Numerical methods for stochastic Volterra integral equations with weakly singular kernels

In this paper, we first establish the existence, uniqueness and H\"older continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a $\theta$-Euler-Maruyama scheme and a Milstein scheme to solve the equations numerically and we obtain the strong rates of convergence for both schemes in $L^{p}$ norm for any $p\geq 1$. For the $\theta$-Euler-Maruyama scheme the rate is $\min\{1-\alpha,\frac{1}{2}-\beta\}~$ and for the Milstein scheme the rate is $\min\{1-\alpha,1-2\beta\}$ when $\alpha\neq \frac 12$, where $(0<\alpha<1, 0< \beta<\frac{1}{2})$. These results on the rates of convergence are significantly different from that of the similar schemes for the stochastic Volterra integral equations with regular kernels. The difficulty to obtain our results is the lack of It\^o formula for the equations. To get around of this difficulty we use instead the Taylor formula and then carry a sophisticated analysis on the equation the solution satisfies.