Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods

We introduce two approaches by modifying split-step exponential schemes to study stochastic differential equations. Under the Lipschitz condition and linear-growth bounds, it is shown that our explicit schemes converge to the solution of the corresponding stochastic differential equations with the order 1.0 in the mean-square sense. The mean-square stability of our methods is investigated through some linear stochastic test systems. Additionally, asymptotic mean-square stability is analyzed for the two-dimensional system with symmetric and asymmetric coefficients and driven by two commutative noise terms. In particular, we prove that our methods are mean-square stable for any step-size. Finally, some numerical experiments are carried out to confirm the theoretical results