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Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach

Author

Listed:
  • Claude Lefèvre

    (Mathématique, Université Libre de Bruxelles, Campus de la Plaine C.P. 210, Bruxelles B-1050, Belgium)

  • Philippe Picard

    (Institut de Science Financière et d'Assurances, Université de Lyon, 50 Avenue Tony Garnier, Lyon F-69007, France)

Abstract

This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserves prior to ruin, and the Laplace transform of the ruin time. Interestingly, the usual net profit condition will be essentially relaxed. Most results generalize those known for the compound Poisson claim process.

Suggested Citation

  • Claude Lefèvre & Philippe Picard, 2013. "Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach," Risks, MDPI, vol. 1(3), pages 1-21, December.
    • Handle: RePEc:gam:jrisks:v:1:y:2013:i:3:p:192-212:d:31342
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    References listed on IDEAS

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    1. Willmot, Gordon E. & Sheldon Lin, X., 1998. "Exact and approximate properties of the distribution of surplus before and after ruin," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 91-110, October.
    2. Rulliere, Didier & Loisel, Stephane, 2004. "Another look at the Picard-Lefevre formula for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 187-203, October.
    3. Dickson, David C. M., 1992. "On the distribution of the surplus prior to ruin," Insurance: Mathematics and Economics, Elsevier, vol. 11(3), pages 191-207, October.
    4. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. Morales, Manuel, 2007. "On the expected discounted penalty function for a perturbed risk process driven by a subordinator," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 293-301, March.
    6. Dufresne, François & Gerber, Hans U. & Shiu, Elias S. W., 1991. "Risk Theory with the Gamma Process," ASTIN Bulletin, Cambridge University Press, vol. 21(2), pages 177-192, November.
    7. Dickson,David C. M., 2005. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521846400.
    8. Dufresne, Francois & Gerber, Hans U., 1988. "The surpluses immediately before and at ruin, and the amount of the claim causing ruin," Insurance: Mathematics and Economics, Elsevier, vol. 7(3), pages 193-199, October.
    9. Gerber, Hans U., 1988. "Mathematical fun with ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 7(1), pages 15-23, January.
    10. Dickson, David C. M. & dos Reis, Alfredo Egidio, 1994. "Ruin problems and dual events," Insurance: Mathematics and Economics, Elsevier, vol. 14(1), pages 51-60, April.
    11. Morales, Manuel, 2004. "Risk Theory with the Generalized Inverse Gaussian Lévy Process," ASTIN Bulletin, Cambridge University Press, vol. 34(2), pages 361-377, November.
    12. Gerber, Hans U. & Shiu, Elias S. W., 1997. "The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 129-137, November.
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    Cited by:

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