The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

TL;DR

This paper introduces a tamed Milstein scheme for commutative SDEs with non-globally Lipschitz coefficients, achieving higher convergence order than the tamed Euler method and demonstrating computational efficiency.

Contribution

A new explicit tamed Milstein method for SDEs with commutative noise that improves convergence order over existing tamed Euler schemes.

Findings

Achieves strong convergence order one.

More computationally efficient than tamed Euler.

Successfully overcomes difficulties in convergence proof.

Abstract

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit Euler scheme fails to converge strongly to the exact solution. Recently, an explicit strongly convergent numerical scheme, called the tamed Euler method, is proposed in [Hutzenthaler, Jentzen & Kloeden, Ann. Appl. Probab., 22 (2012), pp. 1611-1641.] for such SDEs. Motivated by their work, we here introduce a tamed version of the Milstein scheme for SDEs with commutative noise. The proposed method is also explicit and easily implementable, but achieves higher strong convergence order than the tamed Euler method does. In recovering the strong convergence order one of the new method, new difficulties arise and kind of a bootstrap argument is developed to overcome them. Finally, an illustrative example confirms the computational efficiency of the tamed Milstein method compared to the tamed Euler method.