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

Haiyan Yuan,Jihong Shen,Cheng Song

doi:10.1155/2016/7397941

Abstract

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 withθ∈(1/2,1]is asymptotically mean square stable for all positive step sizes, and the split-step theta method withθ∈[0,1/2]is asymptotically mean 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.

Highlights

Stochastic functional differential equations (SFDEs) play important roles in science and engineering applications, especially for systems whose evolutions in time are influenced by random forces as well as their history information
When the time delays in SFDEs are constants, they turn into stochastic delay differential equations (SDDEs)
Many dynamical systems depend on the present and the past states and involve derivatives with delays; they are described as the neutral stochastic delay differential equations (NSDDEs)

Summary

Introduction

Stochastic functional differential equations (SFDEs) play important roles in science and engineering applications, especially for systems whose evolutions in time are influenced by random forces as well as their history information. When the time delays in SFDEs are constants, they turn into stochastic delay differential equations (SDDEs) Both the theory and numerical methods for SDDEs have been well developed in the recent decades; see [1,2,3,4,5,6,7,8]. The analytical solutions of NSDDEs are hard to obtain; many authors have to study the numerical methods for NSDDEs. Wu and Mao [10] studied the convergence of the Euler-Maruyama method for neutral stochastic functional differential equations under the one-side Lipschitz conditions and the linear growth conditions.

Exponential Mean Square Stability of Analytic Solution

The Stability of the Split-Step Theta Method

Mean Square Dissipativity

The Numerical Experiment