Least Squares Type Estimation for Discretely Observed Non-Ergodic Gaussian Ornstein-Uhlenbeck Processes

Abstract

In this article, we consider the drift parameter estimation problem for the nonergodic Ornstein-Uhlenbeck process defined as dXt = θXtdt + dGt, t ≥ 0 with an unknown parameter θ > 0, where G is a Gaussian process. We assume that the process {Xt,t ≥ 0} is observed at discrete time instants t1 = Δn, …, tn = nΔn, and we construct two least squares type estimators $${\hat \theta _n}$$ and $${\check \theta _n}$$ for θ on the basis of the discrete observations { $${X_{{t_i}}},\;i = 1, \cdots ,n$$ } as n → ∞. Then, we provide sufficient conditions, based on properties of G, which ensure that $${\hat \theta _n}$$ and $${\check \theta _n}$$ are strongly consistent and the sequences $$\sqrt {n{{\rm{\Delta }}_n}} ({\hat \theta _n} - \theta )$$ and $$\sqrt {n{{\rm{\Delta }}_n}} ({\check \theta _n} - \theta )$$ 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 ∈ (½, 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.