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

摘要

We consider a least square-type method to estimate the drift parameters for the mean- reverting Ornstein-Uhlenbeck process of the second kind {X-t, t >= 0} defined as dX(t) = (theta(mu + X-t)dt + dY(t,G)(()(1)), t >= 0, with unknown parameters theta > 0 and mu is an element of R, where Y-t,G((1)) := integral(t)(0) e(-s)dG(as) with a(t) = He (H) over bar (t), and {G(t), t >= 0} is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of theta and mu based on the continuous-time observations {X-t, t is an element of [0, T]} as T -> infinity, we impose some technical conditions on the process C, which are satisfied, for instance, if C is a fractional Brownian motion with Hurst parameter H is an element of (0, 1), G is a subfractional Brownian motion with Hurst parameter H is an element of (0, 1) or G is a bifractional Brownian motion with Hurst parameters (H, K) is an element of (0, 1) x (0, 1]. Our method is based on pathwise properties of {X-t, t >= 0} and (Y-t,G((1)), t >= 0} proved in the sequel.