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

摘要

We discuss the optimal dividend and capital injection strategies in the Cramér-Lundberg risk model. The value functionV(x)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 thatV(x)can be characterized by the Hamilton-Jacobi-Bellman equation. We find the optimal dividend barrierb, the optimal upper capital injection barrier 0, and the optimal lower capital injection barrier-z*. In the case of exponential claim size especially, we give an explicit procedure to obtainb,-z*, and the value functionV(x).