We consider the problem of maximizing the discounted utility of dividend payments of an insurance company whose reserves are modeled as a classical Cram\'er-Lundberg risk process. We investigate this optimization problem under the constraint that dividend rate is bounded. We prove that the value function , defined in this model, fulfills the Hamilton-Jacobi-Bellman equation and  identify the optimal dividend strategy . Eventually we extend our results for the reserve process modeled as a classical Cram\'er-Lundberg risk process with capital injections. For the extended model we also prove some results regarding asymptotic analysis of the value function .
We consider the problem of maximizing the discounted utility of dividend payments of an insurance company whose reserves are modeled as a classical Cram\'er-Lundberg risk process. We investigate this optimization problem under the constraint that dividend rate is bounded. We prove that the value function fulfills the Hamilton-Jacobi-Bellman equation and we identify the optimal dividend strategy .
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, defined in this model,
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we
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. Eventually we extend our results for the reserve process modeled as a classical Cram\'er-Lundberg risk process with capital injections. For the extended model we also prove some results regarding asymptotic analysis of the value function