Ruin probabilities in perturbed risk models

Abstract 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.

[1]  Predrag R. Jelenkovic,et al.  Multiplexing on-off sources with subexponential on periods , 1997, Proceedings of INFOCOM '97.

[2]  Cramér-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion , 1995 .

[3]  Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion , 1994 .

[4]  E. Pitman.  Subexponential distribution functions , 1980 .

[5]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[6]  Hans U. Gerber,et al.  An extension of the renewal equation and its application in the collective theory of risk , 1970 .

[7]  Asymptotic estimates for the probability of ruin in a Poisson model with diffusion , 1993 .

[8]  Charles M. Goldie,et al.  On convolution tails , 1982 .

[9]  S. Asmussen,et al.  Tail probabilities for non-standard risk and queueing processes with subexponential jumps , 1999, Advances in Applied Probability.

[10]  Zbigniew Palmowski,et al.  The superposition of alternating on-off flows and a fluid model , 1998 .

[11]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[12]  Daren B. H. Cline,et al.  Convolution tails, product tails and domains of attraction , 1986 .

[13]  Zbigniew Palmowski,et al.  A note on martingale inequalities for fluid models , 1996 .

[14]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[15]  P. Embrechts,et al.  Estimates for the probability of ruin with special emphasis on the possibility of large claims , 1982 .

[16]  Hansjörg Furrer,et al.  Risk processes perturbed by α-stable Lévy motion , 1998 .

[17]  V. Chistyakov.  A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes , 1964 .

[18]  C. Klüppelberg,et al.  Large claims approximations for risk processes in a Markovian environment , 1994 .

[19]  S. Resnick.  Adventures in stochastic processes , 1992 .