Stability analysis for semi-linear stochastic differential equation with piecewise continuous arguments

Abstract

In this paper, the exponential stability in mean square of the exponential Euler method for semi-linear stochastic differential equation with piecewise continuous arguments is obtained. Firstly, the exponential Euler scheme of the equation is given. Secondly, under Lipschitz condition, sufficient conditions of exponential stability to the exact solution are achieved in mean square. Furthermore, it is proved that exponential Euler method preserves exponential stability of the exact solution without any restriction on the step-size. Finally, an example is provided to illustrate our theories.