Limit distributions for the discretization error of stochastic Volterra equations

TL;DR

This paper characterizes the asymptotic distribution of discretization errors in stochastic Volterra equations with fractional kernels, showing convergence to a linear Volterra equation's solution, extending known results from standard SDEs.

Contribution

It establishes the limit law for discretization errors in stochastic Volterra equations, a novel extension of classical results for standard stochastic differential equations.

Findings

Discretization error converges in law to a linear Volterra equation solution.

Normalized error follows the same fractional kernel as the original equation.

Results extend classical SDE discretization error analysis to Volterra equations.

Abstract

Our study aims to specify the asymptotic error distribution in the discretization of a stochastic Volterra equation with a fractional kernel. It is well-known that for a standard stochastic differential equation, the discretization error, normalized with its rate of convergence $1/\sqrt{n}$, converges in law to the solution of a certain linear equation. Similarly to this, we show that a suitably normalized discretization error of the Volterra equation converges in law to the solution of a certain linear Volterra equation with the same fractional kernel.