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On the depletion problem for an insurance risk process: new non-ruin quantities in collective risk theory

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Abstract

The field of risk theory has traditionally focused on ruin-related quantities. In particular, the so-called expected discounted penalty function (Gerber and Shiu. N Am Actuar J, 2(1):48–78, 1998) has been the object of a thorough study over the years. Although interesting in their own right, ruin related quantities do not seem to capture path-dependent properties of the reserve. In this article we aim at presenting the probabilistic properties of drawdowns and the speed at which an insurance reserve depletes as a consequence of the risk exposure of the company. These new quantities are not ruin related yet they capture important features of an insurance position and we believe it can lead to the design of a meaningful risk measures. Studying drawdowns and speed of depletion for Lévy insurance risk processes represent a novel and challenging concept in insurance mathematics. In this paper, all these concepts are formally introduced in an insurance setting. Moreover, we specialize recent results in fluctuation theory for Lévy processes (Mijatovic and Pistorius. Stoch Process Their Appl, 22:3812–3836, 2012) in order to derive expressions for the distribution of several quantities related to the depletion problem. Of particular interest are the distribution of drawdowns and the Laplace transform for the speed of depletion. These expressions are given for several examples of Lévy insurance risk processes for which they can be calculated, in particular for the classical Cramér–Lundberg model.

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Authors and Affiliations

  1. Department of Mathematics and Actuarial Science, The American University in Cairo, PO Box 74, New Cairo 11835, Cairo, Canada

    Zied Ben-Salah

  2. IRMAR. UMR CNRS 6625., University of Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Hélène Guérin

  3. Laboratory of Excellence for Financial Regulation (LABEX ReFi), Université Paris 1, 81 avenue de la Republique, Paris, 75543 Paris cedex 11, France

    Manuel Morales & Hassan Omidi Firouzi

Authors
  1. Zied Ben-Salah
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  2. Hélène Guérin
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  3. Manuel Morales
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  4. Hassan Omidi Firouzi
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Corresponding author

Correspondence to Manuel Morales.

Additional information

The authors acknowledge the initial input from Prof. Erhan Bayraktar of the University of Michigan that instigated this work. The authors would also like to thank the CNRS, its UMI 3457 and the CRM for providing the research infrastructure. This research was funded in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Ben-Salah, Z., Guérin, H., Morales, M. et al. On the depletion problem for an insurance risk process: new non-ruin quantities in collective risk theory. Eur. Actuar. J. 5, 381–425 (2015). https://doi.org/10.1007/s13385-015-0112-9

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  • DOI: https://doi.org/10.1007/s13385-015-0112-9

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