Martingale Inequalities for the Maximum via Pathwise Arguments
Probability
2014-09-23 v1
Abstract
We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry_Labordere et al. (2013) 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 then use our inequalities to recover Doob's L^p inequalities. For p in (0,1] we obtain new, or refined, inequalities.
Cite
@article{arxiv.1409.6255,
title = {Martingale Inequalities for the Maximum via Pathwise Arguments},
author = {Jan Obloj and Peter Spoida and Nizar Touzi},
journal= {arXiv preprint arXiv:1409.6255},
year = {2014}
}