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A two-dimensional risk model with proportional reinsurance. (English) Zbl 1239.91073
Summary: In this paper we consider an extension of the two-dimensional risk model introduced in F. Avram, Z. Palmowski and M. Pistorius [Insur. Math. Econ. 42, No. 1, 227–234 (2008; Zbl 1141.91482)]. 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.

MSC:
91B30 Risk theory, insurance (MSC2010)
60G51 Processes with independent increments; Lévy processes
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60J75 Jump processes (MSC2010)
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[1] Avram, F., Palmowski, Z. and Pistorius, M. (2008a). A two-dimensional ruin problem on the positive quadrant. Insurance Math. Econom. 42 , 227-234. · Zbl 1141.91482
[2] 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. · Zbl 1163.60010
[3] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7 , 156-169. · Zbl 0880.60077
[4] Cai, J. and Li, H. (2005). Multivariate risk model of phase type. Insurance Math. Econom. 36 , 137-152. · Zbl 1122.60049
[5] Cai, J. and Li, H. (2007). Dependence properties and bounds for ruin probabilities in multivariate compound risk models. J. Multivariate Anal. 98 , 757-773. · Zbl 1280.91090
[6] 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. · Zbl 1055.91041
[7] 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. · Zbl 1231.91156
[8] Dang, L., Zhu, N. and Zhang, H. (2009). Survival probability for a two-dimensional risk model. Insurance Math. Econom. 44 , 491-496. · Zbl 1162.91405
[9] 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. · Zbl 1169.91386
[10] 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. · Zbl 1097.62113
[11] Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2 , 48-78. · Zbl 1081.60550
[12] Gong, L., Badescu, A. L. and Cheung, E. C. K. (2010). Recursive methods for a two-dimensional risk process with common shocks. Submitted. · Zbl 1235.91090
[13] 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.
[14] Li, J., Liu, Z. and Tang, Q. (2007). On the ruin probabilities of a bidimensional perturbed risk model. Insurance Math. Econom. 41 , 185-195. · Zbl 1119.91056
[15] Lin, X. S. and Willmot, G. E. (1999). Analysis of a defective renewal equation arising in ruin theory. Insurance Math. Econom. 25 , 63-84. · Zbl 1028.91556
[16] Rabehasaina, L. (2009). Risk processes with interest force in Markovian environment. Stoch. Models 25 , 580-613. · Zbl 1222.91025
[17] Suprun, V. N. (1976). Problem of destruction and resolvent of a terminating process with independent increments. Ukrainian Math. J. 28 , 39-45. · Zbl 0349.60075
[18] Tijms, H. C. (1994). Stochastic Models . John Wiley, Chichester. · Zbl 0838.60075
[19] 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.
[20] 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. · Zbl 1095.62120
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