Essential supremum and essential maximum with respect to random preference relations (Q393279)

The authors first study the concepts of supremum and maximum as subsets of a topological space \(X\) equipped with a preference relation (preorder) given by a family of semicontinuous functions. Several general existence theorems are obtained. Then, a study of the concepts of essential supremum and essential maximum is carried out in a setting where \(X\) is a separable metric space and the preference relation in the space of \(X\)-valued random variables is given by a countable family of Carathéodory functions. Special attention is given to the case where \(X\) is a separable Hilbert space and the preorder is given by a random cone. An application to the hedging problem for American-type contingent claims under transaction costs concludes the paper.