Delay dependent asymptotic mean square stability analysis of the stochastic exponential Euler method

Abstract This paper is concerned with the delay dependent stability of the stochastic exponential Euler method for stochastic delay differential equations and stochastic delay partial differential equations. By using root locus technique, the necessary and sufficient condition of the numerical delay dependent stability of the method is derived for a class of stochastic delay differential equations and it is shown that the stochastic exponential Euler method can fully preserve the asymptotic mean square stability of the underlying system. Furthermore, we investigate the delay dependent stability of the semidiscrete and fully discrete systems for a linear stochastic delay partial differential equation. The necessary and sufficient condition for the delay dependent stability of the semidiscrete system based on the standard central difference scheme in space is given. Based on this condition, the delay dependent stability of the fully discrete system by using the stochastic exponential Euler method in time is studied. It is shown that the fully discrete scheme can inherit the delay dependent stability of the semidiscrete system completely. At last, some numerical experiments are given to validate our theoretical results.