Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

Abstract

This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize $\Delta t=\tau /m$ when $1/2\le \theta \le 1$, and they are exponentially mean-square stable if the stepsize $\Delta t\in (0,\Delta t_0)$ when . Finally, some numerical experiments are given to illustrate the theoretical results.