Statistique - Probabilités - Optimisation et Contrôle

An open problem in ruin theory and its diffusion approximation regime

by Christophette Blanchet

Europe/Paris
Description

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 the 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 talk, we will consider the diffusion approximations of both the standard risk model and the associated risk model. We will prove that the associated model, when conveniently renormalized, converges in distribution to a gaussian process satisfying a simple SDE with explicit coefficients. We will 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, which is well known. We will then conclude that the De Vylder and Goovaerts conjecture holds true for these diffusion limits. This is a joint work with Stefan Ankirchner (University of Jena) and Nabil Kazi-Tani (Université Lyon 1 ISFA).

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