Error analysis of numerical methods on graded meshes for stochastic Volterra equations

TL;DR

This paper analyzes the error of numerical methods on graded meshes for stochastic Volterra equations with singular kernels, providing new regularity estimates and optimal convergence results confirmed by numerical experiments.

Contribution

It introduces a novel regularity estimate for solutions and develops graded mesh methods with proven optimal convergence for stochastic Volterra equations.

Findings

Regularity estimate reveals initial singularity in solutions.

Graded mesh methods achieve improved convergence orders.

Numerical experiments confirm theoretical sharpness.

Abstract

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.