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August 2018 Parameter estimation for discretely observed non-ergodic fractional Ornstein–Uhlenbeck processes of the second kind
Brahim El Onsy, Khalifa Es-Sebaiy, Djibril Ndiaye
Braz. J. Probab. Stat. 32(3): 545-558 (August 2018). DOI: 10.1214/17-BJPS353

Abstract

We use the least squares type estimation to estimate the drift parameter $\theta>0$ of a non-ergodic fractional Ornstein–Uhlenbeck process of the second kind defined as $dX_{t}=\theta X_{t}\,dt+dY_{t}^{(1)},X_{0}=0$, $t\geq0$, where $Y_{t}^{(1)}=\int_{0}^{t}e^{-s}\,dB_{a_{s}}$ with $a_{t}=He^{\frac{t}{H}}$, and $\{B_{t},t\geq0\}$ is a fractional Brownian motion of Hurst parameter $H\in(\frac{1}{2},1)$. We assume that the process $\{X_{t},t\geq0\}$ 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 and we prove that these estimators are $\sqrt{n\Delta_{n}}$-consistent, in the sense that the sequences $\sqrt{n\Delta_{n}}(\hat{\theta}_{n}-\theta)$ and $\sqrt{n\Delta_{n}}(\check{\theta}_{n}-\theta)$ are tight.

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Brahim El Onsy. Khalifa Es-Sebaiy. Djibril Ndiaye. "Parameter estimation for discretely observed non-ergodic fractional Ornstein–Uhlenbeck processes of the second kind." Braz. J. Probab. Stat. 32 (3) 545 - 558, August 2018. https://doi.org/10.1214/17-BJPS353

Information

Received: 1 March 2016; Accepted: 1 January 2017; Published: August 2018
First available in Project Euclid: 8 June 2018

zbMATH: 06930038
MathSciNet: MR3812381
Digital Object Identifier: 10.1214/17-BJPS353

Keywords: Drift parameter estimation , non-ergodic fractional Ornstein–Uhlenbeck process of the second kind

Rights: Copyright © 2018 Brazilian Statistical Association

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Vol.32 • No. 3 • August 2018
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