Essential Supremum with Respect to a Random Partial Order

Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial order in View the MathML source is lifted to the space View the MathML source of d-dimensional random variables. In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. Applications of the introduced notion to a hedging problem under transaction costs and set-valued dynamic risk measures are given.