Reinsurer's optimal reinsurance strategy with upper and lower premium constraints under distortion risk measures

Motivated by Cui et al. (2013) and Zheng and Cui (2014), we study in this paper the optimal (from the reinsurer's point of view) reinsurance problem where the risk is measured by distortion risk measures, the premiums are calculated under the distortion premium principle, and both the upper and lower premium constraints are involved. Our objective is to seek for the optimal reinsurance strategy which minimizes the reinsurer's risk measure of its total loss. Suppose an reinsurer is exposed to the risk f(X) that is transferred from an insurer, who faces a total loss X and decides to buy from our reinsurer the reinsurance contract. The reinsurance contract specifies that the reinsurer covers f(X) and the insurer covers X−f(X). In addition, the insurer is obligated to compensate our reinsurer for undertaking the risk by paying the reinsurance premium under the distortion premium principle. We present a direct method for discussing the optimization problem. Based on our method, the optimal (or, suboptimal) reinsurance strategy is sought out.