The truncated Euler–Maruyama method for stochastic differential equations

Abstract

Influenced by Higham et al. (2003), several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. In this paper we will develop a new explicit method, called the truncated EM method, for the nonlinear SDE dx(t)=f(x(t))dt+g(x(t))dB(t) and establish the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition xTf(x)+p−12∣g(x)∣2≤K(1+∣x∣2). The type of convergence specifically addressed in this paper is strong-Lq convergence for 2≤q<p, and p is a parameter in the Khasminskii-type condition.