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Classification of the Bounds on the Probability of Ruin for L{\'e}vy Processes with Light-tailed Jumps

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  • J'er^ome Spielmann

    (LAREMA)

Abstract

In this note, we study the ultimate ruin probabilities of a real-valued L{\'e}vy process X with light-tailed negative jumps. It is well-known that, for such L{\'e}vy 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 L{\'e}vy 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 Cram{\'e}r-Lundberg model perturbed by Brownian motion.

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  • J'er^ome Spielmann, 2017. "Classification of the Bounds on the Probability of Ruin for L{\'e}vy Processes with Light-tailed Jumps," Papers 1709.10295, arXiv.org, revised Feb 2018.
    • Handle: RePEc:arx:papers:1709.10295
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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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