The Improved Stability Analysis of Numerical Method for Stochastic Delay Differential Equations

In this paper, the improved split-step theta method, named the split-step composite theta method, is proposed to study the mean-square stability for stochastic differential equations with a fixed time delay. Under the global Lipschitz and linear growth conditions, it is proved that the split-step composite theta method with theta >= 0.5 shows mean-square stability. An approach to improving numerical stability is illustrated by choices of parameters of this method. Some numerical examples are presented to show the accordance between the theoretical and numerical results.