We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operator involving a concave envelope. Our results yield an explanation for certain inequalities that arise in mathematical finance in the context of robust hedging.
"Martingale inequalities and deterministic counterparts." Electron. J. Probab. 19 1 - 15, 2014. https://doi.org/10.1214/EJP.v19-3270