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A multidimensional bipolar theorem in L0(Rd;P)

Bouchard, Bruno; Mazliak, Laurent (2003), A multidimensional bipolar theorem in L0(Rd;P), Stochastic Processes and their Applications, 107, 2, p. 213-231. http://dx.doi.org/10.1016/S0304-4149(03)00073-5

Type
Article accepté pour publication ou publié
Date
2003
Journal name
Stochastic Processes and their Applications
Volume
107
Number
2
Publisher
Elsevier
Pages
213-231
Publication identifier
http://dx.doi.org/10.1016/S0304-4149(03)00073-5
Metadata
Show full item record
Author(s)
Bouchard, Bruno
Mazliak, Laurent
Abstract (EN)
In this paper, we prove a multidimensional extension of the so-called Bipolar Theorem proved in Brannath and Schachermayer (Séminaire de Probabilités, vol. XXX, 1999, p. 349), which says that the bipolar of a convex set of positive random variables is equal to its closed, solid convex hull. This result may be seen as an extension of the classical statement that the bipolar of a subset in a locally convex vector space equals its convex hull. The proof in Brannath and Schachermayer (ibidem) is strongly dependent on the order properties of Image . Here, we define a (partial) order structure with respect to a d-dimensional convex cone K of the positive orthant [0,∞)d. We may then use compactness properties to work with the first component and obtain the result for convex subsets of K-valued random variables from the theorem of Brannath and Schachermayer (ibidem). As a byproduct, we obtain an equivalence property for a class of minimization problems in the spirit of Kramkov and Schachermayer (Ann. Appl. Probab 9(3) (1999) 904, Proposition 3.2). Finally, we discuss some applications in the context of duality theory of the utility maximization problem in financial markets with proportional transaction costs.
Subjects / Keywords
Convex analysis; Probability

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