关键词: Drift parameter estimation, non-ergodic Gaussian Ornstein-Uhlenbeck process, discrete observations

Abstract: In this article, we consider the drift parameter estimation problem for the nonergodic Ornstein-Uhlenbeck process defined as dX_t=θX_tdt + dG_t, t ≥ 0 with an unknown parameter θ > 0, where G is a Gaussian process. We assume that the process {X_t, t ≥ 0} is observed at discrete time instants t₁=△_n,…, t_n=n△_n, and we construct two least squares type estimators and for θ on the basis of the discrete observations {X_ti, i=1,…, n} as n → ∞. Then, we provide sufficient conditions, based on properties of G, which ensure that and  are strongly consistent and the sequences √n△_n(-θ) and √n△_n(-θ) are tight. Our approach offers an elementary proof of[11], which studied the case when G is a fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). As such, our results extend the recent findings by[11] to the case of general Hurst parameter H ∈ (0, 1). We also apply our approach to study subfractional Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes.

Key words: Drift parameter estimation, non-ergodic Gaussian Ornstein-Uhlenbeck process, discrete observations