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Ruin probabilities in perturbed risk models. (English) Zbl 0907.90100

Summary: We consider the asymptotical behaviour of the ruin function in perturbed and unperturbed non-standard risk models when the initial risk reserve tends to infinity. We give a characterization of this behaviour in terms of the unperturbed ruin function and the perturbation law provided that at least one of both is subexponential. By a number of examples for the claim arrival process as well as the perturbation process we show that our result is a generalization of previous work on this subject.

MSC:

91B30 Risk theory, insurance (MSC2010)
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[1] Asmussen, S.; Henriksen, L. F.; Klüppelberg, C., Large claims approximations for risk processes in a Markovian environment, Stochastic Processes and Their Applications, 54, 29-43 (1994) · Zbl 0814.60067
[2] Asmussen, S.; Højgaard, B., Ruin probability approximations for Markov-modulated risk processes with heavy tails, (Th. Random Proc. (1996)), to appear · Zbl 0892.62078
[3] Asmussen, S.; Schmidli, H.; Schmidt, V., Tail probabilities for non-standard risk and queueing processes with subexponential jumps, (Preprint (1997), Universities of Aarhus, Lund and Ulm) · Zbl 0942.60033
[4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001
[5] Chistyakov, V. P., A theorem on sums of independent, positive random variables and its applications to branching processes, Theory of Probability and its Applications, 9, 640-648 (1964) · Zbl 0203.19401
[6] Cline, D. B.H., Convolution tails, product tails and domains of attraction, Probability Theory and Related Fields, 72, 529-557 (1986) · Zbl 0577.60019
[7] Debicki, K.; Michna, Z.; Rolski, T., On the supremum of Gaussian processes over infinite horizon, (Preprint (1996), University of Wroclaw) · Zbl 0985.60034
[8] Dufresne, F.; Gerber, H. U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics, 10, 51-59 (1991) · Zbl 0723.62065
[9] Embrechts, P.; Goldie, C. M., On convolution tails, Stochastic Processes and Their Applications, 13, 263-278 (1982) · Zbl 0487.60016
[10] Embrechts, P.; Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics and Economics, 1, 55-72 (1982) · Zbl 0518.62083
[11] Feller, W., (An Introduction to Probability Theory and its Applications, Vol. II (1971), Wiley: Wiley New York) · Zbl 0219.60003
[12] Furrer, H. J., Risk processes perturbed by α-stable Lévy motion, Scandinavian Actuarial Journal (1997), to appear
[13] Furrer, H. J., (Doctoral Thesis (1997), ETH Zürich), under preparation
[14] Furrer, H. J.; Schmidli, H., Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion, Insurance: Mathematics and Economics, 15, 23-26 (1994) · Zbl 0814.62066
[15] Gerber, H. U., An extension of the renewal equation and its application in the collective theory of risk, Skand. Aktuar Tidskr., 205-210 (1970) · Zbl 0229.60062
[16] Jelenković, P. R.; Lazar, A. A., Multiplexing on-off sources with subexponential on periods, (Preprint (1996), Columbia University: Columbia University New York) · Zbl 0952.60098
[17] Karlin, S.; Taylor, H. M., A Second Course in Stochastic Processes (1981), Academic Press: Academic Press New York · Zbl 0469.60001
[18] Palmowski, Z.; Rolski, T., A note on martingale inequalities for fluid models, Statistics & Probability Letters, 31, 13-21 (1996) · Zbl 0879.60093
[19] Palmowski, Z.; Rolski, T., The superposition of alternating on-off flows and a fluid model, (Preprint (1996), University of Wroclaw) · Zbl 0942.60089
[20] Pitman, E. J.G., Subexponential distribution functions, Australian Mathematical Society Journal, Series A, 29, 337-347 (1980) · Zbl 0425.60012
[21] Resnick, S., Adventures in Stochastic Processes (1992), Birkhäuser: Birkhäuser Boston · Zbl 0762.60002
[22] Schmidli, H., Cramér-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion, Insurance: Mathematics and Economics, 16, 135-149 (1995) · Zbl 0837.62087
[23] Veraverbeke, N., Asymptotic estimates for the probability of ruin in a Poisson model with diffusion, Insurance: Mathematics and Economics, 13, 57-62 (1993) · Zbl 0790.62098
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