LAMN property for hidden processes : the case of integrated diffusions

Gloter, Arnaud; Gobet, Emmanuel

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 104-128.

Dans ce papier nous démontrons la propriété LAMN pour le modèle statistique constitué par l’observation des moyennes locales d’une diffusion $X$. Nos données sont définies comme ${\int }_0^1{X}_{(s+i)/n}{d}_{\mu }\left(s\right)$ avec $i=0,...,n-1$ et le paramètre inconnu apparaît seulement dans le coefficient de diffusion du processus $X$. Bien que cette observation ne soit ni gaussienne ni markovienne nous pouvons, par le calcul de Malliavin, obtenir une expression pour la log-vraisemblance du modèle. Nous sommes alors capables de calculer l’information asymptotique et montrons qu’elle est la même que pour l’observation ponctuelle de la diffusion.

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 ${\int }_0^1{X}_{(s+i)/n}{d}_{\mu }\left(s\right)$ for $i=0,...,n-1$ and the unknown parameter appears in the diffusion coefficient of the process $X$ only. Although the data are neither markovian nor 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 $X$.

Classification : 60F99, 60H07, 62F12, 62M09
Mots clés : diffusion processes, parametric estimation, LAMN property, Malliavin calculus, non-markovian data

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