IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v16y2014i4d10.1007_s11009-013-9334-y.html
   My bibliography  Save this article

Ruin Probabilities for Risk Models with Ordered Claim Arrivals

Author

Listed:
  • Claude Lefèvre

    (Université Libre de Bruxelles)

  • Philippe Picard

    (Université de Lyon)

Abstract

Recently, Lefèvre and Picard (Insur Math Econ 49:512–519, 2011) revisited a non-standard risk model defined on a fixed time interval [0,t]. The key assumption is that, if n claims occur during [0,t], their arrival times are distributed as the order statistics of n i.i.d. random variables with distribution function F t (s), 0 ≤ s ≤ t. The present paper is concerned with two particular cases of that model, namely when F t (s) is of linear form (as for a (mixed) Poisson process), or of exponential form (as for a linear birth process with immigration or a linear death-counting process). Our main purpose is to obtain, in these cases, an expression for the non-ruin probabilities over [0,t]. This is done by exploiting properties of an underlying family of Appell polynomials. The ultimate non-ruin probabilities are then derived as a limit.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:4:d:10.1007_s11009-013-9334-y
    DOI: 10.1007/s11009-013-9334-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-013-9334-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-013-9334-y?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Gerber, Hans U., 1988. "Mathematical fun with ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 7(1), pages 15-23, January.
    2. Stéphane Loisel & Claude Lefèvre, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Post-Print hal-00201377, HAL.
    3. Claude Lefèvre & Stéphane Loisel, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 425-441, September.
    4. Picard, Philippe & Lefèvre, Claude, 2003. "On the first meeting or crossing of two independent trajectories for some counting processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 217-242, April.
    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. 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. 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.
    2. 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.
    3. Pierre-Olivier Goffard, 2019. "Fraud risk assessment within blockchain transactions," Working Papers hal-01716687, HAL.
    4. 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.
    5. 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.
    6. 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.

    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. 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.
    2. 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.
    3. 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.
    4. Pierre-Olivier Goffard, 2019. "Fraud risk assessment within blockchain transactions," Working Papers hal-01716687, HAL.
    5. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
    6. Florin Avram & Romain Biard & Christophe Dutang & Stéphane Loisel & Landy Rabehasaina, 2014. "A survey of some recent results on Risk Theory," Post-Print hal-01616178, HAL.
    7. Lefèvre, Claude & Picard, Philippe, 2011. "A new look at the homogeneous risk model," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 512-519.
    8. 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.
    9. Dimitrova, Dimitrina S. & Kaishev, Vladimir K. & Zhao, Shouqi, 2016. "On the evaluation of finite-time ruin probabilities in a dependent risk model," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 268-286.
    10. Pierre-Olivier Goffard & Claude Lefèvre, 2018. "Duality in ruin problems for ordered risk models," Post-Print hal-01398910, HAL.
    11. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2009. "Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 374-381, December.
    12. 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.
    13. 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.
    14. Andrius Grigutis & Jonas Šiaulys, 2020. "Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model," Mathematics, MDPI, vol. 8(2), pages 1-30, January.
    15. Mathieu Bargès & Stéphane Loisel & Xavier Venel, 2011. "On finite-time ruin probabilities with reinsurance cycles influenced by large claims," Post-Print hal-00430178, HAL.
    16. Pierre-O. Goffard, 2019. "Fraud risk assessment within blockchain transactions," Post-Print hal-01716687, HAL.
    17. Pierre-Olivier Goffard, 2019. "Two-sided exit problems in the ordered risk model," Post-Print hal-01528204, HAL.
    18. 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.
    19. Romain Biard & Stéphane Loisel & Claudio Macci & Noel Veraverbeke, 2010. "Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation," Post-Print hal-00372525, HAL.
    20. D. Perry & W. Stadje & S. Zacks, 2005. "A Two-Sided First-Exit Problem for a Compound Poisson Process with a Random Upper Boundary," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 51-62, March.

    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:spr:metcap:v:16:y:2014:i:4:d:10.1007_s11009-013-9334-y. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.