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:** 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*