Double-implicit and split two-step Milstein schemes for stochastic differential equations

In this work, we propose two classes of two-step Milstein-type schemes : the double-implicit Milstein scheme and the split two-step Milstein scheme, to solve stochastic differential equations (SDEs). Our results reveal that the two new schemes are strong convergent with order one. Moreover, with a restriction on stepsize, these two schemes can preserve the exponential mean square stability of the original SDEs, and the decay rate of numerical solution will converge to the decay rate of the exact solution. Numerical experiments are performed to confirm our theoretic findings.