Essential Supremum and Essential Maximum with Respect to Random Preference Relations

Abstract

In the first part of the paper we study concepts of supremum and maximum as subsets of a topological space X endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper we consider partial orders and preference relations "lifted" from a metric separable space X endowed by a random preference relation to the space L^0(X)X-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.

Keywords: Preference Relation, Partial Order, Random cones, Transaction costs, American option, Hedging

JEL Classification: G11, G13