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The De Vylder–Goovaerts conjecture holds within the diffusion limit

Part of: Probability theory on algebraic and topological structures Limit theorems Applications

Published online by Cambridge University Press:  30 July 2019

Stefan Ankirchner ,
Christophette Blanchet-Scalliet  and
Nabil Kazi-Tani
Stefan Ankirchner*
Affiliation:
University of Jena
Christophette Blanchet-Scalliet*
Affiliation:
Université de Lyon
Nabil Kazi-Tani*
Affiliation:
Université de Lyon
*
*Postal address: Institute for Mathematics, University of Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany.
**Postal address: Institut Camille Jordan – Ecole Centrale de Lyon, CNRS UMR 5208, Université de Lyon, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France.
***Postal address: Laboratoire SAF, ISFA, Université de Lyon, 50 Avenue Tony Garnier, 69366 Lyon Cedex 07, France.
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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 than 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 renormalized, 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 for the diffusion limits.

Keywords

Risk theoryruin probabilityequalized claimdiffusion approximation

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 60B10: Convergence of probability measures
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 546 - 557
Copyright
© Applied Probability Trust 2019 

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