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On Computations in Renewal Risk Models—Analytical and Statistical Aspects

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  • Josef Anton Strini

    (Graz University of Technology, Institute of Statistics, Kopernikusgasse 24/III, A-8010 Graz, Austria)

  • Stefan Thonhauser

    (Graz University of Technology, Institute of Statistics, Kopernikusgasse 24/III, A-8010 Graz, Austria)

Abstract

We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show weak convergence of an approximating sequence of piecewise-deterministic Markov processes (PDMPs) for deriving the convergence of the procedures. We will use estimated PDMP characteristics in a subsequent step from simulated sample data and study its effect on the numerically computed Gerber-Shiu functions. It can be seen that the main source of instability stems from the hazard rate estimator. Interestingly, results obtained using MC methods are hardly affected by estimation.

Suggested Citation

  • Josef Anton Strini & Stefan Thonhauser, 2020. "On Computations in Renewal Risk Models—Analytical and Statistical Aspects," Risks, MDPI, vol. 8(1), pages 1-20, March.
    • Handle: RePEc:gam:jrisks:v:8:y:2020:i:1:p:24-:d:328516
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    References listed on IDEAS

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    1. Vatamidou, E. & Adan, I.J.B.F. & Vlasiou, M. & Zwart, B., 2013. "Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 366-378.
    2. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    3. Romain Azaïs & François Dufour & Anne Gégout-Petit, 2014. "Non-Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise-Deterministic Markov Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 950-969, December.
    4. Hansjörg Albrecher & Eleni Vatamidou, 2019. "Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims," Risks, MDPI, vol. 7(4), pages 1-14, October.
    5. 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.
    6. Peter Kritzer & Gunther Leobacher & Michaela Szolgyenyi & Stefan Thonhauser, 2017. "Approximation methods for piecewise deterministic Markov processes and their costs," Papers 1712.09201, arXiv.org, revised Jan 2019.
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    Cited by:

    1. 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.
    2. Simon Pojer & Stefan Thonhauser, 2023. "The Markovian Shot-noise Risk Model: A Numerical Method for Gerber-Shiu Functions," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.

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