Statistical analysis of the non-ergodic fractional Ornstein-Uhlenbeck process with periodic mean

Consider a periodic, mean-reverting Ornstein-Uhlenbeck process \(X=\{X_t,t\geq0\}\) of the form \(d X_{t}=\left(L(t)+\alpha X_{t}\right) d t+ dB^H_{t}, \quad t \geq 0\), where \(L(t)=\sum_{i=1}^{p}\mu_i\phi_i (t)\) is a periodic parametric function, and \(\{B^H_t,t\geq0\}\) is a fractional Brownian...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Belfadli, Rachid, Khalifa Es-Sebaiy, Fatima-Ezzahra Farah
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 31.08.2020
Subjects
Brownian motion
Ergodic processes
Ornstein-Uhlenbeck process
Parameter estimation
Statistical analysis
Online AccessGet full text

Cover

More Information
Summary:Consider a periodic, mean-reverting Ornstein-Uhlenbeck process \(X=\{X_t,t\geq0\}\) of the form \(d X_{t}=\left(L(t)+\alpha X_{t}\right) d t+ dB^H_{t}, \quad t \geq 0\), where \(L(t)=\sum_{i=1}^{p}\mu_i\phi_i (t)\) is a periodic parametric function, and \(\{B^H_t,t\geq0\}\) is a fractional Brownian motion of Hurst parameter \(\frac12\leq H<1\). In the "ergodic" case \(\alpha<0\), the parametric estimation of \((\mu_1,\ldots,\mu_p,\alpha)\) based on continuous-time observation of \(X\) has been considered in Dehling et al. \cite{DFK}, and in Dehling et al. \cite{DFW} for \(H=\frac12\), and \(\frac12<H<1\), respectively. In this paper we consider the "non-ergodic" case \(\alpha>0\), and for all \(\frac12\leq H<1\). We analyze the strong consistency and the asymptotic distribution for the estimator of \((\mu_1,\ldots,\mu_p,\alpha)\) when the whole trajectory of \(X\) is observed.
ISSN:2331-8422