A version of the G - conditional bipolar theorem

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