Optimal dividend strategies in a Cramér–Lundberg model with capital injections

We consider a classical risk model with dividend payments and capital injections. Thereby, the surplus has to stay positive. Like in the classical de Finetti problem, we want to maximise the discounted dividend payments minus the penalised discounted capital injections. We derive the Hamilton–Jacobi–Bellman equation for the problem and show that the optimal strategy is a barrier strategy. We explicitly characterise when the optimal barrier is at 0 and find the solution for exponentially distributed claim sizes.