Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-06T11:47:43.392Z Has data issue: false hasContentIssue false

Some Numerical Aspects in Transient Risk Theory*

Published online by Cambridge University Press:  29 August 2014

J. Janssen
Affiliation:
Université Libre de Bruxelles
Ph. Delfosse
Affiliation:
Royale Belge
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.

We give some actual possibilities for computing numerical values in the classical risk models both in transient and asymptotical cases by introducing the concept of normed model. Some recent approximations are tested on numerical examples.

We also emphasize the interest of these methods to compute waiting time distributions (transient and stationary cases) in queueing theory.

Type
Research Article
Copyright
Copyright © International Actuarial Association 1982

Footnotes

*

Presented at the 16th Astin Colloquium, September 27–30th, 1982, Liège, Belgium.

References

REFERENCES

Arfwedson, G. (1950) Some Problems in the Collective Theory of Risk. Skand. Aktuartidskr. 138.Google Scholar
Arfwedson, G. (1954) Research in Collective Risk Theory. Part I. Skand. Aktuartidskr. 191223.Google Scholar
Baxter, G. and Donsker, M. (1957) On the Distribution of the Supremum Functional for Process with Independent Increments. Transactions of the American Mathematical Society 85, 7387.CrossRefGoogle Scholar
Beekman, John A. (1966) Research on the Collective Risk Stochastic Process. Skand. Aktuartidskr. 6577.Google Scholar
Beekman, John A. and Bowers, Newton L. Jr. (1972) An Approximation to the Finite Time Ruin Function, Part II. Skand. Aktuartidskr. 128137.Google Scholar
Bohman, M. (1971) The Ruin Probability in a Special Case. ASTIN Bull. 6, 6668.CrossRefGoogle Scholar
Bohman, M. and Esscher, F. (1963) Studies in Risk Theory with Numerical Illustrations concerning Distribution Functions and Stop-less Premiums. Skand. Aktuartidskr. 1963 (3–4) and 1964 (1–2).Google Scholar
Cramér, M. (1955) Collective Risk Theory. Jubilee Volume of Försäkringsaktiebogalet Skandia.Google Scholar
Delfosse, Ph. (1980) Calcul de la Probabilité de Non-ruine en Théorie Collective du Risque dans le cas transitoire pour le modèle M/G/1, Mémoire de Licence en Sciences Actuarielles, Université Libre de Bruxelles.Google Scholar
De Vylder, F. (1978) A Practical Solution to the Problem of Ultimate Ruin Probability. Scand. Actuarial Journall 782, 114119.CrossRefGoogle Scholar
Gerber, H. (1973) Martingales in Risk Theory, Mit. Schw. Vers. Mat. 73 (2), 205215.Google Scholar
Gerber, H. (1979) An Introduction to Mathematical Risk Theory. Wharton School, University of Pennsylvania, Philadelphia.Google Scholar
Janssen, J. (1977) The Semi-Markov Model in Risk Theory. In Advances in Operational Research; edited by Roubens, M.. North-Holland: Amsterdam.Google Scholar
Pesonen, E. (1975) N · P-Technique as a Tool in Decision Making. ASTIN Bull. 8, 359363.CrossRefGoogle Scholar
Pfenninger, F. (1974) Eine Neue Methode zur Berechnung der Ruinwahrscheinlichkeit mittels Laguerre-Entwicklung. DGVM-Blätter Band XI Heft 4.Google Scholar
Piessens, R. (1969) New Quadrature Formulas for the Numerical Invasion of the Laplace Transform. Nordisk Tidskrift for Informations behandeling (BIT) 9, 351361.Google Scholar
Prabhu, N. U. (1961) On the Ruin Problem of Collective Risk Theory. Ann. Math. Statist. 32, 757764.CrossRefGoogle Scholar
Seal, H. L. (1969) Stochastic Theory of a Risk Business. J. Wiley: New York.Google Scholar
Seal, H. L. (1974) The Numerical Calculation of U(w, t) the Probability of Non-Ruin in an Interval (0, t). Scand. Actuarial Journal 121129.CrossRefGoogle Scholar
Seal, H. L. (1978) Survival Probabilities—The Goal of Risk Theory. J. Wiley: Chichester-New-York-Brisbane-Toronto.Google Scholar
Stroeymeyt, Ph. (1977) Calcul numérique de la probabilité de non-ruine dans les cas transitoire et asymptotique. Mémoire de Licence en Sciences Actuarielles, Université Libre de Bruxelles.Google Scholar
Stroud, A. H. and Secrest, D. (1966) Gaussian Quadrature Formules. Prentice Hall Inc: Englewood Cliffs, New Jersey.Google Scholar
Taylor, G. C. (1978) Representation and Explicit Calculation of Finite Time Ruin Probabilities. Scand. Actuarial Journal 78 (1), 118.CrossRefGoogle Scholar
Thorin, O. (1968) An Identity in the Collective Risk Theory with Some Applications. Skand. Aktuartidskr. 2644.Google Scholar
Thorin, O. (1970) Some Remarks on the Ruin Problem in Case the Epochs of Claims Form a Renewal Process. Skand. Aktuartidskr. 2950.Google Scholar
Thorin, O. (1977) Ruin Probabilities Prepared for Numerical Calculation. Scand. Actuarial Journal. Suppl. 717.CrossRefGoogle Scholar
Thorin, O. and Wikstad, N. (1977) Calculation of Ruin Probabilities When the Claim Distribution is Lognormal. ASTN Bull. 9, 231246.CrossRefGoogle Scholar
Wikstad, N. (1977) How to Catalogue Ruin Probabilities According to the Classical Risk Theory. Scand. Actuarial Journal Suppl. 1924.CrossRefGoogle Scholar