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.

Continuous dependence of the solution of Ito stochastic differential equation with nonlocal conditions

“…Many authors in the last decades studied nonlocal problems of ordinary differential equations, the reader is referred to [1][2][3][4][5][6][7] , and references therein. Also the theory of stochastic differential equations, random fixed point theory, existence of solutions of stochastic differential equations by using successive approximation method and properties of these solutions have been extensively studied by several authors, especially those contain the Brownian motion as a formal derivative of the Gausian white noise, the Brownian motion W (t), t ∈ R, is defined as a stochastic process such that W (0) = 0; E(W (t)) = 0, E(W (t)) 2 = t and [W (t 1 ) W (t 2 )] is a Gaussian random variable for all t 1 , t 2 ∈ R. The reader is referred to [8,9] and [10][11][12][13][14][15][16] and references therein.…”