Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series

TL;DR

This paper develops new expansions for high-multiplicity iterated Stratonovich stochastic integrals using generalized Fourier series, simplifying their representation and aiding numerical solutions of stochastic differential equations.

Contribution

It introduces simplified, convergent Fourier series expansions for high-order iterated Stratonovich integrals, improving upon previous methods by reducing complexity and limiting the number of limit operations.

Findings

Expansions converge in the mean-square sense.

Only one limit transition is used in the expansions.

Applications to numerical solutions of Ito SDEs.

Abstract

The article is devoted to the construction of expansions of iterated Stratonovich stochastic integrals of fifth, sixth, seventh and eighth multiplicities based on the method of generalized multiple Fourier series converging in the sense of norm in the Hilbert space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ Specifically, we mainly use multiple Fourier-Legendre series and multiple trigonometric Fourier series $(k=1,\ldots,8)$. The case of generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in $L_2([t, T])$ is also considered for $k=1,\ldots,6$. Recently, expansions of iterated Stratonovich stochastic integrals of multiplicity $k,$ $k\in\mathbb{N}$ (the case of continuous weight functions and an arbitrary complete orthonormal system of functions in $L_2([t, T])$) have been obtained (Theorems 42, 44) but under one additional condition. The considered expansions converge in the mean-square sense and contain only one operation of the limit transition in contrast to its existing analogues. Expansions of iterated Stratonovich stochastic integrals turned out much simpler than appropriate expansions of iterated Ito stochastic integrals. We use expansions of the latter as a tool of the proof of expansions for iterated Stratonovich stochastic integrals. Iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion for solutions of Ito stochastic differential equations. That is why the results of the article can be applied to the numerical integrations of Ito stochastic differential equations.