Risk Measures with Comonotonic Subadditivity or Convexity and Respecting Stochastic Orders

In Song and Yan (2006), we introduced risk measures which are comonotonic subadditive or comonotonic convex, and gave their representations in terms of Choquet integrals. Independently, Heyde et al. (2006) proposed a data based risk measure in which the comonotonic subadditivity is taken as an axiom. The present paper proposes an axiomatic approach to some new risk measures and gives their representations in terms of Choquet integrals. These risk measures are not only comonotonically subadditive or convex, but also respect stochastic dominance, or stop-loss order, or convex order, and consequently, are law-invariant. These properties have sound economic meaning in insurance.