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 when , and they are exponentially mean-square stable if the stepsize when . Finally, some numerical experiments are given to illustrate the theoretical results.
"Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps." Abstr. Appl. Anal. 2012 (SI12) 1 - 13, 2012. https://doi.org/10.1155/2012/831082