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.