The Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics …. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify … the special role of ES in insurance and actuarial science. To fill this gap, we study characterization of risk measures …

specify the reinsurance contract by solving a system of coupled Hamilton-Jacobi-Bellman equations. We obtain explicit … solutions for the game problem when both the insurance and the reinsurance premiums are calculated according to the standard … reinsurance contract is either provided with a peak price when the insurer has sufficient cash reserve and with a minimum price …

control sessions, incentives are formulated as a classic gain contract, while in treatment sessions, incentives are framed as … an isomorphic loss contract. Our results show that loss contracts reduce the minimum efforts of groups and worsen …