IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v121y2011i6p1389-1410.html
   My bibliography  Save this article

Random times with given survival probability and their -martingale decomposition formula

Author

Listed:
  • Jeanblanc, Monique
  • Song, Shiqi

Abstract

Given a filtered probability space , an -adapted continuous increasing process [Lambda] and a positive local martingale N such that satisfies Zt =0, we construct probability measures and a random time [tau] on an extension of , such that the survival probability of [tau], i.e., is equal to Zt for t>=0. We show that there exist several solutions and that an increasing family of martingales, combined with a stochastic differential equation, constitutes a natural way to construct these solutions. Our extended space will be equipped with the enlarged filtration where is the [sigma]-field completed with the -negligible sets. We show that all martingales remain -semimartingales and we give an explicit semimartingale decomposition formula. Finally, we show how this decomposition formula is intimately linked with the stochastic differential equation introduced before.

Suggested Citation

  • Jeanblanc, Monique & Song, Shiqi, 2011. "Random times with given survival probability and their -martingale decomposition formula," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1389-1410, June.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:6:p:1389-1410
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(11)00062-7
    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. El Karoui, Nicole & Jeanblanc, Monique & Jiao, Ying, 2010. "What happens after a default: The conditional density approach," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1011-1032, July.
    2. Jeanblanc, Monique & Le Cam, Yann, 2009. "Progressive enlargement of filtrations with initial times," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2523-2543, August.
    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. Aksamit, Anna & Jeanblanc, Monique & Rutkowski, Marek, 2019. "Integral representations of martingales for progressive enlargements of filtrations," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1229-1258.
    2. Li, Libo & Rutkowski, Marek, 2012. "Random times and multiplicative systems," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2053-2077.
    3. Libo Li, 2018. "Characterisation of honest times and optional semimartingales of class-($\Sigma$)," Papers 1801.03873, arXiv.org, revised Dec 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. Jeanblanc, Monique & Song, Shiqi, 2015. "Martingale representation property in progressively enlarged filtrations," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4242-4271.
    2. Fontana, Claudio, 2018. "The strong predictable representation property in initially enlarged filtrations under the density hypothesis," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 1007-1033.
    3. Ying Jiao & Huyên Pham, 2011. "Optimal investment with counterparty risk: a default-density model approach," Finance and Stochastics, Springer, vol. 15(4), pages 725-753, December.
    4. Stéphane Crépey & Shiqi Song, 2014. "Counterparty risk and funding: Immersion and beyond," Working Papers hal-00989062, HAL.
    5. Antonella Calzolari & Barbara Torti, 2022. "A Note on the Strong Predictable Representation Property and Enlargement of Filtration," Mathematics, MDPI, vol. 10(10), pages 1-12, May.
    6. Fontana, Claudio & Grbac, Zorana & Jeanblanc, Monique & Li, Qinghua, 2014. "Information, no-arbitrage and completeness for asset price models with a change point," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3009-3030.
    7. Nicole El Karoui & Monique Jeanblanc & Ying Jiao, 2013. "Density approach in modelling multi-defaults," Working Papers hal-00870492, HAL.
    8. Lijun Bo & Ying Jiao & Xuewei Yang, 2011. "Credit derivatives pricing with default density term structure modelled by L\'evy random fields," Papers 1112.2952, arXiv.org.
    9. Stephane Goutte & Armand Ngoupeyou, 2012. "Optimization problem and mean variance hedging on defaultable claims," Papers 1209.5953, arXiv.org.
    10. Ying Jiao & Yahia Salhi & Shihua Wang, 2022. "Dynamic Bivariate Mortality Modelling," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 917-938, June.
    11. Huayuan Dong & Paolo Guasoni & Eberhard Mayerhofer, 2023. "Rogue traders," Finance and Stochastics, Springer, vol. 27(3), pages 539-603, July.
    12. Paolo Tella, 2022. "On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2194-2216, December.
    13. Claudio Fontana & Thorsten Schmidt, 2016. "General dynamic term structures under default risk," Papers 1603.03198, arXiv.org, revised Nov 2017.
    14. Monique Jeanblanc & Thibaut Mastrolia & Dylan Possamaï & Anthony Réveillac, 2015. "Utility Maximization With Random Horizon: A Bsde Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(07), pages 1-43, November.
    15. Oliver Janke, 2016. "Utility Maximization and Indifference Value under Risk and Information Constraints for a Market with a Change Point," Papers 1610.08644, arXiv.org.
    16. El Karoui & Mrad Mohamed & Caroline Hillairet, 2022. "Bi-revealed utilities in a defaultable universe : a new point of view on consumption," Working Papers hal-03919186, HAL.
    17. Matteo Ludovico Bedini & Rainer Buckdahn & Hans-Jurgen Engelbert, 2016. "Unexpected Default in an Information Based Model," Papers 1611.02952, arXiv.org.
    18. Nicole El Karoui & Monique Jeanblanc & Ying Jiao, 2015. "Dynamics of multivariate default system in random environment," Papers 1509.09133, arXiv.org, revised Nov 2016.
    19. Matteo Ludovico Bedini & Rainer Buckdahn & Hans-Jurgen Engelbert, 2016. "Brownian Bridges on Random Intervals," Papers 1601.01811, arXiv.org.
    20. Claudio Fontana & Zorana Grbac & Monique Jeanblanc & Qinghua Li, 2013. "Information, no-arbitrage and completeness for asset price models with a change point," Papers 1304.0923, arXiv.org, revised Apr 2014.

    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:6:p:1389-1410. 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.