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On the optimal dividend problem for insurance risk models with surplus-dependent premiums. (English) Zbl 1344.49029
Summary: This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximizing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton-Jacobi-Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber-Shiu functions. A number of concrete examples are analyzed.

49J55 Existence of optimal solutions to problems involving randomness
49K45 Optimality conditions for problems involving randomness
93E20 Optimal stochastic control
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
49L99 Hamilton-Jacobi theories
60G50 Sums of independent random variables; random walks
91B30 Risk theory, insurance (MSC2010)
Full Text: DOI arXiv
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