Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

Weiyin Fei,Dengfeng Xia,Liangjian Hu,Xuerong Mao

doi:10.3934/cpaa.2020092

Abstract

Guo et al. [GMY17] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations(SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [GMY17] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction.

Highlights

Stochastic differential delay equations (SDDEs) have been used in many branches of science and industry
We reviewed one of the main results of [GMY17] and pointed out a restrictive condition imposed there via an example
The authors would like to thank the referees for their very helpful comments and suggestions

Summary

Introduction

Stochastic differential delay equations (SDDEs) have been used in many branches of science and industry (see, e.g., [Arn,CLM01,DZ92,Kha,LL]). The classical theory on the existence and uniqueness of the solution to the SDDE requires the coefficients of the SDDE satisfy the local Lipschitz condition and the linear growth condition (see, e.g., [KM,M97,M02,Moh]). The numerical solutions under the linear growth condition plus the local Lipschitz condition have been discussed intensively by many authors (see, e.g., [BB,BB05,CKR06, DFLM, HM05,KloP,KP,MS,Mil,Schurz,WM08]). Mathematical Subject Classifications (2000): Primary: 60H10, 60J65. Stochastic differential delay equation, Ito’s formula, truncated Euler–Maruyama, Khasminskii-type condition.

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