Convergence in Dirichlet Law of Certain Stochastic Integrals

Christophe Chorro

doi:10.1214/ejp.v10-272

Abstract

Recently, Nicolas Bouleau has proposed an extension of the Donsker's invariance principle in the framework of Dirichlet forms. He proves that an erroneous random walk of i.i.d random variables converges in Dirichlet law toward the Ornstein-Uhlenbeck error structure on the Wiener space. The aim of this paper is to extend this result to some families of stochastic integrals.

Highlights

The error calculus, based on the theory of Dirichlet forms ([7],[12],[17]), is a natural extension of the seminal ideas of Gauss concerning small errors and their propagation ([6], chap.1)
The following example describes one of the simplest error structure on the Wiener space that is intrinsically linked to the so-called Malliavin calculus ([18])
Thanks to a simple integration by parts, we show that (3) is given by a continuous operator of the process Xn whose properties are compatible with error calculus

Summary

Introduction

The error calculus, based on the theory of Dirichlet forms ([7],[12],[17]), is a natural extension of the seminal ideas of Gauss concerning small errors and their propagation ([6], chap.). The following example describes one of the simplest error structure on the Wiener space that is intrinsically linked to the so-called Malliavin calculus ([18]) The main tools used in this paper are the notion of the vectorial domain of a Dirichlet form (in the sense of Feyel and La Pradelle [11]), that allows an extension of the functional calculus (1) for a class of random variables with values in a Banach space, and the improvement (for the Wasserstein metric) of some purely probabilistic convergence theorems

Preliminaries

First results of convergence

Convergence of multiple integrals given by a multi-measure