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Optimal dividend strategies for a risk process under force of interest. (English) Zbl 1140.91371
Summary: In the classical Cramér-Lundberg model in risk theory the problem of maximizing the expected cumulated discounted dividend payments until ruin is a widely discussed topic. In the most general case within that framework it is proved [H. U. Gerber, Entscheidungskriterien für den zusammengesetzten Poisson-prozess. Schweiz. Aktuarver. Mitt. 1, 185–227 (1968); P. Azcue and N. Muler, Math. Finance 15, No. 2, 261–308 (2005; Zbl 1136.91016); H. Schmidli, Stochastic Control in Insurance. Springer (2008; Zbl 1133.93002)] that the optimal dividend strategy is of band type. In the present paper we discuss this maximization problem in a generalized setting including a constant force of interest in the risk model. The value function is identified in the set of viscosity solutions of the associated Hamilton-Jacobi-Bellman equation and the optimal dividend strategy in this risk model with interest is derived, which in the general case is again of band type and for exponential claim sizes collapses to a barrier strategy. Finally, an example is constructed for Erlang(2)-claim sizes, in which the bands for the optimal strategy are explicitly calculated.

MSC:
91B28 Finance etc. (MSC2000)
91B30 Risk theory, insurance (MSC2010)
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[1] Abramowitz, M.; Stegun, I.A., (), For sale by the Superintendent of Documents · Zbl 0515.33001
[2] Albrecher, H.; Teugels, J.L.; Tichy, R.F., On a gamma series expansion for the time-dependent probability of collective ruin, Insurance: mathematics & economics, 29, 3, 345-355, (2001) · Zbl 1025.62036
[3] Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the cramér – lundberg model, Mathematical finance, 15, 2, 261-308, (2005) · Zbl 1136.91016
[4] Benth, F.E.; Karlsen, K.H.; Reikvam, K., Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach, Finance and stochastics, 5, 3, 275-303, (2001) · Zbl 0978.91039
[5] Cont, R.; Tankov, P., Financial modelling with jump processes, () · Zbl 1052.91043
[6] Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, () · Zbl 0713.60085
[7] Gerber, H.U., Entscheidungskriterien fuer den zusammengesetzten Poisson-prozess, Schweizerische aktuarvereinigung. mitteilungen, 1, 185-227, (1968) · Zbl 0193.20501
[8] Mnif, M.; Sulem, A., Optimal risk control and dividend policies under excess of loss reinsurance, Stochastics, 77, 5, 455-476, (2005) · Zbl 1076.93046
[9] Paulsen, J., Risk theory in a stochastic economic environment, Stochastic process and their applications, 46, 2, 327-361, (1993) · Zbl 0777.62098
[10] Paulsen, J.; Gjessing, H.K., Optimal choice of dividend barriers for a risk process with stochastic return on investments, Insurance: mathematics & economics, 20, 3, 215-223, (1997) · Zbl 0894.90048
[11] Paulsen, J.; Gjessing, H.K., Ruin theory with stochastic return on investments, Advances in applied probability, 29, 4, 965-985, (1997) · Zbl 0892.90046
[12] Sayah, A., Equations d’hamiltonian-Jacobi du premier ordre avec terms intégro-différentiels. partie i: unicité des solutions de viscosité, Communications in partial differential equations, 16, 6-7, 1057-1074, (1991) · Zbl 0742.45004
[13] Schmidli, H., Stochastic control in insurance, (2008), Springer · Zbl 1133.93002
[14] Shreve, S.E.; Lehoczky, J.P.; Gaver, D.P., Optimal consumption for general diffusions with absorbing and reflecting barriers, SIAM journal on control and optimization, 22, 1, 55-75, (1984) · Zbl 0535.93071
[15] Wheeden, R.L.; Zygmund, A., Measure and integral, ()
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.