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

• Main Authors: Qiyong Li, Siqing Gan
• Format: Article
• Language: English
• Published: Hindawi Limited 2012-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://dx.doi.org/10.1155/2012/831082

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 Δt=τ/m when 1/2≤θ≤1, and they are exponentially mean-square stable if the stepsize Δt∈(0,Δt0) when 0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.