Statistics and Probability; Statistics, Probability and Uncertainty; 49N05; Martingale Optimal Transport Kantorovich Duality AMS 2010 Subject Classification 60G42
We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.
Mathematical Methods in Economics: 50% Fundamental Mathematics Research: 50%