Stability Analysis of the Split-Step Theta Method for Nonlinear Regime-Switching Jump Systems

Abstract

In this paper, we investigate the stability of the split-step theta (SST) method for a class of nonlinear regime-switching jump systems–neutral stochastic delay differential equations (NSDDEs) with Markov switching and jumps. As we know, there are few results on the stability of numerical solutions for NSDDEs with Markov switching and jumps. The purpose of this paper is to enrich conclusions in such respect. It first devotes to showing that the trivial solution of the NSDDE with Markov switching and jumps is exponentially mean square stable and asymptotically mean square stable under some suitable conditions. If the drift coefficient also satisfies the linear growth condition, it then proves that the SST method applied to the NSDDE with Markov switching and jumps shares the same conclusions with the exact solution. Moreover, a numerical example is demonstrated to illustrate the obtained results.