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We study an optimal reinsurance problem under a diffusion risk model for aninsurer who aims to minimize the probability of lifetime ruin. To rule outmoral hazard issues, we only consider moral-hazard-free reinsurance contractsby imposing the incentive compatibility constraint on indemnity functions. Thereinsurance premium is calculated under an extended distortion premiumprinciple, in which the distortion function is not necessarily concave. Wefirst show that an optimal reinsurance contract always exists and then derivetwo sufficient and necessary conditions to characterize it. Due to the presenceof the incentive compatibility constraint and the nonconcavity of thedistortion, the optimal contract is obtained as a solution to a double obstacleproblem. At last, we apply the general result to study three examples andobtain the optimal contract in (semi)closed form.

Optimal moral-hazard-free reinsurance under extended distortion premium principles