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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 $dX_t=(μ+θX_t)dt+dG_t,\ t\geq0$ with unknown parameters $θ>0$ and $μ\in\mathbb{R}$, where $G$ is a Gaussian process. We provide least square-type estimators $\widetildeθ_T$ and $\widetildeμ_T$ respectively for the drift parameters $θ$ and $μ$ 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 $\widetildeθ_T$ and $\widetildeμ_T$ are strongly consistent, the limit distribution of $\widetildeθ_T$ is a Cauchy-type distribution and $\widetildeμ_T$ 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 $μ=0$.