Exponential stability of stochastic differential equation with mixed delay

Wenli Zhu*, Jiexiang Huang, Xinfeng Ruan, Zhao Zhao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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.

Original languageEnglish
Article number187037
JournalJournal of Applied Mathematics
Publication statusPublished - 2014
Externally publishedYes


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