An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

Emmanuelle Clement (1) , Sylvain Delattre (2) , Arnaud Gloter (3)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 Département de Mathématiques (Evry)

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.

Mots clés: LAMN property; convolution theorem; diffusion process

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Emmanuelle Clement, Sylvain Delattre, Arnaud Gloter. An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility. Stochastic Processes and their Applications, 2013, 123 (7), pp.2500-2521. ⟨hal-00719460⟩