Nonlinear expectations of random sets

• Published in: Finance and stochastics Vol. 25; no. 1; pp. 5 - 41
• Main Authors: Molchanov, Ilya, Mühlemann, Anja
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2021; Springer Nature B.V
• Subjects: Construction methods; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Finance; Functionals; Insurance; Management; Mathematical analysis; Mathematics; Mathematics and Statistics; Portfolios; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Representations; Statistics for Business; Utilities; Vector spaces; 60D05; 28B20; Selection expectation; C65; Superlinear expectation; Sublinear expectation; C18; Transaction costs; Set-valued function; Utility; 49J53; 91B16; Multiasset portfolio; Random set; 62H99
• ISSN: 0949-2984; 1432-1122
• DOI: 10.1007/s00780-020-00442-3

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.