Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations

Abstract

Influenced by Higham et al. (2002), 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. Recently, we developed a new explicit method in Mao (2015), called the truncated EM method, for the nonlinear SDE dx(t)=f(x(t))dt+g(x(t))dB(t) and established 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). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq-convergence of the truncated EM method for 2≤q<p and show that the order of Lq-convergence can be arbitrarily close to q/2.