Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations

Abstract

The Euler–Maruyama method is presented for linear stochastic Volterra integral equations. Then the strong convergence property is analyzed for convolution kernels and general kernels, respectively. It is well known that for stochastic ordinary differential equations, the strong convergence order of the Euler–Maruyama method is 12. However, the strong superconvergence order of 1 is obtained for linear stochastic Volterra integral equations with convolution kernels if the kernel K2 of the diffusion term satisfies K2(0)=0; and this strong superconvergence property is inherited by linear stochastic Volterra integral equations with general kernels if the kernel K2 of the diffusion term satisfies K2(t,t)=0. The theoretical results are illustrated by extensive numerical examples.