Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T17:15:39.088Z Has data issue: false hasContentIssue false
  • English
  • Français

The De Vylder–Goovaerts conjecture holds within the diffusion limit

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

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ankirchner, S., Dereich, S. and Imkeller, P. (2006). The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Prob. 34, 743–778.10.1214/009117905000000648CrossRefGoogle Scholar
Asmussen, S. (1984). Approximations for the probability of ruin within finite time. Scand. Actuar. J. 1984, 31–57.10.1080/03461238.1984.10413752CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. World Scientific.10.1142/7431CrossRefGoogle Scholar
Atkinson, M.P. and Singham, D. I. (2015). Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries. Statist. Probab. Lett. 100, 85–92.10.1016/j.spl.2015.02.006CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn (Wiley Series in Probability and Statistics: Probability and Statistics). John Wiley.10.1002/9780470316962CrossRefGoogle Scholar
Furrer, H. (1997). Risk theory and heavy-tailed Lévy processes. PhD thesis, ETH, Zurich.Google Scholar
Furrer, H., Michna, Z. and Weron, A. (1997). Stable Lévy motion approximation in collective risk theory. Insurance Math. Econom. 20, 97–114.10.1016/S0167-6687(97)00008-5CrossRefGoogle Scholar
Grandell, J. (1977). A class of approximations of ruin probabilities. Scand. Actuar. J. 1977, 37–52.10.1080/03461238.1977.10405071CrossRefGoogle Scholar
Iglehart, L. D. (1969). Diffusion approximations in collective risk theory. J. Appl. Prob. 6, 285–292.10.2307/3211999CrossRefGoogle Scholar
Lefèvre, C. and Picard, P. (2011). A new look at the homogeneous risk model. Insurance Math. Econom. 49, 512–519.10.1016/j.insmatheco.2011.08.005CrossRefGoogle Scholar
Revuz, D. and Yor, M. (2004). Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften). Springer, Berlin and Heidelberg.Google Scholar
Robert, C. Y. (2014). On the De Vylder and Goovaerts conjecture about ruin for equalized claims. J. Appl. Prob. 51, 874–879.10.1239/jap/1409932679CrossRefGoogle Scholar
Scheike, T. H. (1992). A boundary-crossing result for Brownian motion. J. Appl. Prob. 29, 448–453.10.2307/3214581CrossRefGoogle Scholar
Vylder, F.D. and Goovaerts, M. (2000). Homogeneous risk models with equalized claim amounts. Insurance Math. Econom. 26, 223–238.10.1016/S0167-6687(99)00055-4CrossRefGoogle Scholar
Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5, 67–85.10.1287/moor.5.1.67CrossRefGoogle Scholar