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

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  • Pavel Kříž

    (Department of Mathematics, Faculty of Chemical Engineering, University of Chemistry and Technology Prague, 16628 Prague, Czech Republic)

  • Leszek Szała

    (Department of Mathematics, Faculty of Chemical Engineering, University of Chemistry and Technology Prague, 16628 Prague, Czech Republic)

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.

Suggested Citation

  • Pavel Kříž & Leszek Szała, 2020. "Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes," Mathematics, MDPI, vol. 8(5), pages 1-20, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:716-:d:353646
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    References listed on IDEAS

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    1. Es-Sebaiy, Khalifa & Viens, Frederi G., 2019. "Optimal rates for parameter estimation of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3018-3054.
    2. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    3. Brouste, Alexandre & Fukasawa, Masaaki & Hino, Hideitsu & Iacus, Stefano & Kamatani, Kengo & Koike, Yuta & Masuda, Hiroki & Nomura, Ryosuke & Ogihara, Teppei & Shimuzu, Yasutaka & Uchida, Masayuki & Y, 2014. "The YUIMA Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 57(i04).
    4. Yaozhong Hu & David Nualart & Hongjuan Zhou, 2019. "Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter," Statistical Inference for Stochastic Processes, Springer, vol. 22(1), pages 111-142, April.
    5. Alexandre Brouste & Stefano Iacus, 2013. "Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package," Computational Statistics, Springer, vol. 28(4), pages 1529-1547, August.
    6. Andreas Neuenkirch & Samy Tindel, 2014. "A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise," Statistical Inference for Stochastic Processes, Springer, vol. 17(1), pages 99-120, April.
    7. Rosenbaum, Mathieu, 2008. "Estimation of the volatility persistence in a discretely observed diffusion model," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1434-1462, August.
    8. Ehsan Azmoodeh & Lauri Viitasaari, 2015. "Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 205-227, October.
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