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Recursive calculation of the probability and severity of ruin. (English) Zbl 0682.62083

Summary: In the classic model of collective risk theory, let the function G(u,y) be the probability that ruin occurs from initial reserve level u, and that the deficit at the time of ruin is less than y. In two recent papers [H. U. Gerber, M. J. Goovaerts and R. Kaas, Astin Bull. 17, 151-163 (1987); H. U. Gerber and F. Dufresne, Insur. Math. Econ. 7, No.2, 75-80 (1988; Zbl 0637.62101)] explicit solutions for G(u,y) have been found when the claim amount distribution is a combination of exponential or gamma distributions.
In this paper we consider an alternative approach to the problem of calculating G(u,y) by deriving an equation which can be used to calculate approximate values of G(u,y) by a recursive method.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics

Citations:

Zbl 0637.62101
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References:

[1] Bowers, N. L.; Gerber, H. U.; Hickman, J. C.; Jones, D. A.; Nesbitt, C. J., Actuarial Mathematics (1987), Society of Actuaries: Society of Actuaries Itasca, IL
[2] Dufresne, F.; Gerber, H. U., The probability and severity of ruin for combinations of exponential claim amounts and their translations, Insurance: Mathematics and Economics, 7, 75-80 (1988) · Zbl 0637.62101
[3] Goovaerts, M. J.; De Vylder, F. A., A stable algorithm to compute the probability of ultimate ruin, Astin Bulletin, 14, 53-59 (1984)
[4] Gerber, H. U.; Goovaerts, M. J.; Kaas, R., On the probability and severity of ruin, Astin Bulletin, 17, 151-163 (1987)
[5] Panjer, H. H., Direct calculation of ruin probabilities, The Journal of Risk and Insurance, 53, 521-529 (1986)
[6] Segerdahl, C.-O., A survey of results in the collective theory of risk, Probability and Statistics: The Harald Cramer Volume (1959), Almqvist and Wiksell: Almqvist and Wiksell Uppsala · Zbl 0122.15501
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