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The Density of the Time to Ruin in the Classical Poisson Risk Model

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  • Dickson, David C.M.
  • Willmot, Gordon E.

Abstract

We derive an expression for the density of the time to ruin in the classical risk model by inverting its Laplace transform. We then apply the result when the individual claim amount distribution is a mixed Erlang distribution, and show how finite time ruin probabilities can be calculated in this case.

Suggested Citation

  • Dickson, David C.M. & Willmot, Gordon E., 2005. "The Density of the Time to Ruin in the Classical Poisson Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 45-60, May.
    • Handle: RePEc:cup:astinb:v:35:y:2005:i:01:p:45-60_01
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    Cited by:

    1. Covrig Mihaela & Serban Radu, 2008. "About Risk Process Estimation Techniques Employed By A Virtual Organization Which Is Directed Towards The Insurance Business," Annals of Faculty of Economics, University of Oradea, Faculty of Economics, vol. 2(1), pages 841-847, May.
    2. 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.
    3. Dickson, David C.M., 2016. "A note on some joint distribution functions involving the time of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 120-124.
    4. Gildas Ratovomirija, 2015. "Multivariate Stop loss Mixed Erlang Reinsurance risk: Aggregation, Capital allocation and Default risk," Papers 1501.07297, arXiv.org.
    5. 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.
    6. Dassios, Angelos & Wu, Shanle, 2008. "Ruin probabilities of the Parisian type for small claims," LSE Research Online Documents on Economics 32037, London School of Economics and Political Science, LSE Library.
    7. Vaios Dermitzakis & Konstadinos Politis, 2011. "Asymptotics for the Moments of the Time to Ruin for the Compound Poisson Model Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 749-761, December.
    8. Dickson, David C.M. & Li, Shuanming, 2013. "The distributions of the time to reach a given level and the duration of negative surplus in the Erlang(2) risk model," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 490-497.
    9. Wang, Zijia & Landriault, David & Li, Shu, 2021. "An insurance risk process with a generalized income process: A solvency analysis," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 133-146.
    10. Dickson, David C.M., 2012. "The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 334-337.
    11. Feng, Runhuan & Volkmer, Hans W., 2012. "Modeling credit value adjustment with downgrade-triggered termination clause using a ruin theoretic approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 409-421.
    12. Dickson, David C.M. & Li, Shuanming, 2010. "Finite time ruin problems for the Erlang(2) risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 12-18, February.
    13. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.
    14. 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.
    15. Xie, Jiayi & Cui, Zhenyu & Zhang, Zhimin, 2022. "Some new infinite series expansions for the first passage time densities in a jump diffusion model with phase-type jumps," Applied Mathematics and Computation, Elsevier, vol. 429(C).
    16. Dassios, Angelos & Wu, Shanle, 2008. "Parisian ruin with exponential claims," LSE Research Online Documents on Economics 32033, London School of Economics and Political Science, LSE Library.
    17. 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.
    18. Willmot, Gordon E., 2015. "On a partial integrodifferential equation of Seal’s type," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 54-61.
    19. Krzysztof Burnecki & Zbigniew Palmowski & Marek Teuerle & Aleksandra Wilkowska, 2023. "Ruin probability for the quota share model with~phase-type distributed claims," Papers 2303.07705, arXiv.org.
    20. 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.
    21. Cheung, Eric C.K. & Zhu, Wei, 2023. "Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 84-101.
    22. 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.
    23. Yuguang Fan & Philip S. Griffin & Ross Maller & Alexander Szimayer & Tiandong Wang, 2017. "The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation," Risks, MDPI, vol. 5(1), pages 1-27, January.
    24. Bihao Su & Chenglong Xu & Jingchao Li, 2022. "A Deep Neural Network Approach to Solving for Seal’s Type Partial Integro-Differential Equation," Mathematics, MDPI, vol. 10(9), pages 1-21, May.

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