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 (DSSM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSM methods is investigated. Furthermore, the stability regions of the DSSM methods are compared with those of test equation, and it is proved that the methods with are stochastically A-stable. Second, the nonlinear stability of DSSM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with 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.

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