A Version of the $${\cal G} \rm{-conditional Bipolar Theorem in} {L^0 (\mathbb{R}^{d}_{+};\Omega}, {\cal F},\mathbb{P})$$

B. Bouchard ()
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B. Bouchard: Université Paris VI–Tour 56

Journal of Theoretical Probability, 2005, vol. 18, issue 2, 439-467

Abstract: Abstract Motivated by applications in financial mathematics, Ref. 3 showed that, although $$L^{0}(\mathbb{R}_{+}; \Omega, {\cal F}, \mathbb{P})$$ fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of $$L^{0}(\mathbb{R}_{+}; \Omega, {\cal F}, \mathbb{P})$$ : 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 Ref. 1 in the multidimensional case, replacing $$\mathbb{R}_{+}$$ by a closed convex cone K of [0, ∞) d , and by Ref. 12 who provided a conditional version in the unidimensional case. In this paper, we show that the conditional bipolar theorem of Ref. 12 can be extended to the multidimensional case. Using a decomposition result obtained in Ref. 3 and Ref. 1, we also remove the boundedness assumption of Ref. 12 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.

Keywords: Bipolar theorem; convex analysis; partial order (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/s10959-005-3512-y

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