Mean‐square stability of the backward Euler–Maruyama method for neutral stochastic delay differential equations with jumps

Abstract

This paper is mainly considered whether the mean-square stability of neutral stochastic delay differential equations (NSD-DEs) with jumps is shared with that of the backward Euler-Maruyama method. Under the one-sided Lipschitz condition and the linear growth condition, the trivial solution of NSDDEs with jumps is proved to be mean-square stable by using the functional comparison principle and the Barbalat's lemma. It is shown that the backward Euler-Maruyama method can reproduce the mean-square stability of the trivial solution under the same conditions. The implicit backward Euler-Maruyama method shows better characteristic than the explicit Euler-Maruyama method for the reason that it works without the linear growth condition on the drift coefficient. Compared with some existing results, our results do not need to add extra condition on the neutral part. The conclusions can be applied to NSDDEs and SDDEs with jumps. The effectiveness of the theoretical results is illustrated by an example.

On stability of numerical solutions of neutral stochastic delay differential equations with time‐dependent delay

“…In [4][5][6][7][8], the exponential stability for the Euler-Maruyama (EM), backward Euler-Maruyama (BEM), and theta Euler-Maruyama approximations have been investigated for SDDEs, neutral SDDEs, and SFDEs. In [9][10][11], numerical solutions together with the moment and almost sure exponential stability of numerical solutions of neutral SDDEs with jumps have been investigated. Numerical solutions and the almost sure exponential stability of numerical solutions of highly nonlinear SDDEs under Khasminskiii-type conditions have been studied in [12][13][14][15][16].…”