Approximation of solutions of the stochastic wave equation by using the Fourier series

TL;DR

This paper investigates approximating solutions to a one-dimensional stochastic wave equation driven by stochastic measures using Fourier series expansions, demonstrating that partial sums and Fejér sums provide effective approximations.

Contribution

The paper introduces a method for approximating solutions of stochastic wave equations via Fourier series of stochastic measures, extending existing techniques to stochastic integrators.

Findings

Partial sums of Fourier series approximate the stochastic wave equation solutions.

Fejér sums improve the approximation accuracy.

The approach applies to general stochastic measures.

Abstract

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fej\`{e}r sums we obtain the approximations of mild solution of the equation.