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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 629426, 11 pages
http://dx.doi.org/10.1155/2014/629426
Research Article

The Almost Sure Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Polynomial Growth Condition

1College of Applied Science, Beijing University of Technology, Beijing 100124, China
2College of Science, Hebei United University, Tangshan, Hebei 063009, China

Received 9 May 2014; Revised 2 August 2014; Accepted 4 August 2014; Published 18 August 2014

Academic Editor: Daoyi Xu

Copyright © 2014 Lichao Feng and Shoumei Li. 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

Stability and boundedness are two of the most important topics in the study of stochastic functional differential equations (SFDEs). This paper mainly discusses the almost sure asymptotic stability and the boundedness of nonlinear SFDEs satisfying the local Lipschitz condition but not the linear growth condition. Here we assume that the coefficients of SFDEs are polynomial or dominated by polynomial functions. We give sufficient criteria on the almost sure asymptotic stability and the boundedness for this kind of nonlinear SFDEs. Some nontrivial examples are provided to illustrate our results.