Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes: Non-ergodic Case

TL;DR

This paper investigates the properties of the least squares estimator for the drift parameter in a non-ergodic fractional Ornstein-Uhlenbeck process driven by fractional Brownian motion, focusing on consistency and asymptotic distribution as observation time grows.

Contribution

It provides new results on the consistency and asymptotic behavior of the least squares estimator for the non-ergodic fractional Ornstein-Uhlenbeck process with Hurst index greater than 1/2.

Findings

Estimator is consistent as t approaches infinity.

Asymptotic distribution characterized for the estimator.

Results extend understanding of parameter estimation in non-ergodic fractional processes.

Abstract

We consider the parameter estimation problem for the non-ergodic fractional Ornstein-Uhlenbeck process defined as $dX_t=\theta X_tdt+dB_t,\ t\geq0$, with a parameter $\theta>0$, where $B$ is a fractional Brownian motion of Hurst index $H\in(1/2,1)$. We study the consistency and the asymptotic distributions of the least squares estimator $\hat{\theta}_t$ of $\theta$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$.