A quasi-sure optional decomposition and super-hedging result on the Skorokhod space

We prove a robust super-hedging duality result for path-dependent options on assets with jumps, in a continuous time setting. It requires that the collection of martingale measures is rich enough and that the payoff function satisfies some continuity property. It is a by-product of a quasi-sure version of the optional decomposition theorem, which can also be viewed as a functional version of It{\^o}'s Lemma, that applies to non-smooth functionals (of c{\`a}dl{\`a}g processes) which are only concave in space and non-increasing in time, in the sense of Dupire.