Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 917892, 22 pages
Research Article

Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

Lin Hu1,2 and Siqing Gan1

1School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410075, China
2College of Science, Northeast Forestry University, Harbin, Heilongjiang 150040, China

Received 1 August 2011; Accepted 11 October 2011

Academic Editor: Xiaohua Ding

Copyright © 2011 Lin Hu 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.


A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.