Mean-square stability of split-step theta Milstein methods for stochastic differential equations
Summary: 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 \(\theta \geq 3 / 2\) 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.
scientific article; zbMATH DE number 7018523
scientific article