Parameter estimation for fractional Ornstein-Uhlenbeck processes of general Hurst parameter

This paper studies the least squares estimator (LSE) for the drift parameter of an Ornstein–Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. A central limit theorem is proved when the Hurst parameter H∈(0,3/4]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \in (0, 3/4]$$\end{document} and a noncentral limit theorem is proved for H∈(3/4,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\in (3/4, 1)$$\end{document}. Thus, the open problem left in the previous paper (Hu and Nualart in Stat Probab Lett 80(11–12):1030–1038, 2010) is completely solved, where a central limit theorem for the least squares estimator is proved for H∈[1/2,3/4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\in [1/2, 3/4)$$\end{document}. The LSE is then used to study the asymptotics for other alternative estimators, such as the ergodic type estimator.