Galerkin Approximation for Stochastic Volterra Integral Equations with Doubly Singular Kernels

Abstract

This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is min{2−2(α1+β1),1−2(α2+β2)} (where α1,α2,β1,β2 are positive numbers satisfying 0<α1+β1<1, 0<α2+β2<12), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method.