Quasi-sure essential supremum and applications to finance

Abstract: When uncertainty is modelled by a set of non-dominated and non-compact probability measures, a notion of essential supremum for a family of real-valued functions is developed in terms of upper semi-analytic functions. We show how the properties postulated on the initial functions carry over to their quasi-sure essential supremum. We propose various applications to financial problems with frictions. We analyse super-replication and prove a bi-dual characterization of the super-hedging cost. We also study a weak no-arbitrage condition called Absence of Instantaneous Profit (AIP) under which prices are finite. This requires new results on the aggregation of quasi-sure statements.