Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes

TL;DR

This paper studies the estimation of the drift parameter in non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes, providing conditions for estimator consistency and asymptotic distribution, extending previous results to fractional and other Gaussian processes.

Contribution

It offers a unified, elementary proof for the estimator's properties and extends existing results to fractional Brownian motion and other Gaussian processes.

Findings

Estimator is strongly consistent under certain conditions.

Asymptotic distribution of the estimator is derived.

Results apply to fractional, subfractional, and bifractional Brownian motions.

Abstract

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein-Uhlenbeck process defined as $dX_t=\theta X_tdt+dG_t,\ t\geq0$ with an unknown parameter $\theta>0$, where $G$ is a Gaussian process. We provide sufficient conditions, based on the properties of $G$, ensuring the strong consistency and the asymptotic distribution of our estimator $\widetilde{\theta}_t$ of $\theta$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$. Our approach offers an elementary, unifying proof of \cite{BEO}, and it allows to extend the result of \cite{BEO} to the case when $G$ is a fractional Brownian motion with Hurst parameter $H\in(0,1)$. We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.