Strong Convergence of the Euler-Maruyama Method for Nonlinear Stochastic Volterra Integral Equations with Time-Dependent Delay

Abstract

We consider a nonlinear stochastic Volterra integral equation with time-dependent delay and the corresponding Euler-Maruyama method in this paper. Strong convergence rate (at fixed point) of the corresponding Euler-Maruyama method is obtained when coefficients $f$ and $g$ both satisfy local Lipschitz and linear growth conditions. An example is provided to interpret our conclusions. Our result generalizes and improves the conclusion in [J. Gao, H. Liang, S. Ma, Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay, Appl. Math. Comput., 348 (2019) 385-398.]