Parameter estimation for Gaussian mean-reverting Ornstein–Uhlenbeck processes of the second kind: Non-ergodic case

Abstract

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind [Formula: see text] defined as [Formula: see text], with unknown parameters [Formula: see text] and [Formula: see text], where [Formula: see text] with [Formula: see text], and [Formula: see text] is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of [Formula: see text] and [Formula: see text] based on the continuous-time observations [Formula: see text] as [Formula: see text], we impose some technical conditions on the process [Formula: see text], which are satisfied, for instance, if [Formula: see text] is a fractional Brownian motion with Hurst parameter [Formula: see text], [Formula: see text] is a subfractional Brownian motion with Hurst parameter [Formula: see text] or [Formula: see text] is a bifractional Brownian motion with Hurst parameters [Formula: see text]. Our method is based on pathwise properties of [Formula: see text] and [Formula: see text] proved in the sequel.