Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

Abstract

This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we first show that the SDE is $p$th moment exponentially stable (for $p\in (0,1)$) if and only if the stochastic theta method is $p$th moment exponentially stable for a sufficiently small step size. We then show that the $p$th moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size ${\Delta t}$. If the stochastic theta method is $p$th moment exponentially stable for a sufficiently small...