## On a class of renewal risk models with a constant dividend barrier.(English)Zbl 1122.91345

Summary: We consider a compound renewal (Sparre Andersen) risk process in the presence of a constant dividend barrier in which the claim waiting times are generalized Erlang$$(n)$$ distributed (i.e., convolution of n exponential distributions with possibly different parameters). An integro-differential equation with certain boundary conditions for the Gerber-Shiu function is derived and solved. Its solution can be expressed as the Gerber-Shiu function in the corresponding Sparre Andersen risk model without a barrier plus a linear combination of $$n$$ linearly independent solutions to the associated homogeneous integro-differential equation. Finally, explicit results are given when the claim sizes are exponentially distributed.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.)
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### References:

 [1] Albrecher, H.; Kainhofer, R., Risk theory with a nonlinear dividend barrier, Computing, 68, 289-311, (2002) · Zbl 1076.91521 [2] Andersen, E.S., On the collective theory of risk in case of contagion between claims, Bull. inst. math. appl., 12, 275-279, (1957) [3] Bühlmann, H., Mathematical methods in risk theory, (1970), Springer-Verlag New York · Zbl 0209.23302 [4] Cheng, Y.; Tang, Q., Moments of surplus before ruin and deficit at ruin in the Erlang(2) risk process, North am. actuarial J., 7, 1, 1-12, (2003) · Zbl 1084.60544 [5] De Finetti, B., Su un’impostazione alternativa dell teoria colletiva del rischio, Trans. XV int. congress actuaries, 2, 433-443, (1957) [6] Dickson, D.C.M., On a class of renewal risk process, North am. actuarial J., 2, 3, 60-68, (1998) [7] Dickson, D.C.M.; Drekic, S., The joint distribution of the surplus prior to ruin and the deficit at ruin in some sparre Andersen models, Insurance: math. econ., 34, 97-107, (2004) · Zbl 1043.60036 [8] Dickson, D.C.M.; Hipp, C., Ruin probabilities for Erlang(2) risk process, Insurance: math. econ., 22, 251-262, (1998) · Zbl 0907.90097 [9] Dickson, D.C.M.; Hipp, C., Ruin problems for phase-type(2) risk processes, Scand. actuarial J., 2, 147-167, (2000) · Zbl 0971.91036 [10] Dickson, D.C.M.; Hipp, C., On the time to ruin for Erlang(2) risk process, Insurance: math. econ., 29, 333-344, (2001) · Zbl 1074.91549 [11] Gerber, H.U., Martingales in risk theory, Mitteilungen der schweizer vereinigung der versicherungsmathematiker, 73, 205-206, (1973) · Zbl 0278.60047 [12] Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. Monograph Series 8, Huebner Foundation, Philadelphia. [13] Gerber, H.U., On the probability of ruin in the presence of a linear dividend barrier, Scand. actuarial J., 2, 105-115, (1981) · Zbl 0455.62086 [14] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North am. actuarial J., 2, 1, 48-78, (1998) · Zbl 1081.60550 [15] Gerber, H.U.; Shiu, E.S.W., Discussion of Y. cheng and Q. tang’s “moments of the surplus before ruin and the deficit at ruin”, North am. actuarial J., 7, 3, 117-119, (2003) · Zbl 1084.60545 [16] Gerber, H.U.; Shiu, E.S.W., Discussion of Y. cheng and Q. tang’s “moments of the surplus before ruin and the deficit at ruin”, North am. actuarial J., 7, 4, 96-101, (2003) · Zbl 1084.60546 [17] Gerber, H.U., Shiu, E.S.W., 2004. The time value of ruin in a Sparre Andersen model, submitted for publication. · Zbl 1085.62508 [18] Højgaard, B., Optimal dynamic premium control in non-life insurance: maximizing dividend payouts, Scand. actuarial J., 225-245, (2002) · Zbl 1039.91042 [19] Li, S., Discussion of Y. cheng and Q. tang’s “moments of the surplus before ruin and the deficit at ruin”, North am. actuarial J., 7, 3, 119-122, (2003) · Zbl 1084.60547 [20] Li, S., 2004. On the time value of ruin for insurance risk models. Ph.D. thesis, Concordia University. [21] Li, S.; Garrido, J., On ruin for Erlang(n) risk process, Insurance: math. econ., 34, 391-408, (2004) · Zbl 1188.91089 [22] Lin, X.S., Discussion of Y. cheng and Q. tang’s “moments of the surplus before ruin and the deficit at ruin”, North am. actuarial J., 7, 3, 122-124, (2003) · Zbl 1084.60548 [23] Lin, X.S.; Willmot, G.E.; Drekic, S., The classical risk model with a constant dividend barrier: analysis of the gerber – shiu discounted penalty function, Insurance: math. econ., 33, 551-566, (2003) · Zbl 1103.91369 [24] Paulsen, J.; Gjessing, H., Optimal choice of dividend barriers for a risk process with stochastic return on investments, Insurance: math. econ., 20, 215-223, (1997) · Zbl 0894.90048 [25] Segerdahl, C., 1970. On some distributions in time-connected with the collective theory of risk. Scand. Actuarial J. 167-192. · Zbl 0229.60063 [26] Sun, L.; Yang, H., On the joint distributions of surplus immediately before ruin and the deficit at ruin for Erlang(2) risk processes, Insurance: math. econ., 34, 121-125, (2004) · Zbl 1054.60017 [27] Tsai, C.C.; Sun, L., On the discounted distribution functions for the Erlang(2) risk process, Insurance: math. econ., 35, 5-19, (2004) · Zbl 1215.62114 [28] Willmot, G.E., A Laplace transform representation in a class of renewal queuing and risk process, J. appl. probability, 36, 570-584, (1999) · Zbl 0942.60086 [29] Willmot, G.E.; Dickson, D.C.M., The gerber-shiu discounted penalty function in the stationary renewal risk model., Insureance: math. econ., 32, 403-411, (2003) · Zbl 1072.91027
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