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

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 $\mu=0$.