The maximum maximum of a martingale constrained by an intermediate law

Let (M-t) be any martingale with M-0 = 0, an intermediate law M-1 similar t o mu (1), and terminal law M-2 similar to mu (2), and let (M) over bar (2) = sup(0 less than or equal to1 less than or equal to2) M-t. In this paper w e prove that there exists an upper bound, with respect to stochastic orderi ng of probability measures, on the law of (M) over bar (2). We construct, u sing excursion theory, a martingale which attains this maximum. Finally we apply this result to the robust hedging of a lookback option.