Optimal reinsurance: minimize the expected time to reach a goal

In this paper, we consider an optimal reinsurance problem. The surplus model of the insurance company is approximated by a diffusion model with the drift coefficient μ$ \mu $. The insurance company employs reinsurance to reduce the risk. π$ \pi $ is the proportion of each claim paid by the company and the remainder proportion of the claim is paid by the reinsurer. λ(1-π)$ \lambda (1-\pi ) $ is the rate at which the premiums are diverted to the reinsurer, thus it holds λ≥μ>0$ \lambda \ge \mu >0 $ in general. We discuss two cases: (i) non-cheap reinsurance (when λ>μ$ \lambda >\mu $), (ii) cheap reinsurance (when λ=μ$ \lambda =\mu $). The objective of the insurance company is to make an optimal decision on reinsurance to reach a goal (a given surplus level) in minimal expected time. We disclose that for some case it is not suitable to search optimal decisions by minimizing the expected time to reach a goal. In order to deal with this case, we study two other methodologies (ruin probability minimization and expected discount factor maximization) for the optimal strategy selection problem in reinsurance decision.