Convergence, non-negativity and stability of a new Milstein scheme with applications to finance

Abstract

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