  We propose and analyse a new Milstein type scheme for simulating stochastic
differential equations (SDEs) with highly nonlinear coefficients. Our work is
motivated by the need to justify multi-level Monte Carlo simulations for
mean-reverting financial models with polynomial growth in the diffusion term.
We introduce a double implicit Milstein scheme and show that it possesses
desirable properties. It converges strongly and preserves non-negativity for a
rich family of financial models and can reproduce linear and nonlinear
stability behaviour of the underlying SDE without severe restriction on the
time step. Although the scheme is implicit, we point out examples of financial
models where an explicit formula for the solution to the scheme can be found.
