On the convergence to the multiple Wiener-Ito integral

Xavier Bardina (1) , Maria Jolis (1) , Ciprian A. Tudor (2, 3)

1 UAB - Departament de Matemàtiques [Barcelona]
2 CES - Centre d'économie de la Sorbonne
3 SAMOS - Statistique Appliquée et MOdélisation Stochastique

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.

Mots clés: Multiple Wiener-Itô integrals; Weak convergence; Donsker theorem

Domaines: Probabilités [math.PR]

Xavier Bardina, Maria Jolis, Ciprian A. Tudor. On the convergence to the multiple Wiener-Ito integral. Bulletin des Sciences Mathématiques, 2009, 133 (3), pp.257-271. ⟨10.1016/j.bulsci.2008.09.002⟩. ⟨hal-00200914⟩