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On Finite-Time Ruin Probabilities for Classical Risk Models

Author

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  • 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

This paper is concerned with the problem of ruin in the classical compound binomial and compound Poisson risk models. Our primary purpose is to extend to those models an exact formula derived by Picard and Lefèvre (1997) for the probability of (non-)ruin within finite time. First, a standard method based on the ballot theorem and an argument of Seal-type provides an initial (known) formula for that probability. Then, a concept of pseudo-distributions for the cumulated claim amounts, combined with some simple implications of the ballot theorem, leads to the desired formula. Two expressions for the (non-)ruin probability over an infinite horizon are also deduced as corollaries. Finally, an illustration within the framework of Solvency II is briefly presented.

Suggested Citation

  • Claude Lefèvre & Stéphane Loisel, 2008. "On Finite-Time Ruin Probabilities for Classical Risk Models," Post-Print hal-00168958, HAL.
    • Handle: RePEc:hal:journl:hal-00168958
    DOI: 10.1080/03461230701766882
    Note: View the original document on HAL open archive server: https://hal.science/hal-00168958
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    References listed on IDEAS

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    1. repec:hal:wpaper:hal-00746251 is not listed on IDEAS
    2. Irmina Czarna & Zbigniew Palmowski, 2009. "De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process," Papers 0906.2100, arXiv.org, revised Feb 2011.
    3. 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.
    4. Denuit, Michel & Robert, Christian Y., 2022. "Dynamic conditional mean risk sharing in the compound Poisson surplus model," LIDAM Discussion Papers ISBA 2022034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Kam Pui Wat & Kam Chuen Yuen & Wai Keung Li & Xueyuan Wu, 2018. "On the Compound Binomial Risk Model with Delayed Claims and Randomized Dividends," Risks, MDPI, vol. 6(1), pages 1-13, January.
    6. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2009. "Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 374-381, December.
    7. Lanpeng Ji & Chunsheng Zhang, 2014. "A Duality Result for the Generalized Erlang Risk Model," Risks, MDPI, vol. 2(4), pages 1-11, November.
    8. Romain Biard & Stéphane Loisel & Claudio Macci & Noel Veraverbeke, 2010. "Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation," Post-Print hal-00372525, HAL.
    9. Yi Lu, 2016. "On the Evaluation of Expected Penalties at Claim Instants That Cause Ruin in the Classical Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 237-255, March.
    10. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
    11. Gathy, Maude & Lefèvre, Claude, 2010. "On the Lagrangian Katz family of distributions as a claim frequency model," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 76-83, August.
    12. Stéphane Loisel & Claude Lefèvre, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Post-Print hal-00201377, HAL.
    13. Hamed Amini & Zhongyuan Cao & Andreea Minca & Agn`es Sulem, 2023. "Ruin Probabilities for Risk Processes in Stochastic Networks," Papers 2302.06668, arXiv.org.
    14. Andrius Grigutis & Jonas Šiaulys, 2020. "Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model," Mathematics, MDPI, vol. 8(2), pages 1-30, January.
    15. Li, Shuanming & Lu, Yi, 2017. "Distributional study of finite-time ruin related problems for the classical risk model," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 319-330.
    16. Lee, Wing Yan & Li, Xiaolong & Liu, Fangda & Shi, Yifan & Yam, Sheung Chi Phillip, 2021. "A Fourier-cosine method for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 256-267.
    17. Julien Trufin & Stéphane Loisel, 2013. "Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments," Post-Print hal-00426790, HAL.
    18. Muhsin Tamturk & Sergey Utev, 2019. "Optimal Reinsurance via Dirac-Feynman Approach," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 647-659, June.
    19. Yuen, Fei Lung & Lee, Wing Yan & Fung, Derrick W.H., 2020. "A cyclic approach on classical ruin model," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 104-110.
    20. Edita Kizinevič & Jonas Šiaulys, 2018. "The Exponential Estimate of the Ultimate Ruin Probability for the Non-Homogeneous Renewal Risk Model," Risks, MDPI, vol. 6(1), pages 1-17, March.
    21. Serkan Eryilmaz, 2014. "On Distributions of Runs in the Compound Binomial Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 149-159, March.
    22. Li Qin & Susan M. Pitts, 2012. "Nonparametric Estimation of the Finite-Time Survival Probability with Zero Initial Capital in the Classical Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 919-936, December.

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