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On the Stability of Recursive Formulas

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  • H. Panjer, Harry
  • Shaun Wang,

Abstract

Based on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by Panjer (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

Suggested Citation

  • H. Panjer, Harry & Shaun Wang,, 1993. "On the Stability of Recursive Formulas," ASTIN Bulletin, Cambridge University Press, vol. 23(2), pages 227-258, November.
    • Handle: RePEc:cup:astinb:v:23:y:1993:i:02:p:227-258_01
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    Cited by:

    1. Rulliere, Didier & Loisel, Stephane, 2004. "Another look at the Picard-Lefevre formula for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 187-203, October.
    2. Sundt, Bjorn, 2002. "Recursive evaluation of aggregate claims distributions," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 297-322, June.
    3. Mijatović, Aleksandar & Vidmar, Matija & Jacka, Saul, 2015. "Markov chain approximations to scale functions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3932-3957.
    4. Michel Denuit & Raluca Vernic, 2018. "Bivariate Bernoulli Weighted Sums and Distribution of Single-Period Tontine Benefits," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1403-1416, December.
    5. Franco-Arbeláez, Luis Ceferino & Franco-Ceballos, Luis Eduardo & Murillo-Gómez, Juan Guillermo & Venegas-Martínez, Francisco, 2015. "Riesgo operativo en el sector salud en Colombia [Operational Risk in the Health Sector in Colombia]," MPRA Paper 63149, University Library of Munich, Germany.
    6. Stefan Gerhold & Uwe Schmock & Richard Warnung, 2010. "A generalization of Panjer’s recursion and numerically stable risk aggregation," Finance and Stochastics, Springer, vol. 14(1), pages 81-128, January.
    7. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2009. "Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 374-381, December.
    8. Venegas-Martínez, Francisco & Franco-Arbeláez, Luis Ceferino & Franco-Ceballos, Luis Eduardo & Murillo-Gómez, Juan Guillermo, 2015. "Riesgo operativo en el sector salud en Colombia: 2013," eseconomía, Escuela Superior de Economía, Instituto Politécnico Nacional, vol. 0(43), pages 7-36, segundo s.
    9. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2008. "Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 746-762, April.
    10. Ambagaspitiya, R. S., 1995. "A family of discrete distributions," Insurance: Mathematics and Economics, Elsevier, vol. 16(2), pages 107-127, May.
    11. Muneya Matsui, 2017. "Prediction of Components in Random Sums," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 573-587, June.
    12. Michel Denuit, 2009. "Life Anuities with Stochastic Survival Probabilities: A Review," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 463-489, September.
    13. Ramsay, Colin M., 2003. "A solution to the ruin problem for Pareto distributions," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 109-116, August.
    14. Usabel, Miguel, 1999. "Calculating multivariate ruin probabilities via Gaver-Stehfest inversion technique," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 133-142, November.
    15. Shaun, Wang, 1995. "On two-sided compound binomial distributions," Insurance: Mathematics and Economics, Elsevier, vol. 17(1), pages 35-41, August.
    16. Gathy, Maude & Lefèvre, Claude, 2010. "On the Lagrangian Katz family of distributions as a claim frequency model," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 76-83, August.
    17. Paul Embrechts & Marco Frei, 2009. "Panjer recursion versus FFT for compound distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 497-508, July.
    18. Pavel V. Shevchenko, 2010. "Calculation of aggregate loss distributions," Papers 1008.1108, arXiv.org.

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