Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions

Loukianova, D.; Loukianov, O.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 771-786.

Habituellement le problème de l'estimation du drift pour un processus de diffusion est considéré sous l'hypothèse de l'ergodicité. Il l'est moins souvent sous l'hypothèse de nulle-récurrence, car dans ce cas il y a moins de théorèmes limites, et ceux qui existent ne s'appliquent pas à toute la classe nulle-récurrente. Le but de cet article est de démontrer quelques théorèmes limites pour les fonctionnelles additives et martingales dépendantes d'une diffusion récurrente générale (ergodique ou nulle). Ces théorèmes permettent de donner une approche unifiée au problème de l'estimation non-paramétrique par noyau du drift dans le cas unidimensionnel récurrent. Comme exemple on obtient la vitesse de convergence de l'estimateur de Nadaraya-Watson dans le cas d'un drift localement hölderien.

Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya-Watson estimator in the case of a locally Hölder-continuous drift.

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