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Ruin probabilities in the presence of regularly varying tails and optimal investment. (English) Zbl 1055.91049

Summary: We study the infinite time ruin probability in the classical Cramer-Lundberg model, where the company is allowed to invest their money in a stock, which is described by geometric Brownian motion. Starting from an integro-differential equation for the maximal survival probability, we analyze the case of claim sizes, which have distribution functions \(F\) with regularly varying tails. Our result is: if \(1-F\) is regularly varying with index \(\rho<-1\), then the ruin probability \(\psi\) is also regularly varying with index \(\rho<-1\). This holds under the assumption of zero interest rates.

MSC:

91B30 Risk theory, insurance (MSC2010)
91B28 Finance etc. (MSC2000)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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[1] Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge University Press, Cambridge. · Zbl 0617.26001
[2] Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics of operations research, 20, 937-958, (1995) · Zbl 0846.90012
[3] 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
[4] Feller, W., 1971. An Introduction to Probability Theory and Its Applications. Wiley, New York. · Zbl 0219.60003
[5] Frolova, A.G., Kabanov, Y.M., Pergamenshchikov, S.M., 2001. In the insurance business risky investments are dangerous. Preprint. Universite de Franche Comte Besancon. · Zbl 1002.91037
[6] Hipp, C.; Plum, M., Optimal investment for insurers, Insurance mathematics and economics, 27, 215-228, (2000) · Zbl 1007.91025
[7] Klüppelberg, C., Stadtmüller, U., 1998. Ruin probabilities in the presence of heavy-tails and interest rates. Scandinavian Actuarial Journal 98 (1), 49-58.
[8] Luxemburg, W.A.J., On an asymptotic problem concerning the Laplace transform, Applicable analysis, 8, 61-70, (1978) · Zbl 0394.44001
[9] Paulsen, J., Ruin theory with compounding assets—a survey, Insurance mathematics and economics, 22, 3-16, (1998) · Zbl 0909.90115
[10] Paulsen, J.; Gjessing, H.K., Ruin theory with stochastic return on investments, Advances in applied probability, 29, 965-985, (1997) · Zbl 0892.90046
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