Nonlinear expectations of random sets
将随机变量的次线性期望概念扩展到随机闭集上的集值函数，研究次线性和超线性期望的对偶表示，并应用于多元样本深度修剪和多资产组合效用评估。
Abstract 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) or, equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not alter the direction of the inclusion in the set-valued setting. We identify the extremal expectations as those arising from the primal and dual representations of nonlinear expectations. 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.