Stability Of Numerical Solutions For Stochastic Delay Systems

As is well known, there are rich results on stochastic delay systems and they have been widely applied in many fields such as economics, finance, physics, biol-ogy, medicine, and other science. The stability is the most important topic in the field of stochastic systems since it is the premise condition of all the normal-work systems. However, many stochastic delay systems may not have explicit solutions. Therefore, it seems to be interesting and necessary to study the numerical solutions for stochastic delay systems. In this paper, we consider two representative stochas-tic delay systems. They are neutral stochastic differential delay equations (NSDDEs) and stochastic delay Hopfield neural networks. For above systems, We mainly think about four numerical methods, concluding stochastic linear θ (SLT) method, split-step θ (SST) method, Euler method and backward Euler method.We now introduce our main results as follows:Firstly, we study the mean square stability of two classes of 9 method for neutral stochastic differential delay equations.In this part, a stochastic linear θ (SLT) method is introduced and analyzed for neutral stochastic differential delay equations (NSDDEs). We give some condition-s on the 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∈ [1/2,1] is asymptotically stable in mean square and so is the case θ∈ [0,1/2) under a stronger assumption for Δ ∈ (0, Δt0).Furthermore, we consider the split-step θ (SST) method and obtain a similar but better result. That is, the SST method with θ ∈ [1/2,1] is mean square stable, and so is the case θ∈ [0,1/2) under a stronger assumption for Δ ∈ (0, Δt0).Secondly, we discuss the almost sure exponential stability of numerical solutions for stochastic delay Hopfield neural networks.In this part, we study the almost sure exponential stability of numerical solution-s for stochastic delay Hopfield neural networks by using two approaches:the Euler method and the backward Euler method. Under some simple and reasonable condi-tions, both the Euler scheme and the backward Euler scheme are proved to be almost sure exponential stable. In particular, to establish the stability criteria for the Euler method and the backward Euler method, what we apply is the nonnegative semi-martingale convergence theorem.