Convergence And Stability Of Two Numerical Schemes For Solving Stochastic Delay Differential Equations

Stochastic delay differential equations (SDDEs) are widely applied in many fields, such as economics, biology, environmental engineering, etc. Compared to the SDDEs in Ito sense, the SDDEs in Stratonovich sense holds the ordinary chain rule and has consistency with respect to the approximated equation obtained by replacing the Brownian motion in SDDEs with its spectral expansion. But there are only few results about this topic in literature. The paper focuses on the stability and convergence of two numerical methods for SDDEs in Stratonovich sense.First of all, we summarize numerical methods for SDDEs and their recent de-velopment in this paper. Then we further give the motivation and introduce main work of this paper.Then, we introduce some essential theorems and give some definitions and the-orems related to our research in this paper. Moreover, the conditions of mean-square asymptotic stability of the trivial solution for SDDEs are established with a general form.The main research of this paper is divided into two parts:Firstly, the Midpoint scheme is a fully implicit scheme. There are few results of its convergence for both SDEs and SDDEs. We study the convergence of the Midpoint scheme for SDDEs and prove that the Midpoint scheme is of a half order convergence in mean-square sense.Secondly, we discuss asymptotic mean-square stability of the Predictor-Corrector scheme and the Midpoint scheme for linear equations and systems in Stratonovich sense. It’s proved that the Predictor-Corrector scheme is asymptotically mean-square stable when step size h is less than a given value and the Midpoint scheme is uncon-ditionally mean-square stable. Furthermore, we compare the stability regions of the two numerical schemes with other well-known numerical methods. For the nonlinear SDDEs which satisfy the one-sided Lipschitz condition, we study the mean-square asymptotic stability of the Predictor-corrector scheme and obtain a sufficient condi-tion for its stability.In numerical examples, we solve both a nonlinear equation and a system, which verify the theoretical predictions.