Strong convergence of an A-stable parameterized split-step Milstein method for nonlinear stochastic differential equations

Abstract

In this paper, we consider general nonlinear stochastic differential equations of It\^{o} type, and propose a split-step numerical method, in which parameterized balancing techniques are designed in both steps. The resulting numerical method is called parameterized split-step Milstein method. We present the bounded second moment of the numerical solution, and prove the first order of strong convergence. The mean-square stability of the new method is well investigated, and the range of the proper parameters are given to guarantee the A-stability of the proposed method. Compared with the balanced Milstein method with satisfied stability, the new method has better stability. Several numerical experiments are attached to exhibit the performance of the new method.