Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations with Poisson Jumps

Abstract

In this paper, a split-step θ (SST) method is introduced and used to solve the nonlinear neutral stochastic differential delay equations with Poisson jumps (NSDDEwPJ). The mean square asymptotic stability of the SST method for nonlinear neutral stochastic differential equations with Poisson jumps is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the SST method with θ ∈ (0, 2-$\sqrt{2}$) is asymptotically mean square stable for all positive step sizes, and the SST method with θ ∈ (0, 2-$\sqrt{2}$, 1) is asymptotically mean square stable for some step sizes. It is also proved in this paper that the SST method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.