The De Vylder-Goovaerts conjecture holds true within the diffusion limit

Stefan Ankirchner, Christophette Blanchet-Scalliet (christophette.blanchet@ec-lyon.fr) and Nabil Kazi-Tani (nabil.kazitani@gmail.com)
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Stefan Ankirchner: Institut für Mathematik - Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany]
Christophette Blanchet-Scalliet: PSPM - Probabilités, statistique, physique mathématique - ICJ - Institut Camille Jordan - ECL - École Centrale de Lyon - Université de Lyon - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - INSA Lyon - Institut National des Sciences Appliquées de Lyon - Université de Lyon - INSA - Institut National des Sciences Appliquées - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique
Nabil Kazi-Tani: LSAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon

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Abstract: The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite time ruin probability in a standard risk model is greater or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T. In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renor-malized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds true for the diffusion limits.

Keywords: Risk theory; Equalized claims; Ruin probability; Diffusion approximations (search for similar items in EconPapers)
Date: 2019-06
New Economics Papers: this item is included in nep-rmg
Note: View the original document on HAL open archive server: https://hal.science/hal-01887402v1
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Citations: View citations in EconPapers (2)

Published in Journal of Applied Probability, 2019, 56 (2), pp.546-557. ⟨10.1017/jpr.2019.33⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01887402

DOI: 10.1017/jpr.2019.33

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