Optimal reinsurance strategy under fixed cost and delay

We consider an optimal reinsurance strategy in which the insurance company (1) monitors the dynamics of its surplus process, (2) optimally chooses a time to begin negotiating with a reinsurer to buy quota-share, or proportional, reinsurance, which introduces an implementation delay (denoted by Δ≥0), (3) chooses the optimal proportion at the beginning of the negotiation period, and (4) pays a fixed transaction cost when the contract is signed (Δ units of time after negotiation begins). This setup leads to a combined problem of optimal stopping and stochastic control. We obtain a solution for the value function and the corresponding optimal strategy, while demonstrating the solution procedure in detail. It turns out that the optimal continuation region is a union of two intervals, a rather rare occurrence in optimal stopping. Numerical examples are given to illustrate our results and we discuss relevant economic insights from this model.