Convergence rate and stability of the truncated Euler–Maruyama method for stochastic differential equations

Liangjian Hu,Xuerong Mao,Xiaoyue Li

doi:10.1016/j.cam.2018.01.017

Abstract

Abstract Recently, Mao (2015) developed a new explicit method, called the truncated Euler–Maruyama (EM) method, for the nonlinear SDE and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. In his another follow-up paper (Mao, 2016), he discussed the rates of L q -convergence of the truncated EM method for q ≥ 2 and showed that the order of L q -convergence can be arbitrarily close to q ∕ 2 under some additional conditions. However, there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is to establish the convergence rate without these restrictions. The other aim is to study the stability of the truncated EM method. The advantages of our new results will be highlighted by the comparisons with the results in Mao (2015, 2016) as well as others on the tamed EM and implicit methods.

Highlights

Influenced by Higham, Mao and Stuart [1], the strong convergence theory of numerical methods for nonlinear stochastic differential equations (SDEs) without the global Lipschitz condition has become more and more popular
A nice numerical method should have an acceptable finite-time convergence rate and have the ability to preserve the asymptotic properties of the underlying SDEs. Another aim of this paper is to show the ability of the truncated EM method to preserve the asymptotic stability of the underlying SDEs
In this paper we made a quick review on the main result of [14,15] on the truncated EM method and pointed out that condition (3.5) imposed there to obtain the strong convergence rate is somehow restrictive

Summary

Introduction

Influenced by Higham, Mao and Stuart [1], the strong convergence theory of numerical methods for nonlinear stochastic differential equations (SDEs) without the global Lipschitz condition has become more and more popular. Sabanis in [11] went a further step to propose the modified tamed Euler method approximating the SDEs with superlinearly growing drift and diffusion coefficients, recovered the strong order 1/2 in the estimation of convergence rate. Other explicit methods, such as the stopped EM method, as well as the tamed Milstein method, have been further developed (see, e.g., [12,13] for details). We prove that the classical EM method cannot reproduce asymptotic stability while the truncated method does

Notation and lemmas

Convergence rates

Comparisons with known results

Asymptotic stability

Conclusion