Nonlinear expectations of random sets

Ilya Molchanov () and Anja Mühlemann ()
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Ilya Molchanov: University of Bern
Anja Mühlemann: University of Bern

Finance and Stochastics, 2021, vol. 25, issue 1, No 2, 5-41

Abstract: 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.

Keywords: Multiasset portfolio; Random set; Selection expectation; Sublinear expectation; Superlinear expectation; Set-valued function; Transaction costs; Utility; 28B20; 49J53; 60D05; 62H99; 91B16 (search for similar items in EconPapers)
JEL-codes: C18 C65 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s00780-020-00442-3

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