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Journal of Applied Mathematics
Volume 2014, Article ID 187037, 11 pages
http://dx.doi.org/10.1155/2014/187037
Research Article

Exponential Stability of Stochastic Differential Equation with Mixed Delay

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 24 November 2013; Revised 21 January 2014; Accepted 21 January 2014; Published 4 March 2014

Academic Editor: Oluwole Daniel Makinde

Copyright © 2014 Wenli Zhu et al. 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 focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.