Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes

By Mohamed El Machkouri, Khalifa Es-Sebaiy, Y. Ouknine. 2015

89 citations recorded by the source corpus.

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein-Uhlenbeck process defined as $dX_t=\theta X_tdt+dG_t,\ t\geq0$ with an unknown parameter $\theta>0$, where $G$ is a Gaussian process. We provide sufficient conditions, based on the properties of $G$, ensuring the strong consistency and the asymptotic distribution of our estimator $\widetilde{\theta}_t$ of $\theta$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$. Our approach offers an elementary, unifying proof of \cite{BEO}, and it allows to extend the result of \cite{BEO} to the case when $G$ is a fractional Brownian motion with Hurst parameter $H\in(0,1)$. We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0cce3b9253ec2d442bea7e7aad62ad99caeb1f4a

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Citations in the corpus

ASYMPTOTIC THEORY FOR ESTIMATING DRIFT PARAMETERS IN THE FRACTIONAL VASICEK MODEL (2018, unknown): “1 of El Machkouri et al. (2016), we obtain Z∞ = ∫∞ 0 e κs B H s ds which is well-defined and ZT a.”
Optimal rates for parameter estimation of stationary Gaussian processes (2016, background): “The case of non-ergodic fOU process of the first kind and of the second kind can be found in [3], [11] and [12] respectively.”