Regularity and strong convergence of numerical approximations for stochastic wave equations with multiplicative fractional Brownian motions

Abstract

Stochastic wave equations with multiplicative fractional Brownian motions (fBms) provide a competitive means to describe wave propagation process driven by inner fractional noise. However, regularity theory and approximate solutions of such equations is still an unsolved problem until now. In this paper, we achieve some progress on the regularity and strong convergence of numerical approximations for a class of semilinear stochastic wave equations with multiplicative fBms. Firstly, we impose some suitable assumptions on the nonlinear term multiplied by fBms and its Malliavin derivatives, and analyze the temporal and spatial regularity of stochastic convolution operator under the assumptions for two cases $H\in (0,1/2)$ and $H\in (1/2,1)$. Using the obtained regularity results of the stochastic convolution operator, we further establish the regularity theory of mild solution of the equation, and reveal quantitatively the influence of Hurst parameter on the regularity of the mild solution. Besides that, we give a fully discrete scheme for the stochastic wave equation and analyze its strong convergence. Finally, two numerical examples are carried out to verify the theoretical findings.