Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-19T07:20:07.219Z Has data issue: false hasContentIssue false
  • English
  • Français

The Probability of Eventual Ruin in the Compound Binomial Model

Published online by Cambridge University Press:  07 February 2018

Elias S.W. Shiu
Elias S.W. Shiu*
Affiliation:
University of Manitoba, Canada
*
Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper derives several formulas for the probability of eventual ruin in a discrete-time model. In this model, the number of claims process is assumed to be binomial. The claim amounts, premium rate and initial surplus are assumed to be integer-valued.

Keywords

Compound binomial processProbability of eventual ruinUltimate ruin probabilityInfinite-time ruin probabilityRisk theoryRandom walkGambler's ruinLagrange series
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 19 , Issue 2 , November 1989 , pp. 179 - 190
Copyright
Copyright © International Actuarial Association 1989

References

Burman, J. P. (1946) Sequential sampling formulae for a binomial population. Journal of the Royal Statistical Society B 8, 98–103.Google Scholar
Brown, A. L. and Page, A. (1970) Elements of Functional Analysis. Van Nostrand Reinhold, London.Google Scholar
Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht.Google Scholar
Dufresne, F. (1988) Distributions stationnaires d'un système bonus-malus et probabilité et ruine. ASTIN Bulletin 18, 31–46.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2 (2nd edn). Wiley, New York.Google Scholar
Gerber, H.U. (1988a) Mathematical fun with ruin theory. Insurance: Mathematics and Economics 7, 15–23.Google Scholar
Gerber, H. U. (1988b) Mathematical fun with the compound binomial process. ASTIN Bulletin 18, 161–168.CrossRefGoogle Scholar
Girshick, M.A. (1946) Contributions to the theory of sequential analysis, II, III. Annals of Mathematical Statistics 17, 282–298.Google Scholar
Henrici, P. (1974) Applied and Computational Complex Analysis, Vol. 1: Power Series – Integration – Conformed Mapping – Location of Zeros. Wiley, New York.Google Scholar
Hofri, M. (1987) Probabilistic Analysis of Algorithms: On Computing Methodologies for Computer Algorithms Performance Evaluation. Springer-Verlag, New York.Google Scholar
Knuth, D. E. (1973) The Art of Computer Programming, Vol. 1: Fundamental Algorithms (2nd edn). Addison-Wesley, Reading, Massachusetts.Google Scholar
Melzak, Z. A. (1973) Companion to Concrete Mathematics: Mathematical Techniques and Various Applications. Wiley, New York.Google Scholar
Pólya, G. (1922) Sur les séries entières dont la somme est une fonction algébrique. L'Enseignement Mathémalique 22, 38–47. Reprinted in George Pólya: Collected Papers, Vol. 1: Singularities of Analytic Functions, MIT Press, Cambridge, Massachusetts (1974), pp. 165-174.Google Scholar
Pólya, G. and Szegö, G. (1970) Aufgabe und Lehrsätze aus der Analysis, Vol. 1 (4th edn). Springer-Verlag, Berlin.CrossRefGoogle Scholar
Prabhu, N.U. (1965) Queues and Inventories: A Study of Their Basic Stochastic Processes. Wiley, New York.Google Scholar
Riesz, F. and Sz.-Nagy, B. (1955) Functional Analysis. Ungar, New York. An English translation of Leçons d'analyse fonctionelle (deuxième édition), Akadémiai Kiadó, Budapest (1953).Google Scholar
Riordan, J. (1968) Combinatorial Identities. Wiley, New York. Reprinted with correction by Krieger, Huntington, New York (1979).Google Scholar
Roman, S.M. and Rota, G.-C. (1978) The umbral calculus. Advances in Mathematics 27, 95–188.Google Scholar
Rota, G.-C. (1975) Finite Operator Calculus. Academic Press, New York.Google Scholar
Seal, H. L. (1962) The random walk of a simple risk business. ASTIN Bulletin 4, 19–28.Google Scholar
Seal, H. L. (1969) Stochastic Theory of a Risk Business. Wiley, New York.Google Scholar
Shiu, E.S.W. (1988) Calculation of the probability of eventual ruin by Beekman's convolution series. Insurance: Mathematics and Economics 7, 41–47.Google Scholar
Shiu, E.S.W. (1989a) Ruin probability by operational calculus. Insurance: Mathematics and Economics, forthcoming.Google Scholar
Shiu, E.S.W. (1989b) On Gerber's fun. Scandinavian Actuarial Journal, forthcoming.CrossRefGoogle Scholar
Whittaker, E.T. and Watson, G.N. (1927) A Course of Modern Analysis (4th edn). Cambridge University Press, London.Google Scholar