This paper first constructs a Milstein-type scheme for stochastic Volterra integral equations with doubly singular kernels. Then, we also study the Hölder continuity of the solution to these equations and investigate the convergence rate of the Milstein scheme. More precisely, the mean-square convergence rate is $min\{1-{\alpha }_1-{\beta }_1,1-2{\alpha }_2-2{\beta }_2\}$, where ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2$ are the singularity exponents of the equations. The difficulty in obtaining our convergence results is the lack of Itô formula for the equations. To get around this problem, we adopt the Taylor formula and then perform a complex analysis of the equations satisfied by the solution. Moreover, we apply the multilevel Monte Carlo technique based on the Euler scheme and the fast Euler scheme for the equations to reduce the computational complexity. More concretely, to achieve given desired accuracy $O\left(ϵ\right)$, the multilevel Monte Carlo technique based on the Euler scheme and the fast Euler scheme decrease the computational cost of the standard Euler scheme from $O\left({ϵ}^{-2-\frac{2}{\tilde{\alpha }}}\right)$ to $O\left({ϵ}^{-\frac{2}{\tilde{\alpha }}}\right)$ and $O\left({ϵ}^{-\frac{1}{\tilde{\alpha }}}{|logϵ|}^3\right)$ when $T\approx 1$. Finally, some numerical experiments are given to demonstrate our theoretical results.

本文首先构造了具有双奇异核的随机沃尔泰拉积分方程的米尔斯坦型格式。然后，我们还研究了这些方程解的 Hölder 连续性，并研究了 Milstein 格式的收敛速度。更准确地说，均方收敛速度为 $分钟\{1-{\alpha }_1-{\beta }_1,1-2{\alpha }_2-2{\beta }_2\}$， 在哪里 ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2$是方程的奇异指数。获得我们的收敛结果的困难在于方程缺乏伊藤公式。为了解决这个问题，我们采用泰勒公式，然后对解所满足的方程进行复分析。此外，我们对方程应用基于欧拉格式和快速欧拉格式的多级蒙特卡罗技术，以降低计算复杂度。更具体地说，要达到给定的所需精度 $○\left(\epsilon \right)$，基于欧拉方案和快速欧拉方案的多级蒙特卡罗技术将标准欧拉方案的计算成本从 $○\left({\epsilon }^{-2-\frac{2}{\stackrel{～}{\alpha }}}\right)$至 $○\left({\epsilon }^{-\frac{2}{\stackrel{～}{\alpha }}}\right)$和 $○\left({\epsilon }^{-\frac{1}{\stackrel{～}{\alpha }}}{|日志\epsilon |}^3\right)$什么时候 $吨\approx 1$. 最后，给出了一些数值实验来证明我们的理论结果。