Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations

TL;DR

This paper compares the efficiency of Legendre polynomials and trigonometric functions in numerically integrating Ito stochastic differential equations, showing Legendre polynomials are more computationally efficient for high-order methods.

Contribution

It provides a comparative analysis demonstrating that Legendre polynomial expansions are more efficient than trigonometric functions for stochastic integral approximation in Ito SDEs.

Findings

Legendre polynomial expansions are easier to implement.

Legendre methods require less computational cost.

Results aid in developing high-order numerical methods for Ito SDEs.

Abstract

The article is devoted to comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations in the framework of the method of approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. On the example of iterated Ito stochastic integrals of multiplicities 1 to 3, included in the Taylor-Ito expansion, it is shown that expansions of stochastic integrals based on Legendre polynomials are much easier and require significantly less computational costs compared to their analogues obtained using the trigonometric system of functions. The results of the article can be useful for construction of high-order strong numerical methods for Ito stochastic differential equations.

Tables

Table 1. Table 1. Numbers q trig , subscript 𝑞 trig q_{\rm trig}, q trig ∗ superscript subscript 𝑞 trig q_{\rm trig}^{*} , q pol subscript 𝑞 pol q_{\rm pol}

$T-t$	$2^{-5}$	$2^{-6}$	$2^{-7}$	$2^{-8}$	$2^{-9}$	$2^{-10}$	$2^{-11}$	$2^{-12}$
$q_{\mathrm{trig}}$	3	4	7	14	27	53	105	209
$q_{\mathrm{trig}}^{\ast }$	6	11	20	40	79	157	312	624
$q_{\mathrm{pol}}$	5	9	17	33	65	129	257	513

Table 2. Table 2. Confirmation of the formula ( 67 )

$\epsilon /{\left(T-t\right)}^3$	0.0459	0.0072	$7.5722\cdot {10}^{-4}$	$7.5973\cdot {10}^{-5}$	$7.5990\cdot {10}^{-6}$
$q$	1	10	100	1000	10000

Table 3. Table 3. Numbers q , 𝑞 q, q 1 subscript 𝑞 1 q_{1}

$T-t$	$0.08222$	$0.05020$	$0.02310$	$0.01956$
$q$	19	51	235	328
$q_1$	1	2	5	6

Table 4. Table 4. Numbers p , 𝑝 p, p 1 , subscript 𝑝 1 p_{1}, p ∗ , superscript 𝑝 p^{*}, p 1 ∗ superscript subscript 𝑝 1 p_{1}^{*}

$T-t$	$0.08222$	$0.05020$	$0.02310$	$0.01956$
$p$	8	21	96	133
$p_1$	1	1	3	4
$p^{\ast }$	23	61	286	398
$p_1^{\ast }$	1	2	4	5

Table 5. Table 5. Confirmation of the formula ( 85 )

$\epsilon /{\left(T-t\right)}^3$	$0.0629$	$0.0097$	$0.0010$	$1.0129\cdot {10}^{-4}$	$1.0132\cdot {10}^{-5}$
$q$	1	10	100	1000	10000

Table 6. Table 6. Confirmation of the formula ( 88 )

$4\epsilon /{\left(T-t\right)}^4$	0.0540	0.0082	$8.4261\cdot {10}^{-4}$	$8.4429\cdot {10}^{-5}$	$8.4435\cdot {10}^{-6}$
$q$	1	10	100	1000	10000

Table 7. Table 7. Confirmation of the formula ( 90 )

$16\epsilon /{\left(T-t\right)}^4$	0.3797	0.0581	0.0062	$6.2450\cdot {10}^{-4}$	$6.2495\cdot {10}^{-5}$
$q$	1	10	100	1000	10000

Equations

$${\mathbf{f}}_t-\frac{t}{\Delta }{\mathbf{f}}_{\Delta },t\in [0,\Delta ].$$

$${\mathbf{f}}_t^{(i)}-\frac{t}{\Delta }{\mathbf{f}}_{\Delta }^{(i)}=\frac{1}{2}{a}_{i,0}+\sum _{r=1}^{\infty }\left(a_{i,r}\cos \frac{2\pi rt}{\Delta }+b_{i,r}\sin \frac{2\pi rt}{\Delta }\right),$$

$$a_{i,r}=\frac{2}{\Delta }\underset{0}{\overset{\Delta }{\int }}\left({\mathbf{f}}_s^{(i)}-\frac{s}{\Delta }{\mathbf{f}}_{\Delta }^{(i)}\right)\cos \frac{2\pi rs}{\Delta }ds,b_{i,r}=\frac{2}{\Delta }\underset{0}{\overset{\Delta }{\int }}\left({\mathbf{f}}_s^{(i)}-\frac{s}{\Delta }{\mathbf{f}}_{\Delta }^{(i)}\right)\sin \frac{2\pi rs}{\Delta }ds,$$

$$\mathsf{M}\left\{a_{i,r}{b}_{i,r}\right\}=\mathsf{M}\left\{a_{i,r}{b}_{i,k}\right\}=0,\mathsf{M}\left\{a_{i,r}{a}_{i,k}\right\}=\mathsf{M}\left\{b_{i,r}{b}_{i,k}\right\}=0,$$

$$\mathsf{M}\left\{a_{i_1,r}{a}_{i_2,r}\right\}=\mathsf{M}\left\{b_{i_1,r}{b}_{i_2,r}\right\}=0,\mathsf{M}\left\{a_{i,r}^2\right\}=\mathsf{M}\left\{b_{i,r}^2\right\}=\frac{\Delta }{2{\pi }^2{r}^2},$$

$${\mathbf{f}}_t^{(i)}={\mathbf{f}}_{\Delta }^{(i)}\frac{t}{\Delta }+\frac{1}{2}{a}_{i,0}+\sum _{r=1}^{\infty }\left(a_{i,r}\cos \frac{2\pi rt}{\Delta }+b_{i,r}\sin \frac{2\pi rt}{\Delta }\right),$$

$$J_{({\lambda }_1\dots {\lambda }_k)T,t}^{(i_1\dots i_k)}=\underset{t}{\overset{T}{\int }}\dots \underset{t}{\overset{t_2}{\int }}d{\mathbf{w}}_{t_1}^{(i_1)}\dots d{\mathbf{w}}_{t_k}^{(i_k)},$$

$$J_{({\lambda }_1\dots {\lambda }_k)T,t}^{\ast (i_1\dots i_k)}=\underset{t}{\overset{\ast T}{\int }}\dots \underset{t}{\overset{\ast t_2}{\int }}d{\mathbf{w}}_{t_1}^{(i_1)}\dots d{\mathbf{w}}_{t_k}^{(i_k)},$$

$$J[{\psi }^{(k)}{]}_{T,t}=\underset{t}{\overset{T}{\int }}{\psi }_k(t_k)\dots \underset{t}{\overset{t_2}{\int }}{\psi }_1(t_1)d{\mathbf{w}}_{t_1}^{(i_1)}\dots d{\mathbf{w}}_{t_k}^{(i_k)},$$

$$J^{\ast }[{\psi }^{(k)}{]}_{T,t}=\underset{t}{\overset{\ast T}{\int }}{\psi }_k(t_k)\dots \underset{t}{\overset{\ast t_2}{\int }}{\psi }_1(t_1)d{\mathbf{w}}_{t_1}^{(i_1)}\dots d{\mathbf{w}}_{t_k}^{(i_k)},$$

$$\int and\stackrel{\ast }{\int }$$

$$J_{(11)T,t}^{(i_{1}i_{2})}=\frac{1}{2}(T-t)\Biggl{(}\zeta_{0}^{(i_{1})}\zeta_{0}^{(i_{2})}+\frac{1}{\pi}\sum_{r=1}^{\infty}\frac{1}{r}\left(\zeta_{2r}^{(i_{1})}\zeta_{2r-1}^{(i_{2})}-\zeta_{2r-1}^{(i_{1})}\zeta_{2r}^{(i_{2})}\right.\biggr{.}+\left.\sqrt{2}\left(\zeta_{2r-1}^{(i_{1})}\zeta_{0}^{(i_{2})}-\zeta_{0}^{(i_{1})}\zeta_{2r-1}^{(i_{2})}\right)\right)\Biggr{)},$$

$${\zeta }_j^{(i)}=\underset{t}{\overset{T}{\int }}{\varphi }_j(s)d{\mathbf{f}}_s^{(i)}$$

$${\varphi }_j(s)=\frac{1}{\sqrt{T-t}}\{\begin{array}{ll}1& forj=0\\ & \\ \sqrt{2}\sin (2\pi r(s-t)/(T-t))& forj=2r-1\\ & \\ \sqrt{2}\cos (2\pi r(s-t)/(T-t))& forj=2r\end{array},$$

$$J_{(1)T,t}^{(i_1)}=\sqrt{T-t}{\zeta }_0^{(i_1)},$$

$$J_{(1)T,t}^{(i_1)q}\stackrel{\mathsf{def}}{=}{J}_{(1)T,t}^{(i_1)}=\sqrt{T-t}{\zeta }_0^{(i_1)},$$

$$J_{(11)T,t}^{(i_{1}i_{2})q}=\frac{1}{2}(T-t)\Biggl{(}\zeta_{0}^{(i_{1})}\zeta_{0}^{(i_{2})}+\frac{1}{\pi}\sum_{r=1}^{q}\frac{1}{r}\left(\zeta_{2r}^{(i_{1})}\zeta_{2r-1}^{(i_{2})}-\zeta_{2r-1}^{(i_{1})}\zeta_{2r}^{(i_{2})}\right.\biggr{.}+\left.\sqrt{2}\left(\zeta_{2r-1}^{(i_{1})}\zeta_{0}^{(i_{2})}-\zeta_{0}^{(i_{1})}\zeta_{2r-1}^{(i_{2})}\right)\right)\biggl{.}\Biggr{)},$$

$${\sf M}\left\{\left(J_{(11)T,t}^{(i_{1}i_{2})}-J_{(11)T,t}^{(i_{1}i_{2})q}\right)^{2}\right\}=\frac{3(T-t)^{2}}{2\pi^{2}}\Biggl{(}\frac{\pi^{2}}{6}-\sum_{r=1}^{q}\frac{1}{r^{2}}\Biggr{)}.$$

$$J_{(01)T,t}^{(0i_{1})}=\frac{{(T-t)}^{3/2}}{2}\biggl{(}\zeta_{0}^{(i_{1})}-\frac{\sqrt{2}}{\pi}\sum_{r=1}^{\infty}\frac{1}{r}\zeta_{2r-1}^{(i_{1})}\biggr{)},$$

$$J_{(001)T,t}^{(00i_{1})}=(T-t)^{5/2}\Biggl{(}\frac{1}{6}\zeta_{0}^{(i_{1})}+\frac{1}{2\sqrt{2}\pi^{2}}\sum_{r=1}^{\infty}\frac{1}{r^{2}}\zeta_{2r}^{(i_{1})}-\frac{1}{2\sqrt{2}\pi}\sum_{r=1}^{\infty}\frac{1}{r}\zeta_{2r-1}^{(i_{1})}\Biggr{)},$$

$$J_{(01)T,t}^{(0i_{1})q}=\frac{{(T-t)}^{3/2}}{2}\biggl{(}\zeta_{0}^{(i_{1})}-\frac{\sqrt{2}}{\pi}\sum_{r=1}^{q}\frac{1}{r}\zeta_{2r-1}^{(i_{1})}\biggr{)},$$

$$J_{(001)T,t}^{(00i_{1})q}=(T-t)^{5/2}\Biggl{(}\frac{1}{6}\zeta_{0}^{(i_{1})}+\frac{1}{2\sqrt{2}\pi^{2}}\sum_{r=1}^{q}\frac{1}{r^{2}}\zeta_{2r}^{(i_{1})}-\frac{1}{2\sqrt{2}\pi}\sum_{r=1}^{q}\frac{1}{r}\zeta_{2r-1}^{(i_{1})}\Biggr{)},$$

$$J_{(01)T,t}^{(0i_{1})q}=\frac{{(T-t)}^{3/2}}{2}\biggl{(}\zeta_{0}^{(i_{1})}-\frac{\sqrt{2}}{\pi}\biggl{(}\sum_{r=1}^{q}\frac{1}{r}\zeta_{2r-1}^{(i_{1})}+\sqrt{\alpha_{q}}\xi_{q}^{(i_{1})}\biggr{)}\biggr{)},$$

$$J_{(11)T,t}^{(i_{1}i_{2})q}=\frac{1}{2}(T-t)\Biggl{(}\zeta_{0}^{(i_{1})}\zeta_{0}^{(i_{2})}+\frac{1}{\pi}\sum_{r=1}^{q}\frac{1}{r}\left(\zeta_{2r}^{(i_{1})}\zeta_{2r-1}^{(i_{2})}-\zeta_{2r-1}^{(i_{1})}\zeta_{2r}^{(i_{2})}+\right.\Biggr{.}$$

$$+\Biggl{.}\left.\sqrt{2}\left(\zeta_{2r-1}^{(i_{1})}\zeta_{0}^{(i_{2})}-\zeta_{0}^{(i_{1})}\zeta_{2r-1}^{(i_{2})}\right)\right)+\frac{\sqrt{2}}{\pi}\sqrt{\alpha_{q}}\left(\xi_{q}^{(i_{1})}\zeta_{0}^{(i_{2})}-\zeta_{0}^{(i_{1})}\xi_{q}^{(i_{2})}\right)\Biggr{)},$$

$${\xi }_q^{(i)}=\frac{1}{\sqrt{{\alpha }_q}}\sum _{r=q+1}^{\infty }\frac{1}{r}\text{leavevmode}{\zeta }_{2r-1}^{(i)},{\alpha }_q=\frac{{\pi }^2}{6}-\sum _{r=1}^q\frac{1}{r^2},$$

$$\mathsf{M}\left\{{\left(J_{(01)T,t}^{(0i_1)}-J_{(01)T,t}^{(0i_1)q}\right)}^2\right\}=0,$$

$${\sf M}\left\{\left(J_{(11)T,t}^{(i_{1}i_{2})}-J_{(11)T,t}^{(i_{1}i_{2})q}\right)^{2}\right\}=\frac{(T-t)^{2}}{2\pi^{2}}\Biggl{(}\frac{\pi^{2}}{6}-\sum_{r=1}^{q}\frac{1}{r^{2}}\Biggr{)}.$$

$$J_{(001)T,t}^{(00i_{1})q}=(T-t)^{5/2}\Biggl{(}\frac{1}{6}\zeta_{0}^{(i_{1})}+\frac{1}{2\sqrt{2}\pi^{2}}\Biggl{(}\sum_{r=1}^{q}\frac{1}{r^{2}}\zeta_{2r}^{(i_{1})}+\sqrt{\beta_{q}}\mu_{q}^{(i_{1})}\Biggr{)}-\Biggl{.}$$

$$\Biggl{.}-\frac{1}{2\sqrt{2}\pi}\Biggl{(}\sum_{r=1}^{q}\frac{1}{r}\zeta_{2r-1}^{(i_{1})}+\sqrt{\alpha_{q}}\xi_{q}^{(i_{1})}\Biggr{)}\Biggr{)},$$

Full text

1. 1 Introduction
2. 2 Milstein Approach
3. 3 Method of Generalized Multiple Fourier Series
4. 4 Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 6
5. 5 Exact Calculation of the Mean-Square Error in Theorems 1, 2
6. 6 Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions for the Integral $J_{(11)T,t}^{(i_{1}i_{2})}$
7. 7 Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions for the Integrals $J_{(1)T,t}^{(i_{1})},$ $J_{(11)T,t}^{(i_{1}i_{2})},$ $J_{(01)T,t}^{(0i_{1})},$ $J_{(10)T,t}^{(i_{1}0)},$ $J_{(111)T,t}^{(i_{1}i_{2}i_{3})}$
8. 8 Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions for the Integral $J_{(011)T,t}^{*(0i_{1}i_{2})}$
9. 9 Conclusions
10. 10 Further Development of Multiple Fourier–Legendre Series Approach to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 3 to 5
11. 11 Theorems 1–7 from Point of View of the Wong–Zakai Approximation

1. Introduction

In a lot of author’s publications [2]-[44] the mean-square approximation method for iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series is proposed and developed (see Theorems 1–8 below). Further, we will call this method as the method of generalized multiple Fourier series. Under the term ”generalized multiple Fourier series” we understand the Fourier series constructed using various complete orthonormal systems of functions in the space $L_{2}([t,T])$, and not only using the trigonometric system of functions. Here $[t,T]$ is an interval of integration of iterated Ito or Stratonovich stochastic integrals.

It is well known the another approach to series expansion of stochastic processes using eigenfunctions of their covariance operators (the so-called Karhunen–Loeve expansion) [45]. If the stochastic process is the Brownian bridge process on the time interval $[t,T]$, then the eigenfunctions of its covariance operator will be trigonometric functions which form a complete orthonormal system of functions in the space $L_{2}([t,T])$ [46]. This means that the basis functions in the mentioned approach can only be trigonometric functions. In [46]-[50] the series expansion of the Brownian bridge process was used for the expansion and mean-square approximation of iterated Ito and Stratonovich stochastic integrals. Further, we will call this expansion as the Milstein expansion.

As mentioned above, in contrast to the Milstein expansion the method of generalized multiple Fourier series [2]-[44] (see Theorems 1, 2 below) allows to use different systems of basis functions. Thus, we can set the problem of choice the optimal system of basis functions within the framework of the method of generalized multiple Fourier series. Some ideas on the solution of the mentioned problem were given in a number of the author’s works [5]-[14], [17]-[21].

For example, in [5]-[14], [17], [18] it was shown that expansions for simplest iterated (double) Stratonovich stochastic integrals based on the systems of Haar and Rademacher–Walsh functions are too complex and ineffective in practice. In these works, a very brief comparison of the efficiency of application of Legendre polynomials and trigonometric functions in the framework of the method of generalized multiple Fourier series was also carried out. The subject of this article is the development and refinement of the results obtained in [5]-[14], [17], [18] in this direction.

2. Milstein Approach

Let $(\Omega,$ ${\rm F},$ ${\sf P})$ be a complete probability space. Let $\{{\rm F}_{t},t\in[0,T]\}$ be a nondecreasing right-continous family of $\sigma$-algebras of ${\rm F},$ and let ${\bf f}_{t}$ be a standard $m$-dimensional Wiener stochastic process, which is ${\rm F}_{t}$-measurable for any $t\in[0,T].$ We assume that the components ${\bf f}_{t}^{(i)}$ $(i=1,\ldots,m)$ of this process are independent. Consider the Brownian bridge process [46]

[TABLE]

The componentwise expansion of the stochastic process (1) into converging in the mean-square sense trigonometric Fourier series (version of the so-called Karhunen–Loeve expansion) has the following form [46]

[TABLE]

where

[TABLE]

where $r=0,1,\ldots;$ $i=1,\ldots,m.$

It is easy to demonstrate [46] that the random variables $a_{i,r},b_{i,r}$ are Gaussian ones and they satisfy the following relations

[TABLE]

[TABLE]

where $i,i_{1},i_{2}=1,\ldots,m;$ $r\neq k;$ $i_{1}\neq i_{2};$ ${\sf M}$ denotes a mathematical expectation.

According to (2), we have

[TABLE]

where the series converges in the mean-square sense.

Denote

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where every $\psi_{l}(\tau)$ $(l=1,\ldots,k)$ is a non-random function on $[t,T];$ ${\bf w}_{\tau}^{(i)}={\bf f}_{\tau}^{(i)}$ for $i=1,\ldots,m$ and ${\bf w}_{\tau}^{(0)}=\tau;$ $i_{1},\ldots,i_{k}=0,1,\ldots,m;$ $\lambda_{l}=0$ for $i_{l}=0$ and $\lambda_{l}=1$ for $i_{l}=1,\ldots,m$ $(l=1,\ldots,k);$

[TABLE]

denote Ito and Stratonovich stochastic integrals, respectively. In this paper we use the definition of the Stratonovich stochastic integral from [47], [48].

In [46] Milstein G.N. obtained the following expansion of $J_{(11)T,t}^{(i_{1}i_{2})}$ using the expansion (3)

[TABLE]

where the series converges in the mean-square sense; $i_{1}\neq i_{2};$ $i_{1},i_{2}=1,\ldots,m;$

[TABLE]

are independent standard Gaussian random variables for various $i$ or $j;$

[TABLE]

where $r=1,2,\ldots$ Moreover, [46]

[TABLE]

where $i_{1}=1,\ldots,m.$

In principle for implementing the strong numerical method with the order $1.0$ of accuracy (Milstein method [46]) for Ito stochastic differential equations it is sufficient to take the following approximations

[TABLE]

[TABLE]

where $i_{1}\neq i_{2};$ $i_{1},i_{2}=1,\ldots,m.$

It is not difficult to show that

[TABLE]

However, this approach has an obvious drawback. Indeed, we have too complex formulas for the stochastic integrals with Gaussian distribution

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the sense of notations from (12) is hold.

In [46] Milstein G.N. proposed the following mean-square approximations on the base of the expansions (8), (14)

[TABLE]

[TABLE]

[TABLE]

where $i_{1}\neq i_{2}$ in (16), and

[TABLE]

where $\zeta_{0}^{(i)},$ $\zeta_{2r}^{(i)},$ $\zeta_{2r-1}^{(i)},$ $\xi_{q}^{(i)};$ $r=1,\ldots,q;$ $i=1,\ldots,m$ are independent standard Gaussian random variables.

Obviously, for the approximations (15) and (16) we obtain [46]

[TABLE]

[TABLE]

This idea has been developed in [47]-[49]. For example, the approximation $J_{(001)T,t}^{(00i_{1})q},$ which corresponds to (15), (16) has the form [47]-[49]

[TABLE]

[TABLE]

[TABLE]

where $\xi_{q}^{(i)}$ and $\alpha_{q}$ have the form (17) and

[TABLE]

$\phi_{j}(s)$ is defined by (9); $\zeta_{0}^{(i)},$ $\zeta_{2r}^{(i)},$ $\zeta_{2r-1}^{(i)},$ $\xi_{q}^{(i)},$ $\mu_{q}^{(i)}$ ($r=1,\ldots,q;$ $i=1,\ldots,m$) are independent standard Gaussian random variables.

Nevetheless, the expansions (15), (19) are too complex for approximation of two Gaussian random variables $J_{(01)T,t}^{(0i_{1})},$ $J_{(001)T,t}^{(00i_{1})}.$

Further, we will see that introduction of random variables $\xi_{q}^{(i)}$ and $\mu_{q}^{(i)}$ will sharply complicate the approximation of the stochastic integral $J_{(111)T,t}^{(i_{1}i_{2}i_{3})};$ $i_{1},i_{2},i_{3}=1,\ldots,m$ within the framework of the Milshtein approach. This is due to the fact that the number $q$ is fixed for all stochastic integrals included into the considered collection. However, it is clear that due to the smallness of $T-t$, the number $q$ for $J_{(111)T,t}^{(i_{1}i_{2}i_{3})}$ could be taken significantly less than the number $q$ in the formula (16). This feature is also valid for the formulas (15), (19).

On the other hand, the following very simple formulas are well known

[TABLE]

[TABLE]

[TABLE]

where $\zeta_{0}^{(i)},$ $\zeta_{1}^{(i)},$ $\zeta_{2}^{(i)};$ $i=1,\ldots,m$ are indepentent standard Gaussian random variables.

Looking ahead, we note that the formulas (20)-(22) are part of the method that will be discussed in the next section (see Theorems 1, 2 below).

To obtain the Milstein expansion for (7) the truncated expansions (3) of components of the Wiener process ${\bf f}_{s}$ must be iteratively substituted in the single integrals, and the integrals must be calculated, starting from the innermost integral. This is a complicated procedure that obviously does not lead to a general expansion of (7) valid for an arbitrary multiplicity $k.$ For this reason, only expansions of simplest single, double, and triple integrals (7) were obtained [46]-[51].

At that, in [46], [50] the case $\psi_{1}(s),\psi_{2}(s)\equiv 1$ and $i_{1},i_{2}=0,1,\ldots,m$ is considered. In [47]-[49], [51] the attempt to consider the case $\psi_{1}(s),\psi_{2}(s),\psi_{3}(s)\equiv 1$ and $i_{1},i_{2},i_{3}=0,1,\ldots,m$ is implemented.

Note that generally speaking the mean-square convergence of the approximation

[TABLE]

(obtained by the Milstein approach) to the appropriate iterated Stratonovich stochastic integral

[TABLE]

must be proved separately due to iterated application of passing to the limit in the Milstein approach [46]. However, in [47] (pp. 438-439), [48] (Sect. 5.8, pp. 202–204), [49] (pp. 82-84), [51] (pp. 263-264) the authors use the Wong–Zakai approximation [53]-[55] (without rigorous proof) within the frames of the mentioned approach based on the Karhunen–Loeve expansion of the Brownian bridge process [46] (see discussion in Sect. 11 for details).

3. Method of Generalized Multiple Fourier Series

Let us consider an another approach to the expansion of iterated Ito stochastic integrals [5]-[44] (method of generalized multiple Fourier series).

The idea of this method is as follows: the iterated Ito stochastic integral (6) of multiplicity $k$ is represented as the multiple stochastic integral from the certain non-random discontinuous function of $k$ variables defined on the hypercube $[t,T]^{k}$, where $[t,T]$ is the interval of integration of the iterated Ito stochastic integral. Then, the indicated non-random function is expanded in the hypercube $[t,T]^{k}$ into the generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_{2}([t,T]^{k})$. After a number of nontrivial transformations we come (see Theorems 1, 2 below) to the mean-square convergening expansion of the iterated Ito stochastic integral (6) into the multiple series of products of standard Gaussian random variables. Coefficients of this series are coefficients of generalized multiple Fourier series for the mentioned non-random function of $k$ variables, which can be calculated using the explicit formula regardless of the multiplicity $k$ of the iterated Ito stochastic integral (6).

Suppose that every $\psi_{l}(\tau)$ $(l=1,\ldots,k)$ is a non-random function from the space $L_{2}([t,T])$. Define the following function on the hypercube $[t,T]^{k}$

[TABLE]

and $K(t_{1})\equiv\psi_{1}(t_{1})$ for $t_{1}\in[t,T].$

Suppose that $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of functions in the space $L_{2}([t,T])$.

The function $K(t_{1},\ldots,t_{k})$ belongs to the space $L_{2}([t,T]^{k}).$ At this situation it is well known that the generalized multiple Fourier series of $K(t_{1},\ldots,t_{k})\in L_{2}([t,T]^{k})$ is converging to $K(t_{1},\ldots,t_{k})$ in the hypercube $[t,T]^{k}$ in the mean-square sense, i.e.

[TABLE]

where

[TABLE]

is the Fourier coefficient,

[TABLE]

Consider the partition $\{\tau_{j}\}_{j=0}^{N}$ of $[t,T]$ such that

[TABLE]

Theorem 1 [5] (2006), [6]-[33]. Suppose that every $\psi_{l}(\tau)$ $(l=1,\ldots,k)$ is a continuous non-random function on the interval $[t,T]$ and $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of continuous functions in the space $L_{2}([t,T]).$ Then

[TABLE]

[TABLE]

where $J[\psi^{(k)}]_{T,t}$ is defined by (6),

[TABLE]

[TABLE]

${\rm l.i.m.}$* is a limit in the mean-square sense**,** $i_{1},\ldots,i_{k}=0,1,\ldots,m,$*

[TABLE]

*are independent standard Gaussian random variables for various $i$ or $j$ *(*if $i\neq 0$), $C_{j_{k}\ldots j_{1}}$ is the Fourier coefficient (25), $\Delta{\bf w}_{\tau_{j}}^{(i)}={\bf w}_{\tau_{j+1}}^{(i)}-{\bf w}_{\tau_{j}}^{(i)}$ $(i=0,1,\ldots,m),$ $\left\{\tau_{j}\right\}_{j=0}^{N}$ is a partition of the interval $[t,T],$ which satisfies the condition (26). *

Note that the continuity condition of $\phi_{j}(x)$ can be weakened (see [5]-[22]). Moreover, Theorem 1 can be generalized to the case of an arbitrary complete orthonormal systems of functions in the space $L_{2}([t,T])$ (see Theorem 2 below).

In order to evaluate the significance of Theorem 1 for practice we will demonstrate its transformed particular cases for $k=1,\ldots,6$ [5]-[33]

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[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where ${\bf 1}_{A}$ is the indicator of the set $A$.

As a result we obtain the following new possibilities and advantages compared with the method based on the Milstein approach [46].

There is the explicit formula (see (25)) for calculation of expansion coefficients of the iterated Ito stochastic integral (6) with any fixed multiplicity $k$.

We have new possibilities for exact calculation of the mean-square error of approximation of the iterated Ito stochastic integrals (6) (see Theorem 8 below).

Since the used multiple Fourier series is a generalized in the sense that it is built using various complete orthonormal systems of functions in the space $L_{2}([t,T])$, we have new possibilities for approximation — we can use not only trigonometric functions as in the Milstein approach [46] but Legendre polynomials. As it turned out (see below), it is more convenient to work with Legendre polynomials for constructing approximations of the iterated Ito stochastic integrals (6). We can choose different numbers $q$ (see Sect. 7) for approximations of different iterated Ito stochastic integrals from the family (6). This is impossible for approximations based on the Milstein approach [46]. Approximations based on Legendre polynomials essentially simpler than approximations based on trigonometric functions (see (15), (19), (21), (22)).

As we mentioned before, the Milstein approach [46] based on the Karhunen–Loeve expansion of the Brownian bridge process leads to iterated series (in contrast with multiple series from Theorems 1–7) starting at least from the second or third multiplicity of iterated stochastic integrals. Multiple series are more convenient for approximation than the iterated ones, since partial sums of multiple series converge for any possible case of convergence to infinity of their upper limits of summation (let us denote them as $p_{1},\ldots,p_{k}$). For example, when $p_{1}=\ldots=p_{k}=p\to\infty$. For iterated series, the condition $p_{1}=\ldots=p_{k}=p\to\infty$ obviously does not guarantee the convergence of this series.

However, in [47] (pp. 438-439), [48] (Sect. 5.8, pp. 202–204), [49] (pp. 82-84), [51] (pp. 263-264) the authors use (without rigorous proof) the condition $p_{1}=p_{2}=p_{3}=q\to\infty$ within the frames of the mentioned approach based on the Karhunen–Loeve expansion of the Brownian bridge process [46] together with the Wong–Zakai approximation [53]-[55] (see discussion in Sect. 11 for details).

For further consideration, let us consider the generalization of formulas (29)–(34) for the case of an arbitrary multiplicity $k$ $(k\in\mathbb{N})$ of the iterated Ito stochastic integral $J[\psi^{(k)}]_{T,t}$ defined by (6). In order to do this, let us introduce some notations. Consider the unordered set $\{1,2,\ldots,k\}$ and separate it into two parts: the first part consists of $r$ unordered pairs (sequence order of these pairs is also unimportant) and the second one consists of the remaining $k-2r$ numbers. So, we have

[TABLE]

where

[TABLE]

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

We will say that (35) is a partition and consider the sum with respect to all possible partitions

[TABLE]

Below there are several examples of sums in the form (36)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now we can write (27) as

[TABLE]

[TABLE]

where $[x]$ is an integer part of a real number $x;$ another notations are the same as in Theorem 1.

In particular, from (37) for $k=5$ we obtain

[TABLE]

[TABLE]

[TABLE]

The last equality obviously agrees with (33).

Let us consider the generalization of Theorem 1 for the case of an arbitrary complete orthonormal systems of functions in the space $L_{2}([t,T])$ and $\psi_{1}(\tau),\ldots,\psi_{k}(\tau)\in L_{2}([t,T]).$

Theorem 2 [19] (Sect. 1.11), [22] (Sect. 15). Suppose that $\psi_{1}(\tau),\ldots,\psi_{k}(\tau)\in L_{2}([t,T])$ and $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is an arbitrary complete orthonormal system of functions in the space $L_{2}([t,T]).$ Then the following expansion

[TABLE]

[TABLE]

converging in the mean-square sense is valid, where $[x]$ is an integer part of a real number $x;$ another notations are the same as in Theorem 1.

It should be noted that an analogue of Theorem 2 was considered in [52]. Note that we use another notations [19] (Sect. 1.11), [22] (Sect. 15) in comparison with [52]. Moreover, the proof of an analogue of Theorem 2 from [52] is somewhat different from the proof given in [19] (Sect. 1.11), [22] (Sect. 15).

4. Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 6

In a number the author’s works [9]-[21], [23], [28] Theorems 1, 2 have been adapted for the integrals (7) of multiplicities 2 to 4. Let us collect some old results in the following theorem.

Theorem 3 [9]-[21], [23], [28]. Suppose that $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space $L_{2}([t,T]).$ At the same time $\psi_{2}(\tau)$ is a continuously differentiable function on $[t,T]$ and $\psi_{1}(\tau),\psi_{3}(\tau)$ are twice continuously differentiable functions on $[t,T]$. Then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

*where $J^{*}[\psi^{(k)}]_{T,t}$ is defined by (7), and $\psi_{l}(\tau)\equiv 1$ $(l=1,\ldots,4)$ in (40), (42); another notations are the same as in Theorems 1, 2. *

Recently, a new approach to the expansion and mean-square approximation of iterated Stratonovich stochastic integrals has been obtained [19] (Sect. 2.10–2.16), [23] (Sect. 13–19), [24] (Sect. 7–13), [28] (Sect. 5–11), [43] (Sect. 4–9). Let us formulate four theorems that were proved using this approach.

Theorem 4 [19], [23], [24], [28], [43]. Suppose that $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space $L_{2}([t,T]).$ Furthermore, let $\psi_{1}(\tau),\psi_{2}(\tau),$ $\psi_{3}(\tau)$ are continuously differentiable nonrandom functions on $[t,T].$ Then, for the iterated Stratonovich stochastic integral of third multiplicity

[TABLE]

the following relations

[TABLE]

[TABLE]

are fulfilled, where $i_{1},i_{2},i_{3}=0,1,\ldots,m$ in (43) and $i_{1},i_{2},i_{3}=1,\ldots,m$ in (44), constant $C$ is independent of $p,$

[TABLE]

and

[TABLE]

*are independent standard Gaussian random variables for various $i$ or $j$ *(in the case when $i\neq 0$); another notations are the same as in Theorems 1, 2.

Theorem 5 [19], [23], [24], [28], [43]. Let $\{\phi_{j}(x)\}_{j=0}^{\infty}$ be a complete orthonormal system of Legendre polynomials or trigonometric functions in the space $L_{2}([t,T]).$ Furthermore, let $\psi_{1}(\tau),\ldots,$ $\psi_{4}(\tau)$ be continuously differentiable nonrandom functions on $[t,T].$ Then, for the iterated Stratonovich stochastic integral of fourth multiplicity

[TABLE]

the following relations

[TABLE]

[TABLE]

are fulfilled, where $i_{1},\ldots,i_{4}=0,1,\ldots,m$ in (45), (46) and $i_{1},\ldots,i_{4}=1,\ldots,m$ in (47), constant $C$ does not depend on $p,$ $\varepsilon$ is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space $L_{2}([t,T])$ and $\varepsilon=0$ for the case of complete orthonormal system of trigonometric functions in the space $L_{2}([t,T]),$

[TABLE]

[TABLE]

another notations are the same as in Theorem 4.

Theorem 6 [19], [23], [24], [28], [43]. Assume that $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space $L_{2}([t,T])$ and $\psi_{1}(\tau),\ldots,$ $\psi_{5}(\tau)$ are continuously differentiable nonrandom functions on $[t,T].$ Then, for the iterated Stratonovich stochastic integral of fifth multiplicity

[TABLE]

the following relations

[TABLE]

[TABLE]

are fulfilled, where $i_{1},\ldots,i_{5}=0,1,\ldots,m$ in (48), (49) and $i_{1},\ldots,i_{5}=1,\ldots,m$ in (50), constant $C$ is independent of $p,$ $\varepsilon$ is an arbitrary small positive real number for the case of complete orthonormal system of Legendre polynomials in the space $L_{2}([t,T])$ and $\varepsilon=0$ for the case of complete orthonormal system of trigonometric functions in the space $L_{2}([t,T]),$

[TABLE]

another notations are the same as in Theorems 4, 5.

Theorem 7 [19], [23], [24], [28], [44]. Suppose that $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space $L_{2}([t,T]).$ Then, for the iterated Stratonovich stochastic integral of sixth multiplicity

[TABLE]

the following expansion

[TABLE]

that converges in the mean-square sense is valid, where $i_{1},\ldots,i_{6}=0,1,\ldots,m,$

[TABLE]

another notations are the same as in Theorems 4–6.

Note that an analogue of Theorem 3 for the case of iterated Stratonovich stochastic integrals of multiplicity 1 follows from (29).

5. Exact Calculation of the Mean-Square Error in Theorems 1, 2

As we mentioned above, Theorems 1, 2 give new possibilities for exact calculation of the mean-square error of approximation of iterated Ito stochastic integrals (see Theorem 8 below).

Assume that $J[\psi^{(k)}]_{T,t}^{p_{1}\ldots p_{k}}$ is the approximation of (6), which is the expression before passing to the limit

l.i.m.

$\stackrel{{\scriptstyle}}{{{}_{p_{1},\ldots,p_{k}\to\infty}}}$ on the right-hand side of (38)

[TABLE]

[TABLE]

where $[x]$ is an integer part of a real number $x;$ another notations are the same as in Theorems 1, 2.

Let us denote

[TABLE]

[TABLE]

[TABLE]

In [14]-[22], [29] it was shown that

[TABLE]

if $i_{1},\ldots,i_{k}=1,\ldots,m$ $(0<T-t<\infty$) or $i_{1},\ldots,i_{k}=0,1,\ldots,m$ $(0<T-t<1).$

Moreover [7]-[22],

[TABLE]

where constant $C_{n,k}$ depends only on $n$ and $k$ $(n\in\mathbb{N})$.

The value $E_{k}^{p}$ can be calculated exactly.

Theorem 8 [19] (Sect. 1.12), [29] (Sect. 6). Suppose that $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is an arbitrary complete orthonormal system of functions in the space $L_{2}([t,T])$ and $\psi_{1}(\tau),\ldots,\psi_{k}(\tau)\in L_{2}([t,T]),$ $i_{1},\ldots,i_{k}=1,\ldots,m$. Then

[TABLE]

where $i_{1},\ldots,i_{k}=1,\ldots,m;$ the expression

[TABLE]

*means the sum with respect to all possible permutations $(j_{1},\ldots,j_{k})$. At the same time if $j_{r}$ swapped with $j_{q}$ in the permutation $(j_{1},\ldots,j_{k}),$ then $i_{r}$ swapped with $i_{q}$ in the permutation $(i_{1},\ldots,i_{k});$ another notations are the same as in Theorems 1, 2. *

Note that

[TABLE]

Then from Theorem 8 for pairwise different $i_{1},\ldots,i_{k}$ and for $i_{1}=\ldots=i_{k}$ we obtain

[TABLE]

[TABLE]

Consider some examples of application of Theorem 8 $(i_{1},i_{2},i_{3}=1,\ldots,m)$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

6. Comparative Analysis of the Efficiency of Application

of Legendre Polynomials and Trigonometric Functions for the Integral $J_{(11)T,t}^{(i_{1}i_{2})}$

Using Theorems 1, 2 and complete orthonormal system of Legendre polynomials in the space $L_{2}([t,T])$ it is shown [5]-[44] (also see [2]-[4]) that

[TABLE]

where the series converges in the mean-square sense; $i_{1},i_{2}=1,\ldots,m,$

[TABLE]

are independent standard Gaussian random variables for various $i$ or $j$,

[TABLE]

where $P_{j}(x)$ is the Legendre polynomial.

The formula (57) can also be found in [2]-[4]. It is not difficult to show that [2]-[33]

[TABLE]

where

[TABLE]

Let us compare (60) with (16) and (59) with (18). Consider minimal natural numbers $q_{\rm trig}$ and $q_{\rm pol},$ which satisfy to (see Table 1)

[TABLE]

[TABLE]

Thus, we have

[TABLE]

The formula (16) includes $(4q+4)m$ independent standard Gaussian random variables. At the same time the folmula (60) includes only $(2q+2)m$ independent standard Gaussian random variables. Moreover, the formula (60) is simpler than the formula (16). Thus, in this case we can talk about approximately equal computational costs for the formulas (16) and (60).

There is one important feature. As we mentioned above, further we will see that introduction of random variables $\xi_{q}^{(i)}$ and $\mu_{q}^{(i)}$ will sharply complicate the approximation of the iterated stochastic integral $J_{(111)T,t}^{(i_{1}i_{2}i_{3})};$ $i_{1},i_{2},i_{3}=1,\ldots,m.$ This is due to the fact that the number $q$ is fixed for all stochastic integrals, which included into the considered collection. However, it is clear that due to the smallness of $T-t$, the number $q$ for $J_{(111)T,t}^{(i_{1}i_{2}i_{3})}$ could be chosen significantly less than in the formula (16). This feature is also valid for the formulas (15), (19). However, for the case of Legendre polynomials we can choose different numbers $q$ for different iterated stochastic integrals.

From the other hand, if we will not introduce the random variables $\xi_{q}^{(i)}$ and $\mu_{q}^{(i)},$ then the mean-square error of approximation of the iterated stochastic integral $J_{(11)T,t}^{(i_{1}i_{2})}$ will be three times larger (see (13)). Moreover, in this case the stochastic integrals $J_{(01)T,t}^{(0i_{1})}$, $J_{(001)T,t}^{(00i_{1})}$ (with Gaussian distribution) will be approximated worse.

Consider minimal natural numbers $q_{\rm trig}^{*}$, which satisfy to (see Table 1)

[TABLE]

In this situation we can talk about the advantage of Legendre polynomials ($q_{\rm trig}^{*}>q_{pol}$ and (16) is more complex than (60)).

7. Comparative Analysis of the Efficiency

of Application of Legendre Polynomials and Trigonometric Functions for the Integrals $J_{(1)T,t}^{(i_{1})},$ $J_{(11)T,t}^{(i_{1}i_{2})},$ $J_{(01)T,t}^{(0i_{1})},$ $J_{(10)T,t}^{(i_{1}0)},$ $J_{(111)T,t}^{(i_{1}i_{2}i_{3})}$

It is well known [46]-[50] that for implementation of strong Taylor–Ito numerical methods with the order 1.5 of accuracy for Ito stochastic differential equations we need to approximate the following collection of iterated Ito stochastic integrals

[TABLE]

Using Theorems 1, 2 for the system of trigonometric functions, we have [5]-[33] (also see [2]-[4])

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

and $\zeta_{0}^{(i)},$ $\zeta_{2r}^{(i)},$ $\zeta_{2r-1}^{(i)},$ $\xi_{q}^{(i)},$ $\mu_{q}^{(i)}$ ($r=1,\ldots,q;$ $i=1,\ldots,m$) are independent standard Gaussian random variables. Moreover, in (65) we suppose that $i_{1}\neq i_{2},$ $i_{1}\neq i_{3},$ $i_{2}\neq i_{3}$.

Mean-square errors for the approximations (62)–(65) are represented by the formulas

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $i_{1}\neq i_{2},\ i_{1}\neq i_{3},\ i_{2}\neq i_{3}$.

In Table 2, we can see the numerical confirmation of the formula (67) ($\varepsilon$ is a right-hand side of (67)).

Note that the formulas (61), (62) have been obtained for the first time in [46]. Using (61), (62), we can realize numerically the explicit one-step strong Taylor–Ito numerical method with the order 1.0 of accuracy (Milstein scheme [46]). The analogue of the formula (65) has been obtained for the first time in [47]-[49].

As we mentioned above, the Milstein approach (see Sect. 2) leads to iterated application of the operation of limit transition. The analogue of (65) has been derived in [47]-[49], [51] on the base of the Milstein approach [46]. It means that the authors of the works [47]-[49], [51] could not formally use the double sum with the upper limit $q$ in the analogue of (65) in [47] (pp. 438-439), [48] (Sect. 5.8, pp. 202–204), [49] (pp. 82-84), [51] (pp. 263-264) on the base of the Wong–Zakai approximation [53]-[55] (see discussion in Sect. 11 for details). From the other hand, the correctness of (65) follows directly from Theorems 1, 2. Note that (65) has been obtained reasonably for the first time in [5]. The version of (65) but without using the random variables $\xi_{q}^{(i)}$ and $\mu_{q}^{(i)}$ can be found in [2]-[4].

The formula (66) appears for the first time in [46]. The mean-square error (67) has been obtained for the first time in [5] on the base of the simplified variant of Theorem 8 (the case of pairwise different $i_{1},\ldots,i_{k}$).

As we noted above, the number $q$ must be the same in (62)–(65). This is the main drawback of this approach because really the number $q$ in (65) can be chosen essentially smaller than in (62).

Note that in (65) we can replace $J_{(111)T,t}^{(i_{1}i_{2}i_{3})q}$ with $J_{(111)T,t}^{*(i_{1}i_{2}i_{3})q}$ and (65) will when be valid for any $i_{1},i_{2},i_{3}=0,1,\ldots,m$ (see Theorem 3).

Consider approximations of the iterated Ito stochastic integrals

[TABLE]

on the base of Theorems 1, 2 (the case of Legendre polynomials) [2]-[33]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\phi_{j}(x)$ has the form (58) and $P_{i}(x)$ is the Legendre polynomial $(i=0,1,2,\ldots)$.

Mean-square errors for the approximations (69), (72) are represented by the formulas (see Theorem 8 and (52)) [2]-[33]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Let us compare the efficiency of application of Legendre polynomials and trigonometric functions for the iterated stochastic integrals $J_{(11)T,t}^{(i_{1}i_{2})},$ $J_{(111)T,t}^{(i_{1}i_{2}i_{3})}$.

Consider the following conditions $(i_{1}\neq i_{2},\ i_{1}\neq i_{3},\ i_{2}\neq i_{3})$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

where $P_{i}(x)$ is the Legendre polynomial.

In Tables 3 and 4, we can see minimal numbers $q,$ $q_{1},$ $p,$ $p_{1},$ which satisfy the conditions (80)–(83). As we mentioned above, the numbers $q,$ $q_{1}$ are different. At that $q_{1}\ll q$ (the case of Legendre polynomials). As we saw in the previous sections, we cannot take different numbers $p,$ $p_{1}$ for the case of trigonometric functions. Thus, we have to choose $q=p$ in (62)–(65). This leads to huge computational costs (see very complex formula (65)). From the other hand, we can choose different numbers $q$ in (62)–(65). At that we must exclude the random variables $\xi_{q}^{(i)},$ $\mu_{q}^{(i)}$ from (62)–(65).

At this situation for the case $i_{1}\neq i_{2},$ $i_{2}\neq i_{3},$ $i_{1}\neq i_{3}$ we have

[TABLE]

[TABLE]

where the left-hand sides of (84), (85) correspond to (16), (65) but without $\xi_{q}^{(i)},$ $\mu_{q}^{(i)}$. In Table 4, we can see minimal numbers $p^{*},$ $p_{1}^{*}$, which satisfy the conditions (84), (85).

Moreover,

[TABLE]

[TABLE]

where $J_{(01)T,t}^{(0i_{1})q}$, $J_{(10)T,t}^{(i_{1}0)q}$ are defined by the formulas (63), (64).

It is not difficult to see that the numbers $q_{\rm trig}$ in Table 1 correspond to minimal numbers $q_{\rm trig}$, which satisfy the condition

[TABLE]

From the other hand, the right-hand sides of (70), (71) include only two random variables. In this situation we can again talk about the advantage of Ledendre polynomials.

In Table 5, we can see the numerical confirmation of the formula (85) ($\varepsilon$ is a left-hand side of the formula (85)).

8. Comparative Analysis of the Efficiency of

Application of Legendre Polynomials and Trigonometric Functions for the Integral $J_{(011)T,t}^{*(0i_{1}i_{2})}$

In this section, we compare computational costs for the iterated Stratonovich stochastic integral $J_{(011)T,t}^{*(0i_{1}i_{2})}$ $(i_{1},i_{2}=1,\ldots,m)$ within the frames of the method of generalized multiple Fourier series for the systems of Legendre polynomials and trigomomenric functions.

Using Theorem 3 for the case of trigonometric system of functions, we obtain [5]-[33] (also see [2]-[4])

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For the case $i_{1}\neq i_{2}$ from Theorem 8 we get [5]-[33] (also see [2]-[4])

[TABLE]

[TABLE]

Analogues of the formulas (87), (88) for the case of Legendre polynomials will look as follows [5]-[33] (also see [2]-[4])

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In Tables 6 and 7, we can see the numerical confirmation of the formulas (88) and (90) ($\varepsilon$ is the right-hand side of (88), (90)).

Let us compare the complexity of the formulas (87) and (89). The formula (87) includes the double sum

[TABLE]

Thus, the formula (87) is more complex than the formula (89) even if we take identical numbers $q$ in these formulas. As we noted above, the number $q$ in (87) must be equal to the number $q$ from the formula (16), so it is much larger than the number $q$ from the formula (89). As a result, we have obvious advantage of the formula (89) in computational costs. As we mentioned above, if we will not use the random variables $\xi_{q}^{(i)}$ and $\mu_{q}^{(i)},$ then the number $q$ in (87) can be chosen smaller, but the mean-square error of approximation of the stochastic integral $J_{(11)T,t}^{(i_{1}i_{2})}$ will be three times larger (see (13)). Moreover, in this case the stochastic integrals $J_{(01)T,t}^{(0i_{1})}$, $J_{(10)T,t}^{(i_{1}0)},$ $J_{(001)T,t}^{(00i_{1})}$ (with Gaussian distribution) will be approximated worse. In this situation we can again talk about the advantage of Ledendre polynomials.

9. Conclusions

Summing up the results of previous sections, we can come to the following conclusions.

1. We can talk about approximately equal computational costs for the formulas (16) and (60). This means that computational costs for implementing the Milstein scheme (explicit one-step strong Taylor–Ito numerical method with the order $\gamma=1.0$ of accuracy for Ito stochastic differential equations [46]) for the case of Legendre polynomials and for the case of trigonometric functions are approximately the same.

2. If we will not use the random variables $\xi_{q}^{(i)}$ (see (16)), then the mean-square error of approximation of the stochastic integral $J_{(11)T,t}^{(i_{1}i_{2})}$ will be three times larger (see (13)). In this situation, we can talk about the advantage of Ledendre polynomials in the Milstein method. Moreover, in this case the stochastic integrals $J_{(01)T,t}^{(0i_{1})}$, $J_{(10)T,t}^{(i_{1}0)},$ $J_{(001)T,t}^{(00i_{1})}$ (with Gaussian distribution) will be approximated worse.

3. If we talk about the explicit one-step strong Taylor–Ito scheme with the order $\gamma=1.5$ of accuracy for Ito stochastic differential equations, then the numbers $q,$ $q_{1}$ (see (69), (72)) are different. At that $q_{1}\ll q$ (the case of Legendre polynomials). The number $q$ must be the same in (62)–(65) (the case of trigonometric functions). This leads to huge computational costs (see very complex formula (65)). From the other hand, we can take different numbers $q$ in (62)–(65). At that we should exclude the random variables $\xi_{q}^{(i)},$ $\mu_{q}^{(i)}$ from (62)–(65). This leads to another problems, which we discussed above (see Conclusion 1).

4. In addition, the author supposes that effect described in Conclusion 3 will be more impressive when analyzing more complex sets of iterated Ito and Stratonovich stochastic integrals (when $\gamma=$ $2.0,$ $2.5,$ $3.0,$ $\ldots$; here $\gamma$ has the same meaning as in Conclusion 3). This supposition is based on the fact that the polynomial system of functions has the significant advantage (compared with the trigonometric system) for approximation of iterated stochastic integrals for which not all weight functions are equal to 1.

10. Further Development of Multiple Fourier–Legendre Series Approach to

the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 3 to 5

From Theorems 1, 2 for $k=4$ and $5$ we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

are defined by the formula (4); $q_{1}<q;$ ${\bf 1}_{A}$ is the indicator of the set $A;$ $i_{1},i_{2},i_{3},i_{4},i_{5}=0,1,\ldots,m,$ and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $P_{i}(x)$ $(i=0,1,2,\ldots)$ is the Legendre polynomial,

[TABLE]

are independent standard Gaussian random variables for various $i$ or $j$ (if $i\neq 0$),

[TABLE]

is a complete orthonormal system of Legendre polynomials in the space $L_{2}([t,T]).$

Note that the Fourier–Legendre coefficients $\bar{C}_{j_{4}j_{3}j_{2}j_{1}},$ $\bar{C}_{j_{5}j_{4}j_{3}j_{2}j_{1}},$ and $\bar{C}_{j_{3}j_{2}j_{1}}$ (see (73)) can be calculated exactly using DERIVE or MAPLE (computer algebra systems). Several tables with these coefficients can be found in [5]-[21], [25], [32]. The database with $270,000$ of exactly calculated Fourier–Legendre coefficients is descibed in [35], [36]. Note that the mentioned Fourier–Legendre coefficients not depend on the integration step $T-t$ of numerical methods for Ito stochastic differential equations. So, $T-t$ can be not a constant in this approach.

From (52) ($0<T-t<1$) we obtain

[TABLE]

[TABLE]

Note that in practice the numbers $q,q_{1}$ in (72), (91), (92) can be selected not large. For example, for the case of pairwise different $i_{1},i_{2},i_{3},i_{4},i_{5}=1,\ldots,m$ we obtain

[TABLE]

[TABLE]

[TABLE]

From Theorems 3–6 we have

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

are defined by the formula (5).

The values

[TABLE]

[TABLE]

are equal to the right-hand sides of (97)–(99) for the case of pairwise different $i_{1},i_{2},i_{3},i_{4},i_{5}=1,\ldots,m.$

Note that the optimization of the mean-square approximation procedures for the itertaed Ito stochastic integrals (6) of multiplicities 1 to 5 is carried out in [56], [57].

11. Theorems 1–7 from Point

of View of the Wong–Zakai Approximation

The iterated Ito stochastic integrals and solutions of Ito SDEs are complex and important functionals from the independent components ${\bf f}_{s}^{(i)},$ $i=1,\ldots,m$ of the multidimensional Wiener process ${\bf f}_{s},$ $s\in[0,T].$ Let ${\bf f}_{s}^{(i)p},$ $p\in\mathbb{N}$ be some approximation of ${\bf f}_{s}^{(i)},$ $i=1,\ldots,m$. Suppose that ${\bf f}_{s}^{(i)p}$ converges to ${\bf f}_{s}^{(i)},$ $i=1,\ldots,m$ if $p\to\infty$ in some sense and has differentiable sample trajectories.

A natural question arises: if we replace ${\bf f}_{s}^{(i)}$ by ${\bf f}_{s}^{(i)p},$ $i=1,\ldots,m$ in the functionals mentioned above, will the resulting functionals converge to the original functionals from the components ${\bf f}_{s}^{(i)},$ $i=1,\ldots,m$ of the multidimentional Wiener process ${\bf f}_{s}$? The answere to this question is negative in the general case. However, in the pioneering works of Wong E. and Zakai M. [53], [54], it was shown that under the special conditions and for some types of approximations of the Wiener process the answere is affirmative with one peculiarity: the convergence takes place to the iterated Stratonovich stochastic integrals and solutions of Stratonovich SDEs and not to iterated Ito stochastic integrals and solutions of Ito stochastic differential equations. The piecewise linear approximation as well as the regularization by convolution [53]-[55] relate the mentioned types of approximations of the Wiener process. The above approximation of stochastic integrals and solutions of SDEs is often called the Wong–Zakai approximation.

Let ${\bf f}_{s},$ $s\in[0,T]$ be an $m$-dimensional standard Wiener process with independent components ${\bf f}_{s}^{(i)},$ $i=1,\ldots,m.$ It is well known that the following representation takes place [58], [59]

[TABLE]

where $\tau\in[t,T],$ $t\geq 0,$ $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is an arbitrary complete orthonormal system of functions in the space $L_{2}([t,T]),$ and $\zeta_{j}^{(i)}$ are independent standard Gaussian random variables for various $i$ or $j.$ Moreover, the series (100) converges for any $\tau\in[t,T]$ in the mean-square sense.

Let ${\bf f}_{\tau}^{(i)p}-{\bf f}_{t}^{(i)p}$ be the mean-square approximation of the process ${\bf f}_{\tau}^{(i)}-{\bf f}_{t}^{(i)},$ which has the following form

[TABLE]

From (101) we obtain

[TABLE]

Consider the following iterated Riemann–Stieltjes integral

[TABLE]

where $p_{1},\ldots,p_{k}\in\mathbb{N},$  $i_{1},\ldots,i_{k}=0,1,\ldots,m,$

[TABLE]

and $d{\bf f}_{\tau}^{(i)p}$ in defined by the relation (102).

Let us substitute (102) into (103)

[TABLE]

where

[TABLE]

are independent standard Gaussian random variables for various $i$ or $j$ (in the case when $i\neq 0$), ${\bf w}_{s}^{(i)}={\bf f}_{s}^{(i)}$ for $i=1,\ldots,m$ and ${\bf w}_{s}^{(0)}=s,$

[TABLE]

is the Fourier coefficient.

To best of our knowledge [53]-[55] the approximations of the Wiener process in the Wong–Zakai approximation must satisfy fairly strong restrictions [55] (see Definition 7.1, pp. 480–481). Moreover, approximations of the Wiener process that are similar to (101) were not considered in [53], [54] (also see [55], Theorems 7.1, 7.2). Therefore, the proof of analogs of Theorems 7.1 and 7.2 [55] for approximations of the Wiener process based on its series expansion (100) should be carried out separately. Thus, the mean-square convergence of the right-hand side of (105) to the iterated Stratonovich stochastic integral (7) does not follow from the results of the papers [53], [54] (also see [55], Theorems 7.1, 7.2).

From the other hand, Theorems 1–7 from this paper can be considered as the proof of the Wong–Zakai approximation for the iterated Stratonovich stochastic integrals (7) of multiplicities 1 to 6 based on the approximation (101) of the Wiener process. At that, the Riemann–Stieltjes integrals (103) converge (according to Theorems 1–7) to the appropriate Stratonovich stochastic integrals (7). Recall that $\{\phi_{j}(x)\}_{j=0}^{\infty}$ (see (100), (101), and Theorems 3–7) is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space $L_{2}([t,T])$.

To illustrate the above reasoning, consider two examples for the case $k=2,$ $\psi_{1}(s),$ $\psi_{2}(s)\equiv 1;$ $i_{1},i_{2}=1,\ldots,m.$

The first example relates to the piecewise linear approximation of the multidimensional Wiener process (these approximations were considered in [53]-[55]).

Let ${\bf b}_{\Delta}^{(i)}(t),$ $t\in[0,T]$ be the piecewise linear approximation of the $i$th component ${\bf f}_{t}^{(i)}$ of the multidimensional standard Wiener process ${\bf f}_{t},$ $t\in[0,T]$ with independent components ${\bf f}_{t}^{(i)},$ $i=1,\ldots,m,$ i.e.

[TABLE]

where

[TABLE]

Note that w. p. 1

[TABLE]

Consider the following iterated Riemann–Stieltjes integral

[TABLE]

Using (106) and additive property of the Riemann–Stieltjes integrals, we can write w. p. 1

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Using (107) it is not difficult to show that

[TABLE]

[TABLE]

where $\Delta\to 0$ if $N\to\infty$ ($N\Delta=T$).

Obviously, (108) agrees with Theorem 7.1 (see [55], p. 486).

The next example relates to the approximation of the Wiener process based on its series expansion (100) for $t=0$, where $\{\phi_{j}(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space $L_{2}([0,T])$.

Consider the following iterated Riemann–Stieltjes integral

[TABLE]

where $d{\bf f}_{\tau}^{(i)p}$ is defined by the relation (102).

Let us substitute (102) into (109)

[TABLE]

where

[TABLE]

is the Fourier coefficient; another notations are the same as in (105).

As we noted above, approximations of the Wiener process that are similar to (101) were not considered in [53], [54] (also see Theorems 7.1, 7.2 in [55]). Furthermore, the extension of the results of Theorems 7.1 and 7.2 [55] to the case under consideration is not obvious.

On the other hand, we can apply the theory built in Chapters 1 and 2 of the monographs [19]-[21]. More precisely, using Theorem 3, we obtain from (110) the desired result

[TABLE]

From the other hand, by Theorems 1, 2 (see (30)) for the case $k=2$ we obtain from (110) the following relation

[TABLE]

[TABLE]

[TABLE]

Since

[TABLE]

then from (112) we obtain (111).