Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 217672, 11 pages
http://dx.doi.org/10.1155/2011/217672
Research Article

Almost Surely Asymptotic Stability of Numerical Solutions for Neutral Stochastic Delay Differential Equations

Zhanhua Yu1 and Mingzhu Liu2

1Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 7 March 2011; Accepted 11 April 2011

Academic Editor: Her-Terng Yau

Copyright © 2011 Zhanhua Yu and Mingzhu Liu. 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

We investigate the almost surely asymptotic stability of Euler-type methods for neutral stochastic delay differential equations (NSDDEs) using the discrete semimartingale convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.