IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v121y2011i7p1588-1606.html
   My bibliography  Save this article

Martingale representation for Poisson processes with applications to minimal variance hedging

Author

Listed:
  • Last, Günter
  • Penrose, Mathew D.

Abstract

We consider a Poisson process [eta] on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure [lambda] of [eta]. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with [eta]), which was previously known only in the special case, when [lambda] is the product of Lebesgue measure on and a [sigma]-finite measure on another space . Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.

Suggested Citation

  • Last, Günter & Penrose, Mathew D., 2011. "Martingale representation for Poisson processes with applications to minimal variance hedging," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1588-1606, July.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:7:p:1588-1606
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414911000731
    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. Fred Espen Benth & Giulia Di Nunno & Arne Løkka & Bernt Øksendal & Frank Proske, 2003. "Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 55-72, January.
    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. Marcus C. Christiansen, 2021. "Time-dynamic evaluations under non-monotone information generated by marked point processes," Finance and Stochastics, Springer, vol. 25(3), pages 563-596, July.
    2. Marcus C. Christiansen, 2018. "A martingale concept for non-monotone information in a jump process framework," Papers 1811.00952, arXiv.org, revised Jan 2021.

    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. El-Khatib, Youssef & Goutte, Stephane & Makumbe, Zororo S. & Vives, Josep, 2023. "A hybrid stochastic volatility model in a Lévy market," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 220-235.
    2. Choe, Hi Jun & Lee, Ji Min & Lee, Jung-Kyung, 2018. "Malliavin calculus for subordinated Lévy process," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 392-401.
    3. 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.
    4. Elisa Alòs & Christian-Olivier Ewald, 2005. "A note on the Malliavin differentiability of the Heston volatility," Economics Working Papers 880, Department of Economics and Business, Universitat Pompeu Fabra.
    5. Kristoffer Lindensjo, 2016. "An explicit formula for optimal portfolios in complete Wiener driven markets: a functional It\^o calculus approach," Papers 1610.05018, arXiv.org, revised Dec 2017.
    6. Ekaterina L. Dyachenko, 2016. "Internal Migration of Scientists in Russia and the USA: The Case of Applied Physics," HSE Working papers WP BRP 58/STI/2016, National Research University Higher School of Economics.
    7. St'ephane Goutte & Nadia Oudjane & Francesco Russo, 2013. "Variance optimal hedging for continuous time additive processes and applications," Papers 1302.1965, arXiv.org.
    8. Ewald, Christian-Oliver & Nawar, Roy & Siu, Tak Kuen, 2013. "Minimal variance hedging of natural gas derivatives in exponential Lévy models: Theory and empirical performance," Energy Economics, Elsevier, vol. 36(C), pages 97-107.
    9. St'ephane Goutte & Nadia Oudjane & Francesco Russo, 2012. "Variance Optimal Hedging for discrete time processes with independent increments. Application to Electricity Markets," Papers 1205.4089, arXiv.org.
    10. Takuji Arai & Yuto Imai, 2017. "A closed-form representation of mean-variance hedging for additive processes via Malliavin calculus," Papers 1702.07556, arXiv.org, revised Nov 2017.

    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:121:y:2011:i:7:p:1588-1606. 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.