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.


91B30 Risk theory, insurance (MSC2010)
91B28 Finance etc. (MSC2000)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.