Martingale Inequalities and Deterministic Counterparts
  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. 
Mathias Beiglb\"ock, Marcel Nutz
math.PR q-fin.PR