Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 46N30 Applications in probability theory and statistics
- 46A20 Duality theory
- 46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
- 06A06 Partial order, general

**Résumé:** In the present paper, we prove a multidimensional
extension of the so-called Bipolar Theorem proved in
a paper by W. Brannath and W. Schachermayer
(\cite{BS}), which says that the bipolar of a convex
set of positive random values 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 \cite{BS} is strongly
dependent on the order properties of $\R$. Here, we
define a (partial) order structure with respect to a
$d$-dimensional convex cone $K$ of the positive
orthant $[0,\infty)^d$. 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 \cite{BS}. As a
byproduct, we obtain an equivalence property for a
class of minimization problems. Finally, we discuss
some applications in the context of the duality
theory of utility maximization problem in financial
markets with proportional transaction costs.

**Mots Clés:** *polarity ; convex analysis ; partial order ; convergence in probability ;
dual formulation in mathematical finance*

**Date:** 2002-05-14

**Prépublication numéro:** *PMA-725*

**Pdf file : **PMA-725.pdf