The truncated Milstein method for stochastic differential equations with commutative noise

Qian Guo,Xuerong Mao,Wei Liu,Rongxian Yue

doi:10.1016/j.cam.2018.01.014

Abstract

Inspired by the truncated EulerMaruyama method developed in Mao (2015), we propose the truncated Milstein method in this paper. The strong convergence rate is proved to be close to 1 for a class of highly non-linear stochastic differential equations with commutative noise. Numerical examples are given to illustrate the theoretical results.

Highlights

Stochastic differential equation (SDE), as a power tool to model uncertainties, has been broadly applied to many areas [1,2,3]
Numerical approximations to SDEs become essential in the applications of SDE models
Despite the good performance of the strong convergence, implicit methods have their own disadvantage that some non-linear systems need to be solved in each iteration, which may be computationally expensive and introduce some more errors

Summary

Introduction

Stochastic differential equation (SDE), as a power tool to model uncertainties, has been broadly applied to many areas [1,2,3]. Despite the good performance of the strong convergence, implicit methods have their own disadvantage that some non-linear systems need to be solved in each iteration, which may be computationally expensive and introduce some more errors Another way to tackle SDEs with non-global Lipschitz coefficients is to modify the drift and diffusion coefficients in the numerical methods. In this paper we propose the truncated Milstein method, which is an explicit method and has the strong convergence rate of arbitrarily closing to one In this work, both of the drift and diffusion coefficients of the SDEs under investigation could grow super-linearly. The truncated Milstein method developed in this paper may still be applicable to the case of the non-commutative diffusion coefficient, but more complicated notations and different techniques will be involved The ideas in those works [34,35,36] may provide hints for the proofs.

Mathematical preliminaries

The classical Milstein method

The truncated Milstein method

Boundedness of the moments

Main results

An example and further discussion