Discrete Compound Poisson Process with Curved Boundaries: Polynomial Structures and Recursions

  • Claude LefèvreEmail author


This paper provides a review of recent results, most of them published jointly with Ph. Picard, on the exact distribution of the first crossing of a Poisson or discrete compound Poisson process through a given nondecreasing boundary, of curved or linear shape. The key point consists in using an underlying polynomial structure to describe the distribution, the polynomials being of generalized Appell type for an upper boundary and of generalized Abel–Gontcharoff type for a lower boundary. That property allows us to obtain simple and efficient recursions for the numerical determination of the distribution.


first crossing time compound Poisson process order statistics generalized Appell polynomials generalized Abel–Gontcharoff polynomials recursive methods dam modelling risk theory 

AMS 2000 Subject Classification

60G40 12E10 62P05 


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  1. Appell, P.E.: "Sur une classe de polynômes," Annales des Sciences de l’Ecole Normale Supérieure vol. 9 pp. 119–144, (1880).Google Scholar
  2. F. G. Ball and P. O’Neill, “The distribution of general final state random variables for stochastic epidemic models,” Journal of Applied Probability vol. 36 pp. 473–491, 1999.Google Scholar
  3. R. P. Boas Jr. and R. C. Buck, Polynomials Expansions of Analytic Functions, Springer: Berlin Heidelberg New York, 1958.Google Scholar
  4. P.C. Consul, Generalized Poisson Distributions Properties and Applications, Dekker: New York, 1989.zbMATHGoogle Scholar
  5. H. E. Daniels, “The Poisson process with a curved absorbing boundary,” Bulletin of the International Statistical Institute vol. 40 pp. 994–1008, 1963.Google Scholar
  6. H.E. Daniels, "The distribution of the total size of an epidemic," Proceedings of the Fifth Berkeley Symposium on Mathematics and Statistical Problems vol. 4 pp. 281–293, 1967.Google Scholar
  7. M. Denuit, Cl. Lefèvre and Ph. Picard, “Polynomial structures in order statistics distributions,” Journal of Statistical Planning and Inference vol. 113 pp. 151–178, 2003.Google Scholar
  8. F. E. De Vylder, “Numerical finite-time ruin probabilities by the Picard-Lefèvre formula,” Scandinavian Actuarial Journal vol. 2 pp. 97–105, 1999.Google Scholar
  9. S.F.L. Gallot, "Absorption and first-passage times for a compound Poisson process in a general upper boundary," Journal of Applied Probability vol. 30 pp. 835–850, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  10. W. Gontcharoff, Détermination des Fonctions Entières par Interpolation, Hermann: Paris, 1937.Google Scholar
  11. Z.G. Ignatov and V.K. Kaishev, "A finite-time ruin probability formula for continuous claim severities,Journal of Applied Probability vol. 41 pp. 570–578, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Z. G. Ignatov, V. K. Kaishev and R. S. Krachunov, “An improved finite-time ruin probability formula and its Mathematica implementation,” Insurance: Mathematics and Economics vol. 29 pp. 375–386, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Y. A. Kaz’min, “Appell polynomials.” In M. Hazewinkel (ed.), Encyclopedia of Mathematics, vol. 1 pp. 209–210, Kluwer: Dordrecht, 1988.Google Scholar
  14. Cl. Lefèvre and Ph. Picard, “A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes,” Advances in Applied Probability vol. 22 pp. 25–48, 1990.Google Scholar
  15. Cl. Lefèvre and Ph. Picard, “On the first-crossing of a Poisson process in a lower boundary.” In C. C. Heyde, Yu. V. Prohorov, R. Pyke, S. T. Rachev (eds.), Athens Conference on Applied Probability and Time Series Analysis, Volume 1: Applied Probability in Honor of J.M. Gani, pp. 159–175, Springer: New York, 1996.Google Scholar
  16. Cl. Lefèvre and Ph. Picard, “Abel-Gontcharoff pseudopolynomials and the exact final outcome of SIR epidemic models (III),” Advances in Applied Probability vol. 31 pp. 532–549, 1999.Google Scholar
  17. Cl. Lefèvre and Ph. Picard, “A nonhomogeneous risk model for insurance,” Computers and Mathematics with Applications vol. 51 pp. 325–334, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  18. R. Mullin and G.-C. Rota, “On the foundation of combinatorial theory—III: theory of binomial enumeration.” In Harris (ed.), Graph Theory and its Applications, pp. 167–213, Academic: New York, 1970.Google Scholar
  19. H. Niederhausen, "Sheffer polynomials for computing exact Kolmogorov-Smirnov and Rényi type distributions," Annals of Statistics vol. 9 pp. 923–944, 1981.zbMATHMathSciNetGoogle Scholar
  20. H. H. Panjer and G. E. Willmot, Insurance Risk Models, Society of Actuaries: Schaumburg, 1992.Google Scholar
  21. D. Perry, D., W. Stadje and S. Zacks, "Contributions to the theory of first-exit times of some compound processes in queueing theory," Queueing Systems vol. 33 pp. 369–379, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  22. Ph. Picard and Cl. Lefèvre, “First crossing of basic counting processes with lower non-linear bundaries: a unified approach through pseudopolynomials (I),” Advances in Applied Probability vol. 28 pp. 853–876, 1996.Google Scholar
  23. Ph. Picard and Cl. Lefèvre, “The probability of ruin in finite time with discrete claim size distribution,” Scandinavian Actuarial Journal vol. 1 pp. 58–69, 1997.Google Scholar
  24. Ph. Picard and Cl. Lefèvre, “On the first meeting or crossing of two independent trajectories for some counting processes,” Stochastic Processes and their Applications vol. 104 pp. 217–242, 2003.Google Scholar
  25. Ph. Picard, Cl. Lefèvre and I. Coulibaly, “Problèmes de ruine en théorie du risque à temps discret avec horizon fini,” Journal of Applied Probability vol. 40 pp. 527–542, 2003.Google Scholar
  26. R. Pyke, "The supremum and infimum of the Poisson process," Annals of Mathematical Statistics vol. 30 pp. 568–576, 1959.MathSciNetGoogle Scholar
  27. I. M. Sheffer, "Some properties of polynomials of degree zero," Duke Mathematical Journal vol. 5 pp. 590–622, 1939.zbMATHCrossRefMathSciNetGoogle Scholar
  28. W. Stadje, "Distributions of first exit times for empirical counting and Poisson processes with moving boundaries," Communications in Statistics—Stochastic Models vol. 9 pp. 91–103, 1993.zbMATHMathSciNetGoogle Scholar
  29. W. Stadje and S. Zacks, "Upper first-exit times of compound Poisson processes revisited," Probability in the Engineering and Informational Sciences vol. 17 pp. 459–465, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  30. S. Zacks, "Distributions of stopping times for Poisson processes with linear boundaries," Communications in Statistics—Stochastic Models vol. 7 pp. 233–242, 1991.zbMATHMathSciNetGoogle Scholar
  31. S. Zacks, “Distributions of first exit times for Poisson processes with lower and upper linear boundaries.” In N. L. Johnson and N. Balakrishnan (eds.), A Volume in Honor of Samuel Kotz, pp. 339–350, Wiley: New York, 1997.Google Scholar
  32. S. Zacks, D. Perry, D. Bshouty and S. Bar-Lev, "Distributions of stopping times for compound Poisson processes with positive jumps and linear boundaries," Communications in Statistics — Stochastic Models vol. 15 pp. 89–101, 1999.zbMATHMathSciNetGoogle Scholar
  33. S. Zacks, "Some recent results on the distributions of stopping times of compound Poisson processes with linear boundaries," Journal of Statistical Planning and Inference vol. 130 pp. 95–109, 2005.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium

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