Weak approximation of a fractional SDE: the Donsker setting

Xavier Bardina, Carles Rovira and Samy Tindel

Departament de Matemàtiques, Facultat de Ciències, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain.
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain.
Institut Élie Cartan Nancy, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France.

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.

Published in: Electronic Communications in Probability, 15, 314-329 (2010)