Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Exact asymptotics for estimating the marginal density of discretely observed diffusion processes

Auteur(s):

Code(s) de Classification MSC:

• 62G07 Density estimation
Résumé: We derive sharp asymptotic minimax bounds (that is, bounds which concern the exact asymptotic constant of the risk) for nonparametric density estimation based on discretely observed diffusion processes. We study two particular problems for which there already exist such results in the case of independent and identically distributed observations, namely, minimax density estimation in Sobolev classes with $L_{2}$-loss and in H\"{o}lder classes with $L_\infty$-loss. We derive independently lower and upper bounds for the asymptotic minimax risks and show that they coincide with the classical efficiency bounds. It is proven that these bounds can be attained by usual kernel density estimators. The lower bounds are obtained by analyzing the problem of estimating the marginal density in certain families of processes, $\left\{ \{X_i^f\}, f\in{\cal F}_n \right\}$, which are shrinking neighborhoods of around some central process, $\{X_i^{f_0}\}$, in the sense that the set of densities ${\cal F}_n$ forms a shrinking neighborhood centered around $f_0$.