Skip to main content Accessibility help
×
Home

The De Vylder–Goovaerts conjecture holds within the diffusion limit

  • Stefan Ankirchner (a1), Christophette Blanchet-Scalliet (a2) and Nabil Kazi-Tani (a2)

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.

Copyright

Corresponding author

*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.

References

Hide All
[1]Ankirchner, S., Dereich, S. and Imkeller, P. (2006). The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Prob. 34, 743778.10.1214/009117905000000648
[2]Asmussen, S. (1984). Approximations for the probability of ruin within finite time. Scand. Actuar. J. 1984, 3157.10.1080/03461238.1984.10413752
[3]Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. World Scientific.10.1142/7431
[4]Atkinson, M.P. and Singham, D. I. (2015). Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries. Statist. Probab. Lett. 100, 8592.10.1016/j.spl.2015.02.006
[5]Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn (Wiley Series in Probability and Statistics: Probability and Statistics). John Wiley.10.1002/9780470316962
[6]Furrer, H. (1997). Risk theory and heavy-tailed Lévy processes. PhD thesis, ETH, Zurich.
[7]Furrer, H., Michna, Z. and Weron, A. (1997). Stable Lévy motion approximation in collective risk theory. Insurance Math. Econom. 20, 97114.10.1016/S0167-6687(97)00008-5
[8]Grandell, J. (1977). A class of approximations of ruin probabilities. Scand. Actuar. J. 1977, 3752.10.1080/03461238.1977.10405071
[9]Iglehart, L. D. (1969). Diffusion approximations in collective risk theory. J. Appl. Prob. 6, 285292.10.2307/3211999
[10]Lefèvre, C. and Picard, P. (2011). A new look at the homogeneous risk model. Insurance Math. Econom. 49, 512519.10.1016/j.insmatheco.2011.08.005
[11]Revuz, D. and Yor, M. (2004). Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften). Springer, Berlin and Heidelberg.
[12]Robert, C. Y. (2014). On the De Vylder and Goovaerts conjecture about ruin for equalized claims. J. Appl. Prob. 51, 874879.10.1239/jap/1409932679
[13]Scheike, T. H. (1992). A boundary-crossing result for Brownian motion. J. Appl. Prob. 29, 448453.10.2307/3214581
[14]Vylder, F.D. and Goovaerts, M. (2000). Homogeneous risk models with equalized claim amounts. Insurance Math. Econom. 26, 223238.10.1016/S0167-6687(99)00055-4
[15]Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5, 6785.10.1287/moor.5.1.67

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed