On the convergence to the multiple Wiener-Ito integral

TL;DR

This paper investigates how processes with paths in the Cameron-Martin space can be used to approximate multiple Wiener-Itô integrals, establishing weak convergence results in finite-dimensional distributions and in the space of continuous functions.

Contribution

It introduces a method to construct processes converging to multiple Wiener-Itô integrals from paths in the Cameron-Martin space, extending convergence results to second order integrals.

Findings

Weak convergence of processes to multiple Wiener-Itô integrals in finite-dimensional distributions.

Weak convergence in the space of continuous functions for second order integrals.

Application to families of processes converging to standard Brownian motion.

Abstract

We study the convergence to the multiple Wiener-It\^{o} 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 $\mathcal C_0([0,T])$. Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-It\^{o} integral process of a function $f\in L^2([0,T]^n)$. We prove also the weak convergence in the space $\mathcal C_0([0,T])$ to the second order integral for two important families of processes that converge to a standard Brownian motion.