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On the Depletion Problem for an Insurance Risk Process: New Non-ruin Quantities in Collective Risk Theory

Author

Listed:
  • Zied Ben-Salah

    (American University in Cairo)

  • Hélène Guérin

    (IRMAR - Institut de Recherche Mathématique de Rennes - UR - Université de Rennes - INSA Rennes - Institut National des Sciences Appliquées - Rennes - INSA - Institut National des Sciences Appliquées - ENS Rennes - École normale supérieure - Rennes - UR2 - Université de Rennes 2 - CNRS - Centre National de la Recherche Scientifique - INSTITUT AGRO Agrocampus Ouest - Institut Agro - Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement)

  • Manuel Morales

    (LABEX ReFi - Laboratory of Excellence for Financial Regulation - UP11 - Université Paris-Sud - Paris 11)

  • Hassan Omidi Firouzi

    (LABEX ReFi - Laboratory of Excellence for Financial Regulation - UP11 - Université Paris-Sud - Paris 11)

Abstract

The field of risk theory has traditionally focused on ruin-related quantities. In particular, the socalled Expected Discounted Penalty Function 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, using recent results in fluctuation theory for Lévy processes, we 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 some examples of Lévy insurance risk processes for which they can be calculated, in particular for the classical Cramer-Lundberg model.

Suggested Citation

  • Zied Ben-Salah & Hélène Guérin & Manuel Morales & Hassan Omidi Firouzi, 2015. "On the Depletion Problem for an Insurance Risk Process: New Non-ruin Quantities in Collective Risk Theory," Post-Print hal-01044440, HAL.
    • Handle: RePEc:hal:journl:hal-01044440
    DOI: 10.1007/s13385-015-0112-9
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    Cited by:

    1. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    2. Ben Salah, Zied & Garrido, José, 2018. "On fair reinsurance premiums; Capital injections in a perturbed risk model," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 11-20.

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