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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 numéraire-based admissibility condition in the primal problem to the notion of a local martingale in the dual problem. The second theorem relates a numéraire -free admissibility condition in the primal problem to the notion of a uniformly integrable martingale in the dual problem.