A multidimensional bipolar theorem in L0(Rd;Ω,F,P)

In this paper, we prove a multidimensional extension of the so-called Bipolar Theorem proved in Brannath and Schachermayer (Séminaire de Probabilités, vol. XXX, 1999, p. 349), which says that the bipolar of a convex set of positive random variables 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 Brannath and Schachermayer (ibidem) 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,∞)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 Brannath and Schachermayer (ibidem). As a byproduct, we obtain an equivalence property for a class of minimization problems in the spirit of Kramkov and Schachermayer (Ann. Appl. Probab 9(3) (1999) 904, Proposition 3.2). Finally, we discuss some applications in the context of duality theory of the utility maximization problem in financial markets with proportional transaction costs.