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

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    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.

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