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We study martingale inequalities from an analytic point of view and show thata general martingale inequality can be reduced to a pair of deterministicinequalities in a small number of variables. More precisely, the optimal boundin the martingale inequality is determined by a fixed point of a simplenonlinear operator involving a concave envelope. Our results yield anexplanation for certain inequalities that arise in mathematical finance in thecontext of robust hedging.

Martingale Inequalities and Deterministic Counterparts