A complete characterization of local martingales which are functions of Brownian motion and its maximum

We prove the max-martingale conjecture of Obloj and Yor. We show that for a continuous local martingale (Nt : t ^ 0) and a function H : U X U+ ? U, H(Nt, sup5^, Ns) is a local martingale if and only if there exists a locally integrable function / such that H(x, y) = ^ f(s)ds f(y)(x y) + H(0, 0). This readily implies, via Levy's equivalence theorem, an analogous result with the maximum process replaced by the local time at 0.