  Let $Z$ denote a Hermite process of order $q \geq 1$ and self-similarity
parameter $H \in (\frac{1}{2}, 1)$. This process is $H$-self-similar, has
stationary increments and exhibits long-range dependence. When $q=1$, it
corresponds to the fractional Brownian motion, whereas it is not Gaussian as
soon as $q\geq 2$. In this paper, we deal with a Vasicek-type model driven by
$Z$, of the form $dX_t = a(b - X_t)dt +dZ_t$. Here, $a > 0$ and $b \in
\mathbb{R}$ are considered as unknown drift parameters. We provide estimators
for $a$ and $b$ based on continuous-time observations. For all possible values
of $H$ and $q$, we prove strong consistency and we analyze the asymptotic
fluctuations.
