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.

随机变量的亚线性函数被称为亚线性期望。它们是无限维线性空间上的凸齐次泛函。我们将此概念扩展到在可测量的集合值函数（形成非线性空间）或等效地在随机封闭集合上定义的集合值函数。这就要求对亚线性和超线性期望进行单独研究，因为符号的改变不会改变集合值设置中包含的方向。

我们将极端期望确定为源自非线性期望的原始表示和对偶表示的那些。提出了几种非线性期望的一般构造方法，并得到了相应的对偶表示结果。在应用方面，亚线性期望自然与多变量样本的深度修整有关，而超线性期望可用于评估多资产投资组合的效用。