Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T10:34:38.978Z Has data issue: false hasContentIssue false
  • English
  • Français

A Risk Model with Delayed Claims

Part of: Limit theorems Stochastic processes Mathematical economics

Published online by Cambridge University Press:  30 January 2018

Angelos Dassios  and
Hongbiao Zhao
Angelos Dassios*
Affiliation:
London School of Economics
Hongbiao Zhao*
Affiliation:
London School of Economics
*
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK.
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Delayed claimrisk modelruin probabilityasymptoticsgeneralised Cramér‒Lundberg approximationnonhomogeneous Poisson process

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 60G55: Point processes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 686 - 702
Copyright
© Applied Probability Trust 

References

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