Optimizing the expected utility of dividend payments for a Cramér–Lundberg risk process

We consider the problem of maximizing the discounted utility of dividend payments of an insurance company whose reserves are modeled as a classical Cramér–Lundberg risk process. We investigate this optimization problem under the constraint that the dividend rate is bounded. We prove that the value function satisfies the Hamilton–Jacobi–Bellman equation and we identify the optimal dividend strategy.