Optimal Dividend and Capital Injection Strategies in the Cramér-Lundberg Risk Model

We discuss the optimal dividend and capital injection strategies in the Cramer-Lundberg risk model. The value function is defined by maximizing the discounted value of the dividend payment minus the penalized discounted capital injection until the time of ruin. It is shown that can be characterized by the Hamilton-Jacobi-Bellman equation. We find the optimal dividend barrier , the optimal upper capital injection barrier 0, and the optimal lower capital injection barrier . In the case of exponential claim size especially, we give an explicit procedure to obtain , , and the value function .