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

TL;DR

This paper develops an exact method for calculating the mean-square error in expanding iterated Ito stochastic integrals using generalized Fourier series, improving numerical solutions for stochastic differential equations.

Contribution

It provides exact and approximate formulas for mean-square errors and proves convergence for Fourier-based expansion methods, enhancing stochastic numerical analysis.

Findings

Exact formulas for mean-square approximation errors.

Proved convergence with probability 1 for Fourier series expansions.

Applicable to high-order numerical methods for stochastic differential equations.

Abstract

The article is devoted to the developement of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k)$ ($k$ is the multiplicity of the iterated Ito stochastic integral). We obtain the exact and approximate expressions for the mean-square approximation error of iterated Ito stochastic integrals of multiplicity $k$ ($k\in\mathbb{N}$) from the stochastic Taylor-Ito expansion. As a result, we do not need to use redundant terms of expansions of iterated Ito stochastic integrals that complicate the numerical methods for Ito stochastic differential equations. Moreover, we proved the convergence with propability 1 for the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series for the cases of multiple Fourier-Legendre series and multiple trigonometric Fourier series. Mean-square approximation of iterated Stratonovich stochastic integrals is also considered in the article. The results of the article can be applied to the high-order strong numerical methods for Ito stochastic differential equations as well as for non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise.