Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation

TL;DR

This paper develops a method to expand and approximate iterated Stratonovich stochastic integrals of multiplicity 3 using generalized Fourier series, facilitating numerical solutions of stochastic differential equations.

Contribution

It introduces a new approach for mean-square expansion of iterated Stratonovich integrals of multiplicity 3 using Fourier-Legendre and trigonometric series, applicable to numerical SDE integration.

Findings

Derived main results using triple Fourier-Legendre series.

Extended results to integrals of multiplicities 3 to 6.

Applicable to numerical integration of Ito SDEs with strong convergence.

Abstract

The article is devoted to the development of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the mean. We adapt this method for iterated Stratonovich stochastic integrals of multiplicity 3 from the Taylor-Stratonovich expansion. The main result of the article has been derived with using the triple Fourier-Legendre series and triple trigonometric Fourier series for the general case of series summation. Some recent results on the expansion of iterated Stratonovich stochastic integrals of multiplicities 3 to 6 are given. The results of the article can be applied to the numerical integration of Ito stochastic differential equations in accordance with the strong criterion of convergence.