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Classification of the Bounds on the Probability of Ruin for Lévy Processes with Light-tailed Jumps

Author

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  • Jérôme Spielmann

    (LAREMA - Laboratoire Angevin de Recherche en Mathématiques - UA - Université d'Angers - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this note, we study the ultimate ruin probabilities of a real-valued Lévy process X with light-tailed negative jumps. It is well-known that, for such Lévy processes, the probability of ruin decreases as an exponential function with a rate given by the root of the Laplace exponent, when the initial value goes to infinity. Under the additional assumption that X has integrable positive jumps, we show how a finer analysis of the Laplace exponent gives in fact a complete description of the bounds on the probability of ruin for this class of Lévy processes. This leads to the identification of a case that is not considered in the literature and for which we give an example. We then apply the result to various risk models and in particular the Cramér-Lundberg model perturbed by Brownian motion.

Suggested Citation

  • Jérôme Spielmann, 2018. "Classification of the Bounds on the Probability of Ruin for Lévy Processes with Light-tailed Jumps," Working Papers hal-01597828, HAL.
    • Handle: RePEc:hal:wpaper:hal-01597828
    Note: View the original document on HAL open archive server: https://hal.science/hal-01597828v2
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    References listed on IDEAS

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    1. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
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