Parameter estimation for discretely observed non-ergodic fractional Ornstein–Uhlenbeck processes of the second kind

Abstract

We use the least squares type estimation to estimate the drift parameter $\theta>0$ of a non-ergodic fractional Ornstein–Uhlenbeck process of the second kind defined as $dX_{t}=\theta X_{t}\,dt+dY_{t}^{(1)},X_{0}=0$, $t\geq0$, where $Y_{t}^{(1)}=\int_{0}^{t}e^{-s}\,dB_{a_{s}}$ with $a_{t}=He^{\frac{t}{H}}$, and $\{B_{t},t\geq0\}$ is a fractional Brownian motion of Hurst parameter $H\in(\frac{1}{2},1)$. We assume that the process $\{X_{t},t\geq0\}$ is observed at discrete time instants $t_{1}=\Delta_{n},\ldots,t_{n}=n\Delta_{n}$. We construct two estimators $\hat{\theta}_{n}$ and $\check{\theta}_{n}$ of $\theta$ which are strongly consistent and we prove that these estimators are $\sqrt{n\Delta_{n}}$-consistent, in the sense that the sequences $\sqrt{n\Delta_{n}}(\hat{\theta}_{n}-\theta)$ and $\sqrt{n\Delta_{n}}(\check{\theta}_{n}-\theta)$ are tight.