Drift Estimation for a L\'evy-Driven Ornstein-Uhlenbeck Process with Heavy Tails

TL;DR

This paper develops an efficient maximum likelihood estimator for the drift parameter of a heavy-tailed Lévy-driven Ornstein-Uhlenbeck process observed continuously over a long period, establishing its asymptotic properties.

Contribution

It introduces a novel estimation method for heavy-tailed Lévy-driven processes and proves the asymptotic efficiency of the maximum likelihood estimator.

Findings

The statistical model is locally asymptotic mixed normal.

The maximum likelihood estimator is asymptotically efficient.

The approach applies to ergodic Ornstein-Uhlenbeck processes driven by Lévy processes.

Abstract

We consider the problem of estimation of the drift parameter of an ergodic Ornstein--Uhlenbeck type process driven by a L\'evy process with heavy tails. The process is observed continuously on a long time interval $[0,T]$, $T\to\infty$. We prove that the statistical model is locally asymptotic mixed normal and the maximum likelihood estimator is asymptotically efficient.