Nonlinear expectations of random sets

Ilya Molchanov,Anja Mühlemann

doi:10.1007/s00780-020-00442-3

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.

Highlights

Fix a probability space (, F, P)
A sublinear expectation is a real-valued function e defined on the space Lp(R) of p-integrable random variables such that e(ξ + a) = e(ξ ) + a for each deterministic a, the function e is monotone, i.e., e(ξ ) ≤ e(η) if ξ ≤ η a.s., homogeneous, i.e., e(cξ ) = ce(ξ ), c ≥ 0, and subadditive, i.e., e(ξ + η) ≤ e(ξ ) + e(η); (1.2)
While the notation e reflects the expectation meaning, the choice of notation u is explained by the fact that the superlinear expectation can be viewed as a utility function that assigns a higher utility value to the sum of two random variables in comparison with the sum of their individual utilities; see Delbaen [7, Chap

Summary

Introduction

Nonlinear maps restricted to the family Lp(Rd ) of p-integrable random vectors and sets having the form of a random vector plus a cone have been studied by Cascos and Molchanov [6] and Hamel and Heyde [12, 13]; comprehensive duality results have been proved by Drapeau et al [9] In our terminology, these studies concern the case when the argument of a superlinear expectation is the sum of a random vector and a convex cone, which in Hamel et al [14] is allowed to be random, but is the same for all random vectors involved. The case of random sets having the form of a vector plus a cone is the standard setting in the theory of markets with proportional transaction costs; see Kabanov and Safarian [19] Superlinear expectations make it possible to assess utilities (and risks) of such portfolios and so develop dynamic hedging strategies; see [23].

Integrable random sets and selection expectation

Support function at random directions

Convergence of random closed convex sets

Definitions

Examples

Expectations of singletons

Nonlinear expectations of random convex functions

Minimal extension

Maximal extension

Exact nonlinear expectations

Duality for minimal sublinear expectations

Exact sublinear expectation