IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v37y2024i1d10.1007_s10959-023-01258-5.html
   My bibliography  Save this article

General Transfer Formula for Stochastic Integral with Respect to Multifractional Brownian Motion

Author

Listed:
  • Christian Bender

    (Saarland University)

  • Joachim Lebovits

    (Laboratoire Analyse, Géométrie et Applications C.N.R.S. (UMR 7539), Université Sorbonne Paris Nord)

  • Jacques Lévy Véhel

    (5 rue Olympe de Gouges)

Abstract

In this work we show how results from stochastic integration with respect to multifractional Brownian motion (mBm) can be simply deduced from results of stochastic integration with respect to fractional Brownian motion (fBm), by using a “Transfer Principle”. To illustrate this fact, we prove an Itô formula for integral with respect to mBm by deriving it from Itô formula for integral with respect to fBm, of any Hurst index H in (0, 1).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-023-01258-5
    DOI: 10.1007/s10959-023-01258-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-023-01258-5
    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/s10959-023-01258-5?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. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
    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.
    4. 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. 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.
    2. Yan, Litan, 2004. "Maximal inequalities for the iterated fractional integrals," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 69-79, August.
    3. 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.
    4. Bender, Christian, 2014. "Backward SDEs driven by Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2892-2916.
    5. Akinlar, M.A. & Inc, Mustafa & Gómez-Aguilar, J.F. & Boutarfa, B., 2020. "Solutions of a disease model with fractional white noise," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    6. Xu, Xiao & Wang, Li & Du, Zhenbin & Kao, Yonggui, 2023. "H∞ Sampled-Data Control for Uncertain Fuzzy Systems under Markovian Jump and FBm," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    7. Stoyan V. Stoyanov & Yong Shin Kim & Svetlozar T. Rachev & Frank J. Fabozzi, 2017. "Option pricing for Informed Traders," Papers 1711.09445, arXiv.org.
    8. Grecksch Wilfried & Roth Christian, 2008. "A quasilinear stochastic partial differential equation driven by fractional white noise," Monte Carlo Methods and Applications, De Gruyter, vol. 13(5-6), pages 353-367, January.
    9. Joachim Lebovits & Mark Podolskij, 2016. "Estimation of the global regularity of a multifractional Brownian motion," CREATES Research Papers 2016-33, Department of Economics and Business Economics, Aarhus University.
    10. Alpay, Daniel & Attia, Haim & Levanony, David, 2010. "On the characteristics of a class of Gaussian processes within the white noise space setting," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1074-1104, July.
    11. Rostek, Stefan & Schöbel, Rainer, 2006. "Risk preference based option pricing in a fractional Brownian market," Tübinger Diskussionsbeiträge 299, University of Tübingen, School of Business and Economics.
    12. Wang, Xiao-Tian & Wu, Min & Zhou, Ze-Min & Jing, Wei-Shu, 2012. "Pricing European option with transaction costs under the fractional long memory stochastic volatility model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1469-1480.
    13. Kyaw, NyoNyo A. & Los, Cornelis A. & Zong, Sijing, 2006. "Persistence characteristics of Latin American financial markets," Journal of Multinational Financial Management, Elsevier, vol. 16(3), pages 269-290, July.
    14. Yang, Zhaoqiang, 2020. "Default probability of American lookback option in a mixed jump-diffusion model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    15. Tapiero, Charles S. & Vallois, Pierre, 2018. "Fractional Randomness and the Brownian Bridge," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 835-843.
    16. Loosveldt, L., 2023. "Multifractional Hermite processes: Definition and first properties," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 465-500.
    17. Cornelis A. Los & Rossitsa M. Yalamova, 2004. "Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash," Finance 0409050, University Library of Munich, Germany.
    18. Čoupek, P. & Maslowski, B., 2017. "Stochastic evolution equations with Volterra noise," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 877-900.
    19. Beghin, Luisa & Cristofaro, Lorenzo & Mishura, Yuliya, 2024. "A class of processes defined in the white noise space through generalized fractional operators," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
    20. Axel A. Araneda, 2019. "The fractional and mixed-fractional CEV model," Papers 1903.05747, arXiv.org, revised Jun 2019.

    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:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-023-01258-5. 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.