Abstract
Ruin probability, being one of the important goal functions in collective risk theory, has received considerable attention during the last few years. In this chapter, we treat the following fairly new topics. First, we find the initial capital securing a prescribed risk level when the relative safety loading tends to O. Second, we derive two-sided bounds of ruin probability in the cases where claim sizes have light and heavy tails. Third, we obtain continuity estimates for ruin probabilities with respect to perturbations of governing parameters of the surplus process. All considerations use a representation of ruin probability as the distribution of a geometric sum and the results of Chapters 3 to 5.
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Commentaries
Section 1
Asmussen, S. (1987) Applied Probability and Queues. J. Wiley & Sons, Chichester.
Grandell, J. (1991) Aspects of Risk Theory. Springer—Verlag, New York.
Kalashnikov, V. and Konstantinidis D. (1996) Ruin probability. Fundamental’naya i Prikladnaya Mathematika, 2, 1055–1100 (in Russian).
Nagaev, A. (1996) Limiting distribution for the extreme point visited by random walk and its application to risk theory. Private communication.
Kalashnikov, V. (1996a) Two-sided bounds of ruin probabilities. Scand. Actuarial J., No 1, 1–18.
Kalashnikov, V. (1993) Two-side estimates of geometric convolutions. LN in Math., 1546, 76–88. Springer-Verlag, Berlin.
Kalashnikov, V. (1995) Bounds for Geometric Sums in the Presence of Heavy-Tailed Summands. Prépublications de l’Equipe d’Analyse et de Mathématiques Appliquées, Universite de Marne-La-Vallee, No 27, Septembre 1995.
Kalashnikov, V. and Konstantinidis D. (1996) Ruin probability. Fundamental’naya i Prikladnaya Mathematika, 2, 1055–1100 (in Russian).
Section 2
Nagaev, A. (1996) Limiting distribution for the extreme point visited by random walk and its application to risk theory. Private communication.
Section 3
Kalashnikov, V. (1996a) Two-sided bounds of ruin probabilities. Scand. Actuarial J., No 1, 1–18.
Kalashnikov, V. (1995) Bounds for Geometric Sums in the Presence of Heavy-Tailed Summands. Prépublications de l’Equipe d’Analyse et de Mathématiques Appliquées, Universite de Marne-La-Vallee, No 27, Septembre 1995.
Kalashnikov, V. and Konstantinidis D. (1996) Ruin probability. Fundamental’naya i Prikladnaya Mathematika, 2, 1055–1100 (in Russian).
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© 1997 Springer Science+Business Media Dordrecht
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Kalashnikov, V. (1997). Ruin Probability. In: Geometric Sums: Bounds for Rare Events with Applications. Mathematics and Its Applications, vol 413. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1693-2_6
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DOI: https://doi.org/10.1007/978-94-017-1693-2_6
Publisher Name: Springer, Dordrecht
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