Asymptotic stability and continuity of nonlinear hybrid stochastic differential equation with randomly occurring delay

Abstract

This paper concentrates on discussing several types of asymptotic stability and continuity for the nonlinear hybrid stochastic differential equation (NHSDE) with randomly occurring delay. Different from the time delay considered in previous studies, we discuss the randomly occurring time-varying delay characterized by the random variable subject to the Bernoulli distribution. The existence and uniqueness of the maximal local solution and the global solution to the NHSDE are investigated by using some classical inequalities and the stochastic analysis theory. Then, by means of the delay-dependent Lyapunov functional method and the M-matrix technique, some novel sufficient conditions are given to ensure that the trivial solution can achieve the pth moment asymptotic stability, the almost surely asymptotic stability, the pth moment continuity and the continuity in probability. The stability and continuity criteria established in this paper include the upper bound of the time-varying delay and are less conservative. In the end, two numerical examples and corresponding computer simulations are provided to verify the validity of the presented theoretical results.