Convergence and stability of the semi-tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

TL;DR

This paper introduces a semi-tamed Milstein method for solving commutative SDEs with non-globally Lipschitz coefficients, achieving higher convergence order and improved stability compared to existing schemes.

Contribution

It presents a new explicit semi-tamed Milstein scheme that handles non-Lipschitz coefficients and demonstrates superior stability and convergence properties.

Findings

Achieves strong convergence order 1

Maintains uniform stability with the exact solution

Outperforms existing schemes in stability tests

Abstract

A new explicit stochastic scheme of order 1 is proposed for solving commutative stochastic differential equations (SDEs) with non-globally Lipschitz continuous coefficients. The proposed method is a semi-tamed version of Milstein scheme to solve SDEs with the drift coefficient consisting of non-Lipschitz continuous term and globally Lipschitz continuous term. It is easily implementable and achieves higher strong convergence order. A stability criterion for this method is derived, which shows that the stability condition of the numerical methods and that of the solved equations keep uniform. Compared with some widely used numerical schemes, the proposed method has better performance in inheriting the mean square stability of the exact solution of SDEs. Numerical experiments are given to illustrate the obtained convergence and stability properties.