2019/03/12 by Ilya Molchanov, Anja Mühlemann · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Fuzzy Systems and Optimization #Risk and Portfolio Optimization #Stochastic processes and financial applications #math.FA #math.PR #msc:26E25 #msc:28B20 #msc:28C05 #msc:49J53 #msc:54C60 #msc:58C06 #msc:60D05 #msc:91B30 #q-fin.RM
paper · pdf · doi:10.1007/s00780-020-00442-3
published as Finance and Stochastics, 2021, 25, 5-41 (open access) · 35 pages, 1 figure
arxiv created 2019/03/12 · openalex created_date 2019/03/22 · openalex publication_date 2020/12/30 · arxiv updated 2021/01/15 · openalex updated_date 2026/07/28
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.