Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments

Abstract

<p style='text-indent:20px;'>In this paper, we investigate the strong convergence rate of the split-step theta (SST) method for a kind of stochastic differential equations with piecewise continuous arguments (SDEPCAs) under some polynomially growing conditions. It is shown that the SST method with <inline-formula><tex-math id="M1">\begin{document}$θ∈[\frac{1}{2},1]$ \end{document}</tex-math></inline-formula> is strongly convergent with order <inline-formula><tex-math id="M2">\begin{document}$\frac{1}{2}$ \end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$p$ \end{document}</tex-math></inline-formula>th(<inline-formula><tex-math id="M4">\begin{document}$p≥ 2$ \end{document}</tex-math></inline-formula>) moment if both drift and diffusion coefficients are polynomially growing with regard to the delay terms, while the diffusion coefficients are globally Lipschitz continuous in non-delay arguments. The exponential mean square stability of the improved split-step theta (ISST) method is also studied without the linear growth condition. With some relaxed restrictions on the step-size, it is proved that the ISST method with <inline-formula><tex-math id="M5">\begin{document}$θ∈(\frac{1}{2},1]$ \end{document}</tex-math></inline-formula> is exponentially mean square stable under the monotone condition. Without any restriction on the step-size, there exists <inline-formula><tex-math id="M6">\begin{document}$θ^*∈(\frac{1}{2},1]$ \end{document}</tex-math></inline-formula> such that the ISST method with <inline-formula><tex-math id="M7">\begin{document}$θ∈(θ^*,1]$ \end{document}</tex-math></inline-formula> is exponentially stable in mean square. Some numerical simulations are presented to illustrate the analytical theory.