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A Two-Dimensional Risk Model with Proportional Reinsurance

Part of: Stochastic processes Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Andrei L. Badescu ,
Eric C. K. Cheung  and
Landy Rabehasaina
Andrei L. Badescu*
Affiliation:
University of Toronto
Eric C. K. Cheung*
Affiliation:
University of Hong Kong
Landy Rabehasaina*
Affiliation:
Université de Franche Comté
*
∗ Postal address: Department of Statistics, University of Toronto, 100 St. George Street, Toronto, Ontario, Canada. Email address: badescu@utstat.toronto.edu
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam, Hong Kong. Email address: eckc@hku.hk
∗∗∗ Postal address: Département de Mathématiques, Université de Franche Comté, 16 route de Gray, 25030 Besançon, France. Email address: lrabehas@univ-fcomte.fr
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Abstract

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In this paper we consider an extension of the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008a). To this end, we assume that there are two insurers. The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer, which can be viewed as a different line of business of the same insurer or as a reinsurer, covers a proportion of the claims arising from one of these two compound Poisson processes. We derive the Laplace transform of the time until ruin of at least one insurer when the claim sizes follow a general distribution. The surplus level of the first insurer when the second insurer is ruined first is discussed at the end in connection with some open problems.

Keywords

Two-dimensional risk modelproportional reinsurancegeometric argumentabsorbing settime to ruindeficit at ruin

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60J75: Jump processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 749 - 765
Copyright
Copyright © Applied Probability Trust 2011 

References

Avram, F., Palmowski, Z. and Pistorius, M. (2008a). A two-dimensional ruin problem on the positive quadrant. Insurance Math. Econom. 42, 227–234.Google Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2008b). Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results. Ann. Appl. Prob. 18, 2421–2449.Google Scholar
Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.Google Scholar
Cai, J. and Li, H. (2005). Multivariate risk model of phase type. Insurance Math. Econom. 36, 137–152.Google Scholar
Cai, J. and Li, H. (2007). Dependence properties and bounds for ruin probabilities in multivariate compound risk models. J. Multivariate Anal. 98, 757–773.Google Scholar
Chan, W.-S., Yang, H. and Zhang, L. (2003). Some results on ruin probabilities in a two-dimensional risk model. Insurance Math. Econom. 32, 345–358.Google Scholar
Cheung, E. C. K. and Landriault, D. (2010). A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model. Insurance Math. Econom. 46, 127–134.Google Scholar
Dang, L., Zhu, N. and Zhang, H. (2009). Survival probability for a two-dimensional risk model. Insurance Math. Econom. 44, 491–496.Google Scholar
Dickson, D. C. M. (2008). Some explicit solutions for the Joint density of the time of ruin and the deficit at ruin. ASTIN Bull. 38, 259–276.Google Scholar
Dickson, D. C. M. and Willmot, G. E. (2005). The density of the time to ruin in the classical Poisson risk model. ASTIN Bull. 35, 45–60.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48–78.Google Scholar
Gong, L., Badescu, A. L. and Cheung, E. C. K. (2010). Recursive methods for a two-dimensional risk process with common shocks. Submitted.Google Scholar
Landriault, D. and Willmot, G. E. (2009). On the Joint distributions of the time to ruin, the surplus prior to ruin, and the deficit at ruin in the classical risk model. N. Amer. Actuarial J. 13, 252–279.Google Scholar
Li, J., Liu, Z. and Tang, Q. (2007). On the ruin probabilities of a bidimensional perturbed risk model. Insurance Math. Econom. 41, 185–195.Google Scholar
Lin, X. S. and Willmot, G. E. (1999). Analysis of a defective renewal equation arising in ruin theory. Insurance Math. Econom. 25, 63–84.Google Scholar
Rabehasaina, L. (2009). Risk processes with interest force in Markovian environment. Stoch. Models 25, 580–613.Google Scholar
Suprun, V. N. (1976). Problem of destruction and resolvent of a terminating process with independent increments. Ukrainian Math. J. 28, 39–45.Google Scholar
Tijms, H. C. (1994). Stochastic Models. John Wiley, Chichester.Google Scholar
Willmot, G. E. and Woo, J.-K. (2007). On the class of Erlang mixtures with risk theoretic applications. N. Amer. Actuarial J. 11, 99–115.Google Scholar
Yuen, K. C., Guo, J. and Wu, X. (2006). On the first time of ruin in the bivariate compound Poisson model. Insurance Math. Econom. 38, 298–308.Google Scholar