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Abstract and Applied Analysis
Volume 2014, Article ID 583930, 13 pages
http://dx.doi.org/10.1155/2014/583930
Research Article

The Stochastic -Method for Nonlinear Stochastic Volterra Integro-Differential Equations

1School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
2School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received 6 May 2014; Accepted 28 August 2014; Published 27 October 2014

Academic Editor: Sanling Yuan

Copyright © 2014 Peng Hu and Chengming Huang. 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

The stochastic -method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochastic -method is convergent of order in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochastic -method is mean-square asymptotically stable for every stepsize if and when , the stochastic -method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.