Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4. Combained Approach Based on Generalized Multiple and Iterated Fourier series

TL;DR

This paper develops new Fourier series-based expansions for iterated Stratonovich stochastic integrals of multiplicities 1 to 4, ensuring mean-square convergence without using Ito integrals, aiding numerical solutions of stochastic differential equations.

Contribution

It introduces a combined Fourier series approach for expanding iterated Stratonovich integrals with proven mean-square convergence, avoiding the use of Ito integrals and simplifying approximation procedures.

Findings

Proved mean-square convergence of the expansions.

Considered Fourier-Legendre and trigonometric series.

Ensured only one limit operation in the expansions.

Abstract

The article is devoted to the expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 on the base of the combined approach of generalized multiple and iterated Fourier series. We consider two different parts of the expansion of iterated Stratonovich stochastic integrals. The mean-square convergence of the first part is proved on the base of generalized multiple Fourier series that are converge in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,2,3,4.$ The mean-square convergence of the second part is proved on the base of generalized iterated Fourier series that are converge pointwise. At that, we do not use the iterated Ito stochastic integrals as a tool of the proof and directly consider the iterated Stratonovich stochastic integrals. The cases of multiple Fourier-Legendre series and multiple trigonometric Fourier series are considered in detail. 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.