The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients
For researchers simulating SDEs with superlinear drift, this provides a more accurate explicit scheme than the tamed Euler method.
The paper introduces an explicit tamed Milstein method for commutative SDEs with non-globally Lipschitz drift, achieving strong convergence order 1, which is higher than the tamed Euler method's order 0.5. Numerical experiments confirm its computational efficiency.
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.