IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-01887402.html
   My bibliography  Save this paper

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
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01887402/document
    Download Restriction: no
    File URL: https://libkey.io/10.1017/jpr.2019.33?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani, 2018. "The De Vylder-Goovaerts conjecture holds true within the diffusion limit," Working Papers hal-01887402, HAL.
    2. 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.
    3. Castañer, A. & Claramunt, M.M. & Lefèvre, C., 2013. "Survival probabilities in bivariate risk models, with application to reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 632-642.
    4. Goffard, Pierre-Olivier & Lefèvre, Claude, 2018. "Duality in ruin problems for ordered risk models," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 44-52.
    5. Pierre-Olivier Goffard & Claude Lefèvre, 2018. "Duality in ruin problems for ordered risk models," Post-Print hal-01398910, HAL.
    6. Dimitrina S. Dimitrova & Zvetan G. Ignatov & Vladimir K. Kaishev, 2017. "On the First Crossing of Two Boundaries by an Order Statistics Risk Process," Risks, MDPI, vol. 5(3), pages 1-14, August.
    7. Claude Lefèvre & Philippe Picard, 2014. "Ruin Probabilities for Risk Models with Ordered Claim Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 885-905, December.
    8. Stéphane Loisel & Charles Minier, 2023. "On the Devylder–Goovaerts Conjecture in Ruin Theory," Mathematics, MDPI, vol. 11(6), pages 1-10, March.
    9. Pierre-Olivier Goffard, 2019. "Two-sided exit problems in the ordered risk model," Post-Print hal-01528204, HAL.
    10. Pierre-Olivier Goffard, 2019. "Two-Sided Exit Problems in the Ordered Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 539-549, June.
    11. Stilian A. Stoev & Murad S. Taqqu, 2007. "Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 55-87, March.
    12. 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.
    13. Anatolii A. Puhalskii, 2003. "On Large Deviation Convergence of Invariant Measures," Journal of Theoretical Probability, Springer, vol. 16(3), pages 689-724, July.
    14. Saulius Minkevičius & Igor Katin & Joana Katina & Irina Vinogradova-Zinkevič, 2021. "On Little’s Formula in Multiphase Queues," Mathematics, MDPI, vol. 9(18), pages 1-15, September.
    15. Doruk Cetemen & Can Urgun & Leeat Yariv, 2023. "Collective Progress: Dynamics of Exit Waves," Journal of Political Economy, University of Chicago Press, vol. 131(9), pages 2402-2450.
    16. Dolinsky, Yan & Zouari, Jonathan, 2021. "The value of insider information for super-replication with quadratic transaction costs," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 394-416.
    17. Acciaio, B. & Backhoff-Veraguas, J. & Zalashko, A., 2020. "Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2918-2953.
    18. Macci, Claudio & Pacchiarotti, Barbara, 2015. "Large deviations for a class of counting processes and some statistical applications," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 36-48.
    19. Constantinos Kardaras, 2009. "Num\'{e}raire-invariant preferences in financial modeling," Papers 0903.3736, arXiv.org, revised Nov 2010.
    20. Zhang, Tonglin, 2024. "Variables selection using L0 penalty," Computational Statistics & Data Analysis, Elsevier, vol. 190(C).

    More about this item

    Keywords

    Risk theory; Equalized claims; Ruin probability; Diffusion approximations;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:hal-01887402. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.