Estimating drift parameters in a non-ergodic Gaussian Vasicek-type model

TL;DR

This paper develops estimators for the drift parameters in a non-ergodic Gaussian Vasicek-type model, establishing their consistency and distributional limits under certain conditions, with applications to various fractional processes.

Contribution

It introduces new least squares estimators for the drift parameters in a non-ergodic Gaussian Vasicek model and derives their asymptotic properties, extending previous results to more general Gaussian processes.

Findings

Estimators are strongly consistent under specified conditions.

The limit distribution of the estimator for θ is Cauchy-type.

The estimator for μ is asymptotically normal.

Abstract

We study the problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $dX_t=(\mu+\theta X_t)dt+dG_t,\ t\geq0$ with unknown parameters $\theta>0$ and $\mu\in\mathbb{R}$, where $G$ is a Gaussian process. We provide least square-type estimators $\widetilde{\theta}_T$ and $\widetilde{\mu}_T$ respectively for the drift parameters $\theta$ and $\mu$ based on continuous-time observations $\{X_t,\ t\in[0,T]\}$ as $T\rightarrow\infty$. Our aim is to derive some sufficient conditions on the driving Gaussian process $G$ in order to ensure that $\widetilde{\theta}_T$ and $\widetilde{\mu}_T$ are strongly consistent, the limit distribution of $\widetilde{\theta}_T$ is a Cauchy-type distribution and $\widetilde{\mu}_T$ is asymptotically normal. We apply our result to fractional Vasicek, subfractional Vasicek and bifractional Vasicek processes. In addition, this work extends the result of \cite{EEO} studied in the case where $\mu=0$.