On the convergence to the multiple Wiener-Itô integral

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 C0 ([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 ∈ L2 ([0, T]n). We prove also the weak convergence in the space C0 ([0, T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion. © 2008 Elsevier Masson SAS. All rights reserved.