LAMN property for hidden processes: the case of integrated diffusions

Arnaud Gloter (1) , Emmanuel Gobet (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 MATHFI - Mathématiques financières

In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process $X$. Our data are given by $ \int_0^1 X_{\frac{s+i}{n}} d\mu (s)$ for $i=0,\dots,n-1$ and the unknown parameter appears in the diffusion coefficient of the process $X$ only. Although the data are nor Markovian neither Gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of $X$.

Mots clés: Diffusion processes; parametric estimation; LAMN property; Malliavin calculus; non-Markovian data

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Arnaud Gloter, Emmanuel Gobet. LAMN property for hidden processes: the case of integrated diffusions. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2008, 44 (1), pp.104-128. ⟨10.1214/07-AIHP111⟩. ⟨hal-00159317⟩