The super-replication theorem under proportional transaction costs revisited

We consider a financial market with one riskless and one risky asset. The super-replication theorem states that there is no duality gap in the problem of super-replicating a contingent claim under transaction costs and the associated dual problem. We give two versions of this theorem. The first theorem relates a num\'eraire-based admissibility condition in the primal problem to the notion of a local martingale in the dual problem. The second theorem relates a num\'eraire -free admissibility condition in the primal problem to the notion of a uniformly integrable martingale in the dual problem.