On drift estimation for non-ergodic fractional Ornstein-Uhlenbeck process with discrete observations

Abstract

We consider parameter estimation problems for\nthe non-ergodic fractional Ornstein-Uhlenbeck process defined as $%\ndX_{t}=\\theta X_{t}dt+dB_{t}^{H},\\ t\\geq 0$, with an unknown parameter $%\n\\theta >0$, where $B^{H}$ is a fractional Brownian motion of Hurst index $%\nH\\in (\\frac{1}{2},1)$. We assume that the process $\\{X_{t},t\\geq 0\\}$ is\nobserved at discrete time instants $t_{1}=\\Delta _{n},\\ldots ,t_{n}=n\\Delta\n_{n}$. We construct two estimators $\\hat{\\theta}_{n}$ and $\\check{\\theta}_{n}\n$ of $\\theta $ which are strongly consistent, namely, $\\hat{\\theta}_{n}$ and\n$\\check{\\theta}_{n}$ converge to $\\theta $ almost surely as $n\\rightarrow\n\\infty $. We also prove that $\\sqrt{n\\Delta _{n}}(\\hat{\\theta}_{n}-\\theta )$\nand $\\sqrt{n\\Delta _{n}}(\\check{\\theta}_{n}-\\theta )$ are tight.

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