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In this paper, we consider the problem of optimal reinsurance design, when the risk is measured by a distortion risk measure and the premium is given by a distortion risk premium. First, we show how the optimal reinsurance design for the ceding company, the reinsurance company and the social planner can be formulated in the same way. Second, by introducing the "marginal indemnification functions", we characterize the optimal reinsurance contracts. We show that, for an optimal policy, the associated marginal indemnification function only takes the values zero and one. We will see how the roles of the market preferences and premiums and that of the total risk are separated.

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In this paper, we study the Lebesgue property for convex risk measures on the class of bounded càdlàg processes. For that, we characterize the compact subsets of a family of bounded variation processes, which is, of course, the topological dual of the bounded càdlàg processes, in an appropriate topology. We show that the Lebesgue property can be characterized in several equivalent ways.