Sharp large deviations for the non-stationary Ornstein-Uhlenbeck process

TL;DR

This paper investigates sharp large deviation principles for the maximum likelihood estimator of the drift parameter in the Ornstein-Uhlenbeck process across stable, unstable, and explosive cases, revealing unique rate functions especially in the explosive case.

Contribution

It provides a comprehensive analysis of large deviations for the estimator in all three regimes, highlighting novel features in the explosive case.

Findings

Distinct asymptotic behaviors in stable, unstable, and explosive cases.

Unusual rate function with a flat valley and discontinuity in the explosive case.

Enhanced understanding of estimator deviations in non-stationary processes.

Abstract

For the Ornstein-Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding of this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.