Statistical inference for Vasicek-type model driven by Hermite processes

Let $(Z^{q, H}_t)_{t \geq 0}$ 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 the following Vasicek-type model driven by $Z^{q, H}$: \[ X_0=0,\quad dX_t = a(b - X_t)dt +dZ_t^{q, H}, \qquad t \geq 0, \] where $a > 0$ 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.