Core of convex distortions of a probability on a non atomic space

This paper characterizes the core of a differentiable convex distortion of a probability measure on a non atomic space by identifying it with the set of densities which dominate the derivative of the distortion, for second order stochastic dominance. Furthermore 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 Choquet integral with respect to a distortion of a probability measure (respectively the differentiability of a Yaari’s or Rank Dependent Expected utility function). A Choquet integral is differentiable at x if and only if x has a strictly increasing quantile function. The superdifferential of a Choquet integral at any point is then fully characterized. Examples of uses of these results in simple models where some agent is a Rank Dependent Expected Utility (RDEU) maximizer are then given. In particular, efficient risk sharing among an expected utility maximizer and a RDEU maximizer and among two RDEU maximizers is characterized. We have benefited from stimulating conversations with F. Delbaen and J.F. Mertens. GRAPE, Université Bordeaux 4 Montesquieu, Avenue Léon Duguit, 33608,Pessac, France, [email protected]. CEREMADE, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris cedex 16, France, [email protected].