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

作　　者：Khalifa ES-SEBAIY Fares ALAZEMI Mishari AL-FORAIH 

机构地区：[1]Department of Mathematics, Faculty of Science, Kuwait University

出　　处：《Acta Mathematica Scientia》2019年第4期989-1002,共14页数学物理学报（B辑英文版）

基　　金：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 = OXtdt + dGt, i > 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 = n△n, and we construct two least squares type estimators θn and θn for θ on the basis of the discrete observations ,{xti,i= 1,…, n} as →∞. Then, we provide sufficient conditions, based on properties of G, which ensure that θn and θn are strongly consistent and the sequences √n△n(θn-θ) and √n△n(θn-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.In this article, we consider the drift parameter estimation problem for the nonergodic Ornstein-Uhlenbeck process defined as d Xt= 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= n?n, 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 n?n1/2（■-θ） and n?n1/2（■-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 

分 类 号：O[理学]