Strong convergence of the stopped Euler–Maruyama method for nonlinear stochastic differential equations

Abstract

In this paper, numerical methods for the nonlinear stochastic differential equations (SDEs) with non-global Lipschitz drift coefficient are discussed. The existing known results have only so far shown that the classical (explicit) Euler–Maruyama (EM) approximate solutions converge to the true solution in probability [22,23]. More recently, the authors in [16] proved that the classical EM method will diverge in L2 sense for the underlying SDEs in this paper (and those SDEs with superlinearly growing coefficients). These strongly indicate that the classical EM method is not good enough for the highly nonlinear SDEs. However, in this paper, we introduce a modified EM method using stopping time and show successfully that the discrete version of the modified EM approximate solution converges to the true solution in the strong sense (namely in L2) with a order arbitrarily close to a half.