Large deviations and concentration inequalities for the Ornstein-Uhlenbeck process without tears

TL;DR

This paper develops new large deviation principles and concentration inequalities for the maximum likelihood estimator of the Ornstein-Uhlenbeck process's drift parameter, applicable across stable, unstable, and explosive regimes.

Contribution

It introduces a novel strategy to establish large deviations that overcomes classical non-steepness issues, applicable to all recurrence cases of the process.

Findings

Unified large deviation results for all regimes of the Ornstein-Uhlenbeck process

New concentration inequalities for the maximum likelihood estimator

Applicable to stable, null recurrent, and transient cases

Abstract

Our goal is to establish large deviations and concentration inequalities for the maximum likelihood estimator of the drift parameter of the Ornstein-Uhlenbeck process without tears. We propose a new strategy to establish large deviation results which allows us, via a suitable transformation, to circumvent the classical difficulty of non-steepness. Our approach holds in the stable case where the process is positive recurrent as well as in the unstable and explosive cases where the process is respectively null recurrent and transient. Notwithstanding of this trichotomy, we also provide new concentration inequalities for the maximum likelihood estimator.