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

### A version of the $\cal{G}$-conditional bipolar theorem in $L^0(\mathbb{R}^d_+;\Omega,\cal{F},\mathbb{P})$}

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: Motivated by applications in financial mathematics, Brannath and Schachermayer (1999) showed that, although $L^0(\mathbb{R}^d_+;\Omega,\cal{F},\mathbb{P})$ fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of $L^0(\mathbb{R}^d_+;\Omega,\cal{F},\mathbb{P})$~: if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extended by Bouchard and Mazliak (2003) in the multidimensional case, replacing $\mathbb{R}_+$ by a closed convex cone $K$ of $[0,\infty)^d$, and by Zitkovi\'c (2002), who provided a conditional version in the unidimensional case. In this paper, we show that the conditional bipolar theorem of Zitkovi\'c (2002) can be extended to the multidimensional case. Using a decomposition result obtained in Brannath and Schachermayer (1999) and Bouchard and Mazliak (2003), we also remove the boundedness assumption of Zitkovi\'c (2002) in the one dimensional case and provide less restrictive assumptions in the multidimensional case. These assumptions are completely removed in the case of polyhedral cones $K$.

Mots Clés: bipolar theorem ; convex analysis ; partial order

Date: 2003-09-30

Prépublication numéro: PMA-847