A dynamic optimal reinsurance strategy with capital injections in the Cramér–Lundberg model

In this article we consider the surplus process of an insurance company within the Cramér–Lundberg framework. We study the optimal reinsurance strategy and dividend distribution of an insurance company under proportional reinsurance, in which capital injections are allowed. Our aim is to find a general dynamic reinsurance strategy that maximizes the expected discounted cumulative dividends until the time of passage below a given level, called ruin. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a, and only pay dividends when the reserve reaches an upper barrier b. Using analytical methods, we identify the value function as a particular solution to the associated Hamilton–Jacobi–Bellman equation. This approach leads to an exhaustive and explicit characterization of optimal policy. The proportional reinsurance is given via comprehensive structure equations. Furthermore we give some examples illustrating the applicability of this method for proportional reinsurance treaties.