Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSS theta M) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSS theta M methods is investigated. Furthermore, the stability regions of the DSS theta M methods are compared with those of test equation, and it is proved that the methods with theta >= 3/2 are stochastically A-stable. Second, the nonlinear stability of DSS theta M methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with theta > 1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.