数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (4): 989-1002.doi: 10.1007/s10473-019-0406-0

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LEAST SQUARES TYPE ESTIMATION FOR DISCRETELY OBSERVED NON-ERGODIC GAUSSIAN ORNSTEIN-UHLENBECK PROCESSES

Khalifa ES-SEBAIY, Fares ALAZEMI, Mishari AL-FORAIH   

  1. Department of Mathematics, Faculty of Science, Kuwait University, Kuwait
  • 收稿日期:2017-12-12 修回日期:2018-04-29 出版日期:2019-08-25 发布日期:2019-09-12
  • 通讯作者: Khalifa ES-SEBAIY E-mail:khalifa.essebaiy@ku.edu.kw
  • 作者简介:Fares ALAZEMI,E-mail:fares.alazemi@gmail.com;Mishari AL-FORAIH,E-mail:mishari@alforaih.net
  • 基金资助:
    The second author was supported and funded by Kuwait University, Research Project No. SM01/16.

LEAST SQUARES TYPE ESTIMATION FOR DISCRETELY OBSERVED NON-ERGODIC GAUSSIAN ORNSTEIN-UHLENBECK PROCESSES

Khalifa ES-SEBAIY, Fares ALAZEMI, Mishari AL-FORAIH   

  1. Department of Mathematics, Faculty of Science, Kuwait University, Kuwait
  • Received:2017-12-12 Revised:2018-04-29 Online:2019-08-25 Published:2019-09-12
  • Supported by:
    The second author was supported and funded by Kuwait University, Research Project No. SM01/16.

摘要: 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=△n,…, tn=nn, and we construct two least squares type estimators and for θ on the basis of the discrete observations {Xti, 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 √nn(-θ) and √nn(-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.

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 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=△n,…, tn=nn, and we construct two least squares type estimators and for θ on the basis of the discrete observations {Xti, 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 √nn(-θ) and √nn(-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

中图分类号: 

  • 62F12