Ruin probabilities in the presence of heavy-tails and interest rates. (English) Zbl 1022.60083

Klüppelberg, Claudia; Stadtmüller, Ulrich

Scand. Actuarial J. 1998, No. 1, 49-58 (1998).

Summary: We study the infinite time ruin probability for the classical Cramér-Lundberg model, where the company also receives interest on its reserve. We consider the large claims case, where the claim size distribution \(F\) has a regularly varying tail. Hence our results apply for instance to Pareto, loggamma, certain Benktander and stable claim size distributions. We prove that for a positive force of interest \(\delta\) the ruin probability \(\psi_\delta(u)\sim \kappa_\delta(1- F(u))\) as the initial risk reserve \(u\to\infty\). This is quantitatively different from the non-interest model, where \(\psi(u)\sim\kappa \int^\infty_u (1- F(y)) dy\).

MSC:

60K10	Applications of renewal theory (reliability, demand theory, etc.)
91B30	Risk theory, insurance (MSC2010)

Keywords:

ruin probability; heavy tails; interest rate model; regular variation; Abel-Tauber theorems; modified Laplace transforms

References:

[1]	Asmussen S., Applied probability and queues (1987)
[2]	Asmussen S., Preprint (1996)
[3]	Bingham N. H., Regular variation (1987)
[4]	DOI: 10.1016/0167-6687(87)90016-3 · Zbl 0642.62058 · doi:10.1016/0167-6687(87)90016-3
[5]	DOI: 10.1016/0167-6687(88)90106-0 · Zbl 0683.62059 · doi:10.1016/0167-6687(88)90106-0
[6]	DOI: 10.1016/0167-6687(87)90019-9 · Zbl 0622.62098 · doi:10.1016/0167-6687(87)90019-9
[7]	Embrechts P., Modelling extremal events for insurance and finance (1997) · Zbl 0873.62116
[8]	DOI: 10.1017/S1446788700021224 · doi:10.1017/S1446788700021224
[9]	DOI: 10.1080/15326348908807105 · Zbl 0676.62083 · doi:10.1080/15326348908807105
[10]	DOI: 10.2307/1427443 · Zbl 0811.62096 · doi:10.2307/1427443
[11]	DOI: 10.1016/0167-6687(82)90021-X · Zbl 0518.62083 · doi:10.1016/0167-6687(82)90021-X
[12]	Feller W., An introduction to probability theory and its application, 2. ed. (1971) · Zbl 0219.60003
[13]	Gerber H. U., Mitteilungen der Schweizerischen Vereinigung für Versicherungsmathematiker pp 63– (1979)
[14]	DOI: 10.1007/978-1-4613-9058-9 · doi:10.1007/978-1-4613-9058-9
[15]	DOI: 10.2307/3214240 · Zbl 0651.60020 · doi:10.2307/3214240
[16]	DOI: 10.1016/0167-6687(89)90003-6 · Zbl 0686.62093 · doi:10.1016/0167-6687(89)90003-6
[17]	DOI: 10.1016/0167-6687(94)00023-8 · Zbl 0838.62098 · doi:10.1016/0167-6687(94)00023-8
[18]	Willmot G., Scand. Actuarial J. pp 1– (1989) · Zbl 0679.62094 · doi:10.1080/03461238.1989.10413851