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Integral representations of martingales for progressive enlargements of filtrations

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  • Aksamit, Anna
  • Jeanblanc, Monique
  • Rutkowski, Marek

Abstract

We work in the setting of the progressive enlargement G of a reference filtration F through the observation of a random time τ. We study an integral representation property for some classes of G-martingales stopped at τ. In the first part, we focus on the case where F is a Poisson filtration and we establish a predictable representation property with respect to three G-martingales. In the second part, we relax the assumption that F is a Poisson filtration and we assume that τ is an F-pseudo-stopping time. We establish integral representations with respect to some G-martingales built from F-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two G-martingales.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:4:p:1229-1258
    DOI: 10.1016/j.spa.2018.04.009
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    References listed on IDEAS

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    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. 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.
    3. Xin Guo & Yan Zeng, 2008. "Intensity process and compensator: A new filtration expansion approach and the Jeulin--Yor theorem," Papers 0801.3191, arXiv.org.
    4. Jeanblanc, Monique & Song, Shiqi, 2011. "An explicit model of default time with given survival probability," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1678-1704, August.
    5. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195, April.
    6. Delia Coculescu & Monique Jeanblanc & Ashkan Nikeghbali, 2012. "Default times, no-arbitrage conditions and changes of probability measures," Finance and Stochastics, Springer, vol. 16(3), pages 513-535, July.
    7. 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.
    8. Li, Libo & Rutkowski, Marek, 2012. "Random times and multiplicative systems," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2053-2077.
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    Cited by:

    1. 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.

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