## Journal of Applied Mathematics

### Exponential Stability of Stochastic Differential Equation with Mixed Delay

#### 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.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 187037, 11 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305581

Digital Object Identifier
doi:10.1155/2014/187037

Mathematical Reviews number (MathSciNet)
MR3178952

Zentralblatt MATH identifier
07010564

#### Citation

Zhu, Wenli; Huang, Jiexiang; Ruan, Xinfeng; Zhao, Zhao. Exponential Stability of Stochastic Differential Equation with Mixed Delay. J. Appl. Math. 2014 (2014), Article ID 187037, 11 pages. doi:10.1155/2014/187037. https://projecteuclid.org/euclid.jam/1425305581

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