Optimal reinsurance under convex principles of premium calculation

Suppose an insurer wants to have a reinsurance contract minimizing a convex measure of his retained risk or maximizing a utility function. Suppose the reinsurer's premium is fixed. The premium calculation principle of the reinsurer is a convex functional of his cover. Explicit forms of optimal reinsurance contracts are derived for some classes of convex principles including, among others, the exponential, p-mean value, semi-deviation, semi-variance, Dutch and Wang's principles. The paper is a continuation of the work of Kaluszka [Kaluszka, M., 2004. An extension of Arrow's result on optimality of a stop loss contract. Insur.: Math. Econ. 35, 527-536] which deals with mean-variance premium calculation principles.