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Bounds for classical ruin probabilities. (English) Zbl 0547.62068

This paper derives upper and lower bounds for the ruin probability over infinite time. The key observation is that if \(u=k*(1-u),\) then \(v-u=(v- k*(1-v))*(1-u),\) where \((f*g)(x)=\int^{x}_{0}f(x-y)dg(y)\). Applications to sub-exponential distributions are also given.
Reviewer: E.Shiu

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Athreya, K. B.; Ney, P., Branching Processes (1972), Springer: Springer Berlin-Heidelberg-New York · Zbl 0259.60002
[2] Chistyakov, V. P., A theorem on sums of independent random variables and its applications to branching processes, Theor. Probab. Appl., 9, 640-648 (1964) · Zbl 0203.19401
[3] Embrechts, P.; Goldie, C. M.; Veraverbeke, N., Subexponentially and infinite divisibility, Z. Wahrsch. Verw. Geb., 49, 335-347 (1979) · Zbl 0397.60024
[4] Embrechts, P.; Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance Math. Econom., 1, 1, 55-72 (1982) · Zbl 0518.62083
[5] Taylor, G. C., Use of differential and integral inequalities to bound ruin and queuing probabilities, Scand. Actuarial J., 197-208 (1976) · Zbl 0338.60005
[6] Teugels, J. L., The class of subexponential distributions, Ann. Probab., 3, 1000-1011 (1975) · Zbl 0374.60022
[7] Veraverbeke, N., Asymptotic behaviour of Wiener-Hopf factors of a random walk, Stochastic Process. Appl., 5, 27-37 (1977) · Zbl 0353.60073
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