Risk reducers in convex order

Given a risk position X, a random addition Z is called a risk reducer for X if the new position X+Z is less risky than X+E[Z] in convex order. We utilize the concept of convex hull to give a structural description of risk reducers in the case of an atomless probability space. Then we study risk reducers that are fully dependent on X. Applications to multivariate stochastic ordering, index-linked hedging strategies, and optimal reinsurance are proposed.