Journal of Applied Probability

A risk model with delayed claims

Angelos Dassios and Hongbiao Zhao

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

Abstract

In this paper we introduce a simple risk model with delayed claims, an extension of the classical Poisson model. The claims are assumed to arrive according to a Poisson process and claims follow a light-tailed distribution, and each loss payment of the claims will be settled with a random period of delay. We obtain asymptotic expressions for the ruin probability by exploiting a connection to Poisson models that are not time homogeneous. A finer asymptotic formula is obtained for the special case of exponentially delayed claims and an exact formula is obtained when the claims are also exponentially distributed.

Article information

Source
J. Appl. Probab., Volume 50, Number 3 (2013), 686-702.

Dates
First available in Project Euclid: 5 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1378401230

Digital Object Identifier
doi:10.1239/jap/1378401230

Mathematical Reviews number (MathSciNet)
MR3102509

Zentralblatt MATH identifier
1278.91084

Subjects
Primary: 91B30: Risk theory, insurance
Secondary: 60G55: Point processes 60F05: Central limit and other weak theorems

Keywords
Delayed claim risk model ruin probability asymptotics generalised Cramér‒Lundberg approximation nonhomogeneous Poisson process

Citation

Dassios, Angelos; Zhao, Hongbiao. A risk model with delayed claims. J. Appl. Probab. 50 (2013), no. 3, 686--702. doi:10.1239/jap/1378401230. https://projecteuclid.org/euclid.jap/1378401230


Export citation

References

  • Albrecher, H. and Asmussen, S. (2006). Ruin probabilities and aggregate claims distributions for shot noise Cox processes. Scand. Actuarial J. 2, 86–110.
  • Boogaert, P. and Haezendonck, J. (1989). Delay in claim settlement. Insurance Math. Econom. 8, 321–330.
  • Brémaud, P. (2000). An insensitivity property of Lundberg's estimate for delayed claims. J. Appl. Prob. 37, 914–917.
  • Dassios, A. and Zhao, H. (2011). A dynamic contagion process. Adv. Appl. Prob. 43, 814–846.
  • Dassios, A. and Zhao, H. (2012). A Markov chain model for contagion. Working paper, London School of Economics.
  • Dickson, D. C. M. and Hipp, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance Math. Econom. 29, 333–344.
  • Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory. Huebner, Philadelphia, PA.
  • Grandel, J. (1991). Aspects of Risk Theory. Springer, New York.
  • Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.
  • Lawrance, A. J. and Lewis, P. A. W. (1975). Properties of the bivariate delayed Poisson process. J. Appl. Prob. 12, 257–268.
  • Macci, C. and Torrisi, G. L. (2004). Asymptotic results for perturbed risk processes with delayed claims. Insurance Math. Econom. 34, 307–320.
  • Mirasol, N. M. (1963). The output of an M/G/$\infty$ queuing system is Poisson. Operat. Res. 11, 282–284.
  • Newell, G. F. (1966). M/G/$\infty$ queue. SIAM J. Appl. Math. 14, 86–88.
  • Trufin, J., Albrecher, H. and Denuit, M. (2011). Ruin problems under IBNR dynamics. Appl. Stoch. Models Business Industry 27, 619–632.
  • Waters, H. R. and Papatriandafylou, A. (1985). Ruin probabilities allowing for delay in claims settlement. Insurance Math. Econom. 4, 113–122.
  • Yuen, K. C., Guo, J. and Ng, K. W. (2005). On ultimate ruin in a delayed-claims risk model. J. Appl. Prob. 42, 163–174. \endharvreferences