IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v165y2023icp465-500.html
   My bibliography  Save this article

Multifractional Hermite processes: Definition and first properties

Author

Listed:
  • Loosveldt, L.

Abstract

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener–Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

Suggested Citation

  • Loosveldt, L., 2023. "Multifractional Hermite processes: Definition and first properties," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 465-500.
    • Handle: RePEc:eee:spapps:v:165:y:2023:i:c:p:465-500
    DOI: 10.1016/j.spa.2023.09.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414923001801
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2023.09.003?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Laurent Loosveldt & Samuel Nicolay, 2019. "Some equivalent definitions of Besov spaces of generalized smoothness," Mathematische Nachrichten, Wiley Blackwell, vol. 292(10), pages 2262-2282, October.
    2. Ayache, Antoine, 2020. "Lower bound for local oscillations of Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4593-4607.
    3. Antoine Ayache & Jacques Vehel, 2000. "The Generalized Multifractional Brownian Motion," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 7-18, January.
    4. Nourdin, Ivan & Poly, Guillaume, 2013. "Convergence in total variation on Wiener chaos," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 651-674.
    5. Ayache, Antoine & Esser, Céline & Kleyntssens, Thomas, 2019. "Different possible behaviors of wavelet leaders of the Brownian motion," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 54-60.
    6. Wu, Wei Biao, 2006. "Unit Root Testing For Functionals Of Linear Processes," Econometric Theory, Cambridge University Press, vol. 22(1), pages 1-14, February.
    7. Antoine Ayache, 2013. "Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity," Journal of Theoretical Probability, Springer, vol. 26(1), pages 72-93, March.
    8. Surgailis, Donatas, 2008. "Nonhomogeneous fractional integration and multifractional processes," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 171-198, February.
    9. Lebovits, Joachim & Lévy Véhel, Jacques & Herbin, Erick, 2014. "Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 678-708.
    Full references (including those not matched with items on IDEAS)

    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. Loboda, Dennis & Mies, Fabian & Steland, Ansgar, 2021. "Regularity of multifractional moving average processes with random Hurst exponent," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 21-48.
    2. Angelini, Daniele & Bianchi, Sergio, 2023. "Nonlinear biases in the roughness of a Fractional Stochastic Regularity Model," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    3. Mendy, Ibrahima, 2012. "The two-parameter Volterra multifractional process," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2115-2124.
    4. Aloy Marcel & Dufrénot Gilles & Tong Charles Lai & Peguin-Feissolle Anne, 2013. "A smooth transition long-memory model," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 17(3), pages 281-296, May.
    5. Bai, Shuyang & Taqqu, Murad S. & Zhang, Ting, 2016. "A unified approach to self-normalized block sampling," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2465-2493.
    6. Ayache, Antoine & Lévy Véhel, Jacques, 2004. "On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 119-156, May.
    7. Čoupek, P. & Maslowski, B., 2017. "Stochastic evolution equations with Volterra noise," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 877-900.
    8. Garcin, Matthieu, 2017. "Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 483(C), pages 462-479.
    9. K. J. Falconer & J. Lévy Véhel, 2009. "Multifractional, Multistable, and Other Processes with Prescribed Local Form," Journal of Theoretical Probability, Springer, vol. 22(2), pages 375-401, June.
    10. Nourdin, Ivan & Nualart, David & Peccati, Giovanni, 2021. "The Breuer–Major theorem in total variation: Improved rates under minimal regularity," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 1-20.
    11. 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.
    12. Ayache, Antoine & Bouly, Florent, 2022. "Moving average Multifractional Processes with Random Exponent: Lower bounds for local oscillations," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 143-163.
    13. Antoine Ayache, 2013. "Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity," Journal of Theoretical Probability, Springer, vol. 26(1), pages 72-93, March.
    14. Luca Pratelli & Pietro Rigo, 2018. "Convergence in Total Variation to a Mixture of Gaussian Laws," Mathematics, MDPI, vol. 6(6), pages 1-14, June.
    15. Yu, Z.G. & Anh, V.V. & Wanliss, J.A. & Watson, S.M., 2007. "Chaos game representation of the Dst index and prediction of geomagnetic storm events," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 736-746.
    16. Ting Zhang & Hwai-Chung Ho & Martin Wendler & Wei Biao Wu, 2013. "Block Sampling under Strong Dependence," Papers 1312.5807, arXiv.org.
    17. Thomas Kleyntssens & Samuel Nicolay, 2024. "Generalized Sν$S^\nu$ spaces," Mathematische Nachrichten, Wiley Blackwell, vol. 297(1), pages 266-283, January.
    18. Surgailis, Donatas, 2008. "Nonhomogeneous fractional integration and multifractional processes," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 171-198, February.
    19. Dai, Hongshuai & Li, Yuqiang, 2010. "A weak limit theorem for generalized multifractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 348-356, March.
    20. Davydov, Youri, 2017. "On distance in total variation between image measures," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 393-400.

    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:eee:spapps:v:165:y:2023:i:c:p:465-500. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.