IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v116y2006i5p830-856.html
   My bibliography  Save this article

On bifractional Brownian motion

Author

Listed:
  • Russo, Francesco
  • Tudor, Ciprian A.

Abstract

This paper is devoted to analyzing several properties of the bifractional Brownian motion introduced by Houdré and Villa. This process is a self-similar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K=1). Here, we adopt the strategy of stochastic calculus via regularization. Of particular interest to us is the case . In this case, the process is a finite quadratic variation process with bracket equal to a constant times t and it has the same order of self-similarity as standard Brownian motion. It is a short-memory process even though it is neither a semimartingale nor a Dirichlet process.

Suggested Citation

  • Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:5:p:830-856
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(05)00170-5
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. 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.
    2. Baudoin, Fabrice & Nualart, David, 2003. "Equivalence of Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 327-350, October.
    3. Coutin, Laure & Nualart, David & Tudor, Ciprian A., 2001. "Tanaka formula for the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 94(2), pages 301-315, August.
    4. Errami, Mohammed & Russo, Francesco, 2003. "n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 259-299, April.
    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. Skorniakov, V., 2019. "On a covariance structure of some subset of self-similar Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1903-1920.
    2. Harnett, Daniel & Nualart, David, 2018. "Central limit theorem for functionals of a generalized self-similar Gaussian process," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 404-425.
    3. Benjamin Gess & Cheng Ouyang & Samy Tindel, 2020. "Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths," Journal of Theoretical Probability, Springer, vol. 33(2), pages 611-648, June.
    4. Xu Feng, 2020. "Bifractional Black-Scholes Model for Pricing European Options and Compound Options," Journal of Systems Science and Information, De Gruyter, vol. 8(4), pages 346-355, August.
    5. Yu, Qian & Bajja, Salwa, 2020. "Volatility estimation of general Gaussian Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 163(C).
    6. Tommi Sottinen & Lauri Viitasaari, 2018. "Parameter estimation for the Langevin equation with stationary-increment Gaussian noise," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 569-601, October.
    7. Nualart, David & Xu, Fangjun, 2019. "Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3981-4008.
    8. Daniel Harnett & Arturo Jaramillo & David Nualart, 2019. "Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1105-1144, September.
    9. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 51-104, January.
    10. Slominski, Leszek & Ziemkiewicz, Bartosz, 2009. "On weak approximations of integrals with respect to fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 543-552, February.
    11. Lei, Pedro & Nualart, David, 2009. "A decomposition of the bifractional Brownian motion and some applications," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 619-624, March.
    12. Rosanna Coviello & Cristina Di Girolami & Francesco Russo, 2011. "On stochastic calculus related to financial assets without semimartingales," Papers 1102.2050, arXiv.org.
    13. Zhang, Xili & Xiao, Weilin, 2017. "Arbitrage with fractional Gaussian processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 620-628.
    14. Harnett, Daniel & Nualart, David, 2012. "Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3460-3505.

    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. van Zanten, Harry, 2007. "When is a linear combination of independent fBm's equivalent to a single fBm?," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 57-70, January.
    2. Axel A. Araneda, 2019. "The fractional and mixed-fractional CEV model," Papers 1903.05747, arXiv.org, revised Jun 2019.
    3. Robert Elliott & Leunglung Chan, 2004. "Perpetual American options with fractional Brownian motion," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 123-128.
    4. Fabbri, Giorgio & Russo, Francesco, 2017. "Infinite dimensional weak Dirichlet processes and convolution type processes," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 325-357.
    5. Ehsan Azmoodeh & Lauri Viitasaari, 2015. "Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 28(1), pages 396-422, March.
    6. Yazigi, Adil, 2015. "Representation of self-similar Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 94-100.
    7. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 51-104, January.
    8. Mukeru, Safari, 2017. "Representation of local times of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 131(C), pages 1-12.
    9. Salmerón Garrido, José Antonio & Nunno, Giulia Di & D'Auria, Bernardo, 2022. "Before and after default: information and optimal portfolio via anticipating calculus," DES - Working Papers. Statistics and Econometrics. WS 35411, Universidad Carlos III de Madrid. Departamento de Estadística.
    10. Solesne Bourguin & Ciprian A. Tudor, 2012. "Asymptotic Theory for Fractional Regression Models via Malliavin Calculus," Journal of Theoretical Probability, Springer, vol. 25(2), pages 536-564, June.
    11. Axel A. Araneda, 2021. "Price modelling under generalized fractional Brownian motion," Papers 2108.12042, arXiv.org, revised Nov 2023.
    12. Tommi Sottinen & Ciprian A. Tudor, 2006. "On the Equivalence of Multiparameter Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 19(2), pages 461-485, June.
    13. Davidson, James & Hashimzade, Nigar, 2009. "Representation And Weak Convergence Of Stochastic Integrals With Fractional Integrator Processes," Econometric Theory, Cambridge University Press, vol. 25(6), pages 1589-1624, December.
    14. Wang, Ling & Chiu, Mei Choi & Wong, Hoi Ying, 2021. "Volterra mortality model: Actuarial valuation and risk management with long-range dependence," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 1-14.
    15. Yan, Litan, 2004. "Maximal inequalities for the iterated fractional integrals," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 69-79, August.
    16. Sun, Xichao & Yan, Litan & Yu, Xianye, 2019. "An integral functional driven by fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2249-2285.
    17. Bouchard, Bruno & Loeper, Grégoire & Tan, Xiaolu, 2022. "A ℂ0,1-functional Itô’s formula and its applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 299-323.
    18. Ling Wang & Mei Choi Chiu & Hoi Ying Wong, 2020. "Volterra mortality model: Actuarial valuation and risk management with long-range dependence," Papers 2009.09572, arXiv.org.
    19. Bender, Christian, 2014. "Backward SDEs driven by Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2892-2916.
    20. León, Jorge A. & Nualart, David, 2005. "An extension of the divergence operator for Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 481-492, March.

    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:116:y:2006:i:5:p:830-856. 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.