Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes

Pavel Kříž,Leszek Szała

doi:10.3390/math8050716

Pavel Kříž, Leszek Szała

Open Access

https://doi.org/10.3390/math8050716

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Journal: Mathematics	Publication Date: May 3, 2020
Citations: 3	License type: CC BY 4.0

Affiliation: University of Chemistry and Technology

Abstract

We introduce three new estimators of the drift parameter of a fractional Ornstein–Uhlenbeck process. These estimators are based on modifications of the least-squares procedure utilizing the explicit formula for the process and covariance structure of a fractional Brownian motion. We demonstrate their advantageous properties in the setting of discrete-time observations with fixed mesh size, where they outperform the existing estimators. Numerical experiments by Monte Carlo simulations are conducted to confirm and illustrate theoretical findings. New estimation techniques can improve calibration of models in the form of linear stochastic differential equations driven by a fractional Brownian motion, which are used in diverse fields such as biology, neuroscience, finance and many others.

Highlights

Stochastic models with fractional Brownian motion as the noise source have attained increasing popularity recently. This is because fBm is a continuous Gaussian process, increments of which are positively, or negatively correlated if Hurst parameter H > 1/2, or H < 1/2, respectively
Estimators b λ1 and b λ3 demonstrate good performance in scenarios with far-from-zero initial (x0 = 100) condition and short time horizon (T = 10) This illustrates the fact that these estimators reflect mainly the speed of convergence to zero of the observed process in its initial phase
It is easy to implement, since it can be calculated by a closed formula

Summary

Introduction

Stochastic models with fractional Brownian motion (fBm) as the noise source have attained increasing popularity recently. This is because fBm is a continuous Gaussian process, increments of which are positively, or negatively correlated if Hurst parameter H > 1/2, or H < 1/2, respectively. If H = 1/2 fBm coincides with classical Brownian motion and its increments are independent. The ability of fBm to include memory into the noise process makes it possible to build more realistic models in such diverse fields as biology, neuroscience, hydrology, climatology, finance and many others. Let { Bt }t∈[0,∞) be a fractional Brownian motion with Hurst parameter H defined on an appropriate probability space {Ω, A, P}.

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