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Intensity process and compensator: A new filtration expansion approach and the Jeulin--Yor theorem

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  • Xin Guo
  • Yan Zeng

Abstract

Let $(X_t)_{t\ge0}$ be a continuous-time, time-homogeneous strong Markov process with possible jumps and let $\tau$ be its first hitting time of a Borel subset of the state space. Suppose $X$ is sampled at random times and suppose also that $X$ has not hit the Borel set by time $t$. What is the intensity process of $\tau$ based on this information? This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin--Yor [Lecture Notes in Math. 649 (1978) 78--97] result regarding compensators under a general filtration expansion framework, a novel computation methodology for the intensity process of a stopping time is proposed. En route, an analogous characterization result for martingales of Jacod and Skorohod [Lecture Notes in Math. 1583 (1994) 21--35] under local jumping filtration is derived.

Suggested Citation

  • Xin Guo & Yan Zeng, 2008. "Intensity process and compensator: A new filtration expansion approach and the Jeulin--Yor theorem," Papers 0801.3191, arXiv.org.
    • Handle: RePEc:arx:papers:0801.3191
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    References listed on IDEAS

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    1. Monique Jeanblanc & Stoyan Valchev, 2005. "Partial Information And Hazard Process," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(06), pages 807-838.
    2. Xin Guo & Robert A. Jarrow & Yan Zeng, 2009. "Modeling The Recovery Rate In A Reduced Form Model," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 73-97, January.
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    6. Xin Guo & Robert A. Jarrow & Yan Zeng, 2009. "Credit Risk Models with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 320-332, May.
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    Citations

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    Cited by:

    1. Beatrice Acciaio & Martin Larsson, 2015. "Semi-static completeness and robust pricing by informed investors," Papers 1510.01890, arXiv.org, revised Sep 2016.
    2. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1761-1784.
    3. Sarma, Utpal Jyoti Deba & Selvamuthu, Dharmaraja, 2024. "Study of discrete-time Hawkes process and its compensator," Statistics & Probability Letters, Elsevier, vol. 214(C).
    4. Xin Guo & Robert A. Jarrow & Yan Zeng, 2009. "Credit Risk Models with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 320-332, May.
    5. Philip Protter & Alejandra Quintos, 2021. "Stopping Times Occurring Simultaneously," Papers 2111.09458, arXiv.org, revised Nov 2024.
    6. Beatrice Acciaio & Claudio Fontana & Constantinos Kardaras, 2014. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," Papers 1401.7198, arXiv.org, revised May 2015.
    7. Younes Kchia & Martin Larsson, 2011. "Credit contagion and risk management with multiple non-ordered defaults," Papers 1104.5272, arXiv.org, revised Jun 2011.
    8. Dong, Xin & Zheng, Harry, 2015. "Intensity process for a pure jump Lévy structural model with incomplete information," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1307-1322.
    9. Okhrati, Ramin & Balbás, Alejandro & Garrido, José, 2014. "Hedging of defaultable claims in a structural model using a locally risk-minimizing approach," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2868-2891.
    10. 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.
    11. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," LSE Research Online Documents on Economics 65150, London School of Economics and Political Science, LSE Library.
    12. Frank Gehmlich & Thorsten Schmidt, 2015. "A generalized intensity based framework for single-name credit risk," Papers 1512.03896, arXiv.org.
    13. Ramin Okhrati & Alejandro Balb'as & Jos'e Garrido, 2015. "Hedging of defaultable claims in a structural model using a locally risk-minimizing approach," Papers 1505.03501, arXiv.org.
    14. Caroline Hillairet & Ying Jiao, 2010. "Information Asymmetry in Pricing of Credit Derivatives," Papers 1002.3256, arXiv.org.
    15. Protter, Philip, 2015. "Strict local martingales with jumps," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1352-1367.
    16. Xin Dong & Harry Zheng, 2014. "Intensity Process for a Pure Jump L\'evy Structural Model with Incomplete Information," Papers 1405.3767, arXiv.org.
    17. Caroline Hillairet & Ying Jiao, 2010. "Information Asymmetry in Pricing of Credit Derivatives," Working Papers hal-00457456, HAL.
    18. Xin Guo & Robert A. Jarrow & Yan Zeng, 2009. "Modeling The Recovery Rate In A Reduced Form Model," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 73-97, January.
    19. Acciaio, Beatrice & Larsson, Martin, 2017. "Semi-static completeness and robust pricing by informed investors," LSE Research Online Documents on Economics 68502, London School of Economics and Political Science, LSE Library.
    20. Alberto Gennaro & Thibaut Mastrolia, 2024. "Delegated portfolio management with random default," Papers 2410.13103, arXiv.org.

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