Delay-dependent stability of predictor-corrector methods of Runge-Kutta type for stochastic delay differential equations

The delay-dependent mean square stability of stochastic delay differential equations is in the forefront the structure-preserving numerical algorithms. The sufficient and necessary conditions of mean square stability for a general class of stochastic Runge-Kutta via predictor-corrector methods (SRK-PCMs) are obtained, which perform better than existing schemes. Furthermore, by regulating factor theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} in drift term in corrector step, we could explore the optimal stable regions. Several theorems about convergence and stability are proved for SRK-PCMs. A thoroughgoing system of numerical experiments verify the theorems ans remarks.