Electronic Journal of Statistics

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, and Oleg Seleznjev

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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 $m>\frac{1}{2H}$.

Article information

Electron. J. Statist., Volume 9, Number 2 (2015), 1799-1825.

Received: January 2015
First available in Project Euclid: 25 August 2015

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Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 60F15: Strong theorems 60F25: $L^p$-limit theorems 62F10: Point estimation 62F12: Asymptotic properties of estimators

Fractional Brownian motion fractional Ornstein–Uhlenbeck process short-range dependence drift parameter estimator consistency strong consistency discretization high-frequency data


Kubilius, Kęstutis; Mishura, Yuliya; Ralchenko, Kostiantyn; Seleznjev, Oleg. 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 (2015), no. 2, 1799--1825. doi:10.1214/15-EJS1062. https://projecteuclid.org/euclid.ejs/1440507394

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