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

TL;DR

This paper develops a unified theoretical framework for analyzing additive functionals and martingales in one-dimensional recurrent diffusions, enabling non-parametric drift estimation even under null-recurrence, with specific results on the Nadaraya–Watson estimator.

Contribution

It provides new limit theorems for additive functionals of recurrent diffusions, extending drift estimation methods to null-recurrent cases.

Findings

Established limit theorems for additive functionals in recurrent diffusions.

Derived the convergence rate of the Nadaraya–Watson estimator for locally Hölder-continuous drift.

Unified approach applicable to both ergodic and null-recurrent diffusions.

Abstract

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 H\"{o}lder-continuous drift.