  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.
