Methodology and Computing in Applied Probability

, Volume 11, Issue 3, pp 307–338 | Cite as

Moment Bounds on Discrete Expected Stop-Loss Transforms, with Applications



This paper shows how to make the best possible use of the information contained in the first few moments (mean, variance and skewness, say) of an integer-valued random variable when one is interested in expected stop-loss transforms. This allows to bound various quantities in applied probability, including the ruin probabilities, for instance.


Increasing convex order Stop-loss transform Insurance 

AMS 2000 Subject Classification

60E15 60E10 91B30 


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  1. N. Brouhns, M. Denuit, and J. K. Vermunt, “A Poisson log-bilinear regression approach to the construction of projected lifetables,” Insurance: Mathematics and Economics vol. 31 pp. 373–393, 2002.MATHMathSciNetCrossRefGoogle Scholar
  2. C. Courtois, M. Denuit, and S. Van Bellegem, “Discrete s-convex extremal distributions: Theory and applications,” Applied Mathematics Letters vol. 19 pp. 1367–1377, 2006.MATHMathSciNetCrossRefGoogle Scholar
  3. M. Denuit, and J. Dhaene, “Comonotonic bounds on the survival probabilities in the Lee–Carter model for mortality projection,” Journal of Computational and Applied Mathematics vol. 203 pp. 169–176, 2006.MathSciNetCrossRefGoogle Scholar
  4. M. Denuit, and C. Lefèvre, “Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences,” Insurance: Mathematics and Economics vol. 20 pp. 197–214, 1997.MATHMathSciNetCrossRefGoogle Scholar
  5. M. Denuit, F. E. De Vijlder, and C. Lefèvre, “Extremal generators and extremal distributions for the continuous s-convex stochastic orderings,” Insurance: Mathematics and Economics vol. 24 pp. 201–217, 1999a.MATHMathSciNetCrossRefGoogle Scholar
  6. M. Denuit, C. Lefèvre, and M. Mesfioui, “On s-convex stochastic extrema for arithmetic risks,” Insurance: Mathematics and Economics vol. 25 pp. 143–155, 1999b.MATHMathSciNetCrossRefGoogle Scholar
  7. F. E. De Vijlder, Advanced Risk Theory. A Self-Contained Introduction. Editions de l’Université Libre de Bruxelles, Swiss Association of Actuaries: Bruxelles, 1996.Google Scholar
  8. H. U. Gerber, “Mathematical fun with the compound binomial process,” ASTIN Bulletin vol. 18 pp. 161–168, 1988.CrossRefGoogle Scholar
  9. W. Hürlimann, “Analytical bounds for two value-at-risk functionals,” ASTIN Bulletin vol. 32 pp. 235–265, 2002.MATHMathSciNetCrossRefGoogle Scholar
  10. W. Hürlimann, “Conditional value-at-risk bounds for compound Poisson risks and a normal approximation,” Journal of Applied Mathematics vol. 3 pp. 141–154, 2003.CrossRefGoogle Scholar
  11. W. Hürlimann, “Improved analytical bounds for gambler’s ruin probability,” Methodology and Computing in Applied Probability vol. 7 pp. 79–95, 2005.MATHMathSciNetCrossRefGoogle Scholar
  12. K. Jansen, J., Haezendonck, and M. J. Goovaerts, “Upper bounds on stop-loss premiums in case of known moments up to the fourth order,” Insurance: Mathematics and Economics vol. 5 pp. 315–334, 1986.MATHMathSciNetCrossRefGoogle Scholar
  13. E. S. W. Shiu, “The probability of eventual ruin in the compound binomial process,” ASTIN Bulletin vol. 19 pp. 179–190, 1989.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institut des Sciences ActuariellesUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Institut de StatistiqueUniversité catholique de LouvainLouvain-la-NeuveBelgium

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