# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The compound Poisson risk model with a threshold dividend strategy. (English) Zbl 1157.91383
Summary: In this paper, we present the classical compound Poisson risk model with a threshold dividend strategy. Under such as strategy, no dividends are paid if the insurer’s surplus is below certain threshold level. When the surplus is above this threshold level, dividends are paid at a constant rate that does not exceed the premium rate. Two integro-differential equations for the Gerber-Shiu discounted penalty function are derived and solved. The analytic results obtained are utilized to derive the probability of ultimate ruin, the time of ruin, the distribution of the first surplus drop below the initial level, and the joint distributions and moments of the surplus immediately before ruin and the deficit at ruin. The special cases where the claim size distribution is exponential and a combination of exponentials are considered in some detail.

##### MSC:
 91B30 Risk theory, insurance
Full Text:
##### References:
 [1] Albrecher, H.; Hartinger, J.; Tichy, R.: On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier. Scandinavian actuarial journal 2, 103-126 (2005) · Zbl 1092.91036 [2] Albrecher, H.; Kainhofer, R.: Risk theory with a nonlinear dividend barrier. Computing 68, 289-311 (2002) · Zbl 1076.91521 [3] Asmussen, S.: Ruin probabilities. (2000) · Zbl 0960.60003 [4] Asmussen, S.: Applied probability and queues. (2003) · Zbl 1029.60001 [5] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., 1997. Actuarial Mathematics. Society of Actuaries. [6] Bühlmann, H.: Mathematical methods in risk theory. (1970) · Zbl 0209.23302 [7] De Finetti, B.: Su un’impostazione alternativa Della teoria collettiva del rischio. Proceedings of the transactions of the XV international congress of actuaries, vol. 2 2, 433-443 (1957) [8] Dickson, D. C. M.; Hipp, C.: On the time of ruin for $Erlang(2)$ risk processes. Insurance: mathematics and economics 29, 333-344 (2001) · Zbl 1074.91549 [9] Dickson, D. C. M.; Waters, H. R.: Some optimal dividend problems. ASTIN bulletin 34, No. 1, 49-74 (2004) · Zbl 1097.91040 [10] Drekic, S.; Stafford, J. E.; Willmot, G. E.: Symbolic calculation of the moments of the time of ruin. Insurance: mathematics and economics 34, 109-120 (2004) · Zbl 1087.91028 [11] Gerber, H. U.: Entscheidungskriterien für den zusammengesetzten Poisson-prozess. Mitteilungen der vereinigung schweizerischer versicherungsmathematiker 69, 185-227 (1969) · Zbl 0193.20501 [12] Gerber, H. U.: Games of economic survival with discrete- and continuous-income processes. Operations research 20, 37-45 (1972) · Zbl 0236.90079 [13] Gerber, H. U.: Martingales in risk theory. Mitteilungen der schweizer vereinigung der versicherungsmathematiker, 205-216 (1973) · Zbl 0278.60047 [14] Gerber, H. U.: An introduction to mathematical risk theory. (1979) · Zbl 0431.62066 [15] Gerber, H. U.: On the probability of ruin in the presence of a linear dividend barrier. Scandinavian actuarial journal, 105-115 (1981) · Zbl 0455.62086 [16] Gerber, H. U.; Goovaerts, M. J.; Kaas, R.: On the probability and severity of ruin. ASTIN bulletin 17, No. 2, 151-163 (1987) [17] Gerber, H. U.; Shiu, E. S. W.: On the time value of ruin. North American actuarial journal 2, No. 1, 48-78 (1998) · Zbl 1081.60550 [18] Gerber, H.U., Shiu, E.S.W., 2005a, On optimal dividend strategies in the compound Poisson model. Preprint. [19] Gerber, H. U.; Shiu, E. S. W.: The time value of ruin in a sparre andersen model. North American actuarial journal 9, No. 2, 49-84 (2005) · Zbl 1085.62508 [20] Gerber H.U., Shiu E.S.W., in press. On optimal dividends: From reflection to refraction. Journal of Computational and Applied Mathematics. · Zbl 1089.91023 [21] Højgaard, B.: Optimal dynamic premium control in non-life insurance: maximizing dividend payouts. Scandinavian actuarial journal 4, 225-245 (2002) · Zbl 1039.91042 [22] Klugman, S. A.; Panjer, H. H.; Willmot, G. E.: Loss models: from data to decisions. (2004) · Zbl 1141.62343 [23] Li, S.; Garrido, J.: On ruin for the $Erlang(n)$ risk process. Insurance: mathematics and economics 34, 391-408 (2004) · Zbl 1188.91089 [24] Li, S.; Garrido, J.: On a class of renewal risk models with a constant dividend barrier. Insurance: mathematics and economics 35, 691-701 (2004) · Zbl 1122.91345 [25] Lin, X.S., Pavlova, K.P., 2005. The compound Poisson risk model with multiple barriers, p. 17, submitted for publication. [26] 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 [27] 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 [28] Lin, X. S.; Willmot, G. E.; Drekic, S.: The classical risk model with a constant dicvidend barrier: analysis of the gerber -- shiu discounted penalty function. Insurance: mathematics and economics 33, 551-566 (2003) · Zbl 1103.91369 [29] Panjer, H.H., Willmot, G.E., 1992. Insurance Risk Models. Society of Actuaries, Schaumburg. [30] 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 [31] Segerdahl, C.: On some distributions in time connected with the collective theory of risk. Scandinavian actuarial journal, 167-192 (1970) · Zbl 0229.60063 [32] Willmot, G. E.; Dickson, D. C. M.: The gerber -- shiu discounted penalty function in the stationary renewal risk model. Insurance: mathematics and economics 32, 403-411 (2003) · Zbl 1072.91027 [33] Willmot, G. E.; Lin, X. S.: Lundberg approximations for compound distributions with applications. (2001) · Zbl 0962.62099 [34] Zhou, X., 2004. The risk model with a two-step premium rate, preprint.