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

Abstract

This paper provides several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. First and higher order power variations are used to estimate the volatility parameter. The almost sure convergence of the estimators and the corresponding central limit theorems are obtained for all the Hurst parameter range $H\in (0, 1)$. The least squares estimator is used for the drift parameter. A central limit theorem is proved when the Hurst parameter $H \in (0, 1/2)$ and a noncentral limit theorem is proved for $H\in[3/4, 1)$. Thus, the open problem left in the paper by Hu and Nualart (2010) is completely solved, where a central limit theorem for least squares estimator is proved for $H\in [1/2, 3/4)$.