A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

Abstract

In this paper, the Euler‐Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs), which can be derived from stochastic fractional substantial integro‐differential equations (SFSIDEs). The existence and uniqueness of the solution are proved for the nonlinear SVIEs under the non‐Lipschitz condition. Moreover, the sharp convergence estimate with if if and O(h) if , respectively, is established for the nonlinear SVIEs, which can be revised as supplemented theory results of our recent work (J Comput Appl Math 356 (2019) 377–390). In particular, it also closes a gap that was left on the convergence analysis in (J Comput Appl Math 317 (2017) 447–457). Finally, numerical experiments are given to illustrate the effectiveness of the presented method.