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On the Joint Distributions of the Time to Ruin, the Surplus Prior to Ruin, and the Deficit at Ruin in the Classical Risk Model

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  • David Landriault
  • Gordon Willmot

Abstract

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.

Suggested Citation

  • David Landriault & Gordon Willmot, 2009. "On the Joint Distributions of the Time to Ruin, the Surplus Prior to Ruin, and the Deficit at Ruin in the Classical Risk Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(2), pages 252-270.
    • Handle: RePEc:taf:uaajxx:v:13:y:2009:i:2:p:252-270
    DOI: 10.1080/10920277.2009.10597550
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    Cited by:

    1. Landriault, David & Li, Bin & Shi, Tianxiang & Xu, Di, 2019. "On the distribution of classic and some exotic ruin times," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 38-45.
    2. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    3. Cossette, Hélène & Landriault, David & Marceau, Etienne & Moutanabbir, Khouzeima, 2012. "Analysis of the discounted sum of ascending ladder heights," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 393-401.
    4. Landriault, David & Shi, Tianxiang, 2015. "Occupation times in the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 75-82.
    5. Huynh, Mirabelle & Landriault, David & Shi, Tianxiang & Willmot, Gordon E., 2015. "On a risk model with claim investigation," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 37-45.
    6. Jos'e Miguel Flores-Contr'o, 2024. "The Gerber-Shiu Expected Discounted Penalty Function: An Application to Poverty Trapping," Papers 2402.11715, arXiv.org.
    7. Landriault, David & Lemieux, Christiane & Willmot, Gordon E., 2012. "An adaptive premium policy with a Bayesian motivation in the classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 370-378.
    8. Willmot, Gordon E. & Woo, Jae-Kyung, 2010. "Surplus analysis for a class of Coxian interclaim time distributions with applications to mixed Erlang claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 32-41, February.
    9. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    10. Cheung, Eric C.K. & Landriault, David, 2010. "A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 127-134, February.
    11. Landriault, David & Shi, Tianxiang & Willmot, Gordon E., 2011. "Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 371-379.
    12. 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.
    13. Ramsden, Lewis & Papaioannou, Apostolos D., 2019. "On the time to ruin for a dependent delayed capital injection risk model," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 119-135.
    14. Kim, So-Yeun & Willmot, Gordon E., 2016. "On the analysis of ruin-related quantities in the delayed renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 77-85.
    15. Dimitrina S. Dimitrova & Zvetan G. Ignatov & Vladimir K. Kaishev, 2019. "Ruin and Deficit Under Claim Arrivals with the Order Statistics Property," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 511-530, June.

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