Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes

• Published in: Journal of the Korean Statistical Society Vol. 45; no. 3; pp. 329 - 341
• Main Authors: El Machkouri, Mohamed, Es-Sebaiy, Khalifa, Ouknine, Youssef
• Format: Journal Article
• Language: English
• Published: Singapore Elsevier B.V 01.09.2016; Springer Singapore; Elsevier; 한국통계학회
• Subjects: Applied Statistics; Bayesian Inference; Mathematics; Non-ergodic Gaussian Ornstein–Uhlenbeck process; Parameter estimation; Probability; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; 통계학; secondary; Parameter estimation; Non-ergodic Gaussian Ornstein–Uhlenbeck process; primary; Non-ergodic Gaussian Ornstein-Uhlenbeck process; secondary 60G18; primary 62F12
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1016/j.jkss.2015.12.001

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt=θXtdt+dGt,t≥0 with an unknown parameter θ>0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimator θ˜t of θ based on the observation {Xs,s∈[0,t]} as t→∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H∈(0,1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.