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

Stefan Ankirchner (1) , Christophette Blanchet-Scalliet (2) , Nabil Kazi-Tani (3)

1 Institut für Mathematik
2 PSPM - Probabilités, statistique, physique mathématique
3 SAF - Laboratoire de Sciences Actuarielle et Financière

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.

Mots clés: Risk theory; Equalized claims; Ruin probability; Diffusion approximations

Domaines: Probabilités [math.PR]; Gestion des risques [q-fin.RM]

Stefan Ankirchner, Christophette Blanchet-Scalliet, Nabil Kazi-Tani. The De Vylder-Goovaerts conjecture holds true within the diffusion limit. Journal of Applied Probability, 2019, 56 (2), pp.546-557. ⟨10.1017/jpr.2019.33⟩. ⟨hal-01887402⟩