This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its H\"older continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings.

翻译：本文针对带有弱奇异核的随机Volterra方程，提出了随机网格上数值方法的误差分析。首先，通过分析相关的卷积结构，我们证明了精确解的一个新颖的正则性估计。该估计揭示了解在初始时刻具有奇异性，具体表现为其在$t=0$邻域内的Hölder连续指数低于$(0,T]$任意紧子集上的指数。受初始奇异性的启发，我们随后构建了基于随机网格的Euler-Maruyama方法、快速Euler-Maruyama方法以及Milstein方法。通过建立这些方法的时间点态误差估计，我们给出了能够达到最优一致时间收敛阶的网格分级指数，其中收敛阶相较于均匀网格情况得到了提升。最后，我们报告了数值实验，以证实理论结果的精确性。