Afrika Statistika

On drift estimation for non-ergodic fractional Ornstein-Uhlenbeck process with discrete observations

Khalifa Es-sebaiy and Djibril Ndiaye

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We consider parameter estimation problems for the non-ergodic fractional Ornstein-Uhlenbeck process defined as $% dX_{t}=\theta X_{t}dt+dB_{t}^{H},\ t\geq 0$, with an unknown parameter $% \theta >0$, where $B^{H}$ is a fractional Brownian motion of Hurst index $% H\in (\frac{1}{2},1)$. We assume that the process $\{X_{t},t\geq 0\}$ is observed at discrete time instants $t_{1}=\Delta _{n},\ldots ,t_{n}=n\Delta _{n}$. We construct two estimators $\hat{\theta}_{n}$ and $\check{\theta}_{n} $ of $\theta $ which are strongly consistent, namely, $\hat{\theta}_{n}$ and $\check{\theta}_{n}$ converge to $\theta $ almost surely as $n\rightarrow \infty $. We also prove that $\sqrt{n\Delta _{n}}(\hat{\theta}_{n}-\theta )$ and $\sqrt{n\Delta _{n}}(\check{\theta}_{n}-\theta )$ are tight.


Dans ce travail, nous étudions des problèmes d'estimation paramétriques relatifs au processus d'Ornstein-Uhlenbeck fractionaire non-ergodique défini par $dX_{t} = \theta X_{t}dt + dB^{H}_{t}, t\geq 0$, où $\theta>0$ est un paramétre et $B^{H}$ est un mouvement Brownien fractionaire d'indice de Hurst $H\in]1/2 , 1[$. Le processus $\{X_{t}, t\geq 0\}$ a été observé (de façon réguliére) aux instants $t_1=\Delta_n,\ldots,t_n=n\Delta_n$, c'est-à-dire pour tout $i\in\{0,\cdots,n\}$, $t_{i} = i\Delta_{n}$. Nous avons construit deux estimateurs $\hat{\theta}_{n}$ et $\check{\theta}_{n}$ de $\theta$ fortement consistants, c'est-à-dire, $\hat{\theta}_{n}$ et $\check{\theta}_{n}$ convergent presque surement vers $\theta$ quand $n\rightarrow\infty$. Nous avons aussi prouvé que $\sqrt{n\Delta_n}(\hat{\theta}_{n}-\theta)$ et $\sqrt{n\Delta_n}(\check{\theta}_{n}-\theta)$ sont tendus.

Article information

Afr. Stat., Volume 9, Number 1 (2014), 615-625.

First available in Project Euclid: 11 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 62M05: Markov processes: estimation 62F12: Asymptotic properties of estimators

Drift estimation Discrete observations Ornstein-Uhlenbeck process Non-ergodicity


Ndiaye, Djibril; Es-sebaiy, Khalifa. On drift estimation for non-ergodic fractional Ornstein-Uhlenbeck process with discrete observations. Afr. Stat. 9 (2014), no. 1, 615--625.

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