  Sublinear functionals of random variables are known as sublinear
expectations; they are convex homogeneous functionals on infinite-dimensional
linear spaces. We extend this concept for set-valued functionals defined on
measurable set-valued functions (which form a nonlinear space), equivalently,
on random closed sets. This calls for a separate study of sublinear and
superlinear expectations, since a change of sign does not convert one to the
other in the set-valued setting. We identify the extremal expectations as those
arising from the primal and dual representations of them. Several general
construction methods for nonlinear expectations are presented and the
corresponding duality representation results are obtained. On the application
side, sublinear expectations are naturally related to depth trimming of
multivariate samples, while superlinear ones can be used to assess utilities of
multiasset portfolios.
