Regularity analysis for SEEs with multiplicative fBms and strong convergence for a fully discrete scheme

Abstract

AbstractOne of the open problems in the study of stochastic differential equations is regularity analysis and approximations to stochastic partial differential equations driven by multiplicative fractional Brownian motions (fBms), especially for the case $H\in (0,\frac {1}{2})$. In this paper, we address this problem by considering a class of stochastic evolution equations (SEEs) driven by multiplicative fBms. We analyze the well-posedness and regularity of mild solutions to such equations with $H\in (0,\frac {1}{2})$ and $H\in (\frac {1}{2},1)$ under the Lipschitz conditions and linear growth conditions. The two cases are treated separately. Compared with the standard Brownian motion case, the main difficulty is that fBm is neither a Markov process nor a semimartingale such that some classical stochastic calculus theories are unavailable. As a consequence, we need to explore some new strategies to complete the existence and uniqueness and regularity analysis of the solutions. Especially for the case $H\in (0,\frac {1}{2})$, we utilize some delicate techniques to overcome the difficulties from the singularity of the covariance of fBms. In addition, we give a fully discrete scheme for such equations, carried out by the spectral Galerkin method in space and a time-stepping method in time. The obtained regularity results of the equations help us to examine the strong convergence of the discrete scheme. In final, several numerical examples are done to substantiate the theoretical findings.