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

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 O ( h..+ 12) if... ( 0, 12), O(h center dot v | ln h|) if.. = 12 and O(h) if... ( 12, 1], 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 thatwas left on the convergence analysis in (J ComputAppl Math 317 (2017) 447-457). Finally, numerical experiments are given to illustrate the effectiveness of the presented method.

Automated summary

In this paper, the Euler-Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs). The existence and uniqueness of the solution are proved for the SVIEs under the non-Lipschitz condition. Convergence estimates are established for the SVIEs, and numerical experiments are provided to illustrate the effectiveness of the method.