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Optimal dividend strategies in a Cramér-Lundberg model with capital injections and administration costs. (English) Zbl 1222.91026
The authors consider a classical risk model with dividend payments and capital injections in the presence of both fixed and proportional administration costs. In this model, a dividends-injections strategy is admissible if it does not lead to a negative surplus or ruin. The value of a strategy is the discounted value of the dividends minus the costs. The authors argue that it can not be optimal to make a capital injections unless the surplus is negative. Also, at the time of an injection the company may not only inject the deficit, but inject additional capital \(C\geq 0\) to prevent future capital injections. The associated Hamilton-Jacobi-Bellman equation is derived and it is shown that the optimal strategy is of band type. By using Gerber-Shiu functions, the authors derive a method to determine numerically the solution to the integro-differential equation and the unknown value \(C\).

MSC:
91B30 Risk theory, insurance (MSC2010)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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