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The De Vylder-Goovaerts conjecture holds true within the diffusion limit

Author

Listed:
  • Stefan Ankirchner

    (Institut für Mathematik - Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany])

  • Christophette Blanchet-Scalliet

    (PSPM - Probabilités, statistique, physique mathématique - ICJ - Institut Camille Jordan - ECL - École Centrale de Lyon - Université de Lyon - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - INSA Lyon - Institut National des Sciences Appliquées de Lyon - Université de Lyon - INSA - Institut National des Sciences Appliquées - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

  • Nabil Kazi-Tani

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

Abstract

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite time ruin probability in a standard risk model is greater or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T. In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renor-malized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds true for the diffusion limits.

Suggested Citation

  • Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani, 2019. "The De Vylder-Goovaerts conjecture holds true within the diffusion limit," Post-Print hal-01887402, HAL.
    • Handle: RePEc:hal:journl:hal-01887402
    DOI: 10.1017/jpr.2019.33
    Note: View the original document on HAL open archive server: https://hal.science/hal-01887402
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    References listed on IDEAS

    as
    1. Stefan Ankirchner & Steffen Dereich & Peter Imkeller, 2005. "The Shannon information of filtrations and the additional logarithmic utility of insiders," Papers math/0503013, arXiv.org, revised May 2006.
    2. Christian Yann Robert, 2014. "On the De Vylder and Goovaerts Conjecture About Ruin for Equalized Claims," Post-Print hal-02006620, HAL.
    3. Furrer, Hansjorg & Michna, Zbigniew & Weron, Aleksander, 1997. "Stable Lévy motion approximation in collective risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 20(2), pages 97-114, September.
    4. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
    5. Lefèvre, Claude & Picard, Philippe, 2011. "A new look at the homogeneous risk model," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 512-519.
    6. Atkinson, Michael P. & Singham, Dashi I., 2015. "Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 85-92.
    7. De Vylder, F. & Goovaerts, M., 2000. "Homogeneous risk models with equalized claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 223-238, May.
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    Cited by:

    1. Kim, Bara & Kim, Jeongsim & Kim, Jerim, 2021. "De Vylder and Goovaerts' conjecture on homogeneous risk models with equalized claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 186-201.
    2. Stéphane Loisel & Charles Minier, 2023. "On the Devylder–Goovaerts Conjecture in Ruin Theory," Mathematics, MDPI, vol. 11(6), pages 1-10, March.

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    Keywords

    Risk theory; Equalized claims; Ruin probability; Diffusion approximations;
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