Switching-dominated stability of numerical solutions for hybrid neutral stochastic differential delay equations

Abstract

This paper is concerned with the exponential stability of numerical solutions for neutral stochastic differential delay equations (NSDDEs) with Markovian switching (or hybrid NSDDEs). It is known that Markov chain can work as a stabilizing factor, that is, when some subsystems are stable and others are unstable, the overall system can be stable. In this paper, such property is called switching-dominated stability, and we prove that the switching-dominated stability can be recovered in the stability of numerical solutions based on the switched Lyapunov function method. Firstly, a fundamental stability criterion is established for the numerical solutions. Then under a linear growth condition, we show that the Euler–Maruyama (EM) method can share the mean square exponential stability of the exact solution. When the linear growth condition is defied, but a one-sided Lipschitz condition is satisfied, we show that the backward EM (BEM) method can reproduce the mean square exponential stability of the exact solution. Numerical experiments are carried out to confirm the theoretical results.