Truncated Euler–Maruyama method for hybrid stochastic functional differential equations with infinite time delay

Abstract

Li et al. (2023) developed a new theory to approximate the solution of hybrid stochastic functional differential equations (SFDEs) with infinite time delay via the numerical solution of the corresponding hybrid SFDEs with finite time delay. But hybrid SFDEs were required to be globally Lipschitz continuous. In this paper, we will lift this restriction. Under the local Lipschitz condition and the Khasminskii-type condition, numerical solutions of hybrid SFDEs with infinite time delay will be designed by using the truncated Euler–Maruyama method. The strong convergence and convergence rate of the numerical solutions in $L^q(q\ge 2)$ will be obtained. Finally, an example to stochastic functional volatility model is given to demonstrate the effectiveness of our new theory.