The Convergence Analyzed by Stochastic C-stability and Stochastic B-consistency of Split-step Theta Method for the SDE

Abstract:

      In this paper, the convergence of the split-step theta method for the stochastic differential equation is analyzed by the stochastic C-stability and the stochastic B-consistency. The result of the numerical scheme which is stochastically C-stable and stochastically B-consistent is convergent has been proved in the precious paper. In order to analyze the convergence of the split-step theta method(\theta\in [\frac{1}{2},1]), the stochastic C-stability and the stochastic B-consistency under the global monotonicity condition has been researched and explored the rate \frac{1}{2}. From the conclusion, it can be seen that the convergence which don't need the drift function satisfied the linear growth condition when \theta=\frac{1}{2}. What's more, the convergent rate of the split-step scheme of the stochastic differential equation with additive noise is researched to be 1. Finally, an example is given to illustrate the convergence with the theoretical results.