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On the Stability of Recursive Formulas

Published online by Cambridge University Press:  29 August 2014

Harry H. Panjer  and
Shaun Wang
Harry H. Panjer*
Affiliation:
University of Waterloo, Ontario, Canada
Shaun Wang*
Affiliation:
University of Waterloo, Ontario, Canada
*
Faculty of Mathematics, Institute of Insurance & Pension Research, University of Waterloo, Waterloo Ontario, N2L 3G1, Canada.
Faculty of Mathematics, Institute of Insurance & Pension Research, University of Waterloo, Waterloo Ontario, N2L 3G1, Canada.
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Abstract

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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.

Keywords

Recursive formulacompound distributionprobability of ruindominant solutionsubordinate solutioncongruent recursionindex of error propagationstablestrongly stablestrongly unstablerelative error analysisempirical inflation factor
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 23 , Issue 2 , November 1993 , pp. 227 - 258
Copyright
Copyright © International Actuarial Association 1993

References

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