Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich Expansions Using Legendre Polynomials

TL;DR

This paper develops a method using Legendre polynomial-based Fourier series to approximate iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6, aiding high-order numerical solutions of stochastic differential equations.

Contribution

It introduces a new approach for mean-square approximation of iterated stochastic integrals using Legendre polynomials within the Fourier series framework.

Findings

Proved convergence of the approximation method in the mean-square sense.

Derived explicit expansions for integrals of multiplicities 1 to 6.

Established almost sure convergence for the Fourier-Legendre series.

Abstract

The article is devoted to the practical material on expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 with respect to components of the multidimensional Wiener process on the base of the method of generalized multiple Fourier series. More precisely, we used the multiple Fourier--Legendre series converging in the sense of norm in the space $L_2([t, T]^k)$ $(k=1,\ldots,6)$ for approximation of iterated Ito and Stratonovich stochastic integrals. The considered iterated Ito and Stratonovich stochastic integrals are part of the stochastic Taylor expansions (Taylor-Ito and Taylor-Stratonovich expansions). Therefore, the results of the article can be useful for construction of the high-order strong numerical methods for Ito stochastic differential equations. Expansions of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 using Legendre polynomials are derived. The convergence with probability 1 of the mentioned method of generalized multiple Fourier series is proved for iterated Ito stochastic integrals of multiplicities $k$ $(k\in\mathbb{N})$ for the cases of multiple Fourier-Legendre series and multiple trigonometric Fourier series.