IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i11p4117-4141.html
   My bibliography  Save this article

On the 1H-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter H<12

Author

Listed:
  • Essaky, El Hassan
  • Nualart, David

Abstract

In this paper, we study the 1H-variation of stochastic divergence integrals Xt=∫0tusδBs with respect to a fractional Brownian motion B with Hurst parameter H<12. Under suitable assumptions on the process u, we prove that the 1H-variation of X exists in L1(Ω) and is equal to eH∫0T|us|1Hds, where eH=E[|B1|1H]. In the second part of the paper, we establish an integral representation for the fractional Bessel Process ‖Bt‖, where Bt is a d-dimensional fractional Brownian motion with Hurst parameter H<12. Using a multidimensional version of the result on the 1H-variation of divergence integrals, we prove that if 2dH2>1, then the divergence integral in the integral representation of the fractional Bessel process has a 1H-variation equals to a multiple of the Lebesgue measure.

Suggested Citation

  • Essaky, El Hassan & Nualart, David, 2015. "On the 1H-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter H<12," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4117-4141.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:11:p:4117-4141
    DOI: 10.1016/j.spa.2015.06.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915001404
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.06.001?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. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    2. Guerra, João M.E. & Nualart, David, 2005. "The 1/H-variation of the divergence integral with respect to the fractional Brownian motion for H>1/2 and fractional Bessel processes," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 91-115, January.
    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. Turvey, Calum G., 2001. "Random Walks And Fractal Structures In Agricultural Commodity Futures Prices," Working Papers 34151, University of Guelph, Department of Food, Agricultural and Resource Economics.
    2. Zhang, Wei-Guo & Li, Zhe & Liu, Yong-Jun, 2018. "Analytical pricing of geometric Asian power options on an underlying driven by a mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 402-418.
    3. Gapeev, Pavel V., 2004. "On arbitrage and Markovian short rates in fractional bond markets," Statistics & Probability Letters, Elsevier, vol. 70(3), pages 211-222, December.
    4. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 240-248.
    5. Dorje Brody & Joanna Syroka & Mihail Zervos, 2002. "Dynamical pricing of weather derivatives," Quantitative Finance, Taylor & Francis Journals, vol. 2(3), pages 189-198.
    6. 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.
    7. Loch-Olszewska, Hanna, 2019. "Properties and distribution of the dynamical functional for the fractional Gaussian noise," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 252-271.
    8. Chr. Framstad, Nils, 2011. "On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes," Memorandum 20/2011, Oslo University, Department of Economics.
    9. Matthieu Garcin, 2021. "Forecasting with fractional Brownian motion: a financial perspective," Papers 2105.09140, arXiv.org, revised Sep 2021.
    10. Akihiko Inoue & Yumiharu Nakano, 2005. "Optimal long term investment model with memory," Papers math/0506621, arXiv.org, revised May 2006.
    11. Vasile Brătian & Ana-Maria Acu & Camelia Oprean-Stan & Emil Dinga & Gabriela-Mariana Ionescu, 2021. "Efficient or Fractal Market Hypothesis? A Stock Indexes Modelling Using Geometric Brownian Motion and Geometric Fractional Brownian Motion," Mathematics, MDPI, vol. 9(22), pages 1-20, November.
    12. Onali, Enrico & Goddard, John, 2011. "Are European equity markets efficient? New evidence from fractal analysis," International Review of Financial Analysis, Elsevier, vol. 20(2), pages 59-67, April.
    13. Mishura, Yuliya & Shevchenko, Georgiy & Valkeila, Esko, 2013. "Random variables as pathwise integrals with respect to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2353-2369.
    14. Aleksandr Kuklin & Gennadiy Bystray & Sergey Okhotnikov & Elena Chistova, 2015. "Economic Tomography: Opportunity to Foresee and Respond to Socio-Economic Crises," Economy of region, Centre for Economic Security, Institute of Economics of Ural Branch of Russian Academy of Sciences, vol. 1(4), pages 40-53.
    15. Foad Shokrollahi, 2016. "Subdiffusive fractional Brownian motion regime for pricing currency options under transaction costs," Papers 1612.06665, arXiv.org, revised Aug 2017.
    16. Dorsaf Cherif & Emmanuel Lépinette, 2021. "No-arbitrage conditions and pricing from discrete-time to continuous-time strategies," Working Papers hal-03284660, HAL.
    17. Beran, Jan, 1999. "SEMIFAR Models - A Semiparametric Framework for Modelling Trends, Long Range Dependence and Nonstationarity," CoFE Discussion Papers 99/16, University of Konstanz, Center of Finance and Econometrics (CoFE).
    18. Xiyue Han & Alexander Schied, 2021. "The roughness exponent and its model-free estimation," Papers 2111.10301, arXiv.org, revised Aug 2023.
    19. Cheridito, Patrick, 2004. "Gaussian moving averages, semimartingales and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 47-68, January.
    20. José Carlos Ramirez Sánchez, 2004. "Usos y limitaciones de los procesos estocásticos en el tratamiento de distribuciones de rendimientos con colas gordas," Revista de Analisis Economico – Economic Analysis Review, Universidad Alberto Hurtado/School of Economics and Business, vol. 19(1), pages 51-76, June.

    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:125:y:2015:i:11:p:4117-4141. 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.