Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations

TL;DR

This paper introduces a Fourier series-based approximation method for iterated Ito stochastic integrals, enabling the development of high-order numerical schemes for complex stochastic PDEs with non-commutative noise.

Contribution

It presents a novel Fourier series approach to approximate iterated stochastic integrals, facilitating high-order numerical methods for non-commutative semilinear stochastic PDEs.

Findings

Effective approximation of iterated stochastic integrals of arbitrary multiplicity.

Application to high-order strong numerical methods for stochastic PDEs.

Enhanced accuracy in simulating non-commutative stochastic systems.

Abstract

We consider a method for the approximation of iterated stochastic integrals of arbitrary multiplicity $k$ $(k\in \mathbb{N})$ with respect to the infinite-dimensional $Q$-Wiener process using the mean-square approximation method of iterated Ito stochastic integrals with respect to the scalar standard Wiener processes based on generalized multiple Fourier series. The case of multiple Fourier-Legendre series is considered in details. The results of the article can be applied to construction of high-order strong numerical methods (with respect to the temporal discretization) for the approximation of mild solution for non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise.