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2015 Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac{1}{2})$
Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko, Oleg Seleznjev
Electron. J. Statist. 9(2): 1799-1825 (2015). DOI: 10.1214/15-EJS1062

Abstract

We consider the Langevin equation which contains an unknown drift parameter $\theta$ and where the noise is modeled as fractional Brownian motion with Hurst index $H\in(0,\frac{1}{2})$. The solution corresponds to the fractional Ornstein–Uhlenbeck process. We construct an estimator, based on discrete observations in time, of the unknown drift parameter, that is similar in form to the maximum likelihood estimator for the drift parameter in Langevin equation with standard Brownian motion. It is assumed that the interval between observations is $n^{-1}$, i.e. tends to zero (high-frequency data) and the number of observations increases to infinity as $n^{m}$ with $m>1$. It is proved that for strictly positive $\theta$ the estimator is strongly consistent for any $m>1$, while for $\theta\leq0$ it is consistent when $m>\frac{1}{2H}$.

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Kęstutis Kubilius. Yuliya Mishura. Kostiantyn Ralchenko. Oleg Seleznjev. "Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac{1}{2})$." Electron. J. Statist. 9 (2) 1799 - 1825, 2015. https://doi.org/10.1214/15-EJS1062

Information

Received: 1 January 2015; Published: 2015
First available in Project Euclid: 25 August 2015

zbMATH: 1326.60048
MathSciNet: MR3391120
Digital Object Identifier: 10.1214/15-EJS1062

Subjects:
Primary: 60F15, 60F25, 60G22, 62F10, 62F12

Rights: Copyright © 2015 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.9 • No. 2 • 2015
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