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

Abstract

We study the problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $d{X}_t=(\mu +\theta X_t)dt+d{G}_t,t\ge 0$ with unknown parameters $\theta >0$ and $\mu \in \mathbb{R}$, where $G$ is a Gaussian process. We provide least square-type estimators ${\tilde{\theta }}_T$ and ${\tilde{\mu }}_T$ respectively for the drift parameters $\theta$ and $\mu$ based on continuous-time observations $\{X_t,t\in [0,T]\}$ as $T\to \infty$. Our aim is to derive some sufficient conditions on the driving Gaussian process $G$ in order to ensure that ${\tilde{\theta }}_T$ and ${\tilde{\mu }}_T$ are strongly consistent, the limit distribution of ${\tilde{\theta }}_T$ is a Cauchy-type distribution and ${\tilde{\mu }}_T$ 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 $\mu =0$.