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Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter

Author

Listed:
  • Yaozhong Hu

    (University of Alberta)

  • David Nualart

    (University of Kansas)

  • Hongjuan Zhou

    (University of Kansas)

Abstract

This paper studies the least squares estimator (LSE) for the drift parameter of an Ornstein–Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. A central limit theorem is proved when the Hurst parameter $$H \in (0, 3/4]$$ H ∈ ( 0 , 3 / 4 ] and a noncentral limit theorem is proved for $$H\in (3/4, 1)$$ H ∈ ( 3 / 4 , 1 ) . Thus, the open problem left in the previous paper (Hu and Nualart in Stat Probab Lett 80(11–12):1030–1038, 2010) is completely solved, where a central limit theorem for the least squares estimator is proved for $$H\in [1/2, 3/4)$$ H ∈ [ 1 / 2 , 3 / 4 ) . The LSE is then used to study the asymptotics for other alternative estimators, such as the ergodic type estimator.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sistpr:v:22:y:2019:i:1:d:10.1007_s11203-017-9168-2
    DOI: 10.1007/s11203-017-9168-2
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    References listed on IDEAS

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    1. Alexandre Brouste & Marina Kleptsyna, 2010. "Asymptotic properties of MLE for partially observed fractional diffusion system," Statistical Inference for Stochastic Processes, Springer, vol. 13(1), pages 1-13, April.
    2. Kubilius, K. & Mishura, Y., 2012. "The rate of convergence of Hurst index estimate for the stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3718-3739.
    3. Nualart, D. & Ortiz-Latorre, S., 2008. "Central limit theorems for multiple stochastic integrals and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 614-628, April.
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    Citations

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    Cited by:

    1. El Mehdi Haress & Yaozhong Hu, 2021. "Estimation of all parameters in the fractional Ornstein–Uhlenbeck model under discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 327-351, July.
    2. Wang, Xiaohu & Xiao, Weilin & Yu, Jun, 2023. "Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process," Journal of Econometrics, Elsevier, vol. 232(2), pages 389-415.
    3. Hui Jiang & Jingying Zhou, 2023. "An Exponential Nonuniform Berry–Esseen Bound for the Fractional Ornstein–Uhlenbeck Process," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1037-1058, June.
    4. Xichao Sun & Litan Yan & Yong Ge, 2022. "The Laws of Large Numbers Associated with the Linear Self-attracting Diffusion Driven by Fractional Brownian Motion and Applications," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1423-1478, September.
    5. Katsuto Tanaka & Weilin Xiao & Jun Yu, 2020. "Maximum Likelihood Estimation for the Fractional Vasicek Model," Econometrics, MDPI, vol. 8(3), pages 1-28, August.
    6. Zhou, Hongjuan & Zhou, Kenneth Q. & Li, Xianping, 2022. "Stochastic mortality dynamics driven by mixed fractional Brownian motion," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 218-238.
    7. Katsuto Tanaka, 2020. "Comparison of the LS-based estimators and the MLE for the fractional Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 415-434, July.
    8. Qian Yu, 2021. "Least squares estimator of fractional Ornstein–Uhlenbeck processes with periodic mean for general Hurst parameter," Statistical Papers, Springer, vol. 62(2), pages 795-815, April.
    9. Marko Voutilainen & Lauri Viitasaari & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor, 2022. "Vector‐valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 992-1022, September.
    10. Radomyra Shevchenko & Ciprian A. Tudor, 2020. "Parameter estimation for the Rosenblatt Ornstein–Uhlenbeck process with periodic mean," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 227-247, April.
    11. 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.

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