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Conditional Cores and Conditional Convex Hulls of Random Sets

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cores-arxiv.pdf (419.9Kb)
Date
2017
Publisher city
Paris
Collection title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Link to item file
https://hal.archives-ouvertes.fr/hal-01664625
Dewey
Probabilités et mathématiques appliquées
Sujet
random set; selection; set-valued expectation; sublinear expectation; superlinear expectation; support function; essential supremum
URI
https://basepub.dauphine.fr/handle/123456789/20703
Collections
  • CEREMADE : Publications
Metadata
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Author
Lépinette, Emmanuel
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Molchanov, Ilya
118520 Institute of Mathematical Statistics and Actuarial Science [Bern] [IMSV]
Type
Document de travail / Working paper
Item number of pages
31
Abstract (EN)
We define two non-linear operations with random (not necessarily closed) sets in Banach space: the conditional core and the conditional convex hull. While the first is sublinear, the second one is superlinear (in the reverse set inclusion ordering). Furthermore , we introduce the generalised conditional expectation of random closed sets and show that it is sandwiched between the conditional core and the conditional convex hull. The results rely on measurability properties of not necessarily closed random sets considered from the point of view of the families of their selections. Furthermore, we develop analytical tools suitable to handle random convex (not necessarily compact) sets in Banach spaces; these tools are based on considering support functions as functions of random arguments. The paper is motivated by applications to assessing multivariate risks in mathematical finance.

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