Mean square stability of two classes of theta method for neutral stochastic differential delay equations

In this paper, a stochastic linear theta (SLT) method is introduced and analyzed for neutral stochastic differential delay equations (NSDDEs). We give some conditions on neutral item, drift and diffusion coefficients, which admit that the diffusion coefficient can be highly nonlinear and does not necessarily satisfy a linear growth or global Lipschitz condition. It is proved that, for all positive stepsizes, the SLT method with θ∈[12,1] is asymptotically mean stable and so is θ∈[0,12) under a stronger assumption. Furthermore, we consider the split-step theta (SST) method and obtain a similar but better result. That is, the SST method with θ∈[12,1] is exponentially mean stable and so is θ∈[0,12). Finally, two numerical examples are given to show the efficiency of the obtained results.