Core of convex distortions of a probability.

This paper characterizes the core of a differentiable convex distortion of a probability measure on a nonatomic space by identifying it with the set of densities which dominate the derivative of the distortion, for second order stochastic dominance. The densities that have the same distribution as the derivative of the distortion are the extreme points of the core. These results are applied to the differentiability of a Yaari's or Rank Dependent Expected utility function. The superdifferential of a Choquet integral at any point is fully characterized. Examples of use of these results in simple models where some agent is a RDEU maximizer are given.