On a general class of renewal risk process: analysis of the Gerber-Shiu function
Shuanming Li and José Garrido
Source: Adv. in Appl. Probab. Volume 37, Number 3
(2005), 836-856.
Abstract
We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a Kn distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.
First Page:
Show
Hide
Keywords: Sparre Andersen risk process; K_n family of distributions; martingale; Laplace transform; generalized Lundberg equation; defective renewal equation
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1127483750
Digital Object Identifier: doi:10.1239/aap/1127483750
Zentralblatt MATH identifier: 1077.60063
Mathematical Reviews number (MathSciNet): MR2156563
References
Andersen, S. E. (1957). On the collective theory of risk in case of contagion between claims. Bull. Inst. Math. Appl. 12, 275--279.
Cheng, Y. and Tang, Q. (2003). Moments of the surplus before ruin and the deficit at ruin in the Erlang$(2)$ risk process. N. Amer. Actuarial J. 7, 1--12.
Mathematical Reviews (MathSciNet): MR1986883
Cohen, J. W. (1982). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.
Mathematical Reviews (MathSciNet): MR668697
Zentralblatt MATH: 0481.60003
Cox, D. R. (1955). A use of complex probabilities in the theory of stochastic processes. Proc. Camb. Philos. Soc. 51, 313--319.
Mathematical Reviews (MathSciNet): MR68767
Digital Object Identifier: doi:10.1017/S0305004100030231
Dickson, D. C. M. (1998a). Discussion on `On the time value of ruin', by Gerber, H. U. and Shiu, E. S. W. N. Amer. Actuarial J. 2, 74.
Dickson, D. C. M. (1998b). On a class of renewal risk processes. N. Amer. Actuarial J. 2, 60--68.
Mathematical Reviews (MathSciNet): MR1989918
Dickson, D. C. M. and Hipp, C. (1998). Ruin probabilities for Erlang$(2)$ risk process. Insurance Math. Econom. 22, 251--262.
Mathematical Reviews (MathSciNet): MR1650296
Dickson, D. C. M. and Hipp, C. (2001). On the time to ruin for Erlang$(2)$ risk process. Insurance Math. Econom. 29, 333--344.
Mathematical Reviews (MathSciNet): MR1874628
Dufresne, D. (2001). On a general class of risk models. Austral. Actuarial J. 7, 755--791.
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48--78.
Mathematical Reviews (MathSciNet): MR1988433
Gerber, H. U. and Shiu, E. S. W. (2003). Discussion on `Moments of the surplus before ruin and the deficit at ruin in the Erlang$(2)$ risk process', by Y. Cheng and Q. Tang. N. Amer. Actuarial J. 7, 117--119.
Mathematical Reviews (MathSciNet): MR2062553
Gerber, H. U. and Shiu, E. S. W. (2005). The time value of ruin in a Sparre Andersen model. N. Amer. Actuarial J. 9, 49--69.
Mathematical Reviews (MathSciNet): MR2160108
Li, S. (2003). Discussion on `Moments of the surplus before ruin and the deficit at ruin in the Erlang$(2)$ risk process', by Y. Cheng and Q. Tang. N. Amer. Actuarial J. 7, 119--122.
Mathematical Reviews (MathSciNet): MR2061863
Li, S. and Garrido, J. (2004). On ruin for the Erlang$(n)$ risk process. Insurance Math. Econom. 34, 391--408.
Mathematical Reviews (MathSciNet): MR2063911
Lin, X. S. (2003). Discussion on `Moments of the surplus before ruin and the deficit at ruin in the Erlang$(2)$ risk process', by Y. Cheng and Q. Tang. N. Amer. Actuarial J. 7, 122--124.
Mathematical Reviews (MathSciNet): MR2061864
Lin, X. S. and Willmot, G. E. (1999). Analysis of a defective renewal equation arising in ruin theory. Insurance Math. Econom. 25, 63--84.
Mathematical Reviews (MathSciNet): MR1718539
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.
Mathematical Reviews (MathSciNet): MR618123
Zentralblatt MATH: 0469.60002
Tijms, H. C. (1994). Stochastic Models. An Algorithmic Approach. John Wiley, Chichester.
Mathematical Reviews (MathSciNet): MR1314821
Zentralblatt MATH: 0838.60075
Willmot, G. E. (1999). A Laplace transform representation in a class of renewal queueing and risk process. J. Appl. Prob. 36, 570--584.
Mathematical Reviews (MathSciNet): MR1725431
Digital Object Identifier: doi:10.1239/jap/1032374472
Zentralblatt MATH: 0942.60086
Advances in Applied Probability
