On the convergence to the multiple Wiener-Itô integral

Xavier Bardina, Maria Jolis and Ciprian Tudor

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Spain
SAMOS/MATISSE, Centre d’Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris 1, 90, rue de Tolbiac, 75634 Paris Cedex 13, France.

Abstract

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C_0([0, T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f \in L^2([0, T]^n). We prove also the weak convergence in the space C_0([0, T]) to the second order integral for two important families of processes that converge to a standard Brownian motion.

Keywords: Multiple Wiener-Itô integrals, weak convergence, Donsker theorem.

Published in: Bulletin des Sciences Mathématiques, 133 (2009) 257-271