Quasi-sure essential supremum and applications to finance

When uncertainty is modelled by a non-dominated and non-compact set of 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 superreplication and prove a bidual characterisation of the superhedging cost. We also study a weak no-arbitrage condition called absence of instantaneous profit ( $\mathrm{AIP}$ $\mathrm{AIP}$ ) under which prices are finite. This requires new results on the aggregation of quasi-sure statements.