# Nonlinear expectations of random sets

## TL;DR

This paper extends the concept of sublinear expectations to set-valued functions, providing new theoretical foundations and methods with applications in multivariate data analysis and portfolio utility assessment.

## Contribution

It introduces a framework for set-valued nonlinear expectations, explores their dual representations, and presents new construction methods for these expectations.

## Key findings

- Identified extremal expectations via primal and dual representations
- Developed general construction methods for nonlinear set-valued expectations
- Connected sublinear expectations to depth trimming in multivariate analysis

## 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), 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.