  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, Isaak, Kruse, J. Sci. Comp., 2015]. 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.
