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Recently the investigation on the stability of the numerical solutions to delayed stochastic differential equations has received an increasing attention, but there has been little work on the stability analysis of the numerical solutions to delayed stochastic Hopfield neural networks (DSHNNs) so far. The aim in this paper is to study the mean square exponential stability of the split-step theta (SST) method and the stochastic linear theta (SLT) method for the underlying model. It is proved that, for any θ ∈ [ 0 , 1 2 ) , there exists a constant Δ ∗ > 0 depending on θ such that the numerical schemes produced by the SST method and the SLT method are mean square exponentially stable for Δ ∈ ( 0 , Δ ∗ ) , under the same assumptions as those to guarantee the mean square exponential stability of the underlying continuous model. For the case θ ∈ [ 1 2 , 1 ] , we show the same stability conclusion for all Δ > 0 . To carry out the required conclusion, a novel technique for the stability analysis of discrete numerical schemes with multi time delays, namely the weighted sum Lyapunov functional method, is proposed. Finally, a numerical example is given to illustrate the application of the suggested methods and to verify the stability conclusions obtained.

Stability of the solution to stochastic delay differential equations (SDDEs) have received a great deal of attention, but there is so far little work on stability of numerical solutions to nonlinear stochastic functional differential equations (SFDEs). To close the gap, this paper proposes and studies a numerical scheme called complete backward Euler numerical scheme for general SFDEs. In the paper, we come up with a more general polynomial growth condition with Lyapunov function. Under the generalized polynomial growth condition, the almost sure exponential stability of the underlying continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Besides, the solvability of the implicit scheme is studied specially by introducing the concept of the generalized monotone vectorial functions. To establish the stability criteria for the nonlinear continuous model and the implicit scheme, some necessary lemmas have been established at first. At the end of the paper, a numerical example with simulation is proposed to illustrate the conditions, method and conclusions of the paper.

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 ? ? 1 2 , 1 is asymptotically mean stable and so is ? ? 0 , 1 2 ) 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 ? ? 1 2 , 1 is exponentially mean stable and so is ? ? 0 , 1 2 ) . Finally, two numerical examples are given to show the efficiency of the obtained results.