Optimisation of drawdowns by generalised reinsurance in the classical risk model

We consider a Cramér–Lundberg model representing the surplus of an insurance company under a general reinsurance control process. We aim to minimise the expected time during which the surplus is bounded away from its own running maximum by at least $$d>0$$ d > 0 (discounted at a preference rate $$\delta >0$$ δ > 0 ) by choosing a reinsurance strategy. By analysing the drawdown process (i.e. the absolute distance of the controlled surplus model to its maximum) directly, we prove that the value function fulfils the corresponding Hamilton–Jacobi–Bellman equation and show how one can calculate the value function and the optimal strategy. If the initial drawdown is critically large, the problem corresponds to the maximisation of the Laplace transform of a passage time. We show that a constant retention level is optimal. If the drawdown is smaller than d, the problem can be expressed as an element of a set of Gerber–Shiu optimisation problems. We show how these problems can be solved and that the optimal strategy is of feedback form. We illustrate the theory by examples of the cases of light and heavy tailed claims.