IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v110y2004i1p83-110.html
   My bibliography  Save this article

An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale

Author

Listed:
  • Oblój, Jan
  • Yor, Marc

Abstract

A general methodology allowing to solve the Skorokhod stopping problem for positive functionals of Brownian excursions, with the help of Brownian local time, is developed. The stopping times we consider have the following form: T[mu]=inf{t>0: Ft[greater-or-equal, slanted][phi][mu]F(Lt)}. As an application, the Skorokhod embedding problem for a number of functionals (Ft: t[greater-or-equal, slanted]0), including the age (length) and the maximum (height) of excursions, is solved. Explicit formulae for the corresponding stopping times T[mu], such that FT[mu]~[mu], are given. It is shown that the function [phi][mu]F is the same for the maximum and for the age, [phi][mu]=[psi][mu]-1, where . The joint law of (gT[mu],T[mu],LT[mu]), in the case of the age functional, is characterized. Examples for specific measures [mu] are discussed. Finally, a randomized solution to the embedding problem for Azéma martingale is deduced. Throughout the article, two possible approaches, using excursions and martingale theories, are presented in parallel.

Suggested Citation

  • Oblój, Jan & Yor, Marc, 2004. "An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 83-110, March.
    • Handle: RePEc:eee:spapps:v:110:y:2004:i:1:p:83-110
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(03)00167-4
    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. Philip Protter & Michael Dritschel, 1999. "Complete markets with discontinuous security price," Finance and Stochastics, Springer, vol. 3(2), pages 203-214.
    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. Lim, Adrian P.C. & Yen, Ju-Yi & Yor, Marc, 2013. "Some examples of Skorokhod embeddings obtained from the Azéma–Yor algorithm," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 329-346.
    2. Nikeghbali, Ashkan, 2006. "A class of remarkable submartingales," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 917-938, June.
    3. Oblój, Jan, 2007. "An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 117(4), pages 409-431, April.

    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. Zura Kakushadze, 2016. "Volatility Smile as Relativistic Effect," Papers 1610.02456, arXiv.org, revised Feb 2017.
    2. Wing Yan Yip & Sofia Olhede & David Stephens, 2008. "Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market," Papers 0801.4941, arXiv.org, revised Oct 2008.
    3. Youssef El-Khatib & Abdulnasser Hatemi-J, 2013. "On option pricing in illiquid markets with jumps," Papers 1304.4690, arXiv.org.
    4. Kakushadze, Zura, 2017. "Volatility smile as relativistic effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 475(C), pages 59-76.
    5. Yip, Wing & Stephens, David & Olhede, Sofia, 2008. "Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market," MPRA Paper 11176, University Library of Munich, Germany.
    6. Umut Çetin & Robert Jarrow & Philip Protter & Yildiray Yildirim, 2008. "Modeling Credit Risk With Partial Information," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 23, pages 579-590, World Scientific Publishing Co. Pte. Ltd..
    7. Dieckmann, Stephan & Gallmeyer, Michael, 2013. "Rare event risk and emerging market debt with heterogeneous beliefs," Journal of International Money and Finance, Elsevier, vol. 33(C), pages 163-187.
    8. Protter, Philip, 2001. "A partial introduction to financial asset pricing theory," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 169-203, February.
    9. Thorsten Rheinländer & Jenny Sexton, 2011. "Hedging Derivatives," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8062.

    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:110:y:2004:i:1:p:83-110. 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.