Nonlinear expectations of random sets

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.

Set-valued risk measures as backward stochastic difference inclusions and equations

“…In this case, as studied in Xu [50], a natural candidate that extends (1.1) is a multidimensional BSDE which has the same form as (1.1), but with a vector-valued driver function g and a vector-valued terminal value X. However, the only vector-valued functions that can be seen as risk measures (i.e., monotone, translative, convex) are the trivial ones whose coordinates are risk measures on individual coordinate spaces; this was observed in Xu [50] in the dynamic setting via BSDE arguments and a more direct proof in the static setting was given recently in Molchanov and Mühlemann [43]. Hence, multidimensional BSDEs have very limited use for the representation of dynamic risk measures in the multivariate framework.…”

Convex bodies generated by sublinear expectations of random vectors

“…It is possible to construct a variant of the set E e (ξ) by applying the underlying sublinear expectation e to the positive part ( ξ, u ) + of the scalar product of ξ and u. The obtained function is the support function of a convex closed set, which may be considered a sublinear expectation of the segment [0, ξ], see [41] for a study of sublinear expectations with set-valued arguments.…”

Set-Valued T-Translative Functions and Their Applications in Finance

“…This again confirms that Definition 3 (a1), (a2) (as well as (c1), (c2), (d1), (d2)) is perfectly consistent with and true generalizations of the scalar ones. One may compare [23]: the set-valued upper and lower expectations therein do not share most of these features. Remark 6.…”

“…(2) The definitions of E + , E − are essentially different from the ones in [23] and, in contrast to the latter, maintain many features of the scalar versions. For example, a relation such as E + (X) = −E − (−X) is not true for the concepts from [23];…”

“…A selector of X is a random variable X : Ω → IR d with X ∈ X P-a.s. The set L p (X ) comprises all p-integrable selectors of X (see [23,45]). By the definition of p-integrability, this set is nonempty.…”