zbMATH — the first resource for mathematics

Optimal dividends in the dual model. (English) Zbl 1131.91026
The authors study the optimal dividend problem in case the surplus of a company is modelled as \[ U(t)=u-ct+S(t) \] where \(u\) is the initial surplus, \(c\) is the rate of expenses and \(S\) is a compound Poisson process with positive jumps. This so-called dual model would be appropriate for companies that exhibit occasional gains, such as pharmaceutical or petroleum companies, in contrast to the more classical ’primal’ model that is suitable for insurance companies. The authors describe several methods to find the level of the optimal dividend and provide numerical illustrations.

91G50 Corporate finance (dividends, real options, etc.)
91B30 Risk theory, insurance (MSC2010)
60G51 Processes with independent increments; Lévy processes
Full Text: DOI
[1] Bühlmann, H., Mathematical methods in risk theory, (1970), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0209.23302
[2] Cramér, H., Collective risk theory: A survey of the theory from the point of view of the theory of stochastic processes, (1955), Ab Nordiska Bokhandeln Stockholm
[3] de Finetti, B., 1957. Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries, vol. 2. pp. 433-443
[4] Dufresne, D., 2006. Fitting combinations of exponentials to probability distributions. Applied Stochastic Models in Business and Industry (in press) · Zbl 1142.60321
[5] Dufresne, F.; Gerber, H.U.; Shiu, E.S.W., Risk theory with the gamma process, ASTIN bulletin, 21, 2, 177-192, (1991)
[6] Gerber, H.U., Games of economic survival with discrete- and continuous-income processes, Operations research, 20, 1, 37-45, (1972) · Zbl 0236.90079
[7] Gerber, H.U.; Shiu, E.S.W., Optimal dividends: analysis with Brownian motion, North American actuarial journal, 8, 1, 1-20, (2004) · Zbl 1085.62122
[8] Miyasawa, K., An economic survival game, Journal of the operations research society of Japan, 4, 3, 95-113, (1962)
[9] Pafumi, G., Discussion on H.U. gerber and E.S.W. shiu’s “on the time value of ruin”, North American actuarial journal, 2, 1, 75-76, (1998)
[10] Seal, H.L., Stochastic theory of a risk business, (1969), Wiley New York · Zbl 0196.23501
[11] Tákacs, L., Combinatorial methods in the theory of stochastic processes, (1967), Wiley New York · Zbl 0189.17602
[12] Yang, H.; Zhang, L., Spectrally negative Lévy processes with applications in risk theory, Advances in applied probability, 33, 281-291, (2001) · Zbl 0978.60104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.