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The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function. (English) Zbl 1103.91369

Summary: The classical compound Poisson risk model is considered in the presence of a constant dividend barrier. An integro-differential equation for the Gerber-Shiu discounted penalty function is derived and solved. The solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of the associated homogeneous integro-differential equation. This latter function is proportional to the product of an exponential function and a compound geometric distribution function. The results are then used to find the Laplace transform of the time to ruin, the distribution of the surplus before ruin, and moments of the deficit at ruin. The special cases where the claim size distribution is exponential and a mixture of two exponentials are considered in some detail. The integro-differential equation is then extended to the stationary renewal risk model.

MSC:

91B30 Risk theory, insurance (MSC2010)
34K60 Qualitative investigation and simulation of models involving functional-differential equations
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Albrecher, H.; Kainhofer, R., Risk theory with a nonlinear dividend barrier, Computing, 68, 289-311 (2002) · Zbl 1076.91521
[2] Bühlmann, H., 1970. Mathematical Methods in Risk Theory, Springer, New York.; Bühlmann, H., 1970. Mathematical Methods in Risk Theory, Springer, New York.
[3] Cai, J.; Dickson, D. C.M., On the expected discounted penalty function at ruin of a surplus process with interest, Insurance: Mathematics and Economics, 30, 389-404 (2002) · Zbl 1074.91027
[4] De Finetti, B., Su un’impostazione alternativa dell teoria colletiva del rischio, Transactions of the XV International Congress of Actuaries, 2, 433-443 (1957)
[5] Dickson, D. C.M., On the distribution of surplus prior to ruin, Insurance: Mathematics and Economics, 11, 191-207 (1992) · Zbl 0770.62090
[6] Dickson, D. C.M., On a class of renewal risk processes, North American Actuarial Journal, 2, 3, 60-73 (1998) · Zbl 1081.60549
[7] Dickson, D.C.M., Dos Reis, A.D.E., 1996. On the distribution of the duration of negative surplus. Scandinavian Actuarial Journal, 148-164.; Dickson, D.C.M., Dos Reis, A.D.E., 1996. On the distribution of the duration of negative surplus. Scandinavian Actuarial Journal, 148-164. · Zbl 0864.62069
[8] Dickson, D. C.M.; Hipp, C., Ruin probabilities for Erlang(2) risk processes, Insurance: Mathematics and Economics, 22, 251-262 (1998) · Zbl 0907.90097
[9] Dickson, D. C.M.; Hipp, C., On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29, 333-344 (2001) · Zbl 1074.91549
[10] Dickson, D. C.M.; Waters, H. R., The probability and severity of ruin in finite and in infinite time, ASTIN Bulletin, 22, 177-190 (1992)
[11] Dickson, D. C.M.; Dos Reis, A. D.E.; Waters, H. R., Some stable algorithms in ruin theory and their applications, ASTIN Bulletin, 25, 153-175 (1995)
[12] Dufresne, F.; Gerber, H. U., The probability and severity of ruin for combinations of exponential claim amount distributions and their translations, Insurance: Mathematics and Economics, 7, 75-80 (1988) · Zbl 0637.62101
[13] Gerber, H.U., 1969. Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Ph.D. Thesis. ETHZ.; Gerber, H.U., 1969. Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Ph.D. Thesis. ETHZ. · Zbl 0193.20501
[14] Gerber, H. U., Games of economic survival with discrete- and continuous-income processes, Operations Research, 20, 37-45 (1972) · Zbl 0236.90079
[15] Gerber, H.U., 1973. Martingales in risk theory. Mutteilungen der Schweizer Vereinigung der Versicherungsmathematiker, 205-216.; Gerber, H.U., 1973. Martingales in risk theory. Mutteilungen der Schweizer Vereinigung der Versicherungsmathematiker, 205-216. · Zbl 0278.60047
[16] Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation, University of Pennsylvania, Philadelphia.; Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation, University of Pennsylvania, Philadelphia. · Zbl 0431.62066
[17] Gerber, H.U., 1981. On the probability of ruin in the presence of a linear dividend barrier. Scandinavian Actuarial Journal, 105-115.; Gerber, H.U., 1981. On the probability of ruin in the presence of a linear dividend barrier. Scandinavian Actuarial Journal, 105-115. · Zbl 0455.62086
[18] Gerber, H. U.; Shiu, E. S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 21, 129-137 (1997) · Zbl 0894.90047
[19] Gerber, H. U.; Shiu, E. S.W., On the time value of ruin, North American Actuarial Journal, 2, 1, 48-78 (1998) · Zbl 1081.60550
[20] Gerber, H. U.; Goovaerts, M.; Kaas, R., On the probability and severity of ruin, ASTIN Bulletin, 17, 151-163 (1987)
[21] Højgaard, B., 2002. Optimal dynamic premium control in non-life insurance: maximizing divident payouts. Scandinavian Actuarial Journal 225-245.; Højgaard, B., 2002. Optimal dynamic premium control in non-life insurance: maximizing divident payouts. Scandinavian Actuarial Journal 225-245. · Zbl 1039.91042
[22] Lin, X. S.; Willmot, G. E., Analysis of a defective renewal equation arising in ruin theory, Insurance: Mathematics and Economics, 25, 63-84 (1999) · Zbl 1028.91556
[23] Lin, X. S.; Willmot, G. E., The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 27, 19-44 (2000) · Zbl 0971.91031
[24] Paulsen, J.; Gjessing, H., Optimal choice of dividend barriers for a risk process with stochastic return on investments, Insurance: Mathematics and Economics, 20, 215-223 (1997) · Zbl 0894.90048
[25] Petrovski, I.G., 1966. Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ.; Petrovski, I.G., 1966. Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ.
[26] Resnick, S.I., 1992. Adventures in Stochastic Processes. Birkhäuser, Boston.; Resnick, S.I., 1992. Adventures in Stochastic Processes. Birkhäuser, Boston.
[27] Schmidli, H., On the distribution of the surplus prior to and at ruin, ASTIN Bulletin, 29, 227-244 (1999) · Zbl 1129.62425
[28] Segerdahl, C., 1970. On some distributions in time connected with the collective theory of risk. Scandinavian Actuarial Journal, 167-192.; Segerdahl, C., 1970. On some distributions in time connected with the collective theory of risk. Scandinavian Actuarial Journal, 167-192. · Zbl 0229.60063
[29] Sundt, B.; Teugels, J., Ruin estimates and interest force, Insurance: Mathematics and Economics, 16, 7-22 (1995) · Zbl 0838.62098
[30] Willmot, G.E., Dickson, D.C.M., 2003. The Gerber-Shiu discounted penalty function in the stationary renewal risk model. Insurance: Mathematics and Economics 32, 403-411.; Willmot, G.E., Dickson, D.C.M., 2003. The Gerber-Shiu discounted penalty function in the stationary renewal risk model. Insurance: Mathematics and Economics 32, 403-411. · Zbl 1072.91027
[31] Willmot, G.E., Lin, X.S., 2001. Lundberg Approximations for Compound Distributions with Insurance Applications, Lecture Notes in Statistics 156. Springer, New York.; Willmot, G.E., Lin, X.S., 2001. Lundberg Approximations for Compound Distributions with Insurance Applications, Lecture Notes in Statistics 156. Springer, New York. · Zbl 0962.62099
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