Approximation of the finite dimensional distributions of multiple fractional integrals

Xavier Bardina, Khalifa Es-Sebaiy and Ciprian A.Tudor

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Spain
SAMOS-MATISSE, Centre d’Economie de La Sorbonne, Universit´e de Paris 1 Panth´eon-Sorbonne, 90, rue de Tolbiac, 75634, Paris, France.
Laboratoire Paul Painlev´e, Universit´e de Lille 1, F-59655 Villeneuve d’Ascq, France

Abstract

We construct a family $I_{n_{\eps}}(f)_{t}$ of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-It\^o integral $I_{n}^{H}(f1^{\otimes n}_{[0,t] })$ with respect to the fractional Brownian motion. We assume that $H>\frac{1}{2}$ and we prove our approximation result for the integrands $f$ in a rather general class.

Keywords: multiple stochastic integrals, limit theorems, fractional Brownian motion, weak convergence

Published in: Journal of Mathematical Analysis and Applications, 369 (2), 694-711, 2010