Mean square stability of two classes of theta methods for numerical computation and simulation of delayed stochastic Hopfield neural networks

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∈[0,12), 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∈[12,1], 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.