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 Rd is lifted to the space L0(Rd) 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.