Acta Mathematica Scientia

, Volume 39, Issue 4, pp 989–1002 | Cite as

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

  • Khalifa Es-SebaiyEmail author
  • Fares Alazemi
  • Mishari Al-Foraih


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.

Key words

Drift parameter estimation non-ergodic Gaussian Ornstein-Uhlenbeck process discrete observations 

2010 MR Subject Classification

62F12 60G22 


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Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Khalifa Es-Sebaiy
    • 1
    Email author
  • Fares Alazemi
    • 1
  • Mishari Al-Foraih
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceKuwait UniversityAl-khaldiyaKuwait

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