Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in Beyn et al. (J Sci Comput 67(3):955–987, 2016. doi: 10.1007/s10915-015-0114-4). As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler–Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.