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

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). 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 epsilon (1/2,1] is asymptotically mean square stable for all positive step sizes, and the split-step theta method with theta epsilon [0, 1/2] is asymptoticallymean 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.