Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A multidimensional bipolar theorem in $L^0(\R^d;\Omega,\Fc,P)$}

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