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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 831082, 13 pages
http://dx.doi.org/10.1155/2012/831082
Research Article

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

1School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410075, China
2Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, China

Received 19 November 2011; Accepted 30 November 2011

Academic Editor: Shaher Momani

Copyright ยฉ 2012 Qiyong Li and Siqing Gan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ฮ” ๐‘ก = ๐œ / ๐‘š when 1 / 2 โ‰ค ๐œƒ โ‰ค 1 , and they are exponentially mean-square stable if the stepsize ฮ” ๐‘ก โˆˆ ( 0 , ฮ” ๐‘ก 0 ) when 0 โ‰ค ๐œƒ < 1 . Finally, some numerical experiments are given to illustrate the theoretical results.