Mean square convergence of explicit two-step methods for highly nonlinear stochastic differential equations

In this paper, we propose the projected two-step Euler Maruyama method and the projected two-step Milstein method for highly nonlinear stochastic differential equations. Under a global monotonicity condition, we first examine the strong convergence (in mean square sense) for these two explicit schemes based on the notions of stochastic stability and B-consistency for two-step methods. We prove that the convergence rates of the projected two-step Euler Maruyama method and the projected two-step Milstein method are [Formula presented] and 1, respectively. In particular, our results can be applied to equations with super-linearly growing drift and diffusion coefficients. Finally, we numerically verify the optimal mean square convergence orders of these two schemes by a series of examples.