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

Li, Shuanming; Garrido, José

Insur. Math. Econ. 35, No. 3, 691-701 (2004).

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.)

Keywords:

Sparre Andersen risk process; Integro-differential equation; Generalized Erlang(n) distribution; Time of ruin; Surplus before ruin; Deficit at ruin

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