Abstract
In this paper, we consider the compound Poisson risk model involving two types of dependent claims, namely main claims and by-claims. The by-claim is induced by the main claim with a certain probability and the occurrence of a by-claim may be delayed depending on associated main claim amount. Using Rouché’s theorem, both of the survival probability with zero initial surplus and the Laplace transform of the survival probability are obtained from an integro-differential equations system. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. An exact representation for the solution of this equation is derived through an associated compound geometric distribution. For exponential claim sizes, we present an explicit formula for the survival probability. We also illustrate the influence of model parameters in the dependent risk model on the survival probability by numerical examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albrecher H., Boxma O.J.: A ruin model with dependence between claim sizes and claim intervals. Insurance Math. Econ. 35, 245–254 (2004)
Albrecher H., Teugels J.L.: Exponential behavior in the presence of dependence in risk theory. J. Appl. Probab. 43, 257–273 (2006)
Boogaert P., Haezendonck J.: Delay in claim settlement. Insurance Math. Econ. 8, 321–330 (1989)
Boudreault M., Cossette H., Landrault. , Marceau E.: On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actuar. J. 5, 265–285 (2006)
Dickson D.C.M., Hipp C.: On the time to ruin for Erlang(2) risk processes. Insurance Math. Econ. 29, 333–344 (2001)
Gerber, H.U.: An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation, University of Pennsylvania, Philadelphia (1979)
Lin X.S., Willmot G.E.: Analysis of a defective renewal equation arising in ruin theory. Insurance Math. Econ. 25, 63–84 (1999)
Macci C.: Large deviations for risk models in which each main claim induces a delayed claim. Stoch. Int. J. Probab. Stoch. Process. 78, 77–89 (2006)
Waters H.R., Papatriandafylou A.: Ruin probabilities allowing for delay in claims settlement. Insurance Math. Econ. 4, 113–122 (1985)
Xiao Y.T., Guo J.Y.: The compound binomial risk model with time-correlated claims. Insurance Math. Econ. 41, 124–133 (2007)
Xie J.H., Zou W.: Expected present value of total dividends in a delayed claims risk model under stochastic interest rates. Insurance Math. Econ. 46, 415–422 (2010)
Xie J.H., Zou W.: On the expected discounted penalty function for the compound Poisson risk model with delayed claims. J. Comput. Appl. Math. 235, 2392–2404 (2011)
Yuen K.C., Guo J.Y.: Ruin probabilities for time-correlated claims in the compound binomial model. Insurance Math. Econ. 29, 47–57 (2001)
Yuen K.C., Guo J.Y., Ng K.W.: On ultimate ruin in a delayed-claims risk model. J. Appl. Probab. 42, 163–174 (2005)
Zhang Z.M., Yang H.: On a risk model with stochastic premiums income and dependence between income and loss. J. Comput. Appl. Math. 234, 44–57 (2010)
Zhang Z.M., Yang H.: Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times. J. Comput. Appl. Math. 235, 1189–1204 (2011)
Zou, W.; Xie, J.h.: On the probability of ruin in a continuous risk model with delayed claims. J. Korean Math. Soc. (2012) (in Press)
Zou W., Xie J.h.: On the ruin problem in an Erlang(2) risk model with delayed claims. Commun. Comput. Inf. Sci. 105, 54–61 (2010)
Acknowledgments
This research is supported by the Natural Science Foundation of China under Grant No. 11201217, the Program for New Century Excellent Talents from Ministry of Education 326 of China (NCET-10-0375), the Natural Science Foundation of China under Grant No. 70971039, and the Fundamental Research 327 Funds for the Central Universities under No. 11ZG06.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Xie, Jh., Zou, W. & Gao, Jw. On the probability of ruin in the compound Poisson risk model with potentially delayed claims. Arab. J. Math. 2, 115–127 (2013). https://doi.org/10.1007/s40065-012-0043-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-012-0043-0