Before revision:  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 .
After revision:  We prove that the value function , defined in this model, fulfills the Hamilton-Jacobi-Bellman equation and identify the optimal dividend strategy .