Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps

The exponential mean-square stability of the split-step θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. New conditions for jumps are proposed to ensure the exponential mean-square stability of the trivial solution. If the drift coefficient satisfies the linear growth condition, it is shown that the split-step θ-method can reproduce the exponential mean-square stability of the trivial solution for the constrained stepsize. Then by applying the Chebyshev inequality and the Borel-Cantelli lemma, the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained. Since split-step θ-method covers Euler-Maruyama (EM) method and split-step backward Euler (SSBE) method, the conclusions are valid for these two methods. Moreover, they can adapt to the NSDDEs and the SDDEs with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.