Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels

TL;DR

This paper establishes the optimal convergence rate of the Milstein scheme for stochastic Volterra equations with singular kernels and characterizes the limit distribution of the normalized error.

Contribution

It proves the optimality of the convergence rate and describes the stable law limit of the normalized error for these equations.

Findings

Convergence rate of n^{-2H} is optimal for the Milstein scheme.

Normalized error converges stably in law to a solution of a linear Volterra equation.

Provides a detailed description of the error distribution for stochastic Volterra equations with singular kernels.

Abstract

For stochastic Volterra equations driven by standard Brownian and with singular kernels $K(u)=u^{H-\frac{1}{2}}/\Gamma(H+1/2), H\in (0,1/2)$, it is known that the Milstein scheme has a convergence rate of $n^{-2H}$. In this paper, we show that this rate is optimal. Moreover, we show that the error normalized by $n^{-2H}$ converge stably in law to the (nonzero) solution of a certain linear Volterra equation of random coefficients with the same fractional kernel.