The truncated Euler\u2013Maruyama method for stochastic differential delay equations

Qian Guo,Xuerong Mao,Rongxian Yue

doi:10.1007/s11075-017-0391-0

Qian Guo, Xuerong Mao + Show 1 more

Open Access

https://doi.org/10.1007/s11075-017-0391-0

Copy DOI

Abstract

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512–5524 2011), and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in Lp) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370–384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.

Highlights

In the study of stochastic differential delay equations (SDDEs), the classical existence-and-uniqueness theorem requires that the coefficients of the SDDEs satisfy the local Lipschitz condition and the linear growth condition
The Khasminskii-type theorem enables us to verify if a given nonlinear SDDE has a unique global solution under the local Lipschitz condition but without the linear growth condition
The numerical solutions of SDDEs under the generalized Khasminskii-type condition were discussed by Mao [15], and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability

Summary

Introduction

In the study of stochastic differential delay equations (SDDEs), the classical existence-and-uniqueness theorem requires that the coefficients of the SDDEs satisfy the local Lipschitz condition and the linear growth condition (see, e.g., [1, 6, 9, 12, 20]). The Khasminskii-type theorem enables us to verify if a given nonlinear SDDE has a unique global solution under the local Lipschitz condition but without the linear growth condition. The numerical solutions under the linear growth condition plus the local Lipschitz condition have been discussed intensively by many authors (see, e.g., [3,4,5, 8, 11, 19, 21]). Mao [16] develops a new explicit numerical method, called the truncated EM method, for SDEs under the Khasminskii-type condition plus the local Lipschitz condition and establishes the strong convergence theory. We will use this new truncated EM method to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.

The truncated Euler-Maruyama method

Convergence rates

Conclusion