Ruin estimates under interest force. (English) Zbl 0838.62098

Summary: We discuss infinite time ruin probabilities in continuous time in a compound Poisson process with a constant premium rate and a constant interest rate. We discuss equations for the ruin probability as well as approximations and upper and lower bounds. Two special cases are treated in more detail: the case with zero initial reserve, and the case with exponential claim sizes.


62P05 Applications of statistics to actuarial sciences and financial mathematics
45H05 Integral equations with miscellaneous special kernels
Full Text: DOI


[1] Asmussen, S., Applied Probability and Queues (1987), Wiley: Wiley New York · Zbl 0624.60098
[2] Beard, R. E.; Pentikainen, T.; Pesonen, E., Risk Theory (1984), Chapman & Hall: Chapman & Hall London · Zbl 0532.62081
[3] Beekman, J. A., Two Stochastic Processes (1974), Almqvist and Wiksell: Almqvist and Wiksell Stockholm · Zbl 0137.35601
[4] Boogaert, P.; Crijns, V., Upper bounds on ruin probabilities in case of negative loadings and positive interest rates, Insurance: Mathematics and Economics, 6, 221-232 (1987) · Zbl 0642.62058
[5] Boogaert, P.; Haezendonck, J.; Delbaen, F., Limit theorems for the present value of the surplus of an insurance portfolio, Insurance: Mathematics and Economics, 7, 131-138 (1988) · Zbl 0683.62059
[6] Bühlmann, J., Mathematical Methods in Risk Theory (1970), Springer Verlag: Springer Verlag Heidelberg · Zbl 0209.23302
[7] Delbaen, F.; Haezendonck, J., Classical risk theory in an economic environment, Insurance: Mathematics and Econimics, 6, 85-116 (1987) · Zbl 0622.62098
[8] Embrechts, P.; Jensen, J. L.; Maejima, M.; Teugels, J. L., Approximations for compound Poisson and Pólya processes, Advances in Applied Probability, 17, 623-637 (1985) · Zbl 0576.62098
[9] Feller, W., (An Introduction to Probability Theory and its Applications, Vol. II (1966), Wiley: Wiley New York) · Zbl 0138.10207
[10] Gerber, H. U., Der Einfluss von Zins auf die Ruinwahrscheinlichkeit, Mitteilungen Vereinigung schweizerische Versicherungsmathematiker, 71, 63-70 (1971) · Zbl 0217.26804
[11] Gerber, H. U., The discounted central limit theorem and its Berry-Esséen analogue, Annals of Mathematical Statistics, 42, 389-392 (1971) · Zbl 0224.60012
[12] Gerber, H. U., An introduction to Mathematical Risk Theory (1979), University of Pennsylvania: University of Pennsylvania Philadelphia, PA, SS. Huebner Foundation for Insurance Education · Zbl 0431.62066
[13] Gradshteyn, I. S.; Ryzhik, I. M., (Table of Integrals, Series and Products (1965), Academic Press: Academic Press New York) · Zbl 0918.65002
[14] Harrison, J. M., Ruin problems with compounding assets, Stochastic Processes Appl., 5, 67-79 (1977) · Zbl 0361.60053
[15] Mitrinovic, D. S., Analytic Inequalities (1970), Springer Verlag: Springer Verlag Berlin · Zbl 0199.38101
[16] Ross, S. S., Stochastic Processes (1988), J. Wiley: J. Wiley New York
[17] Segerdahl, C. O., Über einige risikotheoretische Fragestellungen, Skand. Aktuaritidskrift, 61, 43-83 (1942) · Zbl 0026.41901
[18] Segerdahl, C. O., A survey of results in the collective theory of risk, (Probability and Statistics. The Harold Cramér volume (1954), John Wiley: John Wiley Stockholm), 276-299 · Zbl 0122.15501
[19] Sundt, B., An Introduction to Non-Life Insurance Mathematics (1993), Verlag Versicherungswissenschaft: Verlag Versicherungswissenschaft Karlsruhe · Zbl 0811.62098
[20] Sundt, B.; Teugels, J. L., The adjustment function in ruin estimates under interest force (1994) · Zbl 0910.62107
[21] Teugels, J. L., Approximation and estimation of some compound distributions, Insurance: Mathematics and Economics, 4, 143-153 (1985) · Zbl 0583.62091
[22] Teugels, J. L.; Willmot, G., Approximations for stop-loss premiums, Insurance: Mathematics and Economics, 6, 195-202 (1987) · Zbl 0624.62097
[23] Widder, D., The Laplace Transform (1946), Princeton University Press: Princeton University Press Princeton, NJ · JFM 67.0384.01
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