An Explicit Martingale Version of Brenier's Theorem

TL;DR

This paper extends Brenier's theorem to a martingale setting, providing explicit optimal transport plans and hedging strategies for a class of coupling functions, with applications in financial mathematics.

Contribution

It introduces an explicit martingale version of Brenier's theorem, including optimal plans and hedging strategies, extending to multiple marginals.

Findings

Explicit extremal martingale measures for certain coupling functions.

Connection between martingale optimal transport and semi-static hedging strategies.

Extension of results to multiple marginals.

Abstract

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.