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Stopping Times Occurring Simultaneously

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  • Philip Protter
  • Alejandra Quintos

Abstract

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that they are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. We use a modified Cox construction along with the bivariate exponential introduced by Marshall and Olkin (1967) to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We show that our initial construction only allows for positive dependence between stopping times, but we also propose a joint distribution that allows for negative dependence while preserving the property of non-zero probability of equality. We indicate applications to modeling COVID-19 contagion (and epidemics in general), civil engineering, and to credit risk.

Suggested Citation

  • Philip Protter & Alejandra Quintos, 2021. "Stopping Times Occurring Simultaneously," Papers 2111.09458, arXiv.org, revised Nov 2024.
    • Handle: RePEc:arx:papers:2111.09458
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    References listed on IDEAS

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    1. Robert Jarrow & Philip Protter & Alejandra Quintos, 2024. "Computing the probability of a financial market failure: a new measure of systemic risk," Annals of Operations Research, Springer, vol. 336(1), pages 481-503, May.
    2. Xin Guo & Yan Zeng, 2008. "Intensity process and compensator: A new filtration expansion approach and the Jeulin--Yor theorem," Papers 0801.3191, arXiv.org.
    3. Weiping Li, 2016. "Probability of Default and Default Correlations," JRFM, MDPI, vol. 9(3), pages 1-19, July.
    4. Lindskog, Filip & McNeil, Alexander J., 2003. "Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling," ASTIN Bulletin, Cambridge University Press, vol. 33(2), pages 209-238, November.
    5. Svante Janson & Sokhna M'Baye & Philip Protter, 2011. "Absolutely Continuous Compensators," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(03), pages 335-351.
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    1. Robert Jarrow & Philip Protter & Alejandra Quintos, 2024. "Computing the probability of a financial market failure: a new measure of systemic risk," Annals of Operations Research, Springer, vol. 336(1), pages 481-503, May.

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