Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-14T10:34:20.531Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytical steps towards a numerical calculation of the ruin probability for a finite period when the riskprocess is of the Poisson type or of the more general type studied by Sparre Andersen*

Published online by Cambridge University Press:  29 August 2014

Olof Thorin
Olof Thorin*
Affiliation:
Stockholm
Rights & Permissions [Opens in a new window]

Extract

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.

As is well-known, in the early 60's a Swedish committee set to work at the numerical calculation of the distribution function of the total amount of claims and of the related stop loss premiums in the Poisson and Polya cases (Bohman and Esscher [6]). Since the characteristic function for the said distribution function was easily available in terms of the characteristic function for the distribution function of an individual claim, the committee chose to base the numerical calculations on the C-method by H. Bohman (Bohman [5]). The calculation of the ruin probability for a finite or infinite period was not considered by the committee.

The last-mentioned problem has now been taken up by a new committee formed by the Swedish Council for Actuarial Science and Insurance Statistics. The committee—consisting of H. Bohman, J. Jung, N. Wikstad and the present author—has to consider several aspects of the practical applicability of the collective risk theory. However, without possibilities of calculating—at least approximately—the ruin probability for a finite period the applicability of the existing ruin theories seems to be rather limited, so the committee has looked around for such possibilities. At the present stage the committee is considering the classical Poisson theory and Sparre Andersen's generalization of this theory [2]. It is the hope of the committee that, at a later stage, also the Polya theory and the theory recently presented by Segerdahl [11] combining the Sparre Andersen theory and the Polya theory may be treated.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 6 , Issue 1 , September 1971 , pp. 54 - 65
Copyright
Copyright © International Actuarial Association 1971

Footnotes

*

A paper presented to the 9th Astin Colloquium (Randers 1970) appearing in this issue for strictly technical reasons.

References

[1]Almer, B. 1957. Risk analysis in theory and practical statistics. Transactions XVth International Congress of Actuaries, New York, vol. II, 314–370.Google Scholar
[2]Andersen, E. Sparre. 1957. On the- collective theory of risk in the case of contagion between the claims. Transactions XVth International Congres of Actuaries, New York, vol. II, 219–229.Google Scholar
[3]Arfwedson, G. 1950. Some Problems in the Collective Theory of Risk. Skandinavisk Aktuarietidskrift, Nos. 1-2, 1–38.Google Scholar
[4]Beekman, J. 1966. Research on the collective risk stochastic process. Skandinavisk Aktuarietidskrift, Nos. 1-2, 65–77.Google Scholar
[5]Bohman, H. 1963. To Compute the Distribution Function when the Characteristic Function is Known. Skandinavisk Aktuarietidskrift, Nos. 1-2, 41–46.Google Scholar
[6]Bohman, H. & ESSCHER, F. 1963-1964. Studies in Risk Theory with Numerical Illustrations Concerning Distribution Functions and Stop Loss Premiums 1-11. Scandinavisk Aktuarietidskrift, 1963 Nos. 3-4, 173–225, 1964 Nos. 1-2, 1-40.Google Scholar
[7]Cox, D. R. & Miller, H. D. 1965. The Theory of Stochastic Processes. Methuen.Google Scholar
[8]Cramér, H. 1955. Collective risk theory. Jubilee volume of Försäkringsaktiebelaget Skandia.Google Scholar
[9]Feller, W. 1966. An Introduction to Probability Theory and Its Applications. Vol. II. Wiley.Google Scholar
[10]Philipson, C. 1963. On the numerical calculation of the distribution functions defining some compound Poisson processes. The ASTIN Bulletin, vol. III part. I, 20–42.CrossRefGoogle Scholar
[11]Segerdahl, C. O. 1970. Stochastic processes and practical working models or Why is the Polya process approach defective in modern practice and how cope with its deficiences?Institute of Mathematical Statistics and Actuarial Mathematics, Stockholm. Research Report no. 48.Google Scholar
[12]Thorin, O. 1968. An Identity in the Collective Risk Theory with some Applications. Skandinavisk Aktuarietidskrift, Nos. 1-2, 26–44.Google Scholar
[13]Thorin, O. 1969. An outline of a generalization—started by Sparre Andersen—of the classical ruin theory. Report to the ASTIN-Colloquium at Sopot.*Google Scholar
[14]Thorin, O. 1970. Some remarks on the ruin problem in case the epochs of claims form a renewal process. Skandinavisk Aktuarietidskrift, forthcoming paper.Google Scholar
[15]Thorin, O. 1971. Further remarks on the ruin problem in case the epochs of claims form a renewal process. Skandinavisk Aktuarietidskrift, forthcoming paper. * Added in proof: After that the present paper was read at the ASTIN-Colloquium at Randers 1970, the paper [14] has been published in Skandinavisk Aktuarietidskrift 1970, Nos. 1-2, 29-50. With the latter paper in hand it is not necessary to consult the paper [13].Google Scholar