Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T10:09:28.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Problèmes de ruine en théorie du risque à temps discret avec horizon fini

Part of: Applications General field theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Philippe Picard ,
Claude Lefèvre  and
Ibrahim Coulibaly
Philippe Picard*
Affiliation:
Université Claude Bernard Lyon 1
Claude Lefèvre*
Affiliation:
ET
Ibrahim Coulibaly*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Institut de Science Financière et d'Assurances, Université Claude Bernard Lyon 1, 21 Avenue Claude Bernard, F-69622 Villeurbanne Cedex, France
∗∗Postal address: Institut de Statistique et de Recherche Opérationnelle, CP 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgium
∗∗Postal address: Institut de Statistique et de Recherche Opérationnelle, CP 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgium
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a discrete-time risk model which describes the evolution of the reserves of an insurance company at periodic dates fixed in advance. The amount of loss per unit of time corresponds to independent and identically distributed random variables with arithmetic distribution, and the process of the receipt of premiums is assumed to be deterministic, nonnegative but not uniform (instead of being constant and equal to 1 as in the standard, compound binomial model). For this model, we determine the probability of ruin (or of non-ruin), as well as the distribution of the severity of the eventual ruin, with some finite horizon. A compact and efficient exact expression is found by bringing up-to-date a generalised family of Appell polynomials. The method used is illustrated with some numerical examples.

Résumé

Résumé

Ce travail concerne un modèle de risque à temps discret qui décrit l'évolution des réserves d'une compagnie d'assurances à des dates périodiques fixées d'avance. Les montants des sinistres par unité de temps correspondent à des variables aléatoires indépendantes et identiquement distribuées de distribution arithmétique, et le processus de rentrée des primes est supposé déterministe, non négatif mais non uniforme (au lieu d'être constant et égal à 1 comme dans le modèle standard dit binomial composé). On se propose de déterminer pour ce modèle la probabilité de ruine (ou de non ruine), ainsi que la distribution de la sévérité de la ruine éventuelle, sur un horizon fini quelconque. Une expression exacte, compacte et efficace, est obtenue en mettant à jour une structure de famille généralisée de polynômes d'Appell. La méthodologie suivie est illustrée par quelques exemples numériques.

Keywords

Processus des primes non uniformedistribution des sinistres arithmétiquepolynômes d'Appell généralisésprobabilité et sévérité de la ruineNonuniform process of premiumsarithmetic distribution of lossesgeneralised Appell polynomialsprobability and severity of ruin

MSC classification

Primary: 62P05: Applications to actuarial sciences and financial mathematics 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 12E05: Polynomials (irreducibility, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 527 - 542
Copyright
Copyright © Applied Probability Trust 2003 

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

Cheng, S., Gerber, H. U. et Shiu, E. S. W. (2000). Discounted probabilities and ruin theory in the compound binomial model. Insurance Math. Econom. 26, 239–252.Google Scholar
De Vylder, F. E. (1996). Advanced Risk Theory: A Self-Contained Introduction. Editions de l'Université Libre de Bruxelles.Google Scholar
De Vylder, F. E. (1999). Numerical finite-time ruin probabilities by the Picard-Lefèvre formula. Scand. Actuarial J. 1999, 97–105.Google Scholar
De Vylder, F. E. et Marceau, E. (1996). Classical numerical ruin probabilities. Scand. Actuarial J. 1996, 109–123.CrossRefGoogle Scholar
Dickson, D. C. M. (1994). Some comments on the compound binomial model. ASTIN Bull. 24, 33–45.Google Scholar
Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory (S. S. Huebner Foundation Monog. Ser. 8). University of Pennsylvania, Philadelphia, PA.Google Scholar
Gerber, H. U. (1988). Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161–168.CrossRefGoogle Scholar
Ignatov, Z. G., Kaishev, V. K. et Krachunov, R. S. (2001). An improved finite-time ruin probability formula and its Mathematica implementation. Insurance Math. Econom. 29, 375–386.Google Scholar
Picard, Ph. et Lefèvre, Cl. (1996). First crossing of basic counting processes with lower non-linear boundaries: a unified approach through pseudopolynomials (I). Adv. Appl. Prob. 28, 853–876.CrossRefGoogle Scholar
Picard, Ph. et Lefèvre, Cl. (1997). The probability of ruin in finite time with discrete claim size distribution. Scand. Actuarial J. 1997, 58–69.Google Scholar
Picard, Ph., Lefèvre, Cl., et Coulibaly, I. (2002). Multirisks model and finite-time ruin probabilities. Prépublication 185, ISRO, Université Libre de Bruxelles.Google Scholar
Reinhard, J.-M. (1997). On the probability and severity of ruin in the discrete risk model. Prépublication 74, ISRO, Université Libre de Bruxelles.Google Scholar
Shiu, E. S. W. (1989). The probability of eventual ruin in the compound binomial model. ASTIN Bull. 19, 179–190.Google Scholar
Willmot, G. E. (1993). Ruin probabilities in the compound binomial model. Insurance Math. Econom. 12, 133–142.Google Scholar
Yuen, K. C. et Guo, J. Y. (2001). Ruin probabilities for time-correlated claims in the compound binomial model. Insurance Math. Econom. 29, 47–57.Google Scholar