Martingale inequalities for the maximum via pathwise arguments

We study a class of martingale inequalities involving the running max- imum process. They are derived from pathwise inequalities introduced by Henry- Labordère et al. (Ann. Appl. Probab., 2015 [arxiv:1203.6877v3]) and provide an upper bound on the expectation of a function of the running maximum in terms of marginal distributions at n intermediate time points. The class of inequalities is rich and we show that in general no inequality is uniformly sharp—for any two inequalities we specify martingales such that one or the other inequality is sharper. We use our inequalities to recover Doob’s L^p inequalities. Further, for p = 1 we refine the known inequality and for p < 1 we obtain new inequalities.