# zbMATH — the first resource for mathematics

On a dual model with a dividend threshold. (English) Zbl 1163.91441
Summary: In insurance mathematics, a compound Poisson model is often used to describe the aggregate claims of the surplus process. In this paper, we consider the dual of the compound Poisson model under a threshold dividend strategy. We derive a set of two integro-differential equations satisfied by the expected total discounted dividends until ruin and show how the equations can be solved by using only one of the two integro-differential equations. The cases where profits follow an exponential or a mixture of exponential distributions are then solved and the discussion for the case of a general profit distribution follows by the use of Laplace transforms. We illustrate how the optimal threshold level that maximizes the expected total discounted dividends until ruin can be obtained, and finally we generalize the results to the case where the surplus process is a more general skip-free downwards Lévy process.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60H30 Applications of stochastic analysis (to PDEs, etc.)
EMpht
Full Text:
##### References:
 [1] Albrecher, H.; Badescu, A.L.; Landriault, D., On the dual risk model with tax payments, Insurance, mathematics and economics, 42, 1086-1094, (2008) · Zbl 1141.91481 [2] Asmussen, S.; Nerman, O.; Olsson, M., Fitting phase-type distribution via the EM algorithm, Scandinavian journal of statistics, 30, 365-372, (1996) [3] Avanzi, B.; Gerber, H.U.; Shiu, E.S.W., Optimal dividends in the dual model, Insurance, mathematics and economics, 41, 111-123, (2007) · Zbl 1131.91026 [4] Avanzi, B.; Gerber, H.U., Optimal dividends in the dual model with diffusion, ASTIN bulletin, 38, 2, 653-667, (2008) · Zbl 1274.91463 [5] Barndorff-Nielsen, O.E.; Mikosh, T.; Resnick, S.I., Lévy processes — theory and applications, (2001), Birkhäuser Boston [6] Bayraktar, E.; Egami, M., Optimizing venture capital investment in a jump diffusion model, Mathematical methods of operations research, 67, 21-42, (2008) · Zbl 1151.91049 [7] Bühlmann, H., Mathematical methods in risk theory, (1970), Springer New York · Zbl 0209.23302 [8] Cai, J.; Gerber, H.U.; Yang, H., Optimal dividends in an ornstein – uhlenbeck type model with credit and debit interest, North American actuarial journal, 10, 2, 94-108, (2006) [9] Cramér, H., Collective risk theory: A survey of the theory from the point of view of the theory of stochastic process, (1955), Ab Nordiska Bokhandeln Stockholm [10] Dufresne, D., Stochastic life annuities, North American actuarial journal, 11, 1, 136-157, (2007) [11] Dufresne, F.; Gerber, H.U.; Shiu, E.S.W., Risk processes with the gamma process, ASTIN bulletin, 21, 177-192, (1991) [12] Finetti, B.de, 1957. Su Un’impostazione Alternativa della Teoria Collettiva del Rischio. In: Transactions of the XVth International Congress of Actuaries 2, pp. 433-443 [13] Furrer, H.J., Risk processes perturbed by a $$\alpha$$-stable Lévy motion, Scandinavian actuarial journal, 97-114, (1998) [14] Gerber, H.U., Games of economic survival with discrete- and continuous-income processes, Operational research, 20, 1, 37-45, (1972) · Zbl 0236.90079 [15] Gerber, H.U., An introduction to mathematical risk theory, (), distributed by R. D. Irwin, Homewood, Illinois · Zbl 0431.62066 [16] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-72, (1998) [17] Gerber, H.U.; Shiu, E.S.W., On optimal dividends: from reflection to refraction, Journal of computational and applied mathematics, 186, 4-22, (2006) · Zbl 1089.91023 [18] Gerber, H.U.; Shiu, E.S.W., On optimal dividend strategies in the compound Poisson model, North American actuarial journal, 10, 2, 76-93, (2006) [19] Gerber, H.U.; Smith, N., Optimal dividends with incomplete information in the dual model, Insurance, mathematics and economics, 43, 2, 227-233, (2008) · Zbl 1189.91074 [20] Miyasawa, K., An economic survival game, Journal of the operations research society of Japan, 4, 3, 95-113, (1962) [21] Seal, H.L., Stochastic theory of a risk business, (1969), Wiley New York · Zbl 0196.23501 [22] Takács, L., Combinatorial methods in the theory of stochastic processes, (1967), Wiley New York · Zbl 0189.17602 [23] Yang, H.; Zhu, J., Ruin probabilities of a dual Markov-modulated risk model, Communications in statistics — theory and methods, 37, 3298-3307, (2008) · Zbl 1292.91100
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.