A stochastic order for random vectors and random sets based on the Aumann expectation

We define a new stochastic order for random vectors in terms of the inclusion relation for the Aumann expectation of certain random sets. We derive some properties of this order, relate it with other well-known multivariate stochastic convex orders, give a geometrical interpretation in terms of the lift zonoid, build a probability metric based on it and extend it to compare random compact sets. Furthermore, a new characterisation of the linear convex order for random vectors is described and the corresponding concept is extended to compare random compact sets.