On mean square stability and dissipativity of split-step theta method for nonlinear neutral stochastic delay differential equations (Q1677662)

Summary: A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations. The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method with \(\theta \in(1 / 2, 1]\) is asymptotically mean square stable for all positive step sizes, and the split-step theta method with \(\theta \in [0, 1 / 2]\) is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.