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On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing

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  • Christophe Dutang

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon, IRMA - Institut de Recherche Mathématique Avancée - UNISTRA - Université de Strasbourg - CNRS - Centre National de la Recherche Scientifique)

  • Claude Lefèvre

    (ULB - Département de Mathématique [Bruxelles] - ULB - Faculté des Sciences [Bruxelles] - ULB - Université libre de Bruxelles)

  • Stéphane Loisel

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

The purpose of this paper is to point out that an asymptotic rule "A+B/u" for the ultimate ruin probability applies to a wide class of dependent risk models, in discrete and continuous time. Dependence is incorporated through a mixing approach among claim amounts or claim inter-arrival times, leading to a systemic risk behavior. Ruin corresponds here either to classical ruin, or to stopping the activity after realizing that it is not pro table at all, when one has little possibility to increase premium income rate. Several special cases for which closed formulas are derived, are also investigated in some detail.

Suggested Citation

  • Christophe Dutang & Claude Lefèvre & Stéphane Loisel, 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Post-Print hal-00746251, HAL.
    • Handle: RePEc:hal:journl:hal-00746251
    Note: View the original document on HAL open archive server: https://hal.science/hal-00746251v2
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    References listed on IDEAS

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    1. Albrecher, Hansjörg & Constantinescu, Corina & Loisel, Stephane, 2011. "Explicit ruin formulas for models with dependence among risks," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 265-270, March.
    2. Genest, Christian & Nešlehová, Johanna, 2007. "A Primer on Copulas for Count Data," ASTIN Bulletin, Cambridge University Press, vol. 37(2), pages 475-515, November.
    3. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    4. Hans Gerber & Elias Shiu, 2005. "The Time Value of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 49-69.
    5. Julien Trufin & Stéphane Loisel, 2013. "Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments," Post-Print hal-00426790, HAL.
    6. Stéphane Loisel & Claude Lefèvre, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Post-Print hal-00201377, HAL.
    7. Claude Lefèvre & Stéphane Loisel, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 425-441, September.
    8. Cai, Jun & Li, Haijun, 2005. "Multivariate risk model of phase type," Insurance: Mathematics and Economics, Elsevier, vol. 36(2), pages 137-152, April.
    9. Shiu, Elias S.W., 1989. "The Probability of Eventual Ruin in the Compound Binomial Model," ASTIN Bulletin, Cambridge University Press, vol. 19(2), pages 179-190, November.
    10. Albrecher, Hansjorg & Boxma, Onno J., 2004. "A ruin model with dependence between claim sizes and claim intervals," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 245-254, October.
    11. Centeno, Maria de Lourdes, 2002. "Excess of loss reinsurance and Gerber's inequality in the Sparre Anderson model," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 415-427, December.
    12. Claude Lefèvre & Stéphane Loisel, 2008. "On Finite-Time Ruin Probabilities for Classical Risk Models," Post-Print hal-00168958, HAL.
    13. Willmot, Gordon E., 1993. "Ruin probabilities in the compound binomial model," Insurance: Mathematics and Economics, Elsevier, vol. 12(2), pages 133-142, April.
    14. Gerber, Hans U., 1988. "Mathematical Fun with the Compound Binomial Process," ASTIN Bulletin, Cambridge University Press, vol. 18(2), pages 161-168, November.
    15. Embrechts, P. & Veraverbeke, N., 1982. "Estimates for the probability of ruin with special emphasis on the possibility of large claims," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 55-72, January.
    16. Lefèvre, Claude & Picard, Philippe, 2011. "A new look at the homogeneous risk model," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 512-519.
    17. Dimitrova, Dimitrina S. & Kaishev, Vladimir K., 2010. "Optimal joint survival reinsurance: An efficient frontier approach," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 27-35, August.
    18. Constantinescu, Corina & Hashorva, Enkelejd & Ji, Lanpeng, 2011. "Archimedean copulas in finite and infinite dimensions—with application to ruin problems," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 487-495.
    19. Frees, Edward W. & Wang, Ping, 2006. "Copula credibility for aggregate loss models," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 360-373, April.
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    Cited by:

    1. Constantinescu Corina D. & Kozubowski Tomasz J. & Qian Haoyu H., 2019. "Probability of ruin in discrete insurance risk model with dependent Pareto claims," Dependence Modeling, De Gruyter, vol. 7(1), pages 215-233, January.
    2. Badía, F.G. & Sangüesa, C. & Cha, J.H., 2014. "Stochastic comparison of multivariate conditionally dependent mixtures," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 82-94.
    3. Buddana Amrutha & Kozubowski Tomasz J., 2014. "Discrete Pareto Distributions," Stochastics and Quality Control, De Gruyter, vol. 29(2), pages 143-156, December.
    4. Arendarczyk, Marek & Kozubowski, Tomasz. J. & Panorska, Anna K., 2018. "The joint distribution of the sum and maximum of dependent Pareto risks," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 136-156.
    5. Youri Raaijmakers & Hansjörg Albrecher & Onno Boxma, 2019. "The Single Server Queue with Mixing Dependencies," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1023-1044, December.

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