Martingale Inequalities and Deterministic Counterparts

TL;DR

This paper links martingale inequalities to deterministic inequalities through nonlinear operators, providing insights into their optimal bounds and applications in mathematical finance.

Contribution

It introduces a method to reduce martingale inequalities to deterministic inequalities using fixed points of nonlinear operators, explaining their role in robust hedging.

Findings

Martingale inequalities can be characterized by fixed points of a nonlinear operator.

Optimal bounds are determined via concave envelopes in a small variable set.

Results clarify the connection between martingale inequalities and financial applications.

Abstract

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.