Numerical methods for stochastic Volterra integral equations with weakly singular kernels

In this paper we first establish the existence, uniqueness and Holder continuity of the solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, with singularities alpha is an element of(0, 1) for the drift term and beta is an element of (0, 1/2) for the stochastic term. Subsequently, we propose theta-Euler-Maruyama scheme and a Milstein scheme to solve the equations numerically and obtain strong rates of convergence for both schemes in L-p norm for any p >= 1. For the theta-Euler-Maruyama scheme the rate is min {1-alpha, 1/2 - beta} and for the Milstein scheme is min{1 - alpha, 1 - 2 beta}. These results on the rates of convergence are significantly different from those it is similar schemes for the SVIEs with regular kernels. The source of the difficulty is the lack of Ito formula for the equations. To get around this difficulty we use the Taylor formula subsequently carrying out a sophisticated analysis of the equation.