## Brazilian Journal of Probability and Statistics

### Parameter estimation for discretely observed non-ergodic fractional Ornstein–Uhlenbeck processes of the second kind

#### 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.

#### Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 3 (2018), 545-558.

Dates
Accepted: January 2017
First available in Project Euclid: 8 June 2018

https://projecteuclid.org/euclid.bjps/1528444871

Digital Object Identifier
doi:10.1214/17-BJPS353

Mathematical Reviews number (MathSciNet)
MR3812381

Zentralblatt MATH identifier
06930038

#### Citation

El Onsy, Brahim; Es-Sebaiy, Khalifa; Ndiaye, Djibril. Parameter estimation for discretely observed non-ergodic fractional Ornstein–Uhlenbeck processes of the second kind. Braz. J. Probab. Stat. 32 (2018), no. 3, 545--558. doi:10.1214/17-BJPS353. https://projecteuclid.org/euclid.bjps/1528444871

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