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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
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    References listed on IDEAS

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    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.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Pierre-Olivier Goffard, 2019. "Fraud risk assessment within blockchain transactions," Working Papers hal-01716687, HAL.
    5. 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.
    6. 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.

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