Expansions of Iterated Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series: Multiplicities 1 to 8 and Beyond

TL;DR

This paper develops generalized Fourier series expansions for iterated Stratonovich stochastic integrals of any multiplicity, improving approximation methods for stochastic differential equations with multidimensional noises.

Contribution

It extends previous results to arbitrary multiplicities using generalized Fourier series, with only one limit transition, enhancing mean-square approximation accuracy.

Findings

Expansions for multiplicities 1 to 8 using Fourier-Legendre and trigonometric series.

Generalization to arbitrary multiplicities with a single limit transition.

Applications to numerical solutions of multidimensional Ito SDEs.

Abstract

The article is devoted to the expansions of iterated Stratonovich stochastic integrals on the basis of the method of generalized multiple Fourier series that converge in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ Expansions of iterated Stratonovich stochastic integrals are obtained for the case of multiple Fourier-Legendre series and for the case of multiple trigonometric Fourier series $(k=1,\ldots,8)$. Recently, expansions of iterated Stratonovich stochastic integrals of multiplicities $k=1,\ldots,6$ (the case of continuous weight functions and an arbitrary complete orthonormal system of functions in $L_2([t, T])$) have been obtained. These results are generalized to the case of multiplicitity $k,$ $k\in\mathbb{N}$ (Theorems 51, 53) but under one additional condition. The considered expansions contain only one operation of the limit transition in contrast to its existing analogues. This property is very important for the mean-square approximation of iterated stochastic integrals. The results of the article can be applied to the numerical integration of Ito stochastic differential equations with multidimensional non-commutative noises.