Weak approximation of fractional SDES: The Donsker setting

TL;DR

This paper investigates the weak convergence of fractional stochastic differential equations driven by Liouville fractional Brownian motion with Hurst parameter between 1/3 and 1/2, using a novel approximation method.

Contribution

It introduces a new approximation scheme for fractional Brownian motion via rescaled random walks and proves convergence of the associated SDEs in law.

Findings

Approximation of fractional Brownian motion by rescaled random walks with Liouville kernel.

Convergence in law of the approximated SDEs to the fractional SDE driven by B.

Provides a new method for weak approximation of fractional SDEs.

Abstract

In this note, we take up the study of weak convergence for stochastic differential equations driven by a (Liouville) fractional Brownian motion $B$ with Hurst parameter $H\in(1/3,1/2)$. In the current paper, we approximate the $d$-dimensional fBm by the convolution of a rescaled random walk with Liouville's kernel. We then show that the corresponding differential equation converges in law to a fractional SDE driven by $B$.