 = *2 }i{j1=j2}Sj3

1 1 z (i1) 1 1 z(i2)

1{i2 =*3} 1{j2=j3} zj1 i{*1 = *3} i{j1=j3}zj2

(84)

t (*1*2*3)910 T(100 )T,t

910

E

j1,j2,j3=0

C100 I z(*i)z(*2)z(*3) _ 1 1 , z(*3)

Cj3j2j1 I zjl zj2 j i{*1=*2}i{j1=j2}zj3

1 1 z(*1) __________ 1 1 z(*2)

1{*2 =*3} 1{j2=j3} zji 1{*1 = *3} i{j1=j3}zj2

(85)

T

(*1*2*3*4)911 (0001)T,t

911

E

j 1,j 2 ,j3 ,j4 =0

C

0001

j4j3j2j1

z (*1)z (*2) z (*3)z (*4)_

zj1 j j j

__1 1 f z(*3)z(*4) 1 1 f z(*2)z(*4)

i{*1=*2}i{j1=j2}Sj3 j i{*1=*3}i{j1=j3}Sj2 j

1 1 z(*2)z(*3) 1 1 z(*1)z(*4)

i{*1=*4}i{j1=j4}Sj2 j i{*2=*3}i{j2=j3}Sj1 j

—1 ■ ■ E(*1)z(*3) — 1 • • x1( ■ . тz(*1)z(*2)+

X{*2=*4}X{j2=j4}Zj1 j X{*3 = *4}X{j3=j4}Zj1 j +

+1 {*1 =*2 } 1{j1=j2} 1 {*3=*4} 1{j3=j4} + 1{*1 = *3} 1{j1=j3} 1 {*2 =*4 } 1{j2=j4} +

+ 1{*1 = *4} 1{j1=j4} 1{*2 = *3} 1{j2=j3} I ,

(86)

T(*1*2*3*4)912 C0010 z(*1)z(*2)z(*3)z(*4)

T(0010)T.t / V Cj4j3 j2j1 I zj1 j j j

j 1 ,j2 ,j3 ,j4=0 V

1 1 z(*3)z(*4) 1 1 z(*2)z(*4)

'1{*1=*2} 1{j1=j2} zj3 j 1 {*1=*3} 1{j1=j3} zj2 j

1 * =* 1 j =j z(*2)z(*3) 1 * =* 1 j =j z(*1)z(*4)

'1{*1=*4} 1{j1=j4} zj2 zj3 '1{*2=*3} 1{j2=j3} zj1 j

___1 1 # z (*1) z (*3) 1 1 # z (*1)z (*2) I

1 {*2=*4} 1 {j2=j4} zj1 j -1{*3 = *4} j +

+1 {*1 =*2 } 1 {j1 =j2 } 1{*3=*4} 1{j3=j4} + 1{*1 = *3} 1{j1=j3} 1 {*2 =*4 } 1{j2=j4} 1

I1

{*1 = *4} 1{j1=j4} 1{*2 = *3} 1{j2=j3}

(87)

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T(*1*2*3*4)qi3

T(0100)T,t

qi3

E

j1,j2,j3,j4=0

C 0100

Cj4j3j2j1

Z (i1)Z (i2) Z (i3)Z (i4)_

E1 Zj2 ■ Zj4

—1 • • iL■ ■ T.Z(*3)Z(*4) — 1 • • Л;■ . iz(*2)Z(*4) —

X{*1=*2}X{j1=j2}Zj3 Z j4 X{*1=*3}X{j1=j3}Zj2 Z j4

__X X Z(i2)z(i3) ____x 1 Z(i1)Z(i4)___

X{*1=*4}X{j1=j4}Zj2 Z .3 X{*2=*3}X{j2=j3}Zj1 Zj4

—1 • • il ■ ■ iZ(n) Z(*3) — 1 • • T.1 ■ ■ тZ (n)Z (*2)+

X{*2=*4}X{j2=j4}Zj1 Z33 X {*3 = *4} X{^3=^4} Z^1 ■ +

+ X{*1=*2} X{.?1=.?2} X{*3=*4} X{.?3=.?4} + X{*1 = *3} X{.?1=.?3} X{*2=*4} X{.?2=.?4} +

+ X

{*1 = *4} X{j1=.?4} X{*2 = *3} X{.?2=.?3}

(88)

T

(*1*2*3*4)?14

(1000)T,t

914

, E

■ 1J2 J3j4=0

C1000 1 z (i1)z (*2) z (*3)z (*4)

C4.4„4„4. I S71 Z72 Z 73 Z74

■4.?3.?2.?1

___X X . Z(*3)z(*4) ________X X . Z(*2)z(*4)________

X{*1=*2}X{j1=j2}Z7з ZJ4 ■

___X X Z(*2)z(*3) __x X Z(*1)Z(*4)____

X{^1=^4}X{j1=j4}Z72 ■ X{*2=*3}X{j2=j3}Z7'l ZJ4

—1 • • il ■ ■ iZ(n) Z(*3) — 1 • • T.1 ■ ■ тZ (n)Z (*2)+

X{*2=*4}X{j2=j4}Z7l ■ X{*3 = *4}X{j3=j4}Z7l ■ +

+ X{*1=*2} X{.?1=.?2} X{*3=*4} X{.?3=.?4} + X{*1 = *3} X{.?1=.?3} X{*2=*4} X{.?2=.?4} +

+ X{*1 = *4}X{.?’1=.?4}X{*2 = *3} Х{92=.?3} I ,

(89)

915 / 6

T(*1*2*3*4*5*6)915 \ л c000000 I I I z(*i)

T (000000)T,t _ / J M 1 ■

■ 1J2 J3 J4 J5J6=0 V 1=1

X X . Z (*2)Z (*3)Z (*4) Z (*5)

X{j1=j6}X{*1=*6}Z72 Z 93 ■ ■

X X Z (*1)Z (*2) Z (*4) z (*5)

X{j3=j6}X{*3=*6}Z7'l ZJ2 ■ ■

X X r . Z(*1)Z(*2) Z(*3)Z(*4)

X{*5=*6} Eh ZJ2 ■ ■

X X . Z(*2)Z(*4) Z(*5)Z(*6)

■

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X X . Z (*2)Z (*3)Z (*4) Z (*6)

X{j1=j5}X{*1=*5}Z72 ■ ■ ■

X x r. z (*1)Z (*3)Z (*5)Z (*6)

X{j2=j4}X{*2=*4}Z7'l ZJ3 ZJ5 ■

X X . Z (*1)Z (*3)Z (*4) Z (*5)

X{j2=j6}X{*2=*6}Z7'l ZJ3 ZJ4 ■

X X . Z (*1)Z (*2) Z (*3)Z (*5)

X{■ 4 = ■ 6} X{*4=*6} Z■ 1 ZJ2 ■ ■

X X . Z (*3)Z (*4) Z (*5)Z (*6)

ХШ=.ЫХ{*1=*2}Е3 ■ ■ Ч?6

X X . Z(*2)Z(*3)Z(*5)Z(*6)

X{■ 1 = ■ 4} X{*1=*4} Z■ ■ 8?6

X X . Z (*1)Z (*4) Z (*5)z (*6)

X{j2=j3}X{*2=*3}Z7'l ZJ4 ZJ5 8?6

X X . Z (*1)Z (*3)Z (*4) Z (*6)

X{j2=j5}X{*2=*5}Z7'l ■ ZJ4 Ч?6

Electronic Journal. http://diffjournal.spbu.ru/ 129

Differential Equations and Control Processes, N. 1, 2021

1

{j3:

+1{ji=j2 } X{*i + 1{jl=j2} 1{*i + 1{ji=j3} 1{ii + 1{ji=j4} 1{ii + 1{ji=j4} 1{ii +1{ji=j5 } 1{ii + 1 {j2 =j3 } X{*2 + 1{j2=j5} 1{i2 + 1{j6=ji} X{*6 + 1{j6=ji} 1{*6 + 1{j6=ji} 1{*6 + 1{j6=j2} 1{*6 + 1{j6=j2} 1{*6 + 1{j6=j2} 1{*6 + 1{j6=j3} 1{*6 + 1{j6=j3} 1{*6 + 1{j6=j3} 1{*6 + 1{j6=j4} 1{*6 + 1{j6=j4} 1{*6 + 1{j6=j4} 1{*6 + 1{j6=j5} 1{*6 + 1{j6=j5} 1{*6

1 Z(*i)z(*2) Z(*5) Z(*6) _ 1 1 Z(*i) Z(*2) Z(*4) Z(*6)_

=j4}X{*3=*4}Zji Zj2 Zj5 Zj6 X{j3=j5}X{*3=*5}Zji Zj2 Zj4 Zj6

__1 1 _ Z(*i)Z(*2) Z(*3)Z(*6) I

X{j4=j5}X{*4=*5}Zji Z j2 Zj3 Zj6 +

1 1 _ Z (*5) Z (*6) I 1 1 . 1 , -1- . Z (*4)Z (*6) I

=*2} 1{j3=j4} 1{*3=*4} Zj5 Zj6 + X{ji=j2} X{*i = *2} X{j3=j5} X{*3 = *5} Zj4 Zj6 +

1 1 Z(*3) Z(*6) I 1 1 1 1 Z(*5) Z(*6) I

=*2}X{j4=j5}X{*4=*5}Zj3 Zj6 ^ X{ji=j3}X{*i = *3}X{j2=j4}X{*2 = *4} Zj5 Zj6 ^

1 1 _ Z (*4) Z (*6) I 1 1 . 1 , -1- . Z (*2)Z (*6) I

=*3} X{j2=j5} X{*2=*5}Zj4 Zj6 _l_ X{ji=j3} X{*i = *3} X{j4=j5} X{*4 = *5} Zj2 Zj6 _l_

1 1 Z (*5) Z (*6) I 1 1 1 1 Z (*3)Z (*6) I

=*4}X{j2=j3}X{*2=*3}Zj5 Zj6 ^ X{ji=j4}X{*i = *4}X{j2=j5}X{*2 = *5} Zj3 Zj6 ^

1 1 _ Z (*2) Z (*6) I 1 1 . 1 , -1- . Z (*4)Z (*6) I

=*4} X{j3=j5} X{*3=*5}Zj2 Zj6 _l_ X{ji=j5} X{*i = *5} X{j2=j3} X{*2 = *3} Zj4 Zj6 ^

1 1 Z(*3) Z(*6) I 1 1 1 1 Z(*2) Z(*6) I

=*5}X{j2=j4}X{*2=*4}Zj3 Zj6 ' X{ji=j5}X{*i = *5}X{j3=j4}X{*3 = *4} Zj2 Zj6 '

1 1 _ z (*i) z (*6) I 1 .1. . 1 , , 1 . . Z (*i)Z (*6) I

=*3} X{j4=j5} X{*4=*5}Zji Zj6 _l_ X{j2=j4} X{*2 = *4} X{j3=j5} X{*3 = *5} Zji Zj6 ^

1 1 _ Z (*i) Z (*6) I 1 1 -1- . Z (*2)Z (*5) I

=*5} X{j3=j4} X{*3=*4}Zji Zj6 ^ X{j6=ji} X{*6 = *i} X{j3=j4} X{*3 = *4} Zj2 Zj5 ^

1 1 _ Z (*2) Z (*4) I - - - . Z (*3)Z (*4) I

=*i} X{j3=j5} X{*3=*5}Zj2 Zj4 ^ X{j6=ji} X{*6 = *i} X{j2=j5} X{*2 = *5} Zj3 Zj4 ^

1 1 _ Z (*3) Z (*5) I -1- - . Z (*2)Z (*3) I

=*i} X{j2=j4} X{*2=*4}Zj3 Zj5 ^ X{j6=ji} X{*6 = *i} X{j4=j5} X{*4 = *5} Zj2 Zj3 ^

1 1 _ Z (*4) Z (*5) I - - - . Z (*i)Z (*4) I

=*i} X{j2=j3} X{*2=*3}Zj4 Zj5 ^ X{j6=j2} X{*6 = *2} X{j3=j5} X{*3 = *5} Zji Zj4 '

1 1 _ Z (*i) Z (*3) I 1 .1. - . Z (*i)Z (*5) I

=*2} X{j4=j5} X{*4=*5}Zji Zj3 ^ X{j6=j2} X{*6 = *2} X{j3=j4} X{*3 = *4} Zji Zj5 ^

1 1 _ Z (*3) Z (*4) I - -1- . Z (*3)Z (*5) I

=*2} X{ji=j5} X{*i=*5}Zj3 Zj4 ^ X{j6=j2} X{*6 = *2} X{ji=j4} X{*i = *4} Zj3 Zj5 ^

1 1 _ Z (*4) Z (*5) I - - . Z (*i)Z (*4) I

=*2} X{ji=j3} X{*i=*3}Zj4 Zj5 ^ X{j6=j3} X{*6 = *3} X{j2=j5} X{*2 = *5} Zji Zj4 ^

1 1 _ Z (*i) Z (*2) I -1- - . Z (*i)Z (*5) I

=*3} X{j4=j5} X{*4=*5}Zji Zj2 ^ X{j6=j3} X{*6 = *3} X{j2=j4} X{*2 = *4} Zji Zj5 ^

1 1 _ Z (*2) Z (*4) I 1 1 - . Z (*2)Z (*5) I

=*3} X{ji=j5} X{*i=*5}Zj2 Zj4 ^ X{j6=j3} X{*6 = *3} X{ji=j4} X{*i = *4} Zj2 Zj5 _l_

1 1 _ Z (*4) Z (*5) I -1- - . Z (*i)Z (*2) I

=*3} X{ji=j2} X{*i=*2}Zj4 Zj5 ^ X{j6=j4} X{*6 = *4} X{j3=j5} X{*3 = *5} Zji Zj2 ^

1 1 _ Z (*i) Z (*3) I 1 , Z (*i)Z (*5) I

=*4} X{j2=j5} X{*2=*5}Zji Zj3 ^ X{j6=j4} X{*6 = *4} X{j2=j3} X{*2 = *3} Zji Zj5 ^

1 1 _ Z (*2) Z (*3) I - - . Z (*2)Z (*5) I

=*4} X{ji=j5} X{*i=*5}Zj2 Zj3 ^ X{j6=j4} X{*6 = *4} X{ji=j3} X{*i = *3} Zj2 Zj5 ^

1 1 _ Z (*3) Z (*5) I - - . Z (*i)Z (*2) I

=*4} X{ji=j2} X{*i=*2}Zj3 Zj5 ^ X{j6=j5} X{*6 = *5} X{j3=j4} X{*3 = *4} Zji Zj2 ^

1 1 _ Z (*i) Z (*3) I 1 . Z (*i)Z (*4) I

=*5} X{j2=j4} X{*2=*4}Zji Zj3 ^ X{j6=j5} X{*6 = *5} X{j2=j3} X{*2 = *3} Zji Zj4 ^

1 1 _ Z (*2) Z (*3) I 1 1 - . Z (*2)Z (*4) I

=*5} X{ji=j4} X{*i=*4}Zj2 Zj3 ^ X{j6=j5} X{*6 = *5} X{ji=j3} X{*i = *3} Zj2 Zj4 _l_

I 1 1 1 1 _ Z (*3 )Z (*4)_

^ X{j6=j5} X{*6=*5} X{ji=j2} X{*i = *2}Zj3 Zj4

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Differential Equations and Control Processes, N. 1, 2021

-1

1

{j6=j’i}1{*6

{j6=ji}1{*6

{36=j2}1{*6

=*1}1{j2 = =*1}1{j2 = =*l}1{j2 = =*2}1{j1 =

{36=j2}1{*6=*2}1{j’l = {36=j2}1{*6=*2}1{j’l =

{36=33}1{*6=Ы ±{j1=j5} ±{*1=*б} ±{j2=j4} ±{*2 = *4}

1

=j5}1{*2 = =34}1{*2 = =33}1{*2 = 35}1{*1 = 34}1{*1 =

=33}1{*1 = 1

=*5}1{j3 = :*4}1{j3 = =*3}1{j4 = =*5}1{j3 =

34}1{*3 = *4 }" j5}1{*3 = *5 }" j5} 1{*4 = *5 } j4}1{*3 = *4 }"

=*4}1{j3=j5}1{*3 = *5 }" =*3}1{j4=j5}1{*4 = *5 }"

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1

{36=j3}1{*6 {33=j6}1{*3 {36=j4}1{*6 {j6=j4} 1{i6

=*3}1{j’l = =*6}1{j1 = =*4}1{j1 = =*4}1{j1 =

{36=j4}1{*6=*4}1{j’l = {j6=j5}1{*6=*5}1{j1 =

{j6=j5}1{*6=*5} ±{j1=j2} ±{*1=*2} ±{j3=j4} Х{*3 = *4}

1

34}1{*1 = =32}1{*1 = 35}1{*1 = 33}1{*1 = =32}1{*1 =

34}1{*1 = 1

:*4}1{j2 = :*2}1{j4 = =*5}1{j2 = :*3} 1{j2 =

j5}1{*2 = *5 }" j5} 1{*4 = *5 } 33 } 1{*2 = *3 } j5}1{*2 = *5 }"

=*2} 1{j3=j5} 1{*3 = *5 } :*4} 1{j2=j3} 1{*2 = *3 }

1

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

(90)

where 1A is the indicator of the set A and

c000 _ V(2j1 + 1)(2j2 + 1)(2j3 + 1) (T _ , )3/2C000

Cj3j2jl 8 (T b) Cj3j2jl ’

C01 _ V(2j1 + 1)(2j2 + !) (T _ .)2C01

Cj2j1 8 (T t) Cj2j1 ’

C10 _ V(2j1 + 1)(2j2 + !) (T <)2C 10

Cj2j1 8 (T t) Cj2j1 ’

(91)

(92)

(93)

0000 _ л/(2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + !) (t _ ,)2C0000

‘34333231 _ 16 (T l) Cj4j3j2jl ’

(94)

C02 \/(2j1 + 1)(2j2 +T) ,T _ ,>3C.0'2

Cj2ji 16 (T О Cj23i ’

(95)

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Differential Equations and Control Processes, N. 1, 2021

c20 л/(2jl + 1)(2j2 + 1) (T _ ,)3C20

Cj2j1 _ IQ (T Cj2j1 ’

(96)

Cll _ \/(23l + 1)(2j2 + 1) (T _ ,)3cll cj2j1 16 (T b) Cj2j1 ’

(97)

C.

00l

j3j2j1

-y/(2jl + 1)(2j2 + 1)(2j3 + 1) (T_

t)5/2C^'

16

00l

j3j2j1 ,

(98)

C 0l0 -y/(2jl + 1)(2j2 + 1)(2j3 + 1) ,T_

<)5/2C

Cj3j2j1 _ 16 (T _ t) C

0l0,

j3j2j1 ,

(99)

C

l00

j3j2j1

V(2jl + 1)(2j2 + 1)(2j3 + !) (T _ t)5/2(C 16

l00,

j3j2j1 ,

(100)

C

00000

j5j4j3j2j1

\/(2jl + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + !)

32

(T _ t)5/2C

00000,

j5j4j3j2j1-

(101)

C oool ____ л/(2jl + 1)(2j2 + 1)(2j3 + 1)(2j4 + !) (T _ t)3c^000l (102)

j4j3j2j1

32

C ooio л/(2jl + 1)(2j2 + 1)(2j3 + 1)(2j4 + (T _ t)3C ooio (103)

j4 j3 j2 j 1

j3j2j1

32

CO™, _ ^ + 1)(2j2 + 1)(2j3 + 1)(2j4 (T _ t)3C'°™ , (104)

j4j3j2jl

32

C iooo ____ л/(2jl + 1)(2j2 + 1)(2j3 + 1)(2j4 + (T _ t)3C iooo (105)

j4 j3 j2 j1

j4j3j2j1

32

C

000000

j6j5j4j3j2j1

\/(2jl + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + 1)(2j6 + (T _ t)3C000000

64 (T l) Cj6j5j4j3j2j1 ’

(106)

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Differential Equations and Control Processes, N. 1, 2021

where

= / pj3(z) / Pj2 M / Pji {x)dxdydz,

-i -i -i

1 y

j = - I(1 + y)pj2 (yW pji (x)dxdy,

-1 -1 l y

Cj = - J Pj2 (y) J (1 + x)Pn (x)dxdy,

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-1 -1

1 u z y

C0000„^ = f Pj4(u) f Pj3(z) f Pj2 (y) [ Pji(x)dxdydzdu,

j4j3j2ji

-1 -1 -1 -1

1 У

jji = У Pj2 (y)(y + 1)2 j Pji (x)dxdУ, -1 -1

1 У

Cjj = J Pj2 (y) J Pji (x)(x + 1 )2dxdy,

-1 -1

1 y

j = / Pj2(y)(y + 1) I Pji(x)(x + 1)dxdУ,

-1

1

-1

z y

= - I Pj3(z )(z + 1) I Pj2 (y) I Pji (x)dxdУdz, (114)

-1 -1 -1 1 z y

= - / j(z) I Pj2 (y)(y + 1) I Pji (x)dxdУdz, (115)

-1 -1 -1 1 z y

jL = - I j(z) I Pj2 (у) / Pji (x)(x + 1)dxdydz, (116)

(107)

(108)

(109)

(110)

(111)

(112)

(113)

111

v u

= I Pj5 И / Pj4 (u) / Pj3 (z) I Pj2 (y) I Pji (x)dxdУdzdudv, (117)

-1 -1 -1 -1 -1

1

y

z

1

y

z

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Differential Equations and Control Processes, N. 1, 2021

c 1000

Cj4j3j2 j 1

c 0100

Cj4j3j2j'l

1 u z y

I pj4(u) J pj(z) J pj2(y) pji(x)(x + 1)dxdydz, (118)

-1 -1 -1 -1

1 u z y

j Pj4 (U) j Pj3 (Z )j Pj2 (У)(У + 1) Pjl (x)dxdУdz, (119)

1 1 1 1

C 0010

Cj4j3 j2j'l

1 u z y

pj4(u) pj3(z )(z + 1) pj2 (y) pji (x)dxdУdz, (120)

11

= - pj4 (U)(U +1W pj3 (z ) / pj2 Ы / pj1 (x)dxdydz, (121)

1

11

1

000000 _

j6j5j4j3j2jl _

u z

1 w v u z y

= pAM pj5(v) j (u) pj3 (z) pj2(y) pji (x)dxdydzdudvdw\

1 11111

(122)

another notations are the same as in Theorem 1.

1

y

u

z

2.5 Optimization of Approximations of Iterated Ito Stochastic Integrals from the Numerical Schemes (12)—(16)

This section is devoted to the optimization of approximations of iterated Ito stochastic integrals from the numerical schemes (12)-(16). More precisely, we discuss how to minimize the numbers q, q1, q2,..., q15 from Section 2.4.

Suppose that £ > 0 is the mean-square accuracy of approximation of the iterated Ito stochastic integrals (6), i.e.

E(*i...*fc)p _f

E(h...lk)T,t _

M

I (i 1 -..i k) J(li...lk )T,t

I(ii...*k )p 2

J(li...lk )T,t

< £,

where 1((/11"/ifck))pt, P £ N is the approximation of the iterated Ito stochastic integral 1((li11"/ifck))Tt. Then from (74) and (49) we obtain the following conditions

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Differential Equations and Control Processes, N. 1, 2021

for choosing the numbers q,q\,q2,..., q15 for approximations of the iterated Ito stochastic integrals (67)-(69) [26], [67]

d*i*2)q _ (T t)2 ( 1

2(00)T,t _ 2

q

1

(Ъ1Ъ2)Ч

~ 2 4i2 — 1

- e,

E (*i*2*3)qi 6 ■

E(000)T,t < 6 6

i=1 qi

CSE2 < e,

j1 ,j2 ,j3 =0

E616^2 ^ (T t)

4 q2

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(0i)T,t - - \ 4

E (iii2)q2 2 (T — t)

E(10)T,t < 2 12

(T — t)4

Cj)2 - e,

j1,j2=0

q2

j,)2 - e,

j1,j2=0

E(ii...i4)q3 24

E(0000)T,t < 24

E (ii ...i5 )q4 120

E(00000)T,t < 120

q3

24

(T — t)5 120

E (iii2)q5 2 (T — t)6

E(20)T,t < 2 30

E (iii2)q6 2 (T — t)6

E(ii)T,t < 2 18

E (iii2)qr 2 (T — t)6

E(02)T,t < 2 6

(*ii2*3)qe

E(001)T,t < 6 10

(*li2*3)q9

E(010)T,t < 6 20

(iii2i3)qi0 ■

E(100)T,t <6 60

E (CSE) d - e,

j1,j2,j3 ,j4 0

q4

E/C00000 2

Cj5j4j3j2j1 <

j1 ,j2 ,j3 ,j4 ,j5=0

q5

Cj2 - e,

j2jl=°

q6

CE2 - e,

j2,j1=0

q7

Cj2 - e,

j2,j1=0

qs

2 - e,

j1,j2,j3 =0

q9

j0j.)2 - e,

j1,j2,j3 =0

q10

E (CE) d - e,

j1,j2,j3 0

(123)

(124)

(125)

(126)

(127)

(128)

(129)

(130)

(131)

(132)

(133)

(134)

e

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Differential Equations and Control Processes, N. 1, 2021

E (*i...*4)qn 24

E(0001)T,t — 24

(T -t)6 36

y0001 \2

j4 j3 j2 j 1

j1 ,j2 ,j3 ,j4=0

— E

E (*i...*4)qi2 24

E(0010)T,t — 24

(T -t)6 60

y0010 \2

j4 j3 j2 j 1

j1,j2 ,j3 ,j4=0

— E

E (*i...*4)qi3 24

E(0100)T,t — 24

(T -t)6 120

y0100 \2

j4 j3 j2 j 1

j1,j2 ,j3 ,j4=0

— E

E (*1...*4)q14 24

E(1000)T,t — 24

(T -t)6 360

y1000 \2

j4 j3 j2 j 1

j1,j2 ,j3 ,j4=0

— E

(135)

(136)

(137)

(138)

E (*1...*e)q15 E(000000)T,t

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720

(T -t)6 720

E

j1 ,j2 J3j4 ,j5 ,j6

(C

j6j5j4j3 j2jb

2

0

£.

(139)

Taking into account (17) and (91)-(122), (123)-(139), we obtain the following conditions for choosing the numbers q,q1,q2,... ,q15 for the numerical schemes (12)-(16) (constant C is independent of T -t (see below)).

Milstein scheme (12)

1 1 2 2

E 4^ — C(T - ‘).

*=1

Strong Taylor—Ito scheme (13) with convergence order 1.5

6

1

6

1 1 2 2

1

— C(T - t)2,

1

64

q1

E (2 j1 + 1)(2j2 + 1)(2j3 + 1) (jE)

j1,j2,j3=0

— C(T-t). (140)

Strong Taylor—Ito scheme (14) with convergence order 2.0

1 1 2 2

i=1

1

4i2

1

— C(T - t)3,

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Differential Equations and Control Processes, N. 1, 2021

6

24

6

24

1 1

6 - 64

qi

E (2л + 1)(2j2 + 1)(2j3 + 1) (Cj )2 ) < C (T -1)2, (141)

jbj2,j3=0

11

q2

21 4 - 64 (2jl + 1)(2j2 + 1) j2 < C(T - t),

j1,j2=0

11

q2

2'^ - 64 (j + 1)(2j2 + 1) j2 < C (T - t),

j1,j2=0

(142)

(143)

11

q3

УУ (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) (C200O(5'J ) <

24 256

j1,...,j4=0

< C(T - t).

(144)

Strong Taylor—Ito scheme (15) with convergence order 2.5

1 1 q 1

Ёттг^-1 < C(T - t)\

М2 4i2 - 1

i = 1

1 1

6 - 64

q i

E (2ji + 1)(2j2 + 1)(2j3 + 1) (jj)2 < C(T - t)3, (145)

j1j2 j3=0 11

q2

2 1 - ((4 E (2ji + 0(2?2 + 1) (j)2 < C(T - t)\ (l46)

4 64

11

j1,j2=0 q2

2 4; - (1т E 0У1 + 1)(2j2 +1) j)2 < C(T -1)2, (147)

12 64

j1,j2=0

11

q3

24 256

УУ (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) ((Cj)40)O(j'J ) <

j1,...,j4=0

< C(T - t)2,

(148)

11

qs

6

S (2ji +1)(2j2 +1)(2j3 +1) (jj <

10 256

j1,j2,j3=0

< C(T - t),

(149)

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Differential Equations and Control Processes, N. 1, 2021

6

24

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1 1

20 256

(1 1

60 256

1 1

120 322 .

99

(2j1 + l)(2j2 + l)(2j3 + 1) (jj <

j1j2j3=0

< C(T - t), (150)

910

(2jl + 1)(2j2 + 1)(2j3 + 1) ( <

j1j2j3=0

< C(T - t), (151)

94

1201 T^r - (2ji + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + 1)x

j1,...,j5=0

x (CO0‘*)2) < C(T -1).

(152)

Strong Taylor—Ito scheme (16) with convergence order 3.0

9

1 1 1

1 £ -r^l < C(T -1)5,

М2 4i2 - 1

i = 1

1 1

6 - 64

91

£ (2ji + 1)(2j2 + 1)(2j3 + 1) (CO30L) < C(T - t)4, (153)

j1 j2 j3=0

214 i £ (2л + 1)(2j2 +1) (j )2) < C(T -1)3.

j1,j2=0

11

92

2'^ - 64 (2Л + 1)(2j2 + 1) 2 < C(T - t)3.

j1,j2=0

(154)

(155)

11

93

УУ (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) ) <

24 256

j1,..),j4=0

< C(T - t)3,

(156)

11

98

6

X] (2jl + 1)(2j2 + 1)(2j3 + 1) (jj <

10 256

j1 j2 j3=0

< C(T -1)2,

(157)

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Differential Equations and Control Processes, N. 1, 2021

6 (20 - 256 £ (2j1 + 1)(2j2 + 1)(2j3 + ^ ) -

V jl,j2,j3=0 /

- C(T - t)2,

/11 910 \

Мб0 - 256 £ (2j1 + 1)(2j2 + 1)(2j3 + 1) (jj )-

V j1,j2,j3=0 /

- C(T - t)2,

(158)

(159)

120

1

120

94

322

(2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + 1)x

j1,...,j5=0

1

2

1

30

1

18

6

1

256

1

256

1

256

.. I i00000

X lCj5...j1

95

■ £ (2j1 +1)(2j2 +1) (CSJ

j1,j2=0

96

■ £ (2j1 +1)(2j2 +1) (Cj2j 1)

j1,j2=0

97 ^

£ (2j1 + 1)(2j2 + 1) (CjSl)

j1,j2=0 2

Г - C(T - t)2,

- C(T - t),

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- C(T - t),

(160)

(161)

(162)

(163)

24

1

36

24

1

60

24

1

120

1 911 \

322 £ (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) (Cj4[)0jJ j -

j1 ,)..,j4=0 /

- C(T - t), (164)

1 912 \

322 £ (2j1 + 1)(2j2 + 1)(2j3 +1)(2j4 + 1) (Cj4')101) j-

j1 ,)..,j4=0 /

- C(T - t), (165)

1 913 \

322 £ (2j1 + 1)(2j2 +1)(2j3 +1)(2j4 + 1) (j.j j-

j1, . . . ,j4 =0 /

- C(T - t), (166)

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Differential Equations and Control Processes, N. 1, 2021

24

1 1

qi4

УУ (2j1 +1)(2j2 +1)(2j3 +1)(2j4 + 1) (С)™1 <

360 322

720

11

3lvj4 =0

qi5

< C(T - t),

(167)

720 642

У^ (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + 1)(2j6 + 1)x

31 ■ ■ ■ J6=°

x (3T)2 < C(T - t).

(168)

Taking into account Theorem 3 and the results of Listings 5 and |6 (see Section 5) we decided to exclude the multiplier factors k! from the left-hand sides of (140), (141)-(144), (145)-(152), (153)-(168). The detailed numerical confirmation of the mentioned possibility can be found in [73]. This means that we will use the following conditions for choosing the numbers q,q1,q2,... ,q15 for the numerical schemes (12)-(16) (constant C is independent of T — t (see below)).

Milstein scheme (12)

11

q

£

1

М2 4i2 — 1

i=1

< C(T — t).

(169)

Strong Taylor—Ito scheme (13) with convergence order 1.5

1 1 — у y_

2 2 4i2 — 1

i=1

< C(T — t)2,

(170)

1 1

6 — 64

qi

£ (2j1 + 1)<2J2 + 1)(2j3 + 1) (C0L)2 < C(T — t). (171)

3b32j3=0

Strong Taylor—Ito scheme (14) with convergence order 2.0

11 — у !_

2 2 4i2 — 1

i=1

< C(T — t)3,

(172)

1 1

6 — 64

qi

£ (2j1 + 1)(2j2 + 1)(2j3 + 1) (C3 )2 < C(T — t)2, (173)

3i,32,33 =0

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Differential Equations and Control Processes, N. 1, 2021

1 1 42

4 - й E (2ji + l)(2j2 + 1) (C,)2 < C(T - t), (174)

,'l,2=0

11 42

E (2ji + 1)(2J2 + 1) (C,)2 < c(T - t) (Г75)

,'l,2=0

11 43

24 - ^ E (2ji + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) (CT,,.)2 < C(T-1). (176)

,'lv,4=0

Strong Taylor—Ito scheme (15) with convergence order 2.5

1 1 _ 4 1

E 4i^l < c(T -1)4,

2 \ 2 4i2 - 1

i = 1

(177)

1 1

6 - 64

qi

E (2ji + 1)(2j2 + 1)(2j3 + 1) (Cj. )2 < C (T - t)3, (178)

1 1

j 1,2,3 ="

1 1

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4 - 64

1 1

12 - 64

43

42

E (2ji + 1)(2j2 + 1) (C,)2 < C(T - t)2,

,'l,2=0

42

E (2ji + 1)(2j2 + 1) (C2,1 )2 < c(T - t)2,

(179)

(180)

j1,2=0

E (2ji + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) (C”™)2 < C(T-1)2, (181)

24 256

1 1

10 256

1 1

20 256

1 1

60 256

1 1

120 322

,'lv,4=0 48

E (2ji + 1)(2j2 + 1)(2j3 + 1) (C,)2 < C(T - t), (182)

,’l,2,3=0

49

E (2ji + 1)(2j2 + 1)(2j3 + 1) (<30,2< C(T - t), (183)

,’l,2,3=0

410

E (2ji + 1)(2j2 + 1)(2j3 + 1) (C,)2 < C(T - t), (184)

,’l,2,3=0

44

(2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + 1) (C,500°0) <

,'lv,5=0

< C(T - t), (185)

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Differential Equations and Control Processes, N. 1, 2021

Strong Taylor—Ito scheme (16) with convergence order 3.0

q

<C(t-t)i

М2 4 i2 - 1

i = 1

(186)

1 1

6 - 64

q i

E (2ji + 1X2.72 + 1)(2j3 + 1) (Cj )2 < C(T -t)4, (187)

11

jbj2,j3 =0

1 1

4 - 64

1 1 12 - 64

q3

q2

E (2ji + 1)(2j2 + 1) (C0ji)2 < C(T - t)3,

j1,j2=0

q2

E (2ji + 1)(2j2 + 1) (Cj)2 < C(T - t)3,

(188)

(189)

j1,j2=0

E (2ji + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) (jj )2 < C(T -1)3, (190)

24 : 256

jlv

1 1

10 256

1 1

20 256

1 1

60 256

1 1

120 322

qs

E (2ji + 1)(2j2 + 1)(2j3 + 1) (Cj)2 < C(T - t)2, (191)

j’l,j2,j3=0

qo

E (2л + 1)(2j2 + 1)(2j3 + 1) (COj < C (T - t)2, (192)

j’l,j2,j3=0

qio

E (2л + 1)(2j2 + 1)(2j3 + 1) (C?j3j°2j,)2 < C(T - t)2, (193)

j’l,j2,j3=0

q4

УУ (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + 1) (j^)

<

jlv,j5=0

1 1

30 256

1 1

18 256

1 1

6~ 256

< C(T - t)2,

q5

E (2Л + 1)(2j2 + 1) (Cjj)2 < C(T - t),

jl,j2=0

qe

E (2л + 1)(2j2 + 1) (Cj)2 < C(T - t),

jl,j2=0

47

(2ji + 1)(2j2 + 1) (C?°2jJ2 < C(T - t),

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jl,j2=0

(194)

(195)

(196)

(197)

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Differential Equations and Control Processes, N. 1, 2021

1 1 q11

36 - 322 + 1)(2j2 + 1)(2j3 + l)(2j4 + 1) (C2j)4(0OjJ - C(T-t), (I98)

j1,...,j4=0 1 1 qi2

60 - 322 (2j1 +1)(2j2 + 1)(2j3 + 1)(2j4 + 1) — C(T - t), (199)

j1,...,j4=0

1 1 q13

120 - 322 (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) (j.j - C(T-t), (200)

j1,...,j4=0 1 1 q14

360 - 322 (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1) - C (T -t), (201)

j1,...,j4=0 1 1 q15

720 - 642 (2j1 + 1)(2j2 + 1)(2j3 + 1)(2j4 + 1)(2j5 + 1)(2j6 + 1)x

ji,...,j6=0

x (jj0)2 - C (T - t). (202)

2.6 Approximations of Iterated Stratonovich Stochastic Integrals from the Numerical Schemes (24)—(28) Using Legendre Polynomials

This section is devoted to approximation of the Stratonovich stochastic integrals (20) of multiplicities 1 to 6 based on Theorem 2 and Hypothesis 1. At that we will use multiple Fourier-Legendre series for approximation of the mentioned stochastic integrals.

The numerical schemes (24)-(28) contain the following set (see (20)) of iterated Stratonovich stochastic integrals

I*(*1) I*(*1) J(0)T,t, J(1)T,t 1 *(*1) , 1 (2)T,t, 1 *(*1*2) ^TO)^ 1*(*1*2) J(10)T,t

I*(*1*2*3*4*5) 1 (00000)T,t , 1 *(*1*2) 1 (02)T,t, 1 *(*1*2) 1 (20)T,t, 1 *(*1*2) J(11)T,t,

I *(*1*20*4) WnPon) WnPon)

J(0001)T,t , J(0010)T,t , J(0100)T,t ,

i*(* 1*2) i*(* 1*2*3) I * (* 1 i 2 *3 *4) (203)

J(01)T,t, J(000)T,t , J(0000)T,t , (203)

I *(*1*2*3) I *(*1*2*3) r*(*1*2*3) (204)

J(100)T,t , J(010)T,t , J(001)T,t , (204)

1*(*1*2*3*4) r*(*1*2*3*4*5*6) (205)

J(1000)T,t , J(000000)T,t . (205)

Using Theorem 2, Hypothesis 1, and well known properties of the Legendre polynomials, we obtain the following formulas for numerical modeling of the stochastic integrals (203)-(205) [26], [42]-[46], [58], [61], [67]

'68], [70]-[72]

I *(*1)

>)T,t

y/T—zt'K

(206)

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Differential Equations and Control Processes, N. 1, 2021

I *(*1*2)9 i (00 )T,t

i*(* i) _

J(1)T,t _

(T - t)3/^„(i.)

1

/4*1) i /4

Zo + 43Zl )’

_ (T -1)5/2

L(2)T,t

izu. = i c0!l) + -yd'1' + ttNz2

V3 j{

('1)

1

('1)

2л/5

T - 1 t z(*i)z('2) + ^ 1 />(*i)z('2) z(*i)z(*2)

z0 z0 + / j ^ :т V z*-1z* z* z*-1

2 r° w Tii2-!

I*(*i*2*3)qi _ V"^ c000 z(*i)z(*2)z(*3)

i(000)T,t _ / V Cj3j2j1 471 zj2 zj3 ’

j1,j2,j3=0

92

i*(*1*2)92 \ л c 10 z(*i)z(*2)

i(10)T,t _ 2-^ Chk 471 zj2 ’

ji,j2=0

92

i*(*1*2)92 \ л c01 z(*i)z(*2)

i(01)T,t _ 2-^ Chk 471 zj2 ’

ji,j2=0

93

i(*i*2*3*4)93 \ л c0000 z(*i)z(*2)z(*3)z(*4)

i(0000)T,t _ / J Cj4j3j2ji 471 zj2 zj3 zj4 ’

ji,j2,j3,j4=0

94

i (*i*2*3*4*5)94 \ л c00000 z (*i) z (*2 ) z (*3)z (*4)z (*5)

i(00000)T,t _ / V Cj5j4j3j2ji 471 472 zj3 zj4 zj5 ’

ji,j2,j3,j4,j5=0

95

i*(*i*2)95 _ C20 z(*i)z(*2)

i(20)T,t / V Cj2ji 471 zj2 ’

ji,j2=0

96

i(*i*2)96 __ \ л c 11 z(*i)z(*2)

i(11)T,t _ / v Cj2ji 471 472 ’

ji,j2=0

97

i(*i*2)97 _ c02 z(*i)z(*2)

i(02)T,t Cj2ji 471 472 ’

ji,j2=0

98

i*(*i*2*3)98 \ л c001 z(*i) z (*2) z(*3)

i(001)T,t _ / J cj3j2ji 47i 472 473 ’

ji,j2,j3=0

99

i(*i*2*3)99 \ л c010 z (*i)z (*2) z (*3)

i(010)T,t _ / J C7'3j2ji 471 472 473 ’

ji,j2,j3=0

(207)

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(208)

, (209) (210) (211) (212)

(213)

(214)

(215)

(216)

(217)

(218) (219)

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Differential Equations and Control Processes, N. 1, 2021

qw

I(n*2*3)qio \ л c 100 z(*!)z

j (100)T,t _ Cj3j2ji Zjl Zj2 Zj3 ’

j1 ,j2 j3=0

qn

j(*1*2*3*4)qn _ V"^ c0001 z(i^Z(i2)z(i3)z(*4)

1 (0001)T,t _ Cj4j3j2jl Zjl zj2 Zj3 Z j4 ’

j1 ,j2 J3j4=0

qi2

j (*1*2*3*4)qi2 c0010 Z (*l) z (*2) z (*3)z (*4)

J(0010)T,t Cj4j3j2j1 zjl z j2 zj3 zj4 ’

j1,j2 ,j3 ,j4=0

qi3

j(*1 *2*3*4)qi3 __________ \ л c0100 z(*1)z(*2)z(*3)z(*4)

J(0100)T,t _ Cj4j3j2j1 zj1 zj2 zj3 z j4 ’

j1 ,j2 J3j4=0

q14

j (*1 * 2 * 3 * 4 ) q 14 _ \ л c 1000 z (*1) z (*2 ) z (*3)z (*4)

J(1000)T,t _ Cj4j3j2j1 zj1 zj2 zj3 z j4 ’

j1 ,j2 J3j4=0

q15

j(*1*2 *3*4*5*6^15 \ л C000000 z(*1)z(*2)z(*3)z(*4)z(*5)z(*6)

J(000000)T,t _ Cj6j5j4j3j2j1 zj1 z j2 zj3 z j4 zj5 j ’

j1 ,j2 J3j4j5 j6=0

(220)

(221)

(222)

(223)

(224)

(225)

where 1A is the indicator of the set A; another notations are the same as in Section 2.4.

The question on choosing the numbers q1 , q2 ,..., q15 in (210)-(225) turned out to be nontrivial [26] (Chapter 5). The expansions (210)-(225) for iterated Stratonovich stochastic integrals are simpler than their analogues (75)-(90) for iterated Ito stochastic integrals. However, the calculation of the mean-square approximation error for iterated Stratonovich stochastic integrals turns out to be much more difficult than for iterated Ito stochastic integrals [26] (Chapter 5). Below we give some reasoning regarding this problem.

Denote

E*(*1...*fc)p _f у j*(*1...*fc) _ j*(*1...*fc)p 2

E(h...ik)T,t _ M 1(h..h)T,t j(h...ik)T,t

where .^Tt, P £ N is the approximation of the iterated Stratonovich stochas-

tic rntegral /(С/А

From (209) for i1 _ i2 we obtain [26], [67

E *(*1*2)q E(00)T,t

(T - t)2 ^ 1

2 4i2 - 1

*=q+1

<

(T -1)2 1

2 4x2 -

q

1

dx _

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Differential Equations and Control Processes, N. 1, 2021

(T -t)2 8

ln

1

2

2 q + 1

< c , q

where constant C\ is independent of q.

(226)

It is easy to notice that for a sufficiently small T — t (recall that T — t ^ 1 since it is a step of integration for numerical schemes for Ito SDEs) there exists a constant C2 such that

Ec E*(*i*2)q E(h...lk)T,t < C2E(00)T,t •

(227)

From (226) and (227) we finally obtain

E *(*i-*fc )q E(ii...ifc )T,t

< c (T—7!, q

(228)

where constant C does not depend on T—t. The same idea can be found in [2] in the framework of the method of approximation of iterated Stratonovich stochastic integrals based on the trigonometric expansion of the Brownian bridge process.

Obviously, we can get more information about the numbers qi, q2,..., qi5 (these numbers are different for different iterated Stratonovich stochastic integrals) using the another approach. Since

J *

T,t

= J

T,t w. p. 1

for pairwise different i1,... ,ik = 1,... , m, where J[^(k)]T,t, J*[^(k)]T,t are defined by (2) and (3) correspondingly, then for pairwise different i1 ,...,i6 =

1,..., m from (51) we obtain [26], [67

E *(*i*2)q E(00)T,t

E *(*i*2*3)qi

E(000)T,t

E *(*02)92 E(01)T,t

E *(*1*2)42 E(10)T,t

(T -1)2

(T -1)3 6

(T -1)4 “ 4

(T -1)4 “ 12

1

2

1

5

qi

Ei c000 )2

VCj3j2j'i ,

j3,j2,j’i=0

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92

- £ (c&)2,

ji,j2=0

92

- £ (Cj)2,

ji,j2=0

(229)

(230)

(231)

(232)

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Differential Equations and Control Processes, N. 1, 2021

E

s(*i...*4)q3 _ (T -1)‘

13

E

(oooo )T,t 24

(*1...*б)14 _ (T — t)5

(00000)T,t

E

120

Etc 0000 \2

V £74030201/ ’

0b02j3,04=0

14

E(c00000 ) 2

£05*4 *3*201/ ’

E

£*1*2)15 (T — t)6

(20)T,t _ 30

£*1*2)16 (T — t)6

(11)T,t _ 18

£*1*2)17 (T — t)6

(02)T,t _ 6

j1,j2,j3,j4,j5=0 15

£ (£)2.

02 ,01=0

16

£ (0)2.

02 ,01=0

E

*(*1*2*3)18 _ (T — t)5

E

(001)T,t io

(*1*2*3)19 _ (T — *)5

- £ (0)2,

02 ,01=0

18 ( )

E(C 001 )2

£030201/ ’

0b02j3=0

19 )

010 )2

(010)T,t

E

E/C010 у

\C030201/ ’

j1,j2,j3=0

(*1*2*3)110 _ (T — t)5 110

(100)T,t

60

E

*(*1 ...*4)111 _ (T — t)

E

(0001)T,t 36

*(*1 ...*4)112 (t -t)6

(0010)T,t 60

*(*1 ...*4)113 _ (T — t)

E/c 100 \2

£030201/ ’

01,02,03 =0

111 ( )

E(c0001 )2

lc0403020'j ’

01,02,03,04=0

112 ( )

E(c0010 )2

£04030201/ ’

01,02,03,04=0

6 £3 2

0100 2

E

(0100)T,t 120

E*(*1 ...*4)114 _ (T — t)6

E(1000)T,t _ 360

*(*1...*6)115 _ (T — t)6

(000000)T,t

720

E/c0100 \ 2

£04030201/ ’

01,02,03,04=0

114 ( )

E(c 1000 )2

c04030201

01,02,03,04=0

115 ( )

£060504030201/ ■

01,02,03,04,05J6=0

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(233)

(234)

(235)

(236)

(237)

(238)

(239)

(240)

(241)

(242)

(243)

(244)

(245)

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Differential Equations and Control Processes, N. 1, 2021

Table 1. High-order strong Taylor-Stratonovich schemes.

Order of convergence Scheme Conditions for choosing the numbers q, q1, ..., q15

1.0 (24) (169)

1.5 (25) (170), (171)

2.0 (26) (172)-(176)

2.5 (27) (177)-(185)

3.0 (28) (186)-(202)

Taking into account (229)-(245) and the results of paper [73], we use in the SDE-MATH software package the conditions from Table 1 for choosing the numbers q, q1, q2,..., q15 for the numerical schemes (24)-(28).

Note that in the SDE-MATH software package, which is presented in the following sections, we use the following upper bounds b on the numbers qi,..., qi5

b =56 for qi, b =15 for q2, qs, b = 6 for q4, q8, q9, qw,

b = 2 for q5, qe, q7, qn, qu, qis, qi4, qi5.

This means that for the implementing of the numerical methods (13)-(16) and (25)-(28) we use in the SDE-MATH software package the following quantities of the exactly calculated Fourier-Legendre coefficients

573 =185,193 for 33,

163 = 4,096 for each of Cij, C01,,

3231 '

164 = 65,536 for C0000

3

yi00 0i0 00i

34333231'

73 = 343 for each of C%3, , C^,,

75 = 16,807 for C33 3,,

32 = 9 for each of , C02,, ,

C 0i00 c 00i0 c'

, , C34333231, C

34 = 81 for each of Ci00)0 -0100 -00i0 -000i

j4j3j2j1 j4j3j2j1

,6 ^on /-/000000

j4j3j2j1

36 = 729 for C

363534333231■

It should be noted that unlike the method based on Theorems 1-3, existing and well-known approaches to the mean-square approximation of iterated stochastic integrals based on the trigonometric basis functions [2], [3], [7], [27],

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28], [34], [37] do not allow choosing theoretically different numbers q for approximations of different iterated stochastic integrals (starting form the multiplicity 2 of stochastic integrals). Moreover, the noted approaches [2], [3], [7], [27], [28]

34

37] exclude the possibility for obtaining of approximate and exact expressions for the mean-square approximation error similar to the formulas 49), (50).

2.7 Numerical Algorithm for Linear Stationary Systems of Ito SDEs Based on Spectral Decomposition

Consider the following linear stationary system of Ito SDEs

dxt = (Axt + Bu(t)) dt + Fdwt, x0 = x(0), t £ [0,T], (246)

where xt £ Rn is a solution of the system (246), u(t) : [0,T] ^ Rk is a nonrandom function, A £ Rnxn, F £ Rnxm, B £ Rnxk, and wt is a standard m-dimensional Wiener process with independent components wt(i), i = 1,... ,m. Also we suppose that n, m, k > 1. The process yt = Hxt £ R1 is interpreted as an output process of the system (246), where H £ R1xn.

It is well-known that the solution of (246) has the form [4]

xt =

= eA(t-to)xt0 + eA(t—s)Bu(s)ds + eA(t—s)Fdws, 0 < to < t < T, (247)

where eC is a matrix exponent

EC def C

= e

j=0

j!

C is a square matrix, and C° =f I is a unity matrix.

Consider the partition {тр}^=0 of [0,T] such that Tp = pA, A > 0. For simplicity, we will suppose that u(s), s £ [0,T] can be approximated by the step function, i.e. u(s) « u(s), s £ [0,T], where u(s) = u(tp) for s £ [tp,tp+1), p = 0,1,..., N — 1 (more accurate approximations of u(s) are discussed in [62

(also see [58], [61])). Substituting t = Tp+1, t0 = Tp, and u(s) instead of u(s)

into (247), we obtain

xo

p+1

=e

АД

xop

+ A-^ eAД — I)B u(pA) + wp+] (A), x0 = x(0), (248)

t

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where xp is the approximation of xT and

д

еА(Д s)Fdws+pД = wp+i(A).

Also we assume that yp = HXp, where yp is the approximation of y7 random column Wp+1(A) admits the following representation [4]

Wp+i(A) = SD (A)Ad (A)wp+i,

The

(249)

where wp G

n is a column of independent standard Gaussian random variables such that M {WpWT) = O for p = q, O is a zero matrix of size n x n, SD(A) is a matrix of orthonormal eigenvectors of the matrix Df (A) and (A) is a diagonal matrix on the main diagonal of which are the eigenvalues of the matrix Df (A), the matrix Df (A) is defined by

д

Df (A) = M {Wp+i(A)Wp+i(A)} = exp(A(A - s))FF Texp(AT(A - s))ds,

where CT is a transposed matrix C. Moreover, Df (A) = Df (t)\t=^ where Df (t) is a solution of the following Cauchy problem [4]

dD

-f (t) = ADf (t) + Df (t)AT + FFT, Df (0) = O.

In the SDE-MATH software package, we implement the numerical modeling of the system (246) by the formulas (248), (249). At that we use Algorithms 2.3-2.6 from [62] (also see [61], Chapter 11) for the implemetation of

V

3 The Structure of the SDE-MATH SoftWare Package

3.1 Development Tools

The software package was implemented with Python programming language. The main reason to use it is a huge community and significant amount of helpful libraries for calculations and mathematics. The development was performed in free to use Atom text editori.

1 All programs in Python programming language from this paper were written by the first author

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3.2 Dependency Libraries

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In the development of the SDE-MATH software package such libraries as SymPy, NumPy, PyQt5, and Matplotlib were involved. All these libraries and tools are free and open source.

• SymPy is a Python library able to perform symbolic algebra calculations.

• NumPy is a library which specialization is efficient mathematical calculations. Most part of this library is written in C programming language that guarantees high calculation performance.

• The database is SQLite3. This is a tiny database for a local usage on one machine.

• Matplotlib library is a piece of software used to present obtained results in a best way.

• PyQt5 is a library used to build graphical user interface for the SDE-MATH software package.

3.3 Architecture

Taking into account, that the SDE-MATH software package is oriented on a numerical modeling its architecture is clear. There are two main statements. The first is that mathematical formulas are strongly integrated with SymPy library. By that we mean that they completely rely on SymPy. And the second is usage of database to make some calculations able for caching. The architecture itself is provided on Figure 1. Here all parts of the software package can be seen.

The main package is responsible for startup, so it decides which part of the software package must be started. The software package has several modes of operation. The objectives now are

• Run program to calculate and store the Fourier-Legendre coefficients in few text files with further loading in database.

• Run program with graphical user interface. This is the main program entry for the SDE-MATH software package.

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Figure 1: The SDE-MATH software package architecture

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On the current state of development the main entry package booting up PyQt5 library with all necessary widgets. More detailed description of this process will be provided later.

Moving further, the modeling package comes up. This package responsible for all work referenced to modeling including initialization of modeling environment, calculations loops and more. Also, it depends on accuracy calculation module deciding which amount of members in each approximation of iterated stochastic integral should be used in modeling of the Ito SDE (1) solution.

Accuracy calculation module accepts the order of strong numerical scheme for the Ito SDE (1) and its integration step and then calculates necessary amount of members in approximations of iterated Ito and Stratonovich stochastic integrals.

Symbolic algebra module is the construction part which combines many supplementary differential operators with strong numerical schemes for the Ito SDE (1). Having these components combined this module performs simplification of resulting formula so the modeling package can do its modeling work.

Tools module provides some functionality related to bootstrap of runtime environment and external instruments such as database and file system.

3.3.1 Integration with SymPy

Class inheritance tree was extended to implement strong numerical schemes for Ito SDEs. While numerical schemes for Ito SDEs were being implemented it was also necessary to implement supplementary subprograms. SymPy is a Python library able to perform symbolic algebra calculations. This is a core part of the project since it is differentiates input functions, builds and simplifies strong numerical schemes for Ito SDEs to model the Ito SDE (1) solution. Without this part the program package cannot be able to provide such flexible input of data.

3.3.2 Purpose of NumPy

NumPy is a library that helps with calculation optimizations in this project. The library specialization is efficient mathematical calculations. The main usage case is to calculate compiled symbolic formulas with it. It has integration with SymPy to replace symbolic functions with high performance numerical functions.

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3.3.3 Purpose of SQLite Database

The database was used to store the precalculated Fourier-Legendre coefficients, so getting them from there made numerical modeling much faster, because calculation process for these Fourier-Legendre coefficients involve high-cost symbolic operations. The database contains only one table, and might be thought redundant, but modeling needs hundreds (or even thousands) of precalculated coefficients. Obviously, calculation of them at runtime is terribly inefficient, but text files also not the best choice. Text files provide a sequential access memory and combining different accuracy values q1,..., q15 it causes sequential search which extends time to give the result. That is where database comes up. The random access allows to get any Fourier-Legendre coefficient or any quantity of them which makes solution as flexible as it possible.

The download of precalculated Fourier-Legendre coefficients is built in supplemental subprograms to provide fluent calculation pipeline. Having the precalculated Fourier-Legendre coefficient not found, subprogram initiates calculation for it with following store in the database.

Figure 2: Fourier-Legendre coefficients calculations explanation

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It is an interesting note that having mentioned earlier optimization done, the calculations performance were increased in several times. Now the most heavy operation is symbolic simplifications before modeling. The actual modeling takes seconds, for thousands of iterations on m components of stochastic process, so it is not such important how long modeling period of time, as the accuracy that needs to be accomplished.

The scheme of calculation process is presented on Figure 2.

3.3.4 Purpose of Matplotlib

Matplotlib library is a piece of software used to present obtained results in a best way. This library has many features, but feature that needed in this project is to print charts with modeling results in a PyQt5 widget. Thus the data visualization is integrated in graphical user interface.

3.4 Implementation Plan

The implementation of SDE-MATH software package was performed sequentially. The components of SDE-MATH software package were implemented in order of their necessity for calculation pipeline completion.

3.4.1 Calculation of the Fourier—Legendre Coefficients

The Fourier-Legendre coefficients for the approximations of iterated Ito and Stratonovich stochastic integrals were implemented and placed in Listings 0361. This was the first step since the Fourier-Legendre coefficients involved almost in every strong numerical scheme for the Ito SDE (1).

Also it is important to note that the SDE-MATH software package contains a Python script intended for generating of Fourier-Legendre coefficients using multiprocessing. This script placed in Listing 62 and already contains tasks that were performed to generate about 300,000 Fourier-Legendre coefficients. Similarly, user can run and calculate additional Fourier-Legendre coefficients if they are needed. To determine which Fourier-Legendre coefficients will be calculated user must specify pairs of starting and ending values of components in lower multi-index and specify upper multi-index of the Fourier-Legendre coefficient. For example (((0,15), (0,15), (0,15)), (0,1,0)). This means that

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program calculates the Fourier-Legendre coefficients Cjjj, where ji,j2,j3 £ {0,1,..., 14}.

3.4.2 Differential Operators L, L, Gq\ i = 1,... , m

Moving further, strong numerical schemes for ItO SDEs rely on the differential operators (4), (5), and (23). They were implemented and placed in Listings 64-67.

3.4.3 Approximations of Iterated Stochastic Integrals

The next step is implementation of approximations of iterated Ito and Stratono-vich stochastic integrals for the numerical schemes (12)-(16), (24)-(28). They are implemented and definition of their classes are placed in Listings 69-1081.

3.4.4 Strong Numerical Schemes for Ito SDEs

The strong numerical schemes (12)-(16), (24)-(28) for Ito SDEs were implemented. They are placed in Listings 1110-1311.

3.4.5 Graphical User Interface

Finally, the graphical user interface was implemented. The source codes referenced to graphical user interface are placed in Listings 7-42.

4 Software Package Graphical User Interface

For the SDE-MATH software package mentioned above the graphical user interface was developed. The graphical user interface is important and massive part of SDE-MATH software package because it allows user to perform modeling experiments without programming skills and understanding of program package architecture and principles of work.

4.1 Information Model of The Graphical User Interface

The development of graphical user interface was started from consideration of

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SDE math: software package

G

Preparing Database..

2

SDE math: software package |----| I I | X

Welcome to SDE-MATH Software Package

Greeting Message

Z\

SDE malh: software package

The modeling is being performed...

Modeling Logs

SDE math: charts window X

Curves ZtomRsS Save ||Move

SDE math: software package 1 —I □ I X

Charts window

SDE's strong numerical schemes

Taylor-Stratonovich Taylor-Ito Other

1 Convergence Order 1.0 1 1 Euler Linear Systems

1 Convergence Order 1.5 1 1 Milstein

1 Convergence Order 2.0 1 1 Convergence Order 1.5

1 Convergence Order 2.5 1 1 Convergence Order 2.0

1 Convergence Order 3.0 1 1 Convergence Order 2.5

1 Convergence Order 3.0

4

SDE math: software package 1 —1 □ 1 X

X

3

Ok

7

6

X

Charts

window

Ok

Figure 3: Information model of graphical user interface

experiments and routines which can be performed with the SDE-MATH software package. The graphical user interface is aimed on provision of user capabilities to perform nonlinear and linear systems of Ito SDEs modeling experi-

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ments. The information model which schematically describes the graphical user interface structure is presented on Figure 3.

4.1.1 Processing Screens

To represent long duration processes the graphical user interface has two dialogs which can be seen on Figure 3, in Windows 1 and 6. The first one represents database preparing process on application very first run. During this process the Fourier-Legendre coefficients are being loaded into the SQLite database. This screen appears also when user calculates new Fourier-Legendre coefficients. The other screen shows logs during modeling experiment.

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4.1.2 Greetings Dialog

After the SDE-MATH software package has completed the database preparation, it shows greeting dialog which represents short information about its purposes. The greeting dialog can be seen on Figure 3 in Window 2.

4.1.3 Main Menu Dialog

In the main menu of the SDE-MATH software package user can choose one of strong numerical schemes for Ito SDEs to perform modeling experiments. The main menu dialog can be seen on Figure 3 in Window 3.

4.1.4 Visualization Tool

It is important to note that the main SDE-MATH software package window has a checkbox in right upper corner which do switching on and off charts window. In any time user can call this window or hide it if it is not needed. The charts window is universal utility for modeling experiments results visualization. This window has few instruments on it. The left side bar contains all curves labels, and control elements for hiding, showing, and deleting curves. On the right side of the window there are plot which draws the curves. The charts window can be seen on Figure 3, it is Window 3.

4.1.5 Data Input Dialogs

Since the software package has options to perform linear Ito SDEs modeling experiments it is necessary to provide user with input fields for numerical data

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both scalar and matrix. On the other side, for nonlinear Ito SDEs it is necessary to provide symbolic input. The choice of control elements is conditioned by the above obstacles. On Figure 3, and especially in Windows 4 and 5, these input controls can be seen. There are ”LineEditWidget” and ”TableWidget” which are sufficient to provide input abilities. The topic of input data validation is also important but to be more accurate referenced to user experience rather than to information model, so it will be described further.

4.2 The User Experience and Implementation Results

The above part represents the structure of software package but not the dynamics and user experience of it. Let us discuss the SDE-MATH software package user experience on few examples provided further on Figures 141361 This examples represent two scenarios of the SDE-MATH software package use.

The database preparation screen is presented on Figure |4. During the database preparation this screen displays informational message and spinning visualizer of process continuation.

The screen that presented on Figure 5 appears every time when software package runs unless user presses ”Ok” button with marked checkbox. In such case this message screen will not be shown again.

On Figure 6 the main menu dialog is presented. In this dialog user can choose any strong numerical scheme for Ito SDEs to perform modeling experiment.

The tooltip example can be seen on Figure 7. Such tooltips displayed with characteristic icon are placed all over software package interface to help user with explanations.

As noted earlier, the dedicated charts window is universal tool for visualization. The specific examples of such visualization are presented on Figure 8,19, 34-36.

The initial state of input dialogs for nonlinear and linear Ito SDEs are displayed on Figures 9 and 20. At that moment user can start to input the data.

The example of wrong scalar data input is presented on Figures 10, 15, 21, and 31. When user input wrong data the error message appears and ”Next” or ’’Perform modeling” button is blocked. The input field is being checked all the user data input process, and as soon as wrong character is entered notification

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pops up.

If scalar data is correct the ”Next” button is automatically unblocked. On Figures 11, 16, 22, and 32 the examples of scenario are displayed.

On Figures 112, 13, and [28 the example of correct matrix data input is presented. In this particular case the input is symbolic. Symbolic algebra input errors are much harder to determine so this is done on further stages, in modeling runtime.

In the other case when matrix input data are numerical, the validation is performed right after user has finished input. The examples of incorrect matrix numerical input can be found on Figure 24.

When user finishes input with a success the ”Next” or ’’Perform modeling” button is automatically unblocked. On Figures 123,I25H27, Щ and 30 that can be clearly seen.

The Figures 17, 18, and 33 displays sequence of log messages emerged during the modeling process.

After modeling has been done the focus moves to the charts window where obtained modeling results can be seen. The results of modeling is displayed on Figures 19, 34H30. On Figures 35 and 36 the expectations and variances of obtained components of solution are displayed.

Figure 4: Fourier-Legendre coefficients database preparation screen

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Figure 5: Greetings screen

Figure 6: Main menu dialog

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Figure 7: Tooltip

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Figure 9: Nonlinear system of Ito SDEs data input

Figure 10: Wrong data input

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Figure 11: Correct data input

Figure 12: Vector function a(x, t) input

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Figure 13: Matrix function B(x, t) input

Figure 14: Initial data input

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Figure 15: Wrong data input

Figure 16: Correct data input

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Figure 17: Modeling logs

Figure 18: Modeling logs

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Figure 20: Linear system of Ito SDEs data input

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Figure 21: Wrong data input

Figure 22: Correct data input

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Figure 23: Matrix A input

SDE-MATH: software package О Back Linear Systems of Ito SDEs

©Setting of matrix A© Wrong values in matrix!

1 2 3 4

1 -1 0 0 0

2 0 -2 0 0

3 0 0 -3 0

4 0 0 0 a

О Back О Next

Figure 24: Wrong matrix A input

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Figure 25: Correct matrix A input

Figure 26: Matrix B input

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Figure 27: Matrix F input

Figure 28: Vector function u(t) input

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Figure 29: Matrix H input

Figure 30: Initial data input

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Figure 31: Wrong data input

Figure 32: Correct data input

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Figure 33: Modeling logs

Figure 34: Modeling results (components of solution)

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Figure 35: Modeling results (expectations)

Figure 36: Modeling results (variances)

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5 The Results Obtained Using the SDE-MATH Software Package

This section represents the results that were obtained with the SDE-MATH software package at the current stage of the development.

5.1 The Calculated Fourier—Legendre Coefficients

When application runs first time it performs loading of Fourier-Legendre coefficients basic pack in the database from the files. Further, in Listings Ш-Й few examples of them can be seen.

Listing 1: The Fourier—Legendre coefficients Cjj j examples

1 C _0 : 0:0 = 4/3

2 C _0 : 0:1 = -2/3

3 C _0 : 0:2 = 2/15

4 C _0 : 0:3 = 0

5

6 C _0 : 6:4 = -4/429

7 C _0 : 6:5 = 2/143

8 C _0 : 6:6 = 2/2145

9 C _1 : 0:0 = 2/3

10

11 C _47 :33 44 = 3874457388633368/31334948307735906710660485

12 C _47 :33 45 = 0

13 C _47 :33 46 = 52892292737827468/2224781329849249376456894435

14 C _47 :34 :0 = 0

15 C _47 :34 :1 = 0

Listing 2: The Fourier—Legendre coefficients Cjjm examples

1 C_0 : 0:0: 0= = 2/3

2 C_0 : 0:0: 1= = -2/5

3 C_0 : 0:0: 2= = 2/15

4 C_0 : 0:1: 0= = -2/15

5 C_0 : 0:1: 1= = 2/15

6

7 C_1 : 1:0: 1= = -2/35

8 C_1 : 1:0: 2= = 0

9 C_1 : 1:1: 0= = 2/105

10 C_1 : 1:1: 1= = 0

11

12 C_20 :20 20 1 = -2401828/165607444685315115

13 C_20 :20 20 2=0

14 C_20 :20 20 3 = -1241929832/77669891557412788935

15 C_20 :20 20 4=0

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Listing 3: The Fourier—Legendre coefficients J2J1 examples

1 C _0 : 0:0 0: 0 = 4/15

2 C _0 : 0:0 0: 1 = -8/45

3 C _0 : 0:0 1: 0 = -4/45

4 C _0 : 0:0 1: 1 = 8/105

5 C _0 : 0:1 0: 0 = 0

6 C _0 : 0:1 0: 1 = 4/315

7

8 C _1 : 1:0 1: 0 = -4/315

9 C _1 : 1:0 1: 1 = 4/315

10 C _1 : 1:1 0: 0 = 2/105

11 C _1 : 1:1 0: 1 = -8/945

12 C _1 : 1:1 1: 0 = 2/945

13 C _1 : 1:1 1: 1 = 0

14

15 C _20 : 2 0 :20: 20: 1 = 0

16 C _20 : 2 0 :20: 20: 2 = - 249877207023610010/10969028984480026752856704371

17 C _20 : 2 0 :20: 20: 3=0

18 C _20 : 2 0 :20: 20: 4 = - 307937246954575016/571102494076952592887484313076115

Listing 4: The Fourier—Legendre coefficients Cjjjjjj examples

1 C .0 : 0:0: 0: 0:0 = 4/45

2 C .0 : 0:0: 0: 0:1 = -4/63

3 C .0 : 0:0: 0: 0:2 = 2/63

4 C .0 : 0:0: 0: 1:0 = -4/105

5

6 C .2 : 1:0: 1: 0:2 = -2/1575

7 C .2 : 1:0: 1: 1:0 = 38/22275

8 C _2 : 1:0: 1: 1:1 = -2/1575

9 C .2 : 1:0: 1: 1:2 = 68/81081

10

11 C _15 :15 15 : 1 5 15 : 15 = 0

12 C _15 :15 15 : 1 5 15 :16 = - 798538765964/243076352242280511713913783475

13 C _15 :15 15 : 1 5 15 : 17 = 0

14 C _15 :15 15 : 1 5 15 :18 = -59075427603328/17302616709609603697454044769175

5.2 Accuracy Settings

From Theorem 3 (see formulas (51)-(66)) it follows that the number p in the formula (48) should be chosen individually for various combinations of indices i\,... ,ik E {1,...,m}. As follows from Listing 5 (see below) and the results of work [73], these numbers p in the overwhelming majority of cases do not exceed the number p from the formula (51). Moreover, all the mentioned numbers p are many times less than the number p selected using the formula (49) (due to the presence of the multiplier factor k! on the left-hand side of (49)).

In this work, we have replaced the mentioned numbers p for all possible

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combinations of indices i1,... ,ik E {1,...,m} with the number p according to the formula (51). This is possible due to the results of Listing 6. This listing shows that the above replacement does not lead to noticeable accuracy loss of the mean square approximation of iterated Ito stochastic integrals (for more details see [73]).

Thus, in this paper we decided to exclude the multiplier factor k! in the conditions for choosing the numbers q\,... , q15 (see (169)-(202)). Recall that these numbers are used to construct the approximations of iterated Ito and Stratonovich stochastic integrals from the numerical schemes (13)-(16), (25)-(28). The test script was written. The results of its work are presented in Listings 5 and 6, where

1. dt is the integration step;

2. q1(1,2) means p from (54), q1(2,3) means p from (55), q1(1,3) means p from (56), q1 means p from (51) for k = 3;

3. C = 1 (see (17) and (29));

4. error 1 means the left-hand side of (171);

5. error 2 means the left-hand side of (54) divided by (T — t)3;

6. error 3 means the left-hand side of (56) divided by (T — t)3;

7. error 4 means the left-hand side of (55) divided by (T — t)3.

The above idea of calculation of the numbers qi,..., qi5 is described in Listing 109.

Listing 5: Accuracy calculation module

1

2 dt = 0.011

3 q1 = 12

4 q1 (1, 2) = 6

5 q1 (1 , 3) = 12

6 q1 (2, 3) = 6

7

8 dt = 0.008

9 q1 = 16

10 q1 (1, 2) = 8

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11 q1 (1 , 3) = 16

12 q1 (2, 3) = 8

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21

22

Differential Equations and Control Processes, N. 1, 2021

dt = 0.0045 q1 = 28

q1 (1, 2) = 14

q1 (1, 3) = 28

q1 (2, 3) = 14

dt = 0.0035

q1 = 36

q1 (1, 2) = 18

q1 (1, 3) = 36

q1 (2, 3) = 18

dt = 0.0027

q1 = 47

q1 (1, 2) = 23

q1 (1, 3) = 47

q1 (2, 3) = 23

dt = 0.0025

q1 = 50

q1 (1, 2) = 25

q1 (1, 3) = 51

q1 (2, 3) = 25

Process finished with exit code 0

Listing 6: Accuracy calculation module

dt = 0.011 error 1 = 0.010153888451696458

q1 = 12

error 2 = 0.005076944225848201

q1 (1, 2) = 12

error 3 = 0.010307776903394072

q1 (1, 3) = 12

error 4 = 0.005076944225848284

q1 (2, 3) = 12

dt = 0.008 error 1 = 0.007681193827577537

q1 = 16

error 2 = 0.003840596913789046

q1 (1, 2) = 16

error 3 = 0.0077866300793989485

q1 (1, 3) = 16

error 4 = 0.003840596913789157

q1 (2, 3) = 16

dt = 0.0045

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23

24

25

26

27

28

29

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

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60 61 62 63

err or 1= 0. .004432832059862973

q1 = 28

err or 2= 0. .0022164160299319446

q1 (1, 2) = 28

err or 3= 0. .004479699207443705

q1 (1, 3) = 28

err or 4= 0. .002216416029932139

q1 (2, 3) = 28

dt = 0.0035

error 1 = 0.0034564405520411956 q1 = 36

error 2 = 0.0017282202760207088 q1 (1, 2) = 36

error 3 = 0.003488223569838411 q1 (1, 3) = 36

error 4 = 0.0017282202760210141 q1 (2, 3) = 36

dt = 0.0027

error 1 = 0. .0026523659377455377

q1 = 47

error 2 = 0. .00132618296887127

q1 (1, 2) = 47

error 3 = 0. .0026731529281250332

q1 (1, 3) = 47

error 4 = 0. .0013261829688714366

q1 to co = 47

dt = 0.0025 error 1 = q1 = 50 error 2 =

q1 (1, 2) :

error 3 = q1 (1 , 3) error 4 = q1 (2 , 3)

0.002494053620431952

0.0012470268102122428

= 50

0.0025128597161119537

= 50

0.0012470268102123538

= 50

Process finished with exit code 0

5.3 Testing Example (Nonlinear System of Ito SDEs)

The input data for testing of the SDE-MATH software package correspond to the autonomous variant of nonlinear system of Ito SDE (1) with multidimensional non-commutative noise. More precisely, we choose n = 2, m = 2,

x01} = 1, x02) = 1.5,

(2)

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—5x(1)

ax , x

(1) x(2)

0.5 • sin x

x(1)

x

(2)

B (x(1),x(2))

x

x(2) 0.5 • cos (x(1)

Figures 37-92 related to the strong high-order Taylor-Ito and Taylor-Stra-tonovich schemes (T21)-(T6). (241)-(I28) for the Ito SDE (T) represent modeling

Test machine specifications are CPU with maximum core frequency 4.2 GHz and T6GB of RAM.

5.4 Visualization and Numerical Results for Nonlinear System of Ito SDEs Obtained via the SDE-MATH Software Package

This subsection is fully devoted to modeling logs and results visualization. They are presented on Figures 37-9T

results

□ X

The modeling has been completed!

18:19:50 - SQLite Database is successfully connected 18:19:50 SQlite Database Version is: 3.33.0 18:19.54 -10.001 seconds] MJstein start 18:19 54 10001 seconds] Using C = 500 18 19 54 -10001 seconds] Using dt - 0.001 18:1954 - (0 001 seconds) Using q = (3.)

18:19 55 - (0.069 seconds) MJstein subs are finished 18:1955 - (0.825 seconds] MJstein calculations ate fntshed 18:1955 - The SQlite connection is closed

□ X

18:18:52 - SQlite Database is successfiily connected 18:18:52 SQlite Database Version is: 3.33.0 18:18.52 - (0.000 seconds] Euler start 18:18 52 - (0 004 seconds] Euler subs are finished 18:18 52 - (0 168 seconds] Euler calculations are finished 18:18 52 - The SQlite connection is closed

Euler scheme (dt = 0.001)

Milstein scheme (C = 50, dt = 0.001)

Figure 37: Modeling logs

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Differential Equations and Control Processes, N. 1, 2021

SDE-MATH software package - □ X

The modeling has been completed!

18:20:10 SQlite Database is successfully connected

18:2ft 10 - SQlite Database Version is: 3.33.0 18:20-14 10.000 seconds] Strong Taylor-Ito 1.5 start 18:2015 10723 seconds] Using C = 500 18:20 15 - [0.723 seconds] Using dt - 0.001 18:2015 -10728 seconds] Using q = (2500, 3)

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

18:2144 (90.186 seconds) Strong Taylor-Ito 1.5 subs are finished 18:25.20 - [306.154 seconds) Strong Taytor-lto 1.5 calculations are finished 18:25:20 - The SQlite connection is closed

Figure 38: Strong Taylor-Ito scheme of order 1.5 (C = 50, dt = 0.001)

Figure 39: Strong Taylor-Ito schemes of orders 0.5, 1.0, and 1.5 (x(1) component, C = 50, dt = 0.001)

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Figure 40: Strong Taylor-Ito schemes of orders 0.5, 1.0, and 1.5 (x(2) component, C = 50, dt = 0.001)

Euler scheme (dt = 0.005) Milstein scheme (C = 500, dt = 0.005)

Figure 41: Modeling logs

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SDE-MATH software package — □ X

The modeling has been completed!

18:37:10 - SQlite Database is successfiily connected 18:37:10 SQlite Database Version is: 3.33.0 18:37.14 10.000 seconds] strong Taylor-lto 1.5 Start 18:37 14 10001 seconds] Using C = 5000 18:37 14 -10001 seconds] Using dt - 0.005 18:37 14 - (0006 seconds) Using q = (10.0>

18:37:14 - (0.262 seconds) Strong Taylor-lto 1.S subs are finished 18:37:15 - (0.830 seconds) Strong Taylor-lto 1.5 calculations ore finished 18:37:15 - The SQlite connection is closed

✓Ok

SDE-MATH: software package - □ X

The modeling has been completed!

19:34:09 SQlite Database is successfully connected

19:34:09 SQlite Database Version is: 3.33.0 19:34.15 - (0.000 seconds) Strong Taylor-lto 2.0 start 19:3415 (0.475 seconds] Using C = 500.0 19 34 15 - (0 475 seconds] Using dt - 0.005 19:34 15 - (0 481 seconds) Using q = (2000. 10. 0. 0)

19:36:53 - (158.104 seconds] Strong Taylor-lto 2.0 subs are finished 19:38:13 - (238335 seconds] Strong Taylor-lto 2.0 calculations are finshed 19:38:13 - The SQlite connection is closed

✓Ok

Strong Taylor-Ito scheme of order 1.5 (C = 500, dt = 0.005)

Strong Taylor-Ito scheme of order 2.0 (C = 500, dt = 0.005)

Figure 42: Modeling logs

Figure 43: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, and 2.0 (x(1) component, C = 500, dt = 0.005)

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Figure 44: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, and 2.0 (x(2) component, C = 500, dt = 0.005)

Figure 45: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, and 2.5 (x(1) component, C = 7500, dt = 0.01)

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Euler scheme (dt = 0.01) Milstein scheme (C = 7500, dt = 0.01)

SDE-MATH software package — □ X

The modeling has been completed!

18:51 10 - SQlite Database is successfiily connected 18:51:10 SQlite Database Version is: 3.33.0 18.51.14 10.000 seconds] Strong Taylor-lto 1.5 start 18:51 14 10000 seconds] Using C = 7500.0 18 51 14 -10 001 seconds] Using dt - 0.01 18:51 14 - (0005 seconds) Using q = (0. 0)

18:51:14 - (0.141 seconds) Strong Taylor-lto 1.S subs are finished 18:51:15 - (0.338 seconds) Strong Taylor-lto 1.5 calculations ore finished 18:51:15 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

18:51:31 - SQlite Database is successfiily connected 18:51:31 SQlite Database Version is: 3.33.0 18.51 35 - (0.000 seconds) Strong Taylor-lto 2.0 start 18:51 35 (0000 seconds] Using C = 7500.0 18 51 35 - (0000 seconds) Using dt = 0.01 18:51 35 - (0001 seconds) Using q = (17. 0. 0.0)

18:51:36 - (0.910 seconds) Strong Taylor-lto 2.0 subs are finished 18:51:37 - (2.051 seconds) Strong Taylor-lto 2.0 calculations ore finished 18:51:37 - The SQlite connection is closed

✓Ok

Strong Taylor-Ito scheme of order 1.5 (C = Strong Taylor-Ito scheme of order 2.0 (C = 7500, dt = 0.01) 7500, dt = 0.01)

SDE-MATH software package - □ X

The modeling has been completed!

18:52:35 SQlite Database is successfully connected

18:52:35 - SQlite Database Version is: 3.33.0 18:52.39 - (0.000 seconds] Strong Taylor-lto 2.5 start 18:5240 (0.384 seconds] Using C = 7500.0 18 5240 - (0 384 seconds) Using dt = 0.01 18:5240 - (0 384 seconds) Using q = (1667. 17. 0. 0. 0. 0, 0. 0)

18:56:32 - (232.859 secondsj Strong Taylor-lto 2.5 subs are finished 18:57:45 (305.366 seconds! Strong Taylor-lto 2.5 calculations are finshed 18:57 45 The SQlite connection is closed

Strong Taylor-Itoo scheme of order 2.5 (C = 7500, dt = 0.01)

Figure 46: Modeling logs

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SDE-MATH charts window

О Trajectories Expectations variances #|

Taytor-Ito 1.5, xl

v Curves

«- ■> f В

V Mtlstetn,x2

V Teytar-Ito 1.5, x2 *

тауиг 1102 o. xi

V Tavtor-ito 2 o. x2 *

Taytar-Ito 2.5, xl X

✓ Taytor-ItO 2.5, x2 X

Figure 47: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, and 2.5 (x(2) component, C = 7500, dt = 0.01)

Figure 48: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 (x(1) component, C = 14000, dt = 0.025)

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Euler scheme (dt = 0.025) Milstein scheme (C = 14000, dt = 0.025)

SDE-MATH software package — □ X

The modeling has been completed!

19:10:07 - SQlite Database is successfiily connected 19:10:07 SQlite Database Version is: 3.33.0 19:1011 -(0.000 seconds] Strong Taylor-lto 1.5 start 19:1011 [0000 seconds] Using C = 140000 19 Ю 11 - [0000 seconds] Using dt = 0.025 19:Ю 11 - [0005 seconds) Using q = (0. 0)

19:10:12 - (0.145 seconds) Strong Taylor-lto 1.S subs are finished 19:10:12 - (0.224 seconds] Strong Tayior-lto 1.5 calculations are fnished 19:10:12 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

19:1023 - SQlite Database is successfiily connected 19:10:23 SQlite Database Version is: 3.33.0 19:10.27 - [0.000 seconds) Strong Taylor-lto 2.0 start 19:10.27 - [0000 seconds] Using C = 140000 19:1027 - [0 001 seconds] Using dt = 0.025 19:1027 - (0 006 seconds) Using q = (1, 0, 0. 0)

19:10:28 - (0.547 seconds) Strong Taylor-lto 2.0 subs are finished 19:10:28 - (0.899 seconds) Strong Taylor-lto 2.0 calculations are fnished 19:1028 - The SQlite connection is closed

✓Ok

Strong Taylor-Ito scheme of order 1.5 (C = Strong Taylor-Ito scheme of order 2.0 (C = 14000, dt = 0.025) 14000, dt = 0.025)

SDE-MATH software package - □ X

The modeling has been completed!

19:11:05 SQlite Database is successfully connected

19:11:05 SQlite Database Version is: 3.33.0 19:11.09 - [0.000 seconds) Strong Taylor-lto 3.0 start 19:11 10 [0.264 seconds] Using C ^ 140000 19:11 10 - [0.265 seconds) Using dt = 0.025

19:11:10 - (0 265 seconds) Using q = (914. 23.0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0, 0. 0) 19:17:00 - (350.567 seconds; Strong Taylor-lto 3.0 subs are finished 19:17:53 - (403.822 seconds] Strong Taylor-lto 3.0 calculations are finshed 19:17:53 - The SQlite connection is closed

Strong Taylor-Ito scheme of order 2.5 (C = Strong Taylor-Ito scheme of order 3.0 (C = 14000, dt = 0.025) 14000, dt = 0.025)

Figure 49: Modeling logs

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Figure 50: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 (x(2) component, C = 14000, dt = 0.025)

SDE-MATH. software package

□ X

The modeling has been completed!

18:28:36 - SQLite Database is successfully connected 18:28:36 SQlite Database Version is: 3.33.0 18:28.40 -10.000 seconds) Taytor-Stratonovich 1.0 start 18:2840 10000 seconds] Using C = 500 18:2840 -10001 seconds! Using dt = 0.001 18:2840 - (0001 seconds) Using q = (3.)

18:28:40 - (0.080 seconds) Strong Tayfor-Stratonovich 1.0 subs are finished 18:28:41 - (0.916 seconds) Strong Taylor-Sbatonovich 1.0 calculations are finished 18:28:41 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

18:29:04 - SQlite Database is successfiily connected

18:2904 SQlite Database Version is: 3.33.0

18:2908 - (0.000 seconds) Strong Taylor-Stratonovkh 1.5 start

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18:2909 - (0726 seconds) Using C = 500

18:29 09 - (0.727 seconds) Using dt = 0.001

18:29 09 - (0.727 seconds) Using q = (2500. 3)

18:3940 - (91.427 seconds) Strong Taylor-Stratonovich 1.5 subs ore finished 18:34.16 - (307.814 seconds) Strong Tayior-Strotonovich 15 calculations ore finished 18:3416 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order

1.0 (C = 50, dt = 0.001) 1.5 (C = 50, dt = 0.001)

Figure 51: Modeling logs

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Figure 52: Strong Taylor-Stratonovich schemes of orders 1.0 and 1.5 (x(1) component, C = 50, dt = 0.001)

Figure 53: Strong Taylor-Stratonovich schemes of orders 1.0 and 1.5 (x(2) component, C = 50, dt = 0.001)

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Differential Equations and Control Processes, N. 1, 2021

Figure 54: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, and 2.0 (x(1) component, C = 500, dt = 0.005)

SDE-MATH. software package

□ X

The modeling has been completed!

18:42:35 - SQlite Database is successfiily connected 18:42:35 SQlite Database Version is: 3.33.0 18:42.39 -10.000 seconds) Taytor-Stratonovich 1.0 start 18:42 39 *0-000 seconds] Using C = 500 0 18.42 39 -10000 seconds] Using dt = 0005 18:42 39 -(0006 seconds) Using q = (0.)

18:42:39 - (0.043 seconds) Strong Taylor-Stratonovich 1.0 subs are finished 18:42.40 (0.153 seconds) Strong Taylor-Stratonovich 1.0 calculations are finished 18:42 40 - The SQLrte connection is closed

SDE-MATH: software package — □ X

The modeling has been completed!

18:42:58 - SQlite Database is successfiily connected

18:42:58 SQlite Database Version is: 3.33.0

18:43-04 (0.000 seconds) Strong Taylor-Stratonovkh 1.5 start

18:43 04 (0000 seconds] Using C = 500.0

18 43 04 - (0 000 seconds] Using dt - 0005

1843 04 - (0005 seconds) Using q = (10.0>

18:43 04 - (0.227 seconds) Strong Taylor-Stratonovich 1.5 subs are finished 18:43 04 - (0.819 seconds) Strong Taylor-Stratonovich 1.5 calculations are finished 18:43:04 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order

1.0 (C = 500, dt = 0.005)

Strong Taylor-Stratonovich scheme of order

1.5 (C = 500, dt = 0.005)

Figure 55: Modeling logs

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Differential Equations and Control Processes, N. 1, 2021

SDE-MATH software package — □ X

The modeling has been completed!

18:43:36 - SQlite Database is successfiily connected

18:43:36 SQlite Database Version is: 3.33.0

18:43.41 -10.000 seconds] Strong Taylor-Strato no v*ch 2.0 start

18:4342 [0 481 seconds] Using C = 500 0

18 43 42 - [0 481 seconds] Using dt - 0.005

1843 42 * [0.482 seconds] Using q = (2000. 10. 0. 0)

18:46:22 - [160.880 seconds] Strong Taytor-Stiatonovich 2.0 subs are finished 18:47:42 [240279 seconds] Strong Taylor-Strotonovich 2.0 calculations ore finished 18:47 42 - The SQlite connection is closed

✓Ok

Figure 56: Strong Taylor-Stratonovich scheme of order 2.0 (C = 500, dt = 0.005)

(2)

Figure 57: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, and 2.0 (x( ’ component, C = 500, dt = 0.005)

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Figure 58: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, and 2.5 (x(1) component, C = 7500, dt = 0.01)

SDE-MATH software package — □ X

The modeling has been completed!

19:01 25 - SQlite Database is successfiily connected 19:01:25 SQlite Database Version is: 3.33.0 19.01.30 -10.000 seconds] Taytor-Stratonovich 1.0 start 19:01 30 10 001 seconds] Using C = 7500.0 19 01 30 -10001 seconds] Using dt = 0.01 19:01 30 - (0 001 seconds) Using q = (0.)

19:01:30 - (0.039 seconds) Strong Taylor-Stratonovich 1.0 subs are finished 19:01:30 - (0.098 seconds) Strong Taylor-Stratonovich 1.0 calculations are finished 19:01:30 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

19:01 42 - SQlite Database is successfiily connected

19:01:42 SQlite Database Version is: 3.33.0

19.01.46 - (0.000 seconds) Strong Taylor-Stratonovich 1.5 start

19 01 46 - (0.000 seconds] Using C = 7500.0

1901 46 - (0000 seconds] Using dt = 0.01

19:01 46 - (0 006 seconds) Using q = (0. 0)

19:01.46 - (0.110 seconds) Strong Taylor-Stratonovich 1.5 subs are finished 19:01.46 - (0.317 seconds) Strong Taylor-Stratonovich 1.5 cakulafions are finished 19:01 46 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order

1.0 (C = 7500, dt = 0.01) 1.5 (C = 7500, dt = 0.01)

Figure 59: Modeling logs

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SDE-MATH software package — □ X

The modeling has been completed!

19:02.01 - SQlite Database is successfiily connected

19:02:01 SQlite Database Version is: 3.33.0

19.02.05 -10.000 seconds] Strong Taylor-Stratonovkh 2.0 start

19:02 05 [0 000 seconds] Using C = 7500.0

19 02 05 -10 000 seconds] Using dt = 0.01

19:02 05 -10.005 seconds) Using q = (17.0.0.0)

19:0205 - (0.667 seconds) Strong Tayfor-Stratonovich 2.0 subs are finished 19:02:07 - (1.773 seconds) Strong Taylor-Stratonovich 2.0 calculations are finished 19:02:07 - The SQlite connection is closed

✓Ok

SDE-MATH: software package - □ X

The modeling has been completed!

19:0223 SQlite Database is successfully connected

19:02:23 SQlite Database Version is: 3.33.0

19.0227 - (0.000 seconds) Strong Taylor-Strato no vkh 2.5 start

19:02 28 (0.380 seconds] Using C = 7500.0

19 02 28 -10.380 seconds) Using dt = 0.01

19:02 28 • (0.381 seconds) Using q = (1667. 17, 0. 0.0. 0, 0. 0)

19:05:55 - (207.453 seconds; Strong Taylor-Stratonovich 25 subs are fmihed 19:07:06 - (279.034 seconds) Strong Taylor-Stratonovich 25 calculations ore finished 19:07:06 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order

2.0 (C = 7500, dt = 0.01) 2.5 (C = 7500, dt = 0.01)

Figure 60: Modeling logs

(2)

Figure 61: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, and 2.5 (x( ’ component, C = 7500, dt = 0.01)

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Differential Equations and Control Processes, N. 1, 2021

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

SDE-MATH software package — □ X

The modeling has been completed!

19:21:13 - SQlite Database is successfiily connected 19:21:13 - SQlite Database Version is: 3.33.0 19:21.17 -10.000 seconds) Taytor-Stratonovich 1.0 start 19:21 17 - [0 000 seconds] Using C = 14000 0 19:21 17 -10001 seconds) Using dt = 0.025 1921 17 - (0001 seconds) Using q = (0.)

19:21:17 - (0.037 seconds) Strong Taylor-Stratonovich 1.0 subs are finished 19:21:17 - (0.061 seconds) Strong Taytor-Sbatonovich 1.0 calculations are finished 19:21:17 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

19:21:34 - SQlite Database is successfiily connected

19:21:34 SQlite Database Version is: 3.33.0

19:21.38 - (0.000 seconds) Strong Taylor-Stratonovkh 1.5 start

19:21 38 (0000 seconds] Using C = 140000

19:21 38 - (0 000 seconds] Using dt = 0.025

1921 38 - (0.007 seconds) Using q = (0. 0)

19:21:38 (0.113 seconds) Strong Taylor-Stratonovich 1.5 subs are finished 19:21:38 - (0.196 seconds) Strong Taytor-Sbatonovich 1.5 calculations are finished 19:21:38 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order 1.0 (C = 14000, dt = 0.025) 1.5 (C = 14000, dt = 0.025)

SDE-MATH: software package — □ X

The modeling has been completed!

19:21 56 - SQlite Database is successfiily connected

19:21:56 SQlite Database Version is: 3.33.0

19:22.00 - (0.000 seconds) Strong Taylor-Strato no vkh 2.0 start

19:22 00 (0001 seconds] Using C = 140000

19:22 00 - (0001 seconds) Using dt = 0.025

19 22 00 - (0 006 seconds) Using q = (1, 0, 0. 0)

19:22:00 - (0.331 seconds) Strong Taylor-Stratonovich 2.0 subs are finished 19:22:01 - (0.669 seconds) Strong Taylor-Stratonovich 2.0 calculations are finished 19:22:01 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

19:22:29 - SQlite Database is successfiily connected

19:22:29 - SQlite Database Version is: 3.33.0

19:22.33 -10.000 seconds) Strong Taylor-Strato no vkh 2.5 start

19:22 33 - (0001 seconds] Using C = 140000

19:22 33 - (0001 seconds) Using dt = 0.025

19:22 33 • (0006 seconds) Using q = (23,0.0.0. 0. 0.0. 0)

19:22:35 - (1.773 seconds) Strong Tayfor-Stratonovich 2.5 subs are finished 19:22:37 (3.471 seconds) Strong Taytor-Sbatonovich 2.5 calculations are finished 19:22:37 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order 2.0 (C = 14000, dt = 0.025) 2.5 (C = 14000, dt = 0.025)

SDE-MATH: software package — □ X

The modeling has been completed!

19:23:17 - SQlite Database is successfiily connected

19:23:17 - SQlite Database Version is: 3.33.0

19:23:21 -10.001 seconds) Strong Taylor-Strato no vkh 3.0 start

19:23 21 (0.261 seconds] Using C = 140000

19:2321 * (0.262 seconds) Using dt = 0.025

19 23 21 - (0.266 seconds) Using q = (914. 23.0. 0.0. 0. 0. 0. 0. 0. 0. 0. 0. 0, 0. 0)

19:27:47 - (266.455 seconds; Strong Taylor-Stratonovich 3.0 subs are (inched 19:28:37 - (316.214 seconds] Strong Taylor-Stratonovich 3.0 calculations are finished 19:28:37 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order 3.0 (C = 14000, dt = 0.025)

Figure 62: Modeling logs

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Figure 63: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, 2.5, and 3.0 (x(1) component, C = 14000, dt = 0.025)

Figure 64: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, 2.5, and 3.0 (x( ’ component, C = 14000, dt = 0.025)

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Figure 65: Strong Taylor-Ito schemes of orders 0.5, 1.0, and 1.5 (x(1) component, C = 0.1, dt = 0.07)

Euler scheme (dt = 0.07) Milstein scheme (C = 0.1, dt = 0.07)

Figure 66: Modeling logs

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SDE-MATH software package — □ X

The modeling has been completed!

19:4029 - SQlite Database is successfiily connected 19:4029 SQlite Database Version is: 3.33.0 19:40.34 -10.000 seconds] Strong Taylor-lto 1.5 start 19 40 34 {0 074 seconds] Using C = 01 19 40 34 - Ю 074 seconds] Using dt - 0.07 19:40 34 - (0.074 seconds] Using q = (25S. 18)

19:41.47 - [73.360 seconds) Strong Taylor-lto 15 subs are finished 19:41:51 - (77.021 seconds) Strong Taylor-lto 1.5 calculations are finished 19:41:51 - The SQlite connection is closed

✓Ok

Figure 67: Strong Taylor-Ito scheme of order 1.5 (C = 0.1, dt = 0.07)

(2)

Figure 68: Strong Taylor-Ito schemes of orders 0.5, 1.0, and 1.5 (x() component, C = 0.1, dt = 0.07)

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Figure 69: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, and 2.0 (x(1) component, C = 0.5, dt = 0.15)

Euler scheme (dt = 0.15)

Milstein scheme (C = 0.5, dt = 0.15)

Figure 70: Modeling logs

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Strong Taylor-Ito scheme of order 1.5 (C = Strong Taylor-Ito scheme of order 2.0 (C 0.5, dt = 0.15) 0.5, dt = 0.15)

Figure 71: Modeling logs

(2)

Figure 72: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, and 2.0 (x( ’ component, C = 0.5, dt = 0.15)

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Euler scheme (dt = 0.2) Milstein scheme (C = 0.8, dt = 0.2)

SDE-MATH software package — □ X

The modeling has been completed!

19:54:10 - SQlite Database is successfiily connected 19:54:10 SQlite Database Version is: 3.33.0 19.54.14 10.000 seconds] Strong Taylor-lto 1.5 start 19 5414 10000 seconds] Using C = 0.8 19:54 14 -10 000 seconds] Using dt - 0.2 19:54 14 - (0 000 seconds] Using q = (4. 0)

19:54.14 - (0.205 seconds) Strong Taylor-lto 1.5 subs are finished 19:54.14 - (0.217 seconds) Strong Taylor-lto 1.5 calculations ore finished 19:5414 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

19:54:36 - SQlite Database is successfiily connected 19:54:36 SQlite Database Version is: 3.33.0 19.54.40 - (0.000 seconds) Strong Taylor-lto 2.0 start 19:5440 (0001 seconds] Using C =• 0.8 19:5440 - (0001 seconds] Using dt = 0.2 19:54 40 - (0 001 seconds) Using q = (20. 4. 0.0)

19:54:42 - (1.860 seconds) Strong Taylor-lto 2.0 subs are finished 19:54.42 - (1.958 seconds) Strong Taylor-lto 2.0 calculations ore fnished 19:54:42 - The SQlite connection is closed

✓Ok

Strong Taylor-Ito scheme of order 1.5 (C = Strong Taylor-Ito scheme of order 2.0 (C = 0.8, dt = 0.2) 0.8, dt = 0.2)

SDE-MATH: software package — □ X

The modeling has been completed!

19:55:19 - SQlite Database is successfiily connected

19:55:19 - SQlite Database Version is: 3.33.0

19:55.23 - (0.000 seconds) Strong Taylor-lto 2.5 start

19:55 23 - (0 101 seconds] Using C =• 0.8

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

19:55 23 - (0 101 seconds) Using dt = 0.2

19:55 23 - (0 Ю2 seconds) Using q = (98. 20. г 1.0. 0. 0.0)

19:58:39 - (196-216 seconds; Strong Taylor-lto 2.5 subs are finished 19:58:43 - (199.719 seconds] Strong Taylor-lto 2.5 calculations are finshed 19:58:43 - The SQlite connection is closed

✓Ok

Strong Taylor-Itoo scheme of order 2.5 (C = 0.8, dt = 0.2)

Figure 73: Modeling logs

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Figure 74: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, and 2.5 (x(1) component, C = 0.8, dt = 0.2)

Figure 75: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, and 2.5 (x(2) component, C = 0.8, dt = 0.2)

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Euler scheme (dt = 0.2)

Milstein scheme (C = 4, dt = 0.2)

SDE-MATH software package — □ X

The modeling has been completed!

20:10:08 - SQlite Database is successfiily connected 20:10:08 SQlite Database Version is: 3.33.0 20:Ю12 -10.000 seconds] Strong Taylor-lto 1.5 start 20:1012 -10000 seconds] Using C = 4.0 20:1012 -10000 seconds] Using dt = 0.2 20:1012 - (0000 seconds) Using q = (1,0)

20:10:12 - (0.167 seconds] Strong Taylor-lto 1.S subs are finished 20:10:12 (0.178 seconds) Strong Taylor-lto 1.5 calculations ore finished 20:10:12 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

20:10:41 - SQlite Database is successfiily connected 20:10:41 SQlite Database Version is: 3.33.0 20:10.46 - (0.000 seconds) Strong Taylor-lto 2.0 start 20:1046 (0001 seconds] Using C = 4.0 20.Ю46 - (0006 seconds] Using dt = 0.2 20:1046 - (0006 seconds) Using q = (4, 0. 0. 0)

20:1046 - (0.583 seconds] Strong Taylor-lto 2.0 subs are finished 20:1046 - (0.630 seconds) Strong Taylor-lto 2.0 calculabons ore finished 20:1046 - The SQlite connection is closed

✓Ok

Strong Taylor-Ito scheme of order 1.5 (C = 4, Strong Taylor-Ito scheme of order 2.0 (C = 4, dt = 0.2) dt = 0.2)

SDE-MATH: software package — □ X

The modeling has been completed!

20:11:17 - SQlite Database is successfiily connected

20:11:17 - SQlite Database Version is: 3.33.0

20:11.21 - (0.000 seconds) Strong Taylor-lto 2.5 start

20:11 21 (0 001 seconds] Using C = 4.0

20.11 21 - (0 002 seconds] Using dt = 0.2

20:11 21 - (0 002 seconds) Using q = (20.4. 0,0. 0.0, 0. 0)

20:11:25 - (4.343 seconds) Strong Taylor-lto 2.5 subs are finished 20:11:26 - (4.670 seconds) Strong Taylor-lto 2.5 calculabons ore finished 20:11 26 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

20:12:38 - SQlite Database is successfiily connected 20:12:38 SQlite Database Version is: 3.33.0 20:12.43 - (0.000 seconds) Strong Taylor-lto 3.0 start 20:1243 (0 100 seconds] Using C = 4.0 20:1243 - (0 100 seconds) Using dt = 0.2

20:1243 - (0.101 seconds) Using q = (98. 20. г 1. 0, 0. 0. 0. 0. 0. 0. 0.0. 0. 0. 0)

20:16:16 - (213.005 secondsj Strong Taylor-lto 3.0 subs are finished 20:16:20 - (217.631 seconds] Strong Taylor-lto 3.0 calculations are finshed 20:16:20 - The SQlite connection is closed

✓Ok

Strong Taylor-Ito scheme of order 2.5 (C = 4, Strong Taylor-Ito scheme of order 3.0 (C = 4, dt = 0.2) dt = 0.2)

Figure 76: Modeling logs

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Figure 77: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 (x(1) component, C = 4, dt = 0.2)

Figure 78: Strong Taylor-Ito schemes of orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 (x(2) component, C = 4, dt = 0.2)

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Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order

1.0 (C = 0.1, dt = 0.07) 1.5 (C = 0.1, dt = 0.07)

Figure 79: Modeling logs

Figure 80: Strong Taylor-Stratonovich schemes of orders 1.0 and 1.5 (x(1) component, C = 0.1, dt = 0.07)

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Figure 81: Strong Taylor-Stratonovich schemes of orders 1.0 and 1.5 (x(2) component, C = 0.1, dt = 0.07)

SDE-MATH software package — □ X

The modeling has been completed!

19:49:24 - SQLite Database is successfiily connected 19:4924 SQlite Database Version is: 3.33.0 19:49.28 -10.000 seconds) Taytor-Stratonovich 1.0 start 19 4928 )0 000 seconds] Using C =• 0.5 194928 -10001 seconds) Using dt = 0 15 1949 28 -10.006 seconds) Using q = (2.)

19:49:28 - (0.066 seconds) Strong Taylor-Stratonovich 1.0 subs are finished 19:49.28 - (0.071 seconds) Strong Taylor-Stratonovich 1.0 calculations are finished 19:4928 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

19:4956 - SQlite Database is successfiily connected

19:4956 SQlite Database Version is: 3.33.0

19.50.00 - (0.000 seconds) Strong Taylor-Stratonovich 1.5 start

19:5000 (0001 seconds] Using C = 0.5

19:5000 - (0001 seconds) Using dt = 0 15

19:50 00 - (0.005 seconds) Using q * (11.2)

19:50:01 - (0.431 seconds) Strong Taylor-Stratonovich 1.5 subs are finished 19:5001 - (0.464 seconds) Strong Taylor-Stratonovich 1.5 calculations are finished 19:5001 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order

1.0 (C = 0.5, dt = 0.15) 1.5 (C = 0.5, dt = 0.15)

Figure 82: Modeling logs

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SDE-MATH software package — □ X

The modeling has been completed!

19:50:53 - SQlite Database is successfiily connected

19:50:53 SQlite Database Version is: 3.33.0

19:50.57 -10.000 seconds] Strong Taylor-Strato no v*ch 2.0 start

19:5057 - [0013 seconds] Using C = 0.5

19:5057 - [0013 seconds] Using dt = 0 15

19:5057 - [0013 seconds] Using q = (74.11,1.0)

19:51:14 - [16.905 seconds) Strong Taylor-Stratonovich 2.0 subs are finished 19:51:14 - (17.509 seconds) Strong Taylor-Stratonovidi 2.0 calculations are finished 19:51:14 - The SQlite connection is closed

✓Ok

Figure 83: Strong Taylor-Stratonovich scheme of order 2.0 (C = 0.5, dt

0.15)

Figure 84: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, and 2.0 (x(1) component, C = 0.5, dt = 0.15)

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Figure 85: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, and 2.0 (x(2) component, C = 0.5, dt = 0.15)

SDE-MATH software package — □ X

The modeling has been completed!

19:59:58 - SQlite Database is successfiily connected 19:59:58 SQlite Database Version is: 3.33.0 20:00.02 -10.001 seconds) Taytor-Stratonovich 1.0 start 20:0002 [0001 seconds] Using C =• 0.8 20:0002 - [0001 seconds] Using dt - 0.2 200002 - [0007 seconds) Using q = (1.)

20:00:02 - (0.058 seconds] Strong Taylor-Stratonovich 1.0 subs are finished 20:0002 - [0.060 seconds) Strong Taylor-Stratonovich 1.0 calculations are finished 20:0002 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

20:0036 - SQlite Database is successfiily connected

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

20:0036 SQlite Database Version is: 3.33.0

20:00.41 [0.000 seconds] Strong Taylor-Stratonovkh 1.5 start

200041 [0 001 seconds] Using C =• 0.8

200041 - [0001 seconds] Using dt = 0.2

20 0041 - [0.001 seconds) Using q = (4. 0)

20:0041 - (0.164 seconds] Strong Taylor-Stratonovich 1.5 subs are finished 20:0041 - (0.176 seconds) Strong Taylor-Stratonovich 1.5 calculations are finished 20:0041 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order

1.0 (C = 0.8, dt = 0.2) 1.5 (C = 0.8, dt = 0.2)

Figure 86: Modeling logs

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SDE-MATH software package — □ X

The modeling has been completed!

20:02:54 - SQLite Database is successfiily connected

20:02:54 SQlite Database Version is: 3.33.0

20:02.59 -10.000 seconds] Strong Taylor-Stratonovkh 2.0 start

20:02 59 - [0 001 seconds] Using C = 0.8

20:02 59 - (0.001 seconds] Using dt = 0.2

20:02 59 - (0 001 seconds) Using q = (20.4. 0.0)

20:03:00 - (1.723 seconds) Strong Taylor-Stratonovich 2.0 subs are finished 20:03:00 - (1.817 seconds) Strong Tayior-Stratonovich 2.0 calculations are finished 20:03:00 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

20:04:01 - SQlite Database is successfiily connected

20:04:01 SQlite Database Version is: 3.33.0

20:04.06 - (0.000 seconds) Strong Taylor-Strato no vkh 2.5 start

20 04 06 (0 099 seconds] Using C = 0.8

20:04 06 - (0 099 seconds) Using dt = 0.2

20 04 06 - (0.100 seconds) Using q = (98. 20. 2. V 0, 0. 0.0)

20:06:39 - (153.082 seconds; Strong Taytor-Stiatonovich 22b subs are finished 20:06:42 - (156.496 seconds] Strong Taylor-Stratonovich 2.5 calculations ore finished 20:06:42 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order

2.0 (C = 0.8, dt = 0.2) 2.5 (C = 0.8, dt = 0.2)

Figure 87: Modeling logs

Figure 88: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, and 2.5 (x(1) component, C = 0.8, dt = 0.2)

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Figure 89: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, and 2.5 (x(2) component, C = 0.8, dt = 0.2)

Figure 90: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, 2.5, and 3.0 (x(1) component, C = 4, dt = 0.2)

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SDE-MATH software package — □ X

The modeling has been completed!

20:17:57 - SQlite Database is successfiily connected 20:17:57 - SQlite Database Version is: 3.33.0 20:18.01 -(0.000 seconds) Taylor-Stratonovich 1.0 start 20:1801 10001 seconds] Using C = 4 0 20.1801 -10001 seconds] Using dt - 0.2 20:1801 - (0007 seconds) Using q = (0.)

20:18:01 - (0.043 seconds] Strong Tayior-Stratonovich 1.0 subs are finished 20:18:01 - (0.047 seconds) Strong Tayior-Stratonovich 1.0 calculations are finished 20:18:01 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

20:18:48 - SQlite Database is successfiily connected

20:18:48 SQlite Database Version is: 3.33.0

20:18.52 - (0.000 seconds] Strong Taylor-Stratonovkh 1.5 start

20:18 52 - (0 001 seconds] Using C = 4.0

20.18 52 - (0001 seconds] Using dt = 0.2

20:18 52 - (0.007 seconds) Using q = (1, 0)

20:18:52 - (0.131 seconds] Strong Tayior-Stratonovich 1.5 subs are finished 20:18:52 - (0.142 seconds) Strong Tayior-Stratonovich 1.5 calculations are finished 20:18:52 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Tayior-Stratonovich scheme of order 1.0 (C = 4, dt = 0.2) 1.5 (C = 4, dt = 0.2)

SDE-MATH: software package — □ X

The modeling has been completed!

20:19:08 - SQlite Database is successfiily connected

20:19:08 SQlite Database Version is: 3.33.0

20:19.12 - (0.000 seconds] Strong Taylor-Stratonovkh 2.0 start

20:1912 - (0001 seconds] Using C = 4.0

20:1912 - (0001 seconds] Using dt = 0.2

20:1912 - (0.005 seconds) Using q = (4, 0, 0. 0)

20:19:12 - (0.380 seconds] Strong Tayior-Stratonovich 2.0 subs are finished 20:1912 - (0.426 seconds) Strong Tayior-Stratonovich 2.0 calculations are finished 20:1912 - The SQlite connection is closed

✓Ok

SDE-MATH: software package — □ X

The modeling has been completed!

20:1926 - SQlite Database is successfiily connected

20:1926 SQlite Database Version is: 3.33.0

20:1930 - (0.000 seconds] Strong Taylor-Stratonovkh 2.5 start

20:19 30 (0001 seconds] Using C = 4.0

20:19 30 - (0001 seconds] Using dt = 0.2

20:19 30 - (0 001 seconds) Using q = (20.4, 0.0. 0. 0.0. 0)

20:1934 - (3.549 seconds) Strong Tayior-Stratonovich 2.5 subs are finished 20:1934 - (3.833 seconds) Strong Tayior-Stratonovich 2.5 calculations are finished 20:1934 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order Strong Taylor-Stratonovich scheme of order 2.0 (C = 4, dt = 0.2) 2.5 (C = 4, dt = 0.2)

SDE-MATH: software package — □ X

The modeling has been completed!

20:20:06 - SQlite Database is successfiily connected

20:2906 SQlite Database Version is: 3.33.0

20:20.10 - (0.000 seconds] Strong Taylor-Stratonovkh 3.0 start

20:2010 - (0 104 seconds] Using C = 4.0

20:2010 - (0 104 seconds] Using dt = 0.2

20:2010 - (0.105 seconds) Using q = (98. 20, 2. 1. 0, 0. 0. 0. 0. 0. 0. 0. 0, 0. 0. 0)

20:22:47 - (157.030 secondsj Strong Taylor-Stiatonovich 3.0 subs are (inched 20:22:51 - (161.379 seconds] Strong Tayior-Stratonovich 3.0 calculations are finished 20:22:51 - The SQlite connection is closed

✓Ok

Strong Taylor-Stratonovich scheme of order 3.0 (C = 4, dt = 0.2)

Figure 91: Modeling logs

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Figure 92: Strong Taylor-Stratonovich schemes of orders 1.0, 1.5, 2.0, 2.5, and 3.0 (x(2) component, C = 4, dt = 0.2)

5.5 Example of Linear System of Ito SDEs (Solar Activity)

Consider a mathematical model of the solar activity without its average value in a form of the system of linear Ito SDEs (246) [4]. In (246) we choose [4]

n = 2, m = 1, k = 2, x^ = 7, x02) = -0.25,

A

0 1 00

, B =

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

-0.3205 -0.14 00

0 0 \

u(t) = , F =

0 5.08

(250)

(251)

5.6 Visualization and Numerical Results for Solar Activity Model

This subsection is devoted to the visualization and numerical results for the model of solar activity (246), (250), (251).

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Figure 93: Modeling logs for solar activity model

Figure 94: Solar activity model (x

(1)

t

component)

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Figure 95: Solar activity model (x

(2)

t

component)

Figure 96: Solar activity model (expectations)

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Figure 97: Solar activity model (variances)

5.7 Example of Abstract Linear System of Ito SDEs

Now consider the system of linear ItO SDEs (246) with the following data

n = 4, m = 5, k = 3, x0:) = 1, x02) = 2, x03)

1, x04) = -2, (252)

A

-1 0 0 0 1 1 1 0.2 0.1 0.1 0.1 0.1

0 -2 0 0 1 1 1 0.1 0.2 0.1 0.1 0.1

, B = , F =

0 0 -3 0 1 1 1 0.1 0.1 0.2 0.1 0.1

0 0 0 -4 1 1 1 0.1 0.1 0.1 0.2 0.1

(253)

u(t) = (O 0 ^, H = (q.1 0.1 0.1 0.l) . (254)

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5.8 Visualization and Numerical Results for Abstract Linear System of Ito SDEs Obtained via the SDE-MATH Software Package

This subsection is devoted to the visualization and numerical results for the model (246), (252)-(254).

Figure 98: Modeling logs (linear system of Ito SDEs (246), (252)-(254))

Figure 99: Linear system of Ito SDEs (l246l). (252)-(254) (components of solution)

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Figure 100: Linear system of Ito SDEs (246), (252)-(254) (expectations)

Figure 101: Linear system of Ito SDEs (246), (252)-(254) (variances)

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6 Source Codes of the SDE-MATH Software Package in the Python Programming Language

6.1 Source Codes of Graphical User Interface

6.1.1 Source Codes of Main Menu

Listing 7: Configuration file example

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

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app = QApplication(sys.argv)

app . setWindowIcon (QtGui. QIcon(os . path . join (images , ” function .png” ) ) ) app . setStyle (QtWidgets . QStyleFactory . create ( ’ Fusion ’) )

main_window = MainWindow()

main_window . taskbar-button = QWinTaskbarButton()

main .window . taskbar-button . setOverlayIcon ( QtGui. QIcon(” resources / function . svg” ) ) exit(app.exec())

if___name___== ”__main__”:

main ()

Listing 9: Main window

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self . nonlinear-modeling = NonlinearModelingWidget ( self . stack_widget)

self . stack_widget . addWidget ( self. main_menu)

self . stack_widget . addWidget ( self. nonlinear_modeling)

self . stack_widget . addWidget ( self. linear_modeling)

self . stack_widget . addWidget ( self. greetings)

self . stack_widget . addWidget ( self. simple_progress)

self . stack_widget . addWidget ( self . complex_progress)

self . setCentralWidget ( self . stack_widget)

self. exec _ init()

self . setWindowTitle (”SDE—MATH: software package”) self . setMinimumSize (640, 480) self.resize(800, 600)

self.show ()

self . main_menu . group1 . show_nonlinear_dialog . connect ( self . show_nonlinear) self . mai^menu . group2 . show_nonlinear_dialog . connect ( self . show_nonlinear) self . main_menu . group3 . show_linear_dialog . connect (

lambda: self . stack_widget . setCurrentWidget( self . linear_modeling) )

self . nonlinear_modeling . showmai^menu . connect (

lambda: self . stack_widget . setCurrentWidget ( self . main_menu) ) self . nonlinear_modeling . star^progress . connect (

lambda: self . stack_widget . setCurrentWidget( self . complex_progress) )

self . linear_modeling . sho^mai^menu . connect (

lambda: self . stack_widget . setCurrentWidget ( self . main_menu) ) self . linear_modeling . start_progress . connect (

lambda: self . stack_widget . setCurrentWidget( self . complex_progress) )

self . greetings . show_main_menu . connect (

lambda: self . stack_widget . setCurrentWidget ( self . main_menu) )

self . greetings . show_main_menu . connect (

lambda: self . stack_widget . setCurrentWidget ( self . main_menu) )

self . greetings . show_main_menu . connect (

lambda: self . stack_widget . setCurrentWidget ( self . main_menu) )

self . complex_progress . back_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget ( self . main_menu) )

# plot events

self . main_window_close . connect ( self . plot-window . close)

self. main _menu . charts_check . clicked . connect ( self . plot-window . setVisible) self . plot .window . charts_show . connect (

lambda: self. main_menu . charts_check . setChecked( True) )

self . plot .window . charts_hide . connect (

lambda: self. main_menu . charts_check . setChecked( False) )

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self . nonlinear-modeling . draw_chart . connect ( self . plot-window . charts_list .new_items) self . nonlinear_modeling . draw_chart . connect ( self . plot-window . plot_widget . new_items) self . nonlinear_modeling . draw_chart . connect ( self . plot-window . show)

self . nonlinear_modeling . charts_check . stateChanged . connect ( self . plot-window . setVisible

)

self . plot .window . charts_show . connect (

lambda: self . nonlinear_modeling . charts_check . setChecked (True) )

self . plot .window . charts_hide . connect (

lambda: self . nonlinear_modeling . charts_check . setChecked ( False ) )

self . linear_modeling . draw-chart . connect ( self . plot-window . charts_list .new_items)

self . linear_modeling . draw_chart . connect ( self . plot .window . plot_widget . new. items)

self . linear_modeling . draw .chart . connect ( self . plot .window . show)

self . linear_modeling . charts_check . clicked . connect ( self . plot .window . setVisible) self . plot .window . charts_show . connect (

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lambda: self . linear.modeling . charts_check . setChecked ( True) )

self . plot .window . charts_hide . connect (

lambda: self . linear.modeling . charts_check . setChecked( False) )

self . linear_modeling . start-progress . connect ( self . complex-progress . spin ) self . nonlinear.modeling . start-progress . connect ( self . complex_progress . spin) self . linear_modeling . stop_progress . connect ( self . complex .progress . stop ) self . nonlinear.modeling . stop.progress . connect ( self . complex_progress . stop)

def closeEvent(self, event): self . main_window_close . emit ()

def exec.init ( self ) :

self . simple.progress . spin (’’Preparing the database...”) self. stack_widget .setCurrentWidget( self. simple.progress)

worker = Worker( initialization )

worker . signals . finished . connect ( self . init.done )

QThreadPool .globalInstance() .start (worker)

def init.done ( self ) :

if not is_locked (” . welcome . lock”) :

self . stack.widget . setCurrentWidget ( self . greetings ) else :

self . stack.widget . setCurrentWidget ( self . main_menu) self . simple.progress . stop ()

def show.nonlinear ( self , scheme_id) :

self . nonlinear.modeling . set_scheme (scheme_id)

self. stack_widget .setCurrentWidget( self. nonlinear.modeling)

Listing 10: Greetings window

from PyQt5.QtCore import pyqtSignal , Qt

from PyQt5. QtWidgets import QPushButton , QVBoxLayout, QWidget, QSizePolicy , \

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QSpacerltem , QHBoxLayout, QLabel , QCheckBox, QApplication , QStyle

from tools . fsys import lock , unlock from ui . main . svg import SVG

class GreetingsWidget (QWidget) :

show_main_menu = pyqtSignal ()

def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

header = QLabel(”Welcome to SDE—MATH Software Package for ”

”the Numerical Solution of Systems of Ito SDEs”) font = header.font() font . setPointSize(15)

header . setAlignment (Qt. AlignJustify) header . setWordWrap (True) header.setFont(font)

welcome = QLabel (

”Exact solutions of Ito SDEs are known in rare cases. For this ”

” reason , it becomes necessary to construct numerical methods for ”

” Ito SDEs. Moreover , the problem of numerical solution of Ito SDEs ”

”often occurs even in cases when the exact solution of Ito SDE is known. ”

” This means that in some cases , knowing the exact solution to the Ito ”

”SDE does not allow us to simulate it numerically in a simple way.” , self

)

font = welcome . font () welcome . setFont ( font)

welcome . setAlignment (Qt. AlignJustify) welcome . setWordWrap ( True)

welcome . setSizePolicy( QSizePolicy( QSizePolicy . Expanding , QSizePolicy. Minimum) )

check_again = QCheckBox(”Do not show again” , self) next_btn = QPushButton(”Ok” , self)

next _btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_DialogApplyButton) ) check_again.clicked.connect(self.check_lock)

next_btn . clicked . connect (lambda: self. show_main_menu . emit () )

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controls = QHBoxLayout ()

controls . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) controls . addWidget( check_again ) controls . addWidget (next_btn)

controls . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq1 = QHBoxLayout ()

eq1 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq1 . addWidget (SVG(” equation1.svg” , scale_factor = 1.))

eq1 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) )

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eq2 = QHBoxLayout ()

eq2 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq2 . addWidget (SVG(” equation2.svg” , scale_factor = 1.))

eq2 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) column = QVBoxLayout ()

column . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) column . addWidget ( header )

column . addItem(QSpacerItem(0 , 15, QSizePolicy . Expanding , QSizePolicy .Minimum) )

column . addLayout (eq1)

column . addItem(QSpacerItem(0 , 15, QSizePolicy . Expanding , QSizePolicy .Minimum) )

column . addLayout (eq2)

column . addItem(QSpacerItem(0 , 15, QSizePolicy . Expanding , QSizePolicy .Minimum) )

column . addWidget (welcome)

column . addItem(QSpacerItem(0 , 30, QSizePolicy . Expanding , QSizePolicy .Minimum) ) column . addLayout ( controls )

column . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) layout = QHBoxLayout ()

layout . addItem(QSpacerItem(50 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) layout . addLayout (column)

layout . addItem(QSpacerItem(50 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) self . setLayout(layout)

def check_lock ( self ) :

if self.sender() .isChecked() : lock (” . welcome . lock” ) else :

unlock(” . welcome . lock” )

Listing 11: Info icon

from PyQt5.QtCore import QSize

from PyQt5. QtWidgets import QWidget, QApplication , QStyle , QLabel

class InfoIcon (QLabel) :

def _ _ i n i t __ ( self , text: str , parent=None) :

super (QWidget, self) . __init__ (parent)

self . setToolTip(text)

self . setStyleSheet (”QToolTip { background: white; } ”)

self. setPixmap ( QApplication . style () . standardIcon (

QStyle. SP_MessageBoxInformation) . pixmap (QSize(16, 16)))

Listing 12: Error widget

from PyQt5.QtCore import QSize

from PyQt5. QtWidgets import QWidget, QApplication, QStyle, QLabel, QHBoxLayout,

QSpacerItem , QSizePolicy

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class ErrorWidget(QWidget) :

def _ _init__(self , text: str , parent=None) :

super (QWidget, self) . __init__ (parent)

msg_m = QLabel(text)

msg_m. setStyleSheet (”QLabel { color: rgb(230, 0, 0); } ”) ms^i = QLabel ()

ms^i . setStyleSheet (”QToolTip { background: white; } ”)

ms^i .setPixmap(QApplication . style () . standardIcon(

QStyle.SP_MessageBoxCritical) . pixmap (QSize(16, 16)))

layout = QHBoxLayout ()

layout . setContentsMargins(0 , 0, 0, 0)

layout . addWidget ( msg_i)

layout .addWidget( msg_m)

layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum)) self . setLayout(layout)

Listing 13: Svg picture

import os

from PyQt5.QtCore import QSize from PyQt5 . QtSvg import QSvgWidget from PyQt5. QtWidgets import QSizePolicy

from config import images

class SVG(QSvgWidget) :

def __init__(self, name: str , scale_factor = 1.):

super (QSvgWidget , self).__init__()

self . load(os .path . join (images , name) )

self.scale_factor = scale_factor

self . setSizePolicy ( QSizePolicy . Fixed , QSizePolicy . Fixed)

def sizeHint(self):

size = self.renderer().defaultSize() return QSize(size.width() * self.scale_factor, size.height() * self.scale_factor)

Listing 14: Main menu (base part)

from PyQt5. QtWidgets import QWidget, QSizePolicy, QSpacerItem , QHBoxLayout, QVBoxLayout,

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QCheckBox, QLabel from ui . main . info import InfoIcon

from ui .main.menu. linear import LinearGroupWidget from ui .main.menu. taylor_ito import ItoGroupWidget

from ui .main.menu. taylor_stratonovich import StratonovichGroupWidget

class MainMenuWidget(QWidget) :

def __i n it __ ( self , parent=None) :

super (QWidget, self) . __init__ (parent)

self . charts_check = QCheckBox(”Charts window”, self)

icon = InfoIcon(”This is charts window checkbox , it will \ n”

” follow you on every application dialog , so\n”

”you can easily open or close window with available charts”)

bar_layout = QHBoxLayout ()

bar_layout . addItem(QSpacerItem (0 , 35, QSizePolicy . Expanding , QSizePolicy .Minimum))

bar_layout . addWidget ( icon )

bar_layout . addWidget ( self . charts_check)

header = QLabel(”Strong Numerical Schemes for Ito SDEs” , parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

icon = InfoIcon(”You are now in main menu, you can choose \ n” ”any scheme to perform modeling” )

header .layout = QHBoxLayout ()

header .layout addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy Expanding)

)

header .layout . addWidget ( icon )

header .layout addWidget ( header)

header .layout add Item ( Q Spacer Item (0 , 0, QSizePolicy . Expanding , QSizePolicy Expanding)

)

self.group3 = LinearGroupWidget( self ) self.group1 = ItoGroupWidget( self) self.group2 = StratonovichGroupWidget( self)

menu_layout = QHBoxLayout () menu-layout . addWidget ( self . group1) menu-layout . addWidget ( self . group2) menu-layout . addWidget ( self . group3)

layout = QVBoxLayout () layout . addLayout ( bar_layout)

layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) layout . addLayout ( header_layout)

layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) layout . addLayout (menu-layout)

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layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) self . setLayout(layout)

Listing 15: Main menu (linear part)

from PyQt5.QtCore import pyqtSignal

from PyQt5. QtWidgets import QPushButton , QVBoxLayout, QSizePolicy, QSpacerItem , QGroupBox

class LinearGroupWidget(QGroupBox) :

show_linear_dialog = pyqtSignal ()

def __i n it __ ( self , parent=None) :

super (QGroupBox, self) . __init__ (parent)

linear_btn = QPushButton(”Dispersion Spectral Decomposition”)

layout = QVBoxLayout () layout . addWidget ( linear_btn )

layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) self . setLayout(layout)

self . setTitle (”Linear Ito SDEs Systems Modeling”)

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self . setSizePolicy ( QSizePolicy ( QSizePolicy . Expanding , QSizePolicy . Expanding) ) linear_btn . clicked .connect (lambda: self . show_linear_dialog .emit () )

Listing 16: Main menu (Taylor-ItO part)

from PyQt5.QtCore import pyqtSignal from PyQt5. QtWidgets import QPushButton,

QVBoxLayout, QSizePolicy, QSpacerItem, QGroupBox

class ItoGroupWidget(QGroupBox) :

show_nonlinear_dialog = pyqtSignal (int)

def __i n it __ ( self , parent=None) :

super (QGroupBox, self) . __init__ (parent)

btn1 = QPushButton(”Euler”)

btn2 = QPushButton(”Milstein”)

btn3 = QPushButton(”Convergence Order 1.5”)

btn4 = QPushButton(”Convergence Order 2.0”)

btn5 = QPushButton(”Convergence Order 2.5”)

btn6 = QPushButton(”Convergence Order 3.0”)

layout = QVBoxLayout () layout .addWidget(btn1)

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Listing 17: Main menu (Taylor—Stratonovich part)

from PyQt5.QtCore import pyqtSignal

from PyQt5. QtWidgets import QPushButton , QVBoxLayout, QSizePolicy, QSpacerItem , QGroupBox

class StratonovichGroupWidget(QGroupBox) :

show_nonlinear_dialog = pyqtSignal (int)

def __init__(self , parent=None) :

super (QGroupBox, self) . __init__ (parent)

btn1 = QPushButton(”Convergence Order 1.0”)

btn2 = QPushButton(”Convergence Order 1.5”)

btn3 = QPushButton(”Convergence Order 2.0”)

btn4 = QPushButton(”Convergence Order 2.5”)

btn5 = QPushButton(”Convergence Order 3.0”)

layout = QVBoxLayout () layout .addWidget(btn1) layout .addWidget(btn2) layout .addWidget(btn3) layout .addWidget(btn4) layout .addWidget(btn5)

layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) self . setLayout(layout)

self. setTitle (”Taylor—Stratonovich Schemes”)

self . setSizePolicy ( QSizePolicy ( QSizePolicy . Expanding , QSizePolicy . Expanding) )

btn1 . clicked . connect (lambda: self . show_nonlinear_dialog . emit (6) ) btn2. clicked . connect (lambda: self . show_nonlinear_dialog . emit (7) )

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btn3 . clicked . connect (lambda self . show_nonlinear_dialog . emit (8) )

btn4 . clicked . connect (lambda self . show_nonlinear_dialog . emit (9) )

btn5 . clicked . connect (lambda self . show_nonlinear_dialog . emit (10) )

Listing 18: Complex progress view

import logging

from PyQt5. QtWidgets import QWidget, QHBoxLayout, QVBoxLayout, QSpacerltem , QSizePolicy ,

\

QListWidget , QLabel , QPushButton , QApplication , QStyle from pyqtspinner . spinner import WaitingSpinner

from ui . main . progress . log_handler import LogHandler

class ComplexProgressWidget(QWidget) :

def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

self .spinner = WaitingSpinner ( self , radius =5.0, lines =10, line_length = 5.0, centerOnParent=False ) self.list_widget = QListWidget ( self) self. handler = LogHandler ( self . handle_message)

self . label = QLabel ( self) font = self . label . font () font . setPointSize(15) self.label.setFont(font)

self. back _btn = QPushButton ( ”Ok” )

self . back_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_DialogApplyButton) ) self.back_btn.hide()

spinner_layout = QHBoxLayout () spinner_layout . addWidget ( self . spinner) spinner_layout . addWidget ( self . label)

spinner_layout . addSpacerItem ( QSpacerItem (0 , 0 , QSizePolicy . Expanding ,

QSizePolicy .Minimum) )

bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget ( self . back_btn)

layout = QVBoxLayout () layout . addLayout ( spinner_layout) layout . addWidget ( self . list_widget ) layout . addLayout (bottom_bar)

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Listing 19: Simple progress view

from PyQt5.QtCore import Qt

from PyQt5. QtWidgets import QVBoxLayout, QWidget, QLabel , QSpacerltem , QSizePolicy from pyqtspinner . spinner import WaitingSpinner

class SimpleProgressWidget(QWidget) :

def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

self.spinner = WaitingSpinner(self , radius = 15.0, lines = 10, line_length = 15.0) self . label = QLabel ()

self. label . setAlignment(Qt.AlignCenter) font = self . label . font () font . setPointSize(15) self.label.setFont(font)

layout = QVBoxLayout ()

layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) ) layout . addWidget ( self . spinner)

layout . addItem(QSpacerItem(0 , 50, QSizePolicy .Minimum, QSizePolicy .Minimum)) layout . addWidget (self. label)

layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding) )

self . setLayout(layout)

def spin(self, text): self.spinner.start() self.label.setText(text)

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def stop(self):

self . spinner.stop()

Listing 20: Log handler for application

import logging

class LogHandler( logging . Handler) :

def __init__(self, callback): super() . __init__ () self. callback = callback

self. setFormatter( logging. Formatter(”%(asctime)s — %(message)s” , ”%H:%M:%S” ) )

def handle(self , record) :

self . callback( self . format (record))

Listing 21: Matrix widget

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if item is not None:

item. setText( self .m[ i ] [j ]) else :

self.setItem(i, j, CustomItem (self.m[i][j]))

self. blockSignals(False)

def resize_h(self, h: int):

self . blockSignals (True)

old_h = self. rowCount () w = self . columnCount() self. setRowCount (h)

self .m = [[self.m[i][j] if i < old _h and j < w else ”0” for j in range (w) ] for i in range(h)]

for i in range(h): for j in range(w):

item = self.item(i, j) if item is not None:

item. setText( self .m[ i ] [j ]) else :

self.setItem(i, j, CustomItem (self.m[i][j]))

self. blockSignals(False)

def item_changed(self, item):

self .m[ item . row ()][item.column()] = item. text ()

class CustomItem(QTableWidgetItem) :

def __init__(self, value: str):

super ( QTableWidgetItem , self) . __init__(value)

self.valid = True

6.1.2 Source Codes of Charts Window

Listing 22: Charts window

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from PyQt5.QtCore import pyqtSignal

from PyQt5.QtWidgets import QWidget, QHBoxLayout, QMainWindow, QSplitter

from ui . charts . side . available_charts_widget import AvailableChartsWidget from ui . charts . visuals . charts_widget import ChartsWidget

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Listing 23: Curves list

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on_hide_all = pyqtSignal() on_show_all = pyqtSignal() on_remove_all = pyqtSignal()

def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

self. items = diet ()

self. spacer = QSpacerItem(0 , 0, QSizePolicy .Minimum, QSizePolicy . Expanding) self.plot_widget = self.parent().plot_widget

remove_all = QPushButton() remove_all . setFlat (True) remove_all . setIcon (

QApplication . style () . standardIcon (QStyle . SP_DialogResetButton) )

hide_all = QPushButton() hide _all . setFlat (True)

hide_all . setIcon (QtGui .QIcon(os .path . join (images , ” crossed .png” ) ) )

show_all = QPushButton() show_all . setFlat (True)

show_all . setIcon (QtGui .QIcon(os .path . join (images , ”eye .png” ) ) )

header_layout = QHBoxLayout () header_layout .addWidget(

InfoIcon (” Here You will see all modeling series \ n”

”You can hide them or delete , if you need to”)) header_layout .addItem(

QSpacerItem(5 , 0, QSizePolicy .Minimum, QSizePolicy .Minimum) ) header_layout . addWidget ( QLabel (” Curves” ) ) header_layout .addItem(

QSpacerItem(5 , 0, QSizePolicy .Minimum, QSizePolicy .Minimum) ) header_layout.setContentsMargins(0, 0, 0, 0) header_layout . setSpacing (0) header_layout .addItem(

QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum)) header_layout . addWidget (show_all) header_layout . addWidget (hide_all) header_layout . addWidget ( remove_all)

self. layout = QVBoxLayout ()

self. layout.setContentsMargins(3, 3, 3, 3)

self . layout.setSpacing(2)

scro ll_widget = QWidget( self ) scroll_widget . setLayout ( self . layout)

scroll_area = QScrollArea ( self ) scrolLarea . setWidgetResizable(True) scroll_area . setWidget ( scroll_widget )

self . layout . addItem( self . spacer)

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layout = QVBoxLayout () layout . setContentsMargins(0 , 0, 0, 0) layout . addLayout ( header_layout) layout . addWidget ( scrolLarea )

self . setLayout(layout)

self . setSizePolicy (

QSizePolicy( QSizePolicy . MinimumExpanding ,

QSizePolicy . MinimumExpanding) )

show_all . clicked . connect ( self . show_all) hide_all . clicked . connect ( self . hide_all) remove_all . clicked . connect ( self . delete_all )

def new_items(self, lines: list ):

self . layout . removeItem ( self . spacer)

for i in range (len (lines)) :

item_widget = ItemWidget ( lines [ i ] . name , lines [i ]. color , parent=self) item_widget . uid = lines [ i ] . uid

item_widget . on_show . connect ( self . plot_widget . show_item) item_widget . on_hide . connect ( self . plot_widget . hide_item) item_widget . on_delete . connect ( self . plot_widget . delete_item ) item_widget . on_delete . connect ( self . delete_item ) self.items [lines [i ]. uid] = item_widget

self . plot_widget . hide_label . connect (lambda uid: self. items [uid]. hide () )

self . plot_widget . show_label . connect (lambda uid: self. items [uid]. show () )

self . layout . addWidget (item_widget) self . layout . addItem( self . spacer)

def delete_item ( self ) : s = self.sender () s . setParent (None) self . items . pop (s . uid) self . layout . removeWidget (s)

def delete_all(self):

fo r i t e m i n s e l f . i t e m s . v a l u e s ( ) : item . setParent (None) self . layout . removeWidget (item) self . items . clear () self . on_remove_all . emit ()

def hide_all(self):

fo r i t e m i n s e l f . i t e m s . v a l u e s ( ) : item . checkbox . blockSignals (True) item . checkbox . setChecked ( False) item . checkbox . blockSignals(False) self . on_hide_all . emit ()

def show_all(self):

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for item in self. items. values (): item . checkbox . blockSignals (True) item . checkbox . setChecked (True) item . checkbox . blockSignals(False) self . on_show_all . emit ()

Listing 24: Curves list item

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Listing 25: Charts area

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self . btn_to_fn . setFlat (True)

self . btn_to_mx = QPushButton(”Expectations”) self . btn_to_mx . setFlat (True) self . btn_to_dx = QPushButton(”Variances”) self . btn_to_dx . setFlat (True)

toolbar-layout = QHBoxLayout () toolbar_layout.setContentsMargins(0, 0, 0, 0) toolbar_layout . setSpacing (0)

toolbar-layout . addItem(QSpacerItem(15 , 0, QSizePolicy .Minimum, QSizePolicy .Minimum) ) toolbar-layout . addWidget( InfoIcon (”Click this buttons to switch plot modes \ n”

”between trajectories , expectations and variances”)) toolbar_layout . addItem(QSpacerItem(15 , 0, QSizePolicy .Minimum, QSizePolicy .Minimum) ) toolbar_layout . addWidget ( self . btn_to_fn) toolbar_layout . addWidget ( self .btn_to_mx) toolbar_layout . addWidget ( self . btn_to_dx) toolbar_layout . addWidget ( self . toolbar)

toolbar-layout . addItem(QSpacerItem(15 , 0, QSizePolicy .Minimum, QSizePolicy .Minimum) )

layout = QVBoxLayout ()

layout . setContentsMargins(0 , 0, 0, 0)

layout . setSpacing (0)

layout . addLayout (toolbar-layout)

layout . addWidget ( self . canvas)

self . setLayout(layout)

self . setSizePolicy ( QSizePolicy ( QSizePolicy . Expanding , QSizePolicy . Expanding) )

self . btn_to_fn . pressed . connect ( self . fn_mode) self . btn_to_mx . pressed . connect ( self . mx_mode) self . btn_to_dx . pressed . connect ( self . dx_mode)

def rescale(self):

self .ax. relim ( visible_only=True) self .ax.autoscale() self.canvas.draw ()

def clear(self):

for f in self . plots.values() : self . hide_label . emit( f . uid) if f. l ine_fn is not None:

f. line _f n . set_visible( False) if f.line_mx is not None:

f.line_mx. set_visible( False) if f.line_dx is not None:

f. line _d x . set_visible (False)

def fn_mode( self) :

self. mode = 0

self . clear ()

fo r f i n self . plots.values() :

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if f. l ine_fn is not None:

self . show_label . emit ( f. uid) if f .visible:

f. line _f n . set_visible( True) self.rescale() def mx_mode( self ) : self. mode = 1 self . clear ()

for f in self . plots.values() : if f.line_mx is not None:

self . show_label . emit ( f. uid) if f .visible:

f. line_mx. set_visible ( True) self.rescale() def dx_mode(self): self. mode = 2 self . clear ()

fo r f i n self . plots.values() : if f.line_dx is not None:

self . show_label . emit ( f. uid) if f .visible:

f. line_dx . set_visible(True) self.rescale()

def new_items(self, lines: list ):

for line in lines :

self.plots [line.uid] = line

fo r f i n self . plots.values() :

if f. l ine_fn is None and f.fn is not None:

f.line_fn = self .ax. plot ( f .t , f . fn , linewidth = 1, color=f . color ) [0] if f.line_mx is None and f.mx is not None:

f.line_mx = self.ax.plot(f.t, f. mx, linewidth = 1, color=f. color) [0] if f.line_dx is None and f.dx is not None:

f.line_dx = self.ax. plot (f.t, f. dx , linewidth = 1, color=f. color) [0]

if self. mode == 0: self. fn_mode ()

if self. mode == 1: self. mx_mode ()

if self. mode == 2:

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self. dx_mode ()

def delete_item(self, uid: int):

item = self.plots. pop (uid)

Color . free (item . color) if item . line_fn is not None: item . line_fn . remove () if item . line_mx is not None: item . line_mx . remove () if item.line_dx is not None: item . line_dx . remove ()

self.rescale()

def hide_item(self, uid: int):

item = self. plots [uid]

if item. line_fn is not None and self .mode == 0: item . line_fn . set_visible (False) if item . line_mx is not None and self.mode == 1: item . line_mx . set_visible ( False) if item.line_dx is not None and self.mode == 2: item . line_dx . set_visible ( False) item. visible = False

self.rescale()

def show_item (self, uid: int):

item = self. plots [uid]

if item. line_fn is not None and self.mode == 0: item . line_fn . set_visible (True) if item . line_mx is not None and self.mode == 1: item . line_mx . set_visible (True) if item.line_dx is not None and self.mode == 2: item . line_dx . set_visible (True) item. visible = True

self.rescale()

def show_all(self):

for item in self. plots. values ():

if item. line_fn is not None and self.mode == 0: item. line_fn . set_visible(True) if item . line_mx is not None and self.mode == 1: item . line_mx . set_visible (True) if item.line_dx is not None and self.mode == 2: item . li ne _dx . set_visible (True) item. visible = True

self.rescale()

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def hide_all(self):

for item in self. plots. values ():

if item. line_fn is not None and self.mode == 0: item. li ne _fn . set_visible (False) if item . line_mx is not None and self.mode == 1:

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item.line_mx. set_visible (False) if item.line_dx is not None and self.mode == 2: item. line_dx . set_visible (False) item. visible = False

self.rescale()

def delete_all(self):

for item in reversed (self. plots. values ()):

Color . free (item . color ) if item.line_fn is not None: item. li ne _fn . remove () if item . line_mx is not None: item . line_mx . remove () if item.line_dx is not None: item . li ne _dx . remove ()

self.plots.clear()

self.rescale()

def on_resize(self , event): self . figure . tight_layout () self.canvas.draw ()

Listing 26: Curves color

from random import choice

class Color :

reserved_colors = [

”#ff834a” ,

”#ffe100” ,

”#c700c7” ,

”#24e280” ,

”#1100ff” ,

”#f f 1 e2 2” ,

]

a v a i l a b l e_c o lo r s = [

”#ff834a” ,

”#ffe100” ,

”#c700c7” ,

”#24e280” ,

”#1100ff” ,

”#ff 1 e 2 2” ,

]

def __new__(cls, * args, ** kwargs ) : try :

return cls . available_colors .pop ()

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Listing 27: Curve

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6.1.3 Source Codes of Input for Nonlinear Systems of Ito SDEs

from ui . charts . visuals . color import Color

class Line : count = 0

def __init__(self, name, t, fn , mx=None, dx=None) : self.name = name self.t = t self.fn = fn self. mx = mx self.dx = dx self.line_fn = None self.line_mx = None self.line_dx = None self. visible = True self. color = Color ()

self.uid = Line.count Line.count += 1

except IndexError:

return f”# { ’’. join ([ choice (’0123456789ABCDEF’) for j in range(6)]) } @classmethod

def free(cls , code: str) :

if code in cls.reserved_colors:

cls . available_colors .append(code)

Listing 28: Base part of data input for nonlinear systems

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import logging import numpy as np

from PyQt5.QtCore import QThreadPool , pyqtSignal

from PyQt5. QtWidgets import QCheckBox, QPushButton , QStyle , QApplication ,\

QSizePolicy , QHBoxLayout, QSpacerItem , QVBoxLayout, QStackedWidget , \

QWidget , QLabel from sympy import Matrix

import config

from mathematics . sde . nonlinear . drivers . euler import euler

from mathematics . sde . nonlinear . drivers . milstein import milstein

from mathematics . sde . nonlinear . drivers . strong_taylor_ito_1p5 import strong_taylor_ito_1p5

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from mathematics . sde . nonlinear . drivers . strong_taylor_ito_2p0 import strong_taylor_ito_2p0

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from mathematics . sde . nonlinear . drivers . strong_taylor_ito_2p5 import strong_taylor_ito_2p5

from mathematics . sde . nonlinear . drivers . strong_taylor_ito_3p0 import strong_taylor_ito_3p0

from mathematics . sde . nonlinear . drivers . strong_taylor_stratonovich_1p0 import

strong_taylor_stratonovich_1p0

from mathematics . sde . nonlinear . drivers . strong_taylor_stratonovich_1p5 import strong_taylor_stratonovich_1p5

from mathematics . sde . nonlinear . drivers . strong_taylor_stratonovich_2p0 import strong_taylor_stratonovich_2p0

from mathematics . sde . nonlinear . drivers . strong_taylor_stratonovich_2p5 import strong_taylor_stratonovich_2p5

from mathematics . sde . nonlinear . drivers . strong_taylor_stratonovich_3p0 import strong_taylor_stratonovich_3p0

from mathematics . sde . nonlinear . symbolic . coefficients . c import C from tools import database

from ui . async_calls . worker import Worker

from ui . charts . visuals . line import Line

from ui . main . modeling . nonliear . step1 import Step 1

from ui . main . modeling . nonliear . step2 import Step2

from ui . main . modeling . nonliear . step3 import Step3

from ui . main . modeling . nonliear . step4 import Step4

from ui . main . modeling . nonliear . step5 import Step5

class NonlinearModelingWidget(QWidget) :

show_main_menu = pyqtSignal () start_progress = pyqtSignal(str) stop_progress = pyqtSignal(str) draw_chart = pyqtSignal ( list )

def __i n it __ ( self , parent=None) :

super (QWidget, self) . __init__ (parent)

self. logger = logging.getLogger(__name__)

self.scheme_id = 0 self. schemes = [

(euler , ”Euler” , ”Euler Scheme”) ,

(milstein , ”Milstein” , ”Milstein Scheme”) ,

( strong_taylor_ito_1p5 , ”Taylor — Ito 1.5” ,

”Strong Taylor — Ito Scheme with Convergence Order 1.5”) ,

( strong_taylor_ito_2p0 , ”Taylor — Ito 2.0” ,

”Strong Taylor — Ito Scheme with Convergence Order 2.0”) ,

( strong_taylor_ito_2p5 , ”Taylor — Ito 2.5” ,

”Strong Taylor — Ito Scheme with Convergence Order 2.5”) ,

( strong_taylor_ito_3p0 , ”Taylor — Ito 3.0” ,

”Strong Taylor — Ito Scheme with Convergence Order 3.0”) ,

( strong_taylor_stratonovich_1p0 , ”Taylor — Str. 1.0” ,

”Strong Taylor — Stratonovich Scheme with Convergence Order 1.0”) , ( strong_taylor_stratonovich_1p5 , ”Taylor — Str. 1.5” ,

”Strong Taylor — Stratonovich Scheme with Convergence Order 1.5”) , ( strong_taylor_stratonovich_2p0 , ”Taylor — Str. 2.0” ,

”Strong Taylor — Stratonovich Scheme with Convergence Order 2.0”) , ( strong_taylor_stratonovich_2p5 , ”Taylor — Str. 2.5” ,

”Strong Taylor — Stratonovich Scheme with Convergence Order 2.5”) ,

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( strong_taylor_stratonovich_3p0 , ”Taylor — Str . 3.0” ,

’’Strong Taylor — Stratonovich Scheme with Convergence Order 3.0”) ,

]

self . stack_widget = QStackedWidget( self )

self.step1 = Step1() self.step2 = Step2() self.step3 = Step3() self.step4 = Step4() self.step5 = Step5()

back_btn = QPushButton(”Back” , self)

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back_btn . setIcon (QApplication . style () . standardIcon(QStyle. SP_ArrowBack) ) self . charts_check = QCheckBox(”Charts window”, self) self.scheme_name = QLabel ()

self . scheme_name. setSizePolicy ( QSizePolicy . Expanding , QSizePolicy . Minimum)

bar_layout = QHBoxLayout () bar_layout . addWidget( back_btn )

bar_layout . addItem(QSpacerItem(10 , 35, QSizePolicy .Minimum, QSizePolicy .Minimum)) bar_layout . addWidget ( self. scheme_name)

bar_layout . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bar_layout . addWidget ( self . charts_check)

self . stack_widget . addWidget (self.step1) self . stack_widget . addWidget ( self . step2) self . stack_widget . addWidget ( self . step3) self . stack_widget . addWidget ( self . step4) self . stack_widget . addWidget ( self . step5)

layout = QVBoxLayout ()

layout . addLayout ( bar_layout)

layout . addWidget ( self . stack_widget)

self . setLayout(layout)

back_btn . clicked . connect ( self . show_main_menu . emit) back_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget ( self . step1) )

self . step1 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step2) )

self . step2 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget ( self . step1) )

self . step2 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step3) )

self . step3 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step2) )

self . step3 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step4) )

self . step4 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step3) )

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self . step4 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step5) )

self . step5 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step4) )

self . step5 . run_btn . clicked . connect ( lambda: self . run_modeling () )

self . step5 . run_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget ( self . step1) )

self . step1 . n_valid . connect ( self . step2 . matrix . resize_h ) self . step1 . n- valid . connect ( self . step3 . matrix . resize_h ) self . step1 . n_valid . connect ( self . step4 . matrix . resize_h )

self . step1 . m_valid . connect ( self . step3 . matrix . resize_w )

def set_scheme(self, scheme_id): self.scheme_id = scheme_id

self . schem^name. setText( self .schemes [scheme_id ] [2] ) if scheme_id == 0:

self . step5 . count_c ( False) else :

self . step5 . counLc (True)

def run_modeling( self ) :

self.start_progress.emit(”The modeling is being performed.. .”)

worker = Worker( self . routine )

worker . signals . result . connect ( self . on_modeling_finish ) worker . signals . error . connect ( self . on_modeling_corrupted) QThreadPool .globalInstance() .start (worker)

def routine(self):

scheme = self. schemes [ self. scheme_id]

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a = Matrix( self . step2 . matrix .m) b = Matrix( self . step3 . matrix .m)

x0 = np.ndarray(shape = (self.step4. matrix. rowCount () ,

self . step4 . matrix .columnCount() ) , dtype=float) for i in range( self . step4 . matrix . rowCount () ) :

for j in range ( self . step4 . matrix .columnCount() ) : x0[i][j] = float (self.step4.matrix.m[i][j])

if self.step5.s != 0:

np . random . seed ( self . step5 . s)

database.connect(config .database)

if self.scheme_id == 0: result = scheme [0]( x0 , a, b,

(self.step5.t0 , self.step5.dt ,

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Differential Equations and Control Processes, N. 1, 2021

self.step5.t1)

)

else :

C. preload (56 , 56, 56, 56, 56) result = scheme [0](

x0, a, b, self.step5.c,

(self.step5.t0 , self.step5.dt , self.step5.t1)

)

database.disconnect()

lines = [Line(f” {scheme [1] } , x { i + 1} ”,

np.array( result [1]) .astype( float) , np.array(result [ 0 ][ i , :]) .astype( float))

for i in range(len(result [0]) )]

return lines

def on_modeling_finish ( self , result) :

self.stop_progress.emit(”The modeling has been completed!”) self . draw_chart . emit( result )

def on_modeling_corrupted ( self , result ) :

self.logger.error(result [0]) self.logger.error(result [1]) self.logger.error(result [2])

self . stop_progress . emit(”The modeling failed !” )

Listing 29: Step 1 of data input for nonlinear systems

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self.input_stack = self.parent()

info _n = InfoIcon (” Dimension of linear system of Ito SDEs” )

info_m = InfoIcon (” Dimension of vector Wiener process” )

label _n = QLabel(”n”) label_m = QLabel(”m”)

self. lineedit_n = QLineEdit() self.lineedit_m = QLineEdit()

self.msg_n = ErrorWidget(”Wrong value!”) self . ms^n . hide ()

self.msg_m = ErrorWidget(”Wrong value!”) self. ms^m . hide ()

grid_layout = QGridLayout () grid_layout . addWidget( self . ms^n , 0, 2) grid_layout . addWidget( self . ms^m , 2, 2)

grid_layout . addWidget( info_n , 1, 0)

grid_layout . addWidget(info_m , 3, 0) grid_layout . addWidget( labeLn , 1, 1)

grid_layout . addWidget(label_m , 3, 1)

grid_layout . addWidget( self . lineedit _n , 1, 2)

grid_layout . addWidget (self.lineedit_m , 3, 2)

header = QLabel(”Dimensions settings”, parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . next_btn . setEnabled( False)

header_layout = QHBoxLayout ()

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) header_layout . addWidget( header )

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self. next _btn)

eq1 = QHBoxLayout ()

eq1 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq1 . addWidget (SVG(” equation1.svg” , scale_factor = 1.))

eq1 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq2 = QHBoxLayout ()

eq2 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq2 . addWidget (SVG(” equation2.svg” , scale_factor = 1.))

eq2 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) )

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equalities_layout = QVBoxLayout () equalities_layout . addLayout (eq1)

equalities.layout . addItem(QSpacerItem(0 , 20, QSizePolicy .Minimum, QSizePolicy .Minimum

))

equalities_layout . addLayout (eq2) equalities_wrap = QHBoxLayout ()

equalities_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum)

)

equalities_wrap . addLayout ( equalities_layout )

equalities_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum)

)

grid_wrap = QHBoxLayout ()

grid_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) grid_wrap . addLayout ( grid_layout )

grid_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) layout = QVBoxLayout ()

layout . addItem(QSpacerItem(0 , 0, QSizePolicy .Minimum, QSizePolicy . Expanding)) layout . addLayout ( equalities_wrap )

layout . addItem(QSpacerItem(0 , 25, QSizePolicy .Minimum, QSizePolicy .Minimum)) layout . addLayout ( header_layout)

layout . addItem(QSpacerItem(0 , 5, QSizePolicy .Minimum, QSizePolicy .Minimum)) layout . addLayout (grid_wrap)

layout . addItem(QSpacerItem(0 , 0, QSizePolicy .Minimum, QSizePolicy . Expanding)) layout . addItem(QSpacerItem(0 , 25, QSizePolicy .Minimum, QSizePolicy .Minimum)) layout . addLayout (bottom_bar)

self . setLayout(layout)

self . lineedit_n . textChanged . connect ( self . validate_n ) self. lineedit_m . textChanged . connect ( self . validate_m )

def validate_form ( self ) :

if self. n_is_valid and self. m_is_valid : self . next _btn . setEnabled( True) else :

self . next_btn . setEnabled ( False)

def validate_n(self, value): try :

typed_value = int (value) if typed_value <= 0: raise ValueError()

self . n _is-valid = True

self . n - valid . emit (typed_value)

self. msg_n . hide ()

except ValueError :

self.n_is_valid = False self. msg_n . show ()

Electronic Journal. http://diffjournal.spbu.ru/ 248

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Listing 30: Step 2 of data input for nonlinear systems

from PyQt5. QtWidgets import QWidget, QHBoxLayout, QLabel , QSpacerltem , QSizePolicy, \ QVBoxLayout, QPushButton , QApplication , QStyle

from ui .main. error import ErrorWidget from ui . main . info import InfoIcon

from ui .main. modeling . matrix_widget import MatrixWidget

class Step2(QWidget) :

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def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

self.errors = 0

header = QLabel(”Setting of column a(x, t)” , parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

info = InfoIcon(”Elements of column a(x, t) are expected to be functions \ n”

” Size : n x 1 \n”

’’Functions must be set in python and SymPy notation”)

self.msg = ErrorWidget(”Wrong values in matrix!”) self. msg . hide ()

self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) )

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self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next_btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

Listing 31: Step 3 of data input for nonlinear systems

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self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next _btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

Listing 32: Step 4 of data input for nonlinear systems

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self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self.prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next_btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

Listing 33: Step 5 of data input for nonlinear systems

import sys

from PyQt5.QtWidgets import QWidget, QHBoxLayout, QLabel , QLineEdit , \ QGridLayout , QVBoxLayout, QSizePolicy, QSpacerItem , QPushButton , \ QApplication , QStyle

from ui .main. error import ErrorWidget from ui . main . info import InfoIcon

class Step5(QWidget) :

def __i n it __ ( self , parent=None) :

super (QWidget, self) . __init__ (parent)

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

self.t0 = sys.float_info. min self.dt = 0

self.t1 = sys.float.info.max self.s = 0 self.c = 0

self.t0_is_valid = False self.dt_is_valid = False self.t1_is_valid = False self . s_is_valid = True

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self.c_is_valid = False self.input_stack = self.parent()

info _t0 = InfoIcon(”Start point of integration interval\n”

” Must be in [0, t1) range”) info _dt = InfoIcon(”Integration step \ n”

” Must be set in (0, 1) interval”)

info_t1 = InfoIcon (”Final point of integration interval\n”

” Must be more then t0” )

info_s = InfoIcon (”This is random generator seed\n”

” If You do not want to use specific \n”

”seed just leave this field empty” )

self.info_c = InfoIcon (”The constant which defines approximation accuracy”)

label_t0 = QLabel(”t0”) label _dt = QLabel (”dt” ) label_t1 = QLabel(”t1”) labeLs = QLabel(”seed”) self. label_c = QLabel(”C”)

self . lineedit _t0 = QLineEdit() self. lineedit_dt = QLineEdit() self . lineedit _t1 = QLineEdit() self. lineedit_s = QLineEdit() self. lineedit_c = QLineEdit()

self.msg_t0 = ErrorWidget(”Wrong value!”) self . msg_t0 . hide ()

self.msg_dt = ErrorWidget(”Wrong value!”) self. msg_dt . hide ()

self.msg_t1 = ErrorWidget(”Wrong value!”) self . msg_t1 . hide ()

self.msg_s = ErrorWidget(”Wrong value!”) self . ms^s. hide ()

self.msg_c = ErrorWidget(”Wrong value!”) self . ms^c . hide ()

header = QLabel(”Accuracy settings”, parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) ) self.run_btn = QPushButton(”Perform modeling”, self)

self . run_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . run_btn . setEnabled ( False)

header_layout = QHBoxLayout ()

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header_layout . addWidget( header )

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) botto^_bar = QHBoxLayout ()

botto^_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget ( self . run_btn)

grid-layout = QGridLayout ()

grid-layout . addWidget( self . msg_t0 , 0, 2)

grid_layout . addWidget( info_t0 , 1, 0)

grid-layout . addWidget( label_t0 , 1, 1)

grid-layout . addWidget( self . lineedit_t0 , 1, 2)

grid_layout . addWidget( self . msg_dt , 0, 5)

grid-layout . addWidget( info_dt , 1,3)

grid-layout . addWidget( label_dt , 1, 4)

grid-layout . addWidget( self . lineedit_dt , 1, 5)

grid-layout . addWidget( self . msg_t1 , 0, 8)

grid-layout . addWidget( info_t1 , 1, 6)

grid-layout . addWidget( label_t1 , 1, 7)

grid-layout . addWidget( self . lineedit_t1 , 1, 8)

grid-layout . addWidget( self . msg_s , 2, 2)

grid-layout . addWidget( info_s , 3, 0) grid_layout . addWidget( label_s , 3, 1)

grid-layout . addWidget( self . lineedit_s , 3, 2)

grid-layout . addWidget( self . msg_c , 2, 5)

grid-layout . addWidget( self . info_c , 3, 3)

grid-layout . addWidget( self . label_c , 3, 4) grid-layout . addWidget( self . lineedit_c , 3, 5)

column_layout = QVBoxLayout ()

column_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding)

)

column_layout . addLayout ( header_layout)

column_layout . addItem(QSpacerItem(0 , 25, QSizePolicy .Minimum, QSizePolicy .Minimum)) column_layout . addLayout ( grid-layout )

column_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding)

)

control-layout = QHBoxLayout ()

control-layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding

))

control-layout . addLayout (column_layout)

control-layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding

))

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layout = QVBoxLayout ()

layout . addLayout ( control-layout )

layout . addLayout (bottom_bar)

self . setLayout(layout)

self. lineedit_t0 . textChanged . connect ( self . validate_t0 )

self. lineedit_t0 . textChanged . connect ( self . validate_t1 )

self. lineed it _t0 . textChanged . connect ( self . validate_dt )

self. lineed it _dt . textChanged . connect ( self . validate_dt )

self. lineed it _t1 . textChanged . connect ( self . validate_t0 )

self. lineed it _t1 . textChanged . connect ( self . validate_t1 )

self. lineed it _t1 . textChanged . connect ( self . validate_dt )

self. lineedit _s . textChanged . connect ( self . validate_s )

self.lineedit_c. textChanged . connect ( self . validate_c )

def count_c ( self , flag: bool):

if flag :

self . i nfo _c . show () self . label _c . show () self. lineedit_c . show () self . validate_c () else :

self. i nfo _c .hide() self.label_c. hide () self. lineedit_c .hide() self . c _is _ valid = True

self . msg_c . hide ()

def validate_form ( self ) : if self . t0_is_valid \

and self . dt_is_valid \ and self.t1_is_valid \ and self.s_is_valid \ and self. c _is _valid : self . run_btn . setEnabled ( True) else :

self . run_btn . setEnabled( False)

def validate_t0(self): try :

typed _value = float ( self. lineedit_t0. text ()) if typed_value >= self . t1 : raise ValueError()

self.t0_is_valid = True self.t0 = typed_value self . msg_t0 . hide ()

except ValueError :

self.t0_is_valid = False self . msg_t0 . show ()

Electronic Journal. http://diffjournal.spbu.ru/ 255

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def validate_dt(self ) : try :

typed _value = float ( self. lineedit_dt. text ()) if typed_value <= 0 \

or (self.t1 — self.t0) / typed_value < 1 \ or typed_value >= 1: raise ValueError()

self . dt_is_valid = True self.dt = typed_value self . msg_dt . hide () self . validate_form ()

except ValueError :

self.dt_is_valid = False self . msg_dt . show ()

def validate_t1(self): try :

typed-value = float ( self. lineedit_t1.text ()) if typed_value <= self . t0 : raise ValueError ()

self . t1 _is_valid = True self.t1 = typed_value self . ms^t1 . hide ()

except ValueError :

self.t1_is_valid = False self . msg_t1 . show ()

def validate_s ( self ) : try :

if self.lineedit_s.text() == ””: self.s_is_valid = True self.s = 0 else :

typed_value = int (self. lineedit_s.text ()) if typed_value <= 0: raise ValueError () self.s = typed_value

self . s _is _valid = True self . msg_s . hide ()

except ValueError :

self.s_is_valid = False self . msg_s . show ()

finally :

self . validate_form ()

def validate_c(self): try :

typed _value = float ( self. lineedit_c. text ())

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if typed_value <= 0: raise ValueError()

self.c_is_valid = True self.c = typed_value self . msg_c . hide ()

exce pt ValueError :

self.c_is_valid = False self . msg_c . show ()

finally :

self . validate_form ()

6.1.4 Source Codes of Input for Linear Systems of Ito SDEs

Listing 34: Base part of data input for linear systems

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import logging

from time import time

import numpy as np

from PyQt5.QtCore import QThreadPool , pyqtSignal

from PyQt5. QtWidgets import QCheckBox, QPushButton , QStyle , QApplication , \ QSizePolicy , QHBoxLayout, QSpacerItem , QVBoxLayout, QStackedWidget , \ QWidget , QLabel

from mathematics . sde . linear . dindet import dindet

from mathematics . sde . linear . distortions import Symbolic , ComplexDistortion

from mathematics. sde. linear . integration import Integral

from mathematics . sde . linear . stoch import stoch

from ui . async_calls . worker import Worker

from ui . charts . visuals . line import Line

from ui . main . modeling . linear . step1 import Step1

from ui . main . modeling . linear . step2 import Step2

from ui . main . modeling . linear . step3 import Step3

from ui . main . modeling . linear . step4 import Step4

from ui . main . modeling . linear . step5 import Step5

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from ui . main . modeling . linear . step6 import Step6

from ui . main . modeling . linear . step7 import Step7

from ui . main . modeling . linear . step8 import Step8

class LinearModelingWidget(QWidget) :

show_main_menu = pyqtSignal () start_progress = pyqtSignal(str) stop_progress = pyqtSignal(str) draw_chart = pyqtSignal ( list )

def __i n it __ ( self , parent=None) :

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super (QWidget, self) . __init__ (parent)

self. logger = logging.getLogger(__name__)

self . stack_widget = QStackedWidget( self )

self.step1 = Step1() self.step2 = Step2() self.step3 = Step3() self.step4 = Step4() self.step5 = Step5() self.step6 = Step6() self.step7 = Step7() self.step8 = Step8()

back_btn = QPushButton (”Back” , self)

back_btn . setIcon (QApplication . style () . standardIcon(QStyle. SP_ArrowBack) )

self . charts_check = QCheckBox(”Charts window”, self)

self . schem^name = QLabel (”Linear Systems of Ito SDEs” )

bar_layout = QHBoxLayout () bar_layout . addWidget( back_btn )

bar_layout . addItem(QSpacerItem(10 , 35, QSizePolicy .Minimum, QSizePolicy .Minimum)) bar_layout . addWidget ( self. scheme_name)

bar_layout . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bar_layout . addWidget ( self . charts_check)

self . stack_widget . addWidget (self.step1) self . stack_widget . addWidget ( self . step2) self . stack_widget . addWidget ( self . step3) self . stack_widget . addWidget ( self . step4) self . stack_widget . addWidget ( self . step5) self . stack_widget . addWidget ( self . step6) self . stack_widget . addWidget ( self . step7) self . stack_widget . addWidget ( self . step8)

layout = QVBoxLayout ()

layout . addLayout ( bar_layout)

layout . addWidget ( self . stack_widget)

self . setLayout(layout)

back_btn . clicked . connect ( self . show_main_menu . emit) back_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget ( self . step1) )

self . step1 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step2) )

self . step2 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget ( self . step1) )

self . step2 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step3) )

self . step3 . prev_btn . clicked . connect (

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lambda: self . stack_widget . setCurrentWidget( self . step2) )

self . step3 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step4) )

self . step4 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step3) )

self . step4 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step5) )

self . step5 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step4) )

self . step5 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step6) )

self . step6 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step5) )

self . step6 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step7) )

self . step7 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step6) )

self . step7 . next_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step8) )

self . step8 . prev_btn . clicked . connect (

lambda: self . stack_widget . setCurrentWidget( self . step7) )

self . step8 . run_btn . clicked . connect ( lambda: self . run_modeling () )

self.step8.run_btn . clicked .connect(

lambda: self . stack_widget . setCurrentWidget ( self . step1) )

self . step1 . n_valid . connect ( self . step2 . matrix . resize_h ) self . step1 . n _valid . connect ( self . step2 . matrix . resize_w )

self . step1 . n _valid . connect ( self . step3 . matrix . resize_h )

self . step1 . k_valid . connect ( self . step3 . matrix . resize_w )

self . step1 . n_valid . connect ( self . step4 . matrix . resize_h ) self . step1 . m .valid . connect ( self . step4 . matrix . resize_w )

self . step1 . k .valid . connect ( self . step5 . matrix . resize_h )

self . step1 . n .valid . connect ( self . step6 . matrix . resize_w )

self . step1 . n- valid . connect ( self . step7 . matrix . resize_h )

def run_modeling( self ) :

self.start_progress.emit(”The modeling is being performed.. .”)

worker = Worker( self . routine )

worker . signals . result . connect ( self . on_modeling_finish ) worker . signals . error . connect ( self . on_modeling_corrupted) QThreadPool .globalInstance() .start (worker)

def routine(self):

n = int(self.step1.lineedit_n.text()) m = int ( self . step1 . lineedit_m . text () ) k = int ( self. step1.lineedit_k. text ()) t0 = float ( self. step8.lineedit_t0.text ())

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dt = float ( self. step8.lineedit_dt.text ()) t1 = float ( self. step8.lineedit_t1.text ())

self. logger.info(” Reading input data”)

integral = Integral(n)

integral.k, integral.m, integral.dt, integral.t0, integral.tk = \

k, m, dt , t0 , t1

integral . m_a = np.array ( [[ float ( self. step2 . matrix.m[ i ] [j ])

for j in range( self . step2 . matrix . columnCount () ) ]

for i in range ( self . step2 . matrix. rowCount () ) ])

integral . mat_b = np. array ( [[ float ( self. step3 . matrix.m[ i ] [j ])

for j in range ( self . step3 . matrix .columnCount() ) ]

for i in range ( self . step3 . matrix . rowCount () ) ])

integral . mat_f = np. array ( [[ float ( self. step4 . matrix.m[ i ] [j ])

for j in range ( self . step4 . matrix .columnCount() ) ]

for i in range ( self . step4 . matrix . rowCount () ) ])

integral . m_h = np.array ( [[ float ( self. step6 . matrix.m[ i ] [j ])

for j in range( self . step6 . matrix . columnCount () ) ]

for i in range ( self . step6 . matrix. rowCount () ) ])

integral . m_x0 = np.array ( [[ float ( self. step7. matrix.m[ i ] [j ])

for j in range ( self . step7 . matrix .columnCount() ) ] for i in range ( self . step7 . matrix. rowCount () ) ])

integral . m_mx0 = np. array ( [[ float ( self. step7. matrix.m[ i ] [j ])

for j in range ( self . step7 . matrix .columnCount() ) ]

for i in range ( self . step7 . matrix . rowCount () ) ])

integral. m_dx0 = np.zeros((integral .n, integral.n))

self.logger.info(”Input is correct”)

self . logger . info (” Calculation of Ad and Bd (Algorithm 11.2)”)

integral. m_ad, integral . m_bd = dindet(

integral.n, integral .k, integral . m_a, integral.mat_b, integral .dt)

self . logger . info (’’Calculation of Fd (Algorithm 11.6)”)

integral. m_fd = stoch(integral.n, integral. m_a, integral. mat _f, integral.dt)

mat_u = np.array([[ object ]] * self.step5.matrix. rowCount () )

for i in range( self . step5 . matrix . rowCount () ) :

mat_u [ i ] [0] = Symbolic( self .step5 . matrix.m[ i ] [0] )

integral. distortion = ComplexDistortion( self. step5. matrix. rowCount () , mat_u)

self . logger . info (” Starting modeling loop” )

start_time = time ()

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integral.integrate()

self. logger. info(f”Integration took {(time () — start_time):.3f} seconds”)

lines = [ Line( f”Linear , x{ i + 1}” ,

np.array(integral.v_t) .astype(float) , np. array( integral . m_xt [i , :]) .astype( float) ,

mx=np . array ( integral . m_mx [i , :]) .astype( float) , dx=np . array ( integral . m_dx [i, :]) .astype( float)) for i in range(integral . m_xt.shape [0]) ]

name = f” Linear”

lines .append(Line(f” Linear , y” ,

np.array(integral.v_t) . astype(float) , np.array(integral.v_yt) . astype(float) , mx=np.array(integral.v_my).astype(float), dx=np.array(integral.v_dy).astype(float)))

return lines

def on_modeling_finish ( self , result) :

self.stop_progress.emit(”The modeling has been completed!”) self . draw_chart . emit( result )

def on_modeling_corrupted ( self , result ) : self.logger.error(result [0]) self.logger.error(result [1]) self.logger.error(result [2])

self . stop_progress . emit(”The modeling failed !” )

Listing 35: Step 1 of data input for linear systems

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self.m_is_valid = False self.k_is_valid = False

self.input_stack = self.parent()

info_n = InfoIcon(”Dimension of linear system of Ito SDEs” )

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info_m = InfoIcon (’’Dimension of vector Wiener process”)

info_k = InfoIcon(”Dimension of vector function u(t)”)

labeLn = QLabel(”n”) label_m = QLabel(”m”) label _k = QLabel(”k”)

self. lineedit_n = QLineEdit() self.lineedit_m = QLineEdit() self. lineedit_k = QLineEdit()

self.msg_n = ErrorWidget(”Wrong value!”) self . msg_n . hide ()

self.msg_m = ErrorWidget(”Wrong value!”) self. msg_m . hide ()

self.msg_k = ErrorWidget(”Wrong value!”) self . ms^k. hide ()

grid_layout = QGridLayout () grid_layout . addWidget( self .msg_n , 0, 2) grid_layout . addWidget( self . ms^m , 2, 2) grid_layout . addWidget( self . ms^k , 4, 2) grid_layout . addWidget( info_n , 1, 0)

grid_layout . addWidget(info_m , 3, 0) grid_layout . addWidget (info_k , 5, 0) grid_layout . addWidget( labeLn , 1, 1)

grid_layout . addWidget(label_m , 3, 1)

grid_layout . addWidget ( label_k , 5, 1)

grid_layout . addWidget( self . lineedit_n , 1, 2)

grid_layout . addWidget (self.lineedit_m , 3, 2) grid_layout . addWidget( self . lineedit_k , 5, 2)

header = QLabel(”Dimensions settings”, parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . next_btn . setEnabled( False)

header_layout = QHBoxLayout ()

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) header_layout . addWidget( header )

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

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bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self. next _btn)

eq1 = QHBoxLayout ()

eq1 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq1 . addWidget (SVG(” equation3.svg” , scale_factor = 1.))

eq1 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq2 = QHBoxLayout ()

eq2 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) eq2 . addWidget (SVG(” equation4.svg” , scale_factor = 1.))

eq2 . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) )

equalities_layout = QVBoxLayout () equalities_layout . addLayout (eq1)

equalities, layout . addItem(QSpacerItem(0 , 25, QSizePolicy .Minimum, QSizePolicy .Minimum

))

equalities_layout . addLayout (eq2) equalities_wrap = QHBoxLayout ()

equalities_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum)

)

equalities_wrap . addLayout ( equalities_layout )

equalities_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum)

)

grid_wrap = QHBoxLayout ()

grid_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) grid_wrap . addLayout ( grid-layout )

grid_wrap . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) layout = QVBoxLayout ()

layout . addItem(QSpacerItem(0 , 0, QSizePolicy .Minimum, QSizePolicy . Expanding)) layout . addLayout ( equalities_wrap )

layout . addItem(QSpacerItem(0 , 25, QSizePolicy .Minimum, QSizePolicy .Minimum)) layout . addLayout ( header_layout)

layout . addItem(QSpacerItem(0 , 5, QSizePolicy .Minimum, QSizePolicy .Minimum)) layout . addLayout (grid_wrap)

layout . addItem(QSpacerItem(0 , 0, QSizePolicy .Minimum, QSizePolicy . Expanding)) layout . addItem(QSpacerItem(0 , 25, QSizePolicy .Minimum, QSizePolicy .Minimum)) layout . addLayout (bottom_bar)

self . setLayout(layout)

self . lineedit_n . textChanged . connect ( self . validate_n ) self. lineedit_m . textChanged . connect ( self . validate_m ) self. lineedit_k . textChanged . connect ( self . validate_k )

def validate_form ( self ) :

if self. n_is_valid and self. m_is_valid and self . k_is_valid : self . next _btn . setEnabled( True) else :

self . next_btn . setEnabled ( False) def validate_n(self, value):

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try :

typed _value = int (value) if typed_value <= 0: raise ValueError()

self . n _is .valid = True

self . n _valid . emit (typed_value)

self. msg_n . hide ()

except ValueError :

self.n_is_valid = False self. msg_n . show ()

finally :

self . validate_form ()

def validate_m(self , value) : try:

typed _value = int (value) if typed_value <= 0: raise ValueError ( )

self . m_is_valid = True

self . m _valid . emit (typed_value)

self. msg_m . hide ()

except ValueError :

self.m_is_valid = False self. msg_m . show ()

fi n a l l y :

self . validate_form ()

def validate_k(self , value) : try:

typed _value = int (value) if typed_value <= 0: raise ValueError ( )

self . k_is_valid = True

self . k- valid . emit(typed_value)

self. msg_k . hide ()

except ValueError :

self.k_is_valid = False self. msg_k . show ()

fi n a l l y :

self . validate_form ()

Listing 36: Step 2 of data input for linear systems

from PyQt5. QtWidgets import QWidget, QHBoxLayout, QLabel , QSpacerItem , QSizePolicy, \

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QVBoxLayout, QPushButton , QApplication , QStyle

from ui .main. error import ErrorWidget from ui . main . info import InfoIcon

from ui .main. modeling . matrix_widget import MatrixWidget

class Step2(QWidget) :

def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

self.errors = 0

header = QLabel(”Setting of matrix A” , parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

info = InfoIcon(”Elements of matrix A are\n”

”expected to be real values\n”

”Size : n x n”)

self.msg = ErrorWidget(”Wrong values in matrix!”) self. msg . hide ()

self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next_btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

self . matrix . itemChanged . connect ( self . validate_item ) def validate_item(self, item):

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Listing 37: Step 3 of data input for linear systems

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self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self.prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next _btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

self . matrix . itemChanged . connect ( self . validate_item )

def validate_item(self, item):

value = item.text () try :

float ( value) if not item . valid : item. valid = True self.errors —= 1

except ValueError : if item . valid :

item. valid = False self.errors += 1

finally :

self . validate_form ()

def validate_form ( self ) :

if self.errors == 0: self. msg. hide ()

self . next _btn . setEnabled( True) else :

self. msg. show ()

self . next_btn . setEnabled ( False)

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Differential Equations and Control Processes, N. 1, 2021

Listing 38: Step 4 of data input for linear systems

from PyQt5. QtWidgets import QWidget, QHBoxLayout, QVBoxLayout, QLabel , QSpacerltem , \ QSizePolicy , QPushButton , QApplication , QStyle

from ui .main. error import ErrorWidget from ui . main . info import InfoIcon

from ui .main. modeling . matrix_widget import MatrixWidget

class Step4(QWidget) :

def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

self.errors = 0

header = QLabel(”Setting of matrix F” , parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

info = InfoIcon(”Elements of matrix F are\n”

”expected to be real values\n”

”Size : n x m”)

self.msg = ErrorWidget(”Wrong values in matrix!”) self. msg . hide ()

self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next_btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

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self . matrix . itemChanged . connect ( self . validate_item )

def validate_item(self, item):

value = item.text () try :

float ( value) if not item . valid : item. valid = True self.errors —= 1

except ValueError : if item . valid :

item. valid = False self.errors += 1

finally :

self . validate_form () def validate_form ( self ) :

if self.errors == 0: self. msg. hide ()

self . next _btn . setEnabled( True) else :

self. msg. show ()

self . next_btn . setEnabled ( False)

Listing 39: Step 5 of data input for linear systems

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Differential Equations and Control Processes, N. 1, 2021

self.msg = ErrorWidget(”Wrong values in matrix!”) self. msg . hide ()

self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next_btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

Listing 40: Step 6 of data input for linear systems

from PyQt5. QtWidgets import QWidget, QHBoxLayout, QVBoxLayout, QLabel , QSpacerItem , \ QSizePolicy , QPushButton , QApplication , QStyle

from ui .main. error import ErrorWidget from ui . main . info import InfoIcon

from ui .main. modeling . matrix_widget import MatrixWidget

class Step6(QWidget) :

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def __i n it __ ( self , parent=None) :

super (QWidget, self) . __init__ (parent)

self.errors = 0

header = QLabel(”Setting of matrix H” , parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

info = InfoIcon(”Elements of matrix H are\n”

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Differential Equations and Control Processes, N. 1, 2021

”expected to be int values\n”

"Size: 1 x n”)

self.msg = ErrorWidget(”Wrong values in matrix!”) self. msg . hide ()

self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next_btn)

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

self . matrix . itemChanged . connect ( self . validate_item )

def validate_item(self, item):

value = item.text () try :

float ( value) if not item . valid : item. valid = True self.errors —= 1

except ValueError : if item . valid :

item. valid = False self.errors += 1

finally :

self . validate_form ()

def validate_form ( self ) :

if self.errors == 0: self. msg. hide ()

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Differential Equations and Control Processes, N. 1, 2021

self . next _btn . setEnabled( True) else :

self. msg. show ()

self . next_btn . setEnabled ( False)

Listing 41: Step 7 of data input for linear systems

from PyQt5. QtWidgets import QWidget, QHBoxLayout, QVBoxLayout, QLabel , QSpacerItem , \ QSizePolicy , QPushButton , QApplication , QStyle

from ui .main. error import ErrorWidget from ui . main . info import InfoIcon

from ui .main. modeling . matrix_widget import MatrixWidget

class Step7(QWidget) :

def __init__(self , parent=None) :

super (QWidget, self) . __init__ (parent)

self.errors = 0

header = QLabel(”Setting of column x0” , parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

info = InfoIcon(”Elements of column x0 are\n”

”expected to be real values\n”

” Size : n x 1”)

self.msg = ErrorWidget(”Wrong values in matrix!”) self. msg . hide ()

self. matrix = MatrixWidget( self )

self . next_btn = QPushButton(”Next” , self)

self . next_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) )

header_layout = QHBoxLayout () header_layout . addWidget (info) header_layout . addWidget( header ) header_layout . addWidget ( self . msg)

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) bottom_bar = QHBoxLayout ()

bottom_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget (self. next _btn)

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Differential Equations and Control Processes, N. 1, 2021

layout = QVBoxLayout () layout . addLayout ( header_layout) layout . addWidget ( self . matrix) layout . addLayout (bottom_bar)

self . setLayout(layout)

self . matrix . itemChanged . connect ( self . validate_item )

def validate_item(self, item):

value = item.text () try :

fl oat ( value) if not item . valid : item. valid = True self.errors —= 1

exce pt ValueError : if item . valid :

item. valid = False self.errors += 1

finally :

self . validate_form ()

def validate_form ( self ) :

if self.errors == 0: self. msg. hide ()

self . next _btn . setEnabled( True) else :

self. msg. show ()

self . next_btn . setEnabled ( False)

Listing 42: Step 8 of data input for linear systems

import sys

from PyQt5. QtWidgets import QWidget, QHBoxLayout, QLabel , QLineEdit , QGridLayout , \ QVBoxLayout, QSizePolicy , QSpacerItem , QPushButton , QApplication , QStyle

from ui .main. error import ErrorWidget from ui . main . info import InfoIcon

class Step8(QWidget) :

def __i n it __ ( self , parent=None) :

super (QWidget, self) . __init__ (parent)

self.t0 = sys. float_info. min self.dt = 0

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Differential Equations and Control Processes, N. 1, 2021

self.t1 = sys. float-info .max self.s = 0

self.t0_is_valid = False self.dt_is_valid = False self.t1_is_valid = False self . s_is_valid = True

self.input_stack = self.parent()

info_t0 = InfoIcon( ”Start point of integration interval\n”

” Must be in [0, t1) range”)

info _dt = InfoIcon( ”Integration step\n”

” Must be set in (0, 1) interval”)

info_t1 = InfoIcon( ”Final point of integration interval\n”

” Must be more then t0” )

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info _s = InfoIcon (” This is random generator see d\n”

”If You do not want to use specific \n”

” seed just leave this field empty” )

label_t0 = QLabel(”t0”) label_dt = QLabel (”dt” ) label_t1 = QLabel(”t1”) labeLs = QLabel(”seed”)

self . lineedit _t0 = QLineEdit() self. lineedit_dt = QLineEdit() self . lineedit_t1 = QLineEdit() self. lineedit_s = QLineEdit()

self.msg_t0 = ErrorWidget(”Wrong value!”) self . msg_t0 . hide ()

self.msg_dt = ErrorWidget(”Wrong value!”) self. msg_dt . hide ()

self.msg_t1 = ErrorWidget(”Wrong value!”) self . msg_t1 . hide ()

self.msg_s = ErrorWidget(”Wrong value!”) self . ms^s. hide ()

header = QLabel(”Accuracy settings”, parent=self) font = header.font() font . setPointSize(15) header.setFont(font)

self . prev_btn = QPushButton(”Back” , self)

self . prev_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowBack) ) self.run_btn = QPushButton(”Perform modeling”, self)

self . run_btn . setIcon ( QApplication . style () . standardIcon ( QStyle . SP_ArrowForward) ) self . run_btn . setEnabled ( False)

header_layout = QHBoxLayout ()

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header_layout . addWidget( header )

header_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy .Minimum) ) botto^_bar = QHBoxLayout ()

botto^_bar . addItem(QSpacerItem (0 , 0, QSizePolicy .Expanding , QSizePolicy .Minimum)) bottom _bar . addWidget (self.prev_btn) bottom _bar . addWidget ( self . run_btn)

grid-layout = QGridLayout ()

grid-layout . addWidget( self . msg_t0 , 0, 2)

grid_layout . addWidget( info_t0 , 1, 0)

grid-layout . addWidget( label_t0 , 1, 1)

grid-layout . addWidget( self . lineedit_t0 , 1, 2)

grid_layout . addWidget( self . msg_dt , 0, 5)

grid-layout . addWidget( info_dt , 1,3)

grid-layout . addWidget( label_dt , 1, 4)

grid-layout . addWidget( self . lineedit_dt , 1, 5)

grid-layout . addWidget( self . msg_t1 , 0, 8)

grid-layout . addWidget( info_t1 , 1, 6)

grid-layout . addWidget( label_t1 , 1, 7)

grid-layout . addWidget( self . lineedit_t1 , 1, 8)

grid-layout . addWidget( self . msg_s , 2, 2)

grid-layout . addWidget( info_s , 3, 0) grid_layout . addWidget( label_s , 3, 1)

grid-layout . addWidget( self . lineedit_s , 3, 2)

column_layout = QVBoxLayout ()

column_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding)

)

column_layout . addLayout ( header_layout)

column_layout . addItem(QSpacerItem(0 , 25, QSizePolicy .Minimum, QSizePolicy .Minimum)) column_layout . addLayout ( grid-layout )

column_layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding)

)

control-layout = QHBoxLayout ()

control-layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding

))

control-layout . addLayout (column_layout)

control-layout . addItem(QSpacerItem(0 , 0, QSizePolicy . Expanding , QSizePolicy . Expanding

))

layout = QVBoxLayout ()

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layout . addLayout ( control-layout )

layout . addLayout (bottom_bar)

self . setLayout(layout)

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self. lineedit_t0 . textChanged . connect ( self . validate_t0 ) self. lineedit_t0 . textChanged . connect ( self . validate_t1 ) self. lineedit_t0 . textChanged . connect ( self . validate_dt ) self. lineedit_dt . textChanged . connect ( self . validate_dt ) self. lineedit_t1 . textChanged . connect ( self . validate_t0 ) self. lineedit_t1 . textChanged . connect ( self . validate_t1 ) self. lineedit_t1 . textChanged . connect ( self . validate_dt ) self. lineedit_s . textChanged . connect ( self . validate_s )

def count_c ( self , flag: bool):

if flag :

self . i nfo _c . show () self . label _c . show () else :

self. i nfo _c .hide() self . label _c . hide()

self . msg_c . hide ()

def validate-form ( self ) : if self . t0_is_valid \

and self . dt_is_valid \ and self.t1_is_valid \ and self. s_is_valid : self . ru n_btn . setEnabled ( True) else :

self . run_btn . setEnabled( False)

def validate_t0(self): try :

typed_value = float ( self. lineedit_t0. text ()) if typed_value >= self . t1 : raise ValueError()

self . t0_is_valid = True self.t0 = typed_value self . msg_t0 . hide ()

except ValueError :

self.t0_is_valid = False self . msg_t0 . show ()

def validate_dt(self ) : try :

typed_value = float (self. lineedit_dt. text ()) if typed_value <= 0 \

or (self.t1 — self.t0) / typed_value < 1 \ or typed_value >= 1: raise ValueError ()

self . dt_is_valid = True self.dt = typed_value self . msg_dt . hide () self . validate_form ()

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6.2 Source Codes for Nonlinear Systems of Ito SDEs

6.2.1 Source Codes for Calculation of the Fourier—Legendre Coefficients

Listing 43: Symbolic function of the Legendre polinomial

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from sympy import Rational , factorial , diff

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Listing 44: Symbolic function of the Fourier—Legendre coefficient calculation

from sympy import S, integrate

from mathematics . sde . nonlinear . legendre_polynomial import polynomial

def get_c ( indices : tuple , weights : tuple) :

Calculates the Fourier —Legendre coefficient depending on indices and weights Parameters

indices : tuple

indices of the Fourier —Legendre coefficient weights : tuple

weights of the Fourier —Legendre coefficient Returns

sympy. Rational

from sympy. abc import x , y

# multiplicity of iterated integral which is the Fourier —Legendre coefficient n = len ( indices ) w = list ( reversed (weights))

c = S. One

for i in reversed(range(1, n)):

c = integrate(polynomial(indices[i]) * (x + 1) ** w[i] * c, (x, -1, y)).subs(y, x) c = integrate(polynomial(indices[0]) * (x + 1) ** w[0] * c, (x, -1, 1))

if sum(w) % 2 return c else :

return -c

0:

Listing 45: Symbolic function of the Fourier—Legendre coefficient

import logging

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from sympy import sympify , Function import tools . database as db

from mathematics . sde . nonlinear . c import get_c

class C(Function):

Gives the Fourier —Legendre coefficient with requested indices and weights _preloaded = diet ()

def __new__( cls , indices : tuple , weights : tuple , to_float=True , **kwargs) :

Creates the Fourier —Legendre coefficient object with needed indices and weights Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C

calculated value or symbolic expression

if not len ( indices ) == len (weights) :

return super (C, cls).__new__(cls, indices, weights, ** kwargs)

index = f” { ’:’.join([str(i) for i in indices]) }_{ ’:’.join([str(i) for i in weights]) }

try :

return cls . .value (index , to_float) except KeyError :

respond = cls.-download_one(index) if len(respond) != 0:

cls. _preloaded [respond [ 0 ] [ 0 ] ] = respond [0][ 1 ] return cls._value(index, to_float) else :

new_c = cls._calculate(index, indices , weights)

cls._upload_one(new_c)

cls . _preloaded [new_c [0] ] = new_c [ 1 ]

return cls._value(index, to_float)

@classmethod

def _calculate(cls, index, indices, weights): new_c = get_c ( indices , weights) return index, ( new_c , sympify(new_c) . evalf () )

@classmethod

def _upload_one ( cls , c):

logging . info ( f”C: ADDING NEW C_ { c[0] } = { c[1][0] } ”)

db. execute( f”INSERT INTO ‘C‘ (‘index ‘, ‘value ‘, ‘value_f‘) VALUES (’ { c[0] } ’, ’ { c

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[1][0]} ’ , {c[1][1]})”)

@classmethod

def _unpack(cls , rows) :

return [( rows [ i ][0] , ( rows [ i ][1] , rows [ i ] [ 2]) ) for i in range (len ( rows ) ) ]

@classmethod

def _value(cls, index, to_float): c = cls . _preloaded [index] if to_float : return c [1] else :

return sympify(c[0])

@classmethod

def -download _one( cls, index):

logging . info ( f”C: MISSING PRELOADED VERSION OF C_{ index }”) respond = db.execute(

f”SELECT ‘index ‘ , ‘value ‘ , ‘ valu^f ‘ FROM ‘C‘”

f”WHERE REGEXP( ‘index ‘ , { index } $ ’)”

)

return cls._unpack(respond)

@classmethod

def preload(cls , *args) :

Updates dictionary of the preloaded Fourier —Legendre coefficients Note: weights are not accepted , such coefficients are loaded

with all available weights Parameters

args

Indices for the Fourier —Legendre coefficients to download them from database

logging . info ( f”C: PRELOADING COEFFICIENTS { args } ”)

query = []

for q in range(len(args)) :

numbers = [ int (char) for char in str (args[q] + 1)] pattern = []

for i in range (1 , len ( numbers) ) : pattern. append(”[0 — 9]” * i)

for i in range (len ( numbers) ) : p = []

for j in range (len ( numbers) ) :

if j < i:

p.append( str ( numbers [ j ]) ) elif i == j :

p. append( f” [0 — { numbers [ j ] — 1}]”) elif j > i:

p. append(” [0 — 9]”) pattern.append(””.join(p))

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Differential Equations and Control Processes, N. 1, 2021

Listing 46: Calculation of the Fourier—Legendre coefficients CjOjO j

from math import sqrt

from sympy import sympify , Number, Function

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C000 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 4

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C000

calculated value or symbolic expression

j3 , j2 , j1 , dt = sympify ( args )

if not (isinstance (j1 , Number) and isinstance (j2 , Number) and

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Differential Equations and Control Processes, N. 1, 2021

Listing 47: Calculation of the Fourier—Legendre coefficients Cjj

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C10(Function) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C10

calculated value or symbolic expression

j2 , j1 , dt = sympify ( args )

if not (isinstance (j1 , Number) and

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Differential Equations and Control Processes, N. 1, 2021

Listing 48: Calculation of the Fourier—Legendre coefficients C'021?l

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C01 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C01

calculated value or symbolic expression

j2 , j1 , dt = sympify ( args )

if not (isinstance (j1 , Number) and isinstance (j2 , Number) and

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Differential Equations and Control Processes, N. 1, 2021

Listing 49: Calculation of the Fourier—Legendre coefficients C0o°3°2jl

from math import sqrt

from sympy import sympify , Number, Function

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C0000 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C0000

calculated value or symbolic expression

j4 , j3 , j2 , j1 , dt = sympify ( args )

if not (isinstance (j1 , Number) and isinstance (j2 , Number) and isinstance (j3 , Number) and

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Differential Equations and Control Processes, N. 1, 2021

Listing 50: Calculation of the Fourier—Legendre coefficients Cl0,0 •

3 2 1

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C100 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 4

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C100

calculated value or symbolic expression

j3 , j2 , j1 , dt = sympify ( args )

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Differential Equations and Control Processes, N. 1, 2021

Listing 51: Calculation of the Fourier—Legendre coefficients Cjj j

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C010 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 4

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C010

calculated value or symbolic expression

j3 , j2 , j1 , dt = sympify ( args )

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Differential Equations and Control Processes, N. 1, 2021

Listing 52: Calculation of the Fourier—Legendre coefficients C001^

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C001 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 4

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C001

calculated value or symbolic expression

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Differential Equations and Control Processes, N. 1, 2021

Listing 53: Calculation of the Fourier—Legendre coefficients C00000..

from math import sqrt

from sympy import sympify , Number, Function

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C00000 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 6

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

symbolic . Rational or C00000

calculated value or symbolic expression

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Differential Equations and Control Processes, N. 1, 2021

Listing 54: Calculation of the Fourier—Legendre coefficients Cjj

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C20(Function) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tup l e

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Differential Equations and Control Processes, N. 1, 2021

Listing 55: Calculation of the Fourier—Legendre coefficients Cjj

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C11 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights

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Differential Equations and Control Processes, N. 1, 2021

Listing 56: Calculation of the Fourier—Legendre coefficients dj

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C02( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

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Differential Equations and Control Processes, N. 1, 2021

Listing 57: Calculation of the Fourier—Legendre coefficients Cj0.OOj

from math import sqrt

from sympy import sympify , Number, Function

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C1000 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tuple

requested weights Returns

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Differential Equations and Control Processes, N. 1, 2021

Listing 58: Calculation of the Fourier—Legendre coefficients С°41°°2^1

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C0100(Function) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

indices: tuple

requested indices weights : tup l e

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Differential Equations and Control Processes, N. 1, 2021

Listing 59: Calculation of the Fourier—Legendre coefficients C'00,110,- •

4 3 2 1

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C0010 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

indices and weights and calculates it

Parameters

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Differential Equations and Control Processes, N. 1, 2021

Listing 60: Calculation of the Fourier—Legendre coefficients C00°172jl

from math import sqrt

from sympy import sympify , Function , Number

from mathematics . sde . nonlinear . symbolic . coefficients . c import C class C0001 ( Function ) :

Gives the Fourier —Legendre coefficient with requested indices and weights nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates the Fourier —Legendre coefficient object with needed

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Differential Equations and Control Processes, N. 1, 2021

Listing 61: Calculation of the Fourier—Legendre coefficients CejOjOjoim

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Differential Equations and Control Processes, N. 1, 2021

14 def __new__(cls, * args, ** kwargs ) :

15 ” ” ”

16 Creates the Fourier —Legendre coefficient object with needed

17 indices and weights and calculates it

18 Parameters

19

20 indices: tuple

21 requested indices

22 weights : tup l e

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23 requested weights

24 Returns

25

26 symbolic . Rational or C000000

27 calculated value or symbolic expression

28 ” ” ”

29 j6 , j5 , j4 , j3 , j2 , j1 , dt = sympify ( args)

30

31 if not (isinstance (j1 , Number) and

32 isinstance (j2 , Number) and

33 isinstance (j3 , Number) and

34 isinstance (j4 , Number) and

35 isinstance (j5 , Number) and

36 isinstance (dt , Number)):

37 return super (C000000, cls) .__new__(cls , * args , ** kwargs )

38

39 return sqrt (

40 (j1 * 2+1) *

41 (j2 * 2 + 1) *

42 (j3 * 2 + 1) *

43 (j 4 * 2+1) *

44 (j5 * 2 + 1) *

45 (j6 * 2 + 1)) * \

46 dt ** 3 * \

47 C((j6, j 5 , j4 , j3, j 2 , j1), (0, 0, 0, 0, 0, 0)) / 64

48

49 def doit(self, ** hints):

50 ” ” ”

51 Tries to expand or calculate function

52 Returns

53

54 C000000

55 ” ” ”

56 return C000000( * self .args , ** hints)

Listing 62: Program entry for generation of new Fourier—Legendre coefficients

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Differential Equations and Control Processes, N. 1, 2021

from config import csv , new_c_portion_size

from mathematics . sde . nonlinear . new_c import thread_c , split_task

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def main () :

logging . basicConfig( level = logging . INFO,

format=” %(asctime)s — %(levelname)s — %(message) s” , datefmt=”%H:%M:%S”

)

logger = logging. getLogger(___name__)

filename = os . path .join(csv, f”c_ { datetime. now () . strftime( ’%d—%n—%Y_%H—%M—%S ’) } . csv” )

logging. info(f”Writing to file { filename } ”)

tasks = [

(((57, 58) , (57 , 58) , (57 , 58) ) , (0, 0 , 0)) ,

# (((0, 56), (0, 56), (0, 56)), (0, 0, 0)),

#

# (((0, 15), (0, 15)), (0, 1)),

# (((0, 15), (0, 15)), (1, 0)),

# (((0, 15), (0, 15), (0, 15)), (0, 0, 0, 0)),

#

# (((0, 6), (0, 6), (0, 6)), (0, 0, 1)),

# (((0, 6), (0, 6), (0, 6)), (0, 1, 0)),

# (((0, 6), (0, 6), (0, 6)), (1, 0, 0)),

# (((0, 6), (0, 6), (0, 6), (0, 6), (0, 6)), (0, 0, 0, 0, 0)),

#

# (((0, 2), (0, 2)), (0, 2)),

# (((0, 2), (0, 2)), (1, 1)),

# (((0, 2), (0, 2)), (2, 0)),

# (((0, 2), (0, 2), (0, 2), (0, 2)), (0, 0, 0, 1)),

# (((0, 2), (0, 2), (0, 2), (0, 2)), (0, 0, 1, 0)),

# (((0, 2), (0, 2), (0, 2), (0, 2)), (0, 1, 0, 0)),

# (((0, 2), (0, 2), (0, 2), (0, 2)), (1, 0, 0, 0)),

# (((0, 2), (0, 2), (0, 2), (0, 2), (0, 2)), (0, 0, 0, 0, 0)),

# (((0, 2), (0, 2), (0, 2), (0, 2), (0, 2), (0, 2)), (0, 0, 0, 0, 0, 0)),

]

with Pool(cpu_count () ) as p:

for t in tasks:

logger. info(f” Running task {t}”)

for chunk in chunks(split_task(t), new_c_portion_size): c = p. map( thread_c , chunk) c.append(””)

with open (filename , ”a”) as f: f.write(” \ n” .join(c)) f.close ()

logger. info(f”The portion of C has been written {chunk[0] } — { chunk[ — 1] } ”)

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logger. info(”Generation has been done”)

def chunks(lst, n):

for i in range (0, len (lst), n): yield lst [ i : i + n]

if____name___== ”__main__”:

main ()

Listing 63: Module for the Fourier—Legendre coefficients generation

from mathematics . sde . nonlinear . c import get_c

def split_task (tasks): c = len (tasks [1])

i f c == 2:

return s plit _2(tasks)

if c == 3:

return s plit _3(tasks)

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i f c == 4:

return s plit _4(tasks)

i f c == 5:

return s plit _5(tasks)

i f c == 6:

return s plit _6(tasks)

def split _ 2(ranges):

return [( ( i , j ) , ranges [1 ] )

fo r j i n range (ranges [0] [1] [1])

fo r i i n range ( r a n g e s [ 0 ] [ 0 ] [ 1 ] )

if i >= ranges [0][1][0] or j >= ranges [0][0][1]]

def split _ 3(ranges):

return [( ( i , j , k) , ranges [1 ] )

fo r k i n range (ranges [0] [2] [1])

fo r j i n range (ranges [0] [1] [1])

fo r i i n range ( r a n g e s [ 0 ] [ 0 ] [ 1 ] )

if k >= ranges [0][2][0] or j >= ranges [0] [1] [0] or i >= ranges [0][0][0]]

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def split _4 (ranges):

return [(( i , j, k, l), ranges[1]) for j in range(* ranges [0][3]) for i in range(* ranges [0][2]) for k in range(* ranges [0][1 ]) for l in range(* ranges [0][0]) if l >= ranges [0][3][0] or k >= ranges [0] [2][0] or j >= ranges [0] [1] [0] or i >= ranges [0][0][0]]

def split _ 5(ranges):

return [( ( i , j , k, l , m) , ranges [1 ] ) for j in range(* ranges [0][4]) for i in range(* ranges [0][3]) for k in range(* ranges [0][2]) for l in range(* ranges [0][1 ]) for m in range(* ranges [0][0]) if m >= ranges [0][4][0] if l >= ranges [0][3][0] or k >= ranges [0] [2][0] or j >= ranges [0] [1] [0] or i >= ranges [0][0][0]]

def split _ 6(ranges):

return [( ( i , j , k, l , m, n) , ranges [1 ] ) for j in range(* ranges [0][5]) for i in range(* ranges [0][4]) for k in range(* ranges [0][3]) for l in range(* ranges [0][2]) for m in range(* ranges [0][1]) for n in range(* ranges [0][0]) if n >= ranges [0] [5][0] if m >= ranges [0][4][0] if l >= ranges [0][3][0] or k >= ranges [0] [2][0] or j >= ranges [0] [1] [0] or i >= ranges [0][0][0]]

def threa^c ( ranges) : c = len(ranges[1])

i f c == 2:

return gen_2(* ranges [0] , ranges [1])

if c == 3:

return gen_3(* ranges [0] , ranges [1])

i f c == 4:

return gen_4(* ranges [0] , ranges [1])

i f c == 5:

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6.2.2 Source Codes for Supplementary Differential Operators and Functions

Listing 64: Implementation of the differential operator L

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from sympy import Add, sympify , Number, diff

from mathematics . sde . nonlinear . symbolic . operator import Operator

class L(Operator) : nargs = 4

def __new__(cls, * args, ** kwargs ) :

Creates new L object with given args Parameters

args

bunch of necessary arguments Returns

sympy. Expr

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Listing 65: Implementation of the differential operator G

(i)

0

from sympy import Add, sympify , Number, diff

from mathematics . sde . nonlinear . symbolic . operator import Operator

class G(Operator) : nargs = 3

_dict = diet ()

def __new__(cls, * args, ** kwargs ) :

Creates new G object with given args Parameters

args

bunch of necessary arguments Returns

sympy. Expr

formula to simplify and substitute

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Listing 66: Implementation of the function a(x,t)

Listing 67: Implementation of the differential operator L

from sympy import sympify , Number, diff , Add

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Listing 68: Implementation of the indicator function 1 {i1=i2}

from sympy import sympify , Number, Function

class Ind(Function) :

Indicator function nargs = 2

def __new__(cls, * args, ** kwargs ) :

Creates new Ind object with given args

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6.2.3 Source Codes for Iterated Ito Stochastic Integrals Approximations Subprograms

Listing 69: Approximation of Ito stochastic integral I^T + г T

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from math import sqrt

from sympy import Function , sympify , Number

class I0 ( Function ) :

Ito stochastic integral nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates new I0 object with given args Parameters

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Listing 70: Approximation of iterated Ito stochastic integral 00)r+1 T

from math import sqrt

from sympy import sympify , Number, Function, Add from mathematics . sde . nonlinear . symbolic . ind import Ind

class I00 ( Function ) :

Iterated Ito stochastic integral nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates new I00 object with given args Parameters

11 : int

integral index

12 : int

integral index

q : int

amount of terms in approximation of

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Listing 71: Approximation of Ito stochastic integral +i т

from math import sqrt

from sympy import Function , sympify , Number

class 11 ( Function ) :

Ito stochastic integral nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates new I1 object with given args Parameters

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Listing 72: Approximation of iterated Ito stochastic integral loOOh3+i T

from sympy import Function , Number, sympify , Add

from mathematics . sde . nonlinear . symbolic . coefficients . c000 import C000 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I000 ( Function ) :

Iterated Ito stochastic integral nargs = 6

def __new__(cls, * args, ** kwargs ) :

Creates new I000 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

q : int

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Listing 73: Approximation of iterated Ito stochastic integral 1р0)т+1 T

from sympy import Function , sympify , Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c10 import C10 from mathematics . sde . nonlinear . symbolic . ind import Ind

class 110 ( Function ) :

Iterated Ito stochastic integral nargs = 5

def __new__(cls, * args, ** kwargs ) :

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Listing 74: Approximation of iterated Ito stochastic integral I

(iii2)

(01)rp+i ,Tp

from sympy import Function , Number, sympify , Add

from mathematics . sde . nonlinear . symbolic . coefficients . c01 import C01 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I01 ( Function) :

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Differential Equations and Control Processes, N. 1, 2021

Iterated Ito stochastic integral nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates new I01 object with given args Parameters

11 : int

integral index

12 : int

integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral dt : float delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute

i1 , i2 , q, dt , ksi = sympify (args)

if not (isinstance (i1 , Number) and isinstance (i2 , Number) and isinstance (q, Number)):

return super (I01 , cls) . __new__(cls , * args , ** kwargs )

return Add ( * [

C01(j2, j1, dt) *

(ksi[j1, i 1 ] * ksi [j2 , i 2 ] —

Ind(i1, i2) * Ind(j1, j2)) for j2 in range(q + 1) for j1 in range(q + 1)])

def doit(self, ** hints):

Tries to expand or calculate function Returns

I0 1

return I01(* self.args, ** hints)

Listing 75: Approximation of iterated Ito stochastic integral 1((0000)т4)1 Tp

from sympy import Function , sympify, Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c0000 import C0000

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from mathematics . sde . nonlinear . symbolic . ind import Ind

class I0000(Function) :

Iterated Ito stochastic integral nargs = 7

def __new__(cls, * args, ** kwargs ) :

Creates new I0000 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

14 : int

integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral dt : float delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute

i1, i2, i3, i4, q, dt, ksi = sympify(args)

if not (isinstance (i1 , Number) and isinstance (i2 , Number) and isinstance (i3 , Number) and isinstance (i4 , Number) and isinstance (q, Number)):

return super (I0000 , cls) . __new_(cls , * args , ** kwargs)

return Add ( * [

C0000(j4 , j3 , j 2 , j1 , dt) *

( ksi [ j1 , i 1 ] * ksi [ j2 i 2 ] * ksi [ j3 i3 ] * ksi [ j4 i4 ] -

Ind (i1 , i 2 ) * Ind (j1 j 2 ) * ksi [ j3 i3 ] * ksi [ j4 i4 ] -

Ind (i1 , i3 ) * Ind (j1 j3) * ksi [ j2 i2 ] * ksi [ j4 i4 ] -

Ind (i1 , i4 ) * Ind (j1 j 4 ) * ksi [ j2 i2 ] * ksi [ j3 i3 ] -

Ind (i2 , i3 ) * Ind (j2 j3) * ksi [ j1 i1 ] * ksi [ j4 i4 ] -

Ind (i2 , i4 ) * Ind (j2 j 4 ) * ksi [ j1 i1 ] * ksi [ j3 i3 ] -

Ind (i3 , i4 ) * Ind (j3 j 4 ) * ksi [ j1 i1 ] * ksi [ j2 i 2 ] +

Ind (i1 , i 2 ) * Ind (j1 j 2 ) * Ind ( i3 i4 ) * Ind (j3 , j 4 ) +

Ind (i1 , i3 ) * Ind (j1 j3) * Ind (i2 , i4 ) * Ind (j2 , j 4 ) +

Ind (i1 , i4 ) * Ind (j1 j 4 ) * Ind (i2 , i3 ) * Ind (j2 , j3))

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for j4 in range(q + 1) for j3 in range(q + 1) for j2 in range(q + 1) for j1 in range(q + 1)])

def doit(self, ** hints):

Tries to expand or calculate function Returns

I0000

return I0000(* self.args, ** hints)

Listing 76: Approximation of Ito stochastic integral /(2)T +1 Tp

from math import sqrt

from sympy import sympify , Number, Function

class I2(Function) :

Ito stochastic integral nargs = 3

def __new__(cls, * args, ** kwargs ) :

Creates new I2 object with given args Parameters

i1 : int

integral index d t : fl o a t delta time ksi : numpy. ndarray

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

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute

i1 , dt , ksi = sympify( args )

if not isinstance (i1 , Number) :

return super (I2 , cls) . __new__(cls , * args , ** kwargs )

return (ksi[0, i1] + ksi [1 , i1] * sqrt(3) / 2 + ksi [2 , i1] / sqrt(5) /2) * dt ** 2.5 / 3

def doit(self, ** hints):

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Tries to expand or calculate function Returns

I2

return I2(* self.args , ** hints)

Listing 77: Approximation of iterated Ito stochastic integral 1р0с)т)+1 T

from sympy import Function , sympify , Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c100 import C100 from mathematics . sde . nonlinear . symbolic . ind import Ind

class 1100 ( Function ) :

Iterated Ito stochastic integral nargs = 6

def __new__(cls, * args, ** kwargs ) :

Creates new I100 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral d t : fl o a t delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute

i1, i2, i3, q, dt, ksi = sympify (args)

if not (isinstance (i1 , Number) and isinstance (i2 , Number) and isinstance (i3 , Number) and isinstance (q, Number)):

return super (I100 , cls) . __new__(cls , * args , ** kwargs)

return Add ( * [

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Listing 78: Approximation of iterated Ito stochastic integral I(0k))t'+i Tp

from sympy import Function , sympify , Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c010 import C010 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I010 ( Function ) :

Iterated Ito stochastic integral nargs = 6

def __new__(cls, * args, ** kwargs ) :

Creates new I010 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral dt : float delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

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Listing 79: Approximation of iterated Ito stochastic integral I(0 OpT+i Tp

from sympy import Function , sympify , Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c001 import C001 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I00 1 ( Function ) :

Iterated Ito stochastic integral nargs = 6

def __new__(cls, * args, ** kwargs ) :

Creates new I001 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int

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Listing 80: Approximation of iterated Ito stochastic integral 1(000о0)У+1 T

from sympy import Function , Number, sympify , Add

from mathematics . sde . nonlinear . symbolic . coefficients . c00000 import C00000 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I00000 ( Function ) :

Iterated Ito stochastic integral nargs = 8

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def __new__(cls, * args, ** kwargs ) :

Creates new I00000 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

14 : int

integral index

15 : int integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral dt : float delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute i1 , i2 , i3 , i4 , i5 , q, dt , ksi = sympify (args)

if not (isinstance (i1 , Number) and

isinstance (i2 , Number) and

isinstance (i3 , Number) and

isinstance (i4 , Number) and

isinstance (i5 , Number) and

isinstance (q , Number) and

isinstance (dt , Number) ) :

return super (I00000 , cls) new (cls , * args , ** kwargs)

return Add ( * [

C00000 (j5 , j4 , j 3 , j2 , j1 , dt) *

( ksi [ j1 , i 1 ] * ksi [ j2 , i 2 ] * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] * ksi [ j5 , i5 ] -

Ind (i1 , i 2 ) * Ind (j1 , j 2 ) * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] * ksi [ j5 , i5 ] -

Ind (i1 , i3) * Ind (j1 , j3) * ksi [ j2 , i2 ] * ksi [ j4 , i4 ] * ksi [ j5 , i5 ] -

Ind (i1 , i4) * Ind (j1 , j 4 ) * ksi [ j2 , i2 ] * ksi [ j3 , i3 ] * ksi [ j5 , i5 ] -

Ind (i1 , i 5 ) * Ind (j1 , j 5 ) * ksi [ j2 , i2 ] * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] -

Ind (i2 , i3 ) * Ind (j2 , j3) * ksi [ j1 , i1 ] * ksi [ j4 , i4 ] * ksi [ j5 , i5 ] -

Ind (i2 , i4 ) * Ind (j2 , j 4 ) * ksi [ j1 , i1 ] * ksi [ j3 , i3 ] * ksi [ j5 , i5 ] -

Ind (i2 , i 5 ) * Ind (j2 , j 5 ) * ksi [ j1 , i1 ] * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] -

Ind (i3 , i4 ) * Ind (j3 , j 4 ) * ksi [ j1 , i1 ] * ksi [ j2 , i 2 ] * ksi [ j5 , i5 ] -

Ind (i3 , i 5 ) * Ind (j3 , j 5 ) * ksi [ j1 , i1 ] * ksi [ j2 , i 2 ] * ksi [ j4 , i4 ] -

Ind (i4 , i 5 ) * Ind (j4 , j 5 ) * ksi [ j1 , i1 ] * ksi [ j2 , i 2 ] * ksi [ j3 , i3 ] +

Ind (i1 , i 2 ) * Ind (j1 , j 2 ) * Ind ( i3 , i4 ) * Ind (j3 , j 4 ) * ksi [ j5 , i5 ] +

Ind (i1 , i 2 ) * Ind (j1 , j 2 ) * Ind ( i3 , i5 ) * Ind (j3 , j 5 ) * ksi [ j4 , i4 ] +

Ind (i1 , i 2 ) * Ind (j1 , j 2 ) * Ind (i4 , i5 ) * Ind (j4 , j 5 ) * ksi [ j3 , i3 ] +

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Ind(i1, i3 ) * Ind (j 1 , j3) * Ind (i2 , i4 ) * Ind (j2 , j 4 ) * ksi [ j5 , i5 ] +

Ind(i1, i3 ) * Ind (j 1 , j3) * Ind (i2 , i5 ) * Ind (j2 , j 5 ) * ksi [ j4 , i4 ] +

Ind(i1, i3 ) * Ind (j 1 , j3) * Ind (i4 , i5 ) * Ind (j4 , j 5 ) * ksi [ j2 , i2 ] +

Ind(i1, i4 ) * Ind (j 1 , j 4 ) * Ind (i2 , i3 ) * Ind (j2 , j3) * ksi [ j5 , i5 ] +

Ind(i1, i4 ) * Ind (j 1 , j 4 ) * Ind (i2 , i5 ) * Ind (j2 , j 5 ) * ksi [ j3 , i3 ] +

Ind(i1, i4 ) * Ind (j 1 , j 4 ) * Ind ( i3 , i5 ) * Ind(j 3 , j 5 ) * ksi [ j2 , i2 ] +

Ind(i1, i 5 ) * Ind (j 1 , j 5 ) * Ind (i2 , i3 ) * Ind(j 2 , j3) * ksi [ j4 , i4 ] +

Ind(i1, i 5 ) * Ind (j 1 , j 5 ) * Ind (i2 , i4 ) * Ind(j 2 , j 4 ) * ksi [ j3 , i3 ] +

Ind(i1, i 5 ) * Ind (j 1 , j 5 ) * Ind ( i3 , i4 ) * Ind(j 3 , j 4 ) * ksi [ j2 , i2 ] +

Ind (i2 , i3 ) * Ind (j2 , j3) * Ind (i4 , i5 ) * Ind (j4 , j 5 ) * ksi [ j 1 , i1 ] +

Ind (i2 , i4 ) * Ind (j2 , j 4 ) * Ind ( i3 , i5 ) * Ind(j 3 , j 5 ) * ksi [j1 , i1 ] +

Ind (i2 , i 5 ) * Ind (j2 , j 5 ) * Ind ( i3 , i4 ) * Ind(j 3 , j 4 ) * ksi [j1 , i 1 ])

for j5 in range(q + 1) for j4 in range(q + 1) for j3 in range(q + 1) for j2 in range(q + 1) for j1 in range(q + 1)])

def doit(self, ** hints):

Tries to expand or calculate function Returns

I00000

return I00000(* self.args, ** hints)

Listing 81: Approximation of iterated Ito stochastic integral I(20)t+1t

from sympy import Function , Number, sympify , Add

from mathematics . sde . nonlinear . symbolic . coefficients . c20 import C20 from mathematics . sde . nonlinear . symbolic . ind import Ind

class 120 ( Function ) :

Iterated Ito stochastic integral nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates new I20 object with given args Parameters

11 : int

integral index

12 : int

integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral

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Listing 82: Approximation of iterated Ito stochastic integral I(02)t +1 T

from sympy import Function , sympify , Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c02 import C02 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I02 ( Function ) :

Iterated Ito stochastic integral nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates new I02 object with given args Parameters

11 : int

integral index

12 : int

integral index

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Listing 83: Approximation of iterated Ito stochastic integral IppT*+1 Tp

from sympy import Function , sympify , Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c11 import C11 from mathematics . sde . nonlinear . symbolic . ind import Ind

class 111 ( Function ) :

Iterated Ito stochastic integral nargs = 5

def __new__(cls, * args, ** kwargs ) :

Creates new I11 object with given args Parameters

i1 : int

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Listing 84: Approximation of iterated Ito stochastic integral IpOO^r ji T

from sympy import Function , Number, sympify , Add

from mathematics . sde . nonlinear . symbolic . coefficients . c1000 import C1000 from mathematics . sde . nonlinear . symbolic . ind import Ind

class 11000 ( Function ) :

Iterated Ito stochastic integral nargs = 7

def __new__(cls, * args, ** kwargs ) :

Creates new I1000 object with given args

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Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

if : int

integral index q : int

amount of terms in approximation of iterated Ito stochastic integral d t : fl o a t delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

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

sympy. Expr

formula to simplify and substitute

i1, i2, i3, i4, q, dt, ksi = sympify(args)

if not (isinstance (i1 , Number) and isinstance (i2 , Number) and isinstance (i3 , Number) and isinstance (i4 , Number) and isinstance (q, Number)):

return super (I1000 , cls) . __new_(cls , * args , ** kwargs)

return Add ( * [

C1000(j4 , j3 , j2 , j1 , dt) *

( ksi [ j1 , i1 ] * ksi [ j2 , i 2 ] * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] -

Ind (i1 , i 2 ) * Ind (j1 , j 2 ) * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] -

Ind (i1 , i3) * Ind (j1 , j3) * ksi [ j2 , i2 ] * ksi [ j4 , i4 ] -

Ind (i1 , i4) * Ind (j1 , j 4 ) * ksi [ j2 , i2 ] * ksi [ j3 , i3 ] -

Ind (i2 , i3 ) * Ind (j2 , j3) * ksi [ j1 , i1 ] * ksi [ j4 , i4 ] -

Ind (i2 , i4 ) * Ind (j2 , j 4 ) * ksi [ j1 , i1 ] * ksi [ j3 , i3 ] -

Ind (i3 , i4 ) * Ind (j3 , j 4 ) * ksi [ j1 , i1 ] * ksi [ j2 , i 2 ] +

Ind (i1 , i 2 ) * Ind (j1 , j 2 ) * Ind ( i3 , i4 ) * Ind (j3 , j 4 ) +

Ind (i1 , i3 ) * Ind (j1 , j3) * Ind (i2 , i4 ) * Ind (j2 , j 4 ) +

Ind (i1 , i4 ) * Ind (j1 , j 4 ) * Ind (i2 , i3 ) * Ind (j2 , j3))

for j4 in range(q + 1) for j3 in range(q + 1) for j2 in range(q + 1) for j1 in range(q + 1)])

def doit(self, ** hints):

Tries to expand or calculate function Returns

11000

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return I1000(* self.args, ** hints)

Listing 85: Approximation of iterated Ito stochastic integral l(0\00)T

from sympy import Function , Number, sympify , Add

from mathematics . sde . nonlinear . symbolic . coefficients . c0100 import C0100 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I0100 ( Function ) :

Iterated Ito stochastic integral nargs = 7

def __new__(cls, * args, ** kwargs ) :

Creates new I0100 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

14 : int

integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral dt : float delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute

i1, i2, i3, i4, q, dt, ksi = sympify (args)

if not (isinstance (i1 , Number) and isinstance (i2 , Number) and isinstance (i3 , Number) and isinstance (i4 , Number) and isinstance (q, Number)):

return super (I0100 , cls) . __new_(cls , * args , ** kwargs)

r e t u r n Add ( *[

C0100(j4, j 3 , j 2 , j1 , dt) *

(ksi [j 1 , i 1 ] * ksi [ j2 , i2] * ksi[j3, i3 ] * ksi[j4, i4 ] —

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Differential Equations and Control Processes, N. 1, 2021

Listing 86: Approximation of iterated Ito stochastic integral IoOioTt4^ t

from sympy import Function , sympify , Number, Add

from mathematics . sde . nonlinear . symbolic . coefficients . c0010 import C0010 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I0010 ( Function ) :

Iterated Ito stochastic integral nargs = 7

def __new__(cls, * args, ** kwargs ) :

Creates new I0010 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

i3 : int

integral index q : int

amount of terms in approximation of iterated Ito stochastic integral

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dt : float delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute

i1, i2, i3, i4, q, dt, ksi = sympify(args)

if not (isinstance (i1 , Number) and isinstance (i2 , Number) and isinstance (i3 , Number) and isinstance (i4 , Number) and isinstance (q, Number)):

return super (I0010 , cls) . __new_(cls , * args , ** kwargs)

return Add ( * [

C0010(j4 , j3 , j 2 , j1 , dt) *

(ksi [jl , il] * ksi [ j2 , i 2 ] * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] -

Ind(il, i 2 ) * Ind (j l , j 2 ) * ksi [ j3 , i3 ] * ksi [ j4 , i4 ] -

Ind(il, i3 ) * Ind (j l , j3) * ksi [ j2 , i2 ] * ksi [ j4 , i4 ] -

Ind(il, i4 ) * Ind (j l , j 4 ) * ksi [ j2 , i2 ] * ksi [ j3 , i3 ] -

Ind (i2 , i3 ) * Ind (j2 , j3) * ksi [jl , il] * ksi [ j4 , i4 ] -

Ind (i2 , i4 ) * Ind (j2 , j 4 ) * ksi [jl , il] * ksi [ j3 , i3 ] -

Ind (i3 , i4 ) * Ind (j3 , j 4 ) * ksi [jl , il] * ksi [ j2 , i 2 ] +

Ind(il, i 2 ) * Ind (j l , j 2 ) * Ind ( i3 , i4 ) * Ind (j3 , j 4 ) +

Ind(il, i3 ) * Ind (j l , j3) * Ind (i2 , i4 ) * Ind (j2 , j 4 ) +

Ind(il, i4 ) * Ind (j l , j 4 ) * Ind (i2 , i3 ) * Ind (j2 , j3))

for j4 in range(q + 1) for j3 in range(q + 1) for j2 in range(q + 1) for j1 in range(q + 1)])

def doit(self, ** hints):

Tries to expand or calculate function Returns

I0010

return I0010(* self.args, ** hints)

Listing 87: Approximation of iterated Ito stochastic integral ^ООо^т4^ t

from sympy import Function , Number, sympify , Add

from mathematics . sde . nonlinear . symbolic . coefficients . c0001 import C0001 from mathematics . sde . nonlinear . symbolic . ind import Ind

class I0001 (Function):

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Iterated Ito stochastic integral nargs = 7

def __new__(cls, * args, ** kwargs ) :

Creates new I0001 object with given args Parameters

11 : int

integral index

12 : int

integral index

13 : int integral index

14 : int

integral index

q : int

amount of terms in approximation of iterated Ito stochastic integral dt : float delta time ksi : numpy. ndarray

matrix of Gaussian random variables Returns

sympy. Expr

formula to simplify and substitute

i1, i2, i3, i4, q, dt, ksi = sympify(args)

if not (isinstance (i1 , Number) and isinstance (i2 , Number) and isinstance (i3 , Number) and isinstance (i4 , Number) and isinstance (q, Number)):

return super (I0001 , cls) . __new_(cls , * args , ** kwargs)

return Add ( * [

C0001(j4 , j3 , j2 , j1 , dt) *