IDEAS home Printed from https://ideas.repec.org/a/spr/sistpr/v21y2018i1d10.1007_s11203-016-9147-z.html
   My bibliography  Save this article

Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation

Author

Listed:
  • Marco Dozzi

    (Université de Lorraine)

  • Yuriy Kozachenko

    (Taras Shevchenko National University of Kyiv)

  • Yuliya Mishura

    (Taras Shevchenko National University of Kyiv)

  • Kostiantyn Ralchenko

    (Taras Shevchenko National University of Kyiv)

Abstract

We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein–Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.

Suggested Citation

  • Marco Dozzi & Yuriy Kozachenko & Yuliya Mishura & Kostiantyn Ralchenko, 2018. "Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 21-52, April.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:1:d:10.1007_s11203-016-9147-z
    DOI: 10.1007/s11203-016-9147-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11203-016-9147-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11203-016-9147-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Hu, Yaozhong & Nualart, David, 2010. "Parameter estimation for fractional Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1030-1038, June.
    2. Yuliya Mishura & Kostiantyn Ral’chenko & Oleg Seleznev & Georgiy Shevchenko, 2014. "Asymptotic Properties of Drift Parameter Estimator Based on Discrete Observations of Stochastic Differential Equation Driven by Fractional Brownian Motion," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 303-318, Springer.
    3. Stoev, Stilian A. & Taqqu, Murad S., 2006. "How rich is the class of multifractional Brownian motions?," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 200-221, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kostiantyn Ralchenko & Foad Shokrollahi & Tommi Sottinen, 2025. "Discretization of integrals driven by multifractional Brownian motions with discontinuous integrands," Journal of Theoretical Probability, Springer, vol. 38(3), pages 1-26, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Matthieu Garcin, 2019. "Estimation of Hurst exponents in a stationary framework [Estimation d'exposants de Hurst dans un cadre stationnaire]," Post-Print hal-02163662, HAL.
    2. Shi, Shuping & Yu, Jun & Zhang, Chen, 2024. "On the spectral density of fractional Ornstein–Uhlenbeck processes," Journal of Econometrics, Elsevier, vol. 245(1).
    3. Xu, Weijun & Sun, Qi & Xiao, Weilin, 2012. "A new energy model to capture the behavior of energy price processes," Economic Modelling, Elsevier, vol. 29(5), pages 1585-1591.
    4. Katsuto Tanaka, 2013. "Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 16(3), pages 173-192, October.
    5. 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.
    6. Xiaohu Wang & Weilin Xiao & Jun Yu & Chen Zhang, 2025. "Maximum Likelihood Estimation of Fractional Ornstein-Uhlenbeck Process with Discretely Sampled Data," Working Papers 202527, University of Macau, Faculty of Business Administration.
    7. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "The Multivariate Fractional Ornstein-Uhlenbeck Process," CEIS Research Paper 581, Tor Vergata University, CEIS, revised 28 Aug 2024.
    8. Christian Bender & Joachim Lebovits & Jacques Lévy Véhel, 2024. "General Transfer Formula for Stochastic Integral with Respect to Multifractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 37(1), pages 905-932, March.
    9. Khalifa Es-Sebaiy & Mohammed Es.Sebaiy, 2021. "Estimating drift parameters in a non-ergodic Gaussian Vasicek-type model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(2), pages 409-436, June.
    10. Balança, Paul & Herbin, Erick, 2012. "2-microlocal analysis of martingales and stochastic integrals," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2346-2382.
    11. Zhang, Pu & Xiao, Wei-lin & Zhang, Xi-li & Niu, Pan-qiang, 2014. "Parameter identification for fractional Ornstein–Uhlenbeck processes based on discrete observation," Economic Modelling, Elsevier, vol. 36(C), pages 198-203.
    12. Bondarenko, Valeria & Bondarenko, Victor & Truskovskyi, Kyryl, 2017. "Forecasting of time data with using fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 97(C), pages 44-50.
    13. Bardet, Jean-Marc & Surgailis, Donatas, 2013. "Nonparametric estimation of the local Hurst function of multifractional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1004-1045.
    14. Ehsan Azmoodeh & Ozan Hur, 2023. "Generalized Families of Fractional Stochastic Dominance," Papers 2307.08651, arXiv.org, revised Feb 2025.
    15. Katsuto Tanaka, 2015. "Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 315-332, October.
    16. 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.
    17. Cohen, Serge & Lacaux, Céline & Ledoux, Michel, 2008. "A general framework for simulation of fractional fields," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1489-1517, September.
    18. Anderes, Ethan B. & Stein, Michael L., 2011. "Local likelihood estimation for nonstationary random fields," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 506-520, March.
    19. Ayache, Antoine & Louckx, Christophe, 2025. "Harmonizable Multifractional Stable Field: Sharp results on sample path behavior," Stochastic Processes and their Applications, Elsevier, vol. 186(C).
    20. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2009. "Covariance function of vector self-similar processes," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2415-2421, December.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sistpr:v:21:y:2018:i:1:d:10.1007_s11203-016-9147-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.