Supermartingale Brenier's Theorem with full-marginals constraint

TL;DR

This paper constructs a supermartingale version of the Fréchet-Hoeffding coupling with infinite marginals, extending martingale results, and connects it to local Lévy models and supermartingale optimal transport.

Contribution

It introduces an explicit supermartingale coupling with infinitely many marginals, extending previous martingale results and linking to local Lévy processes.

Findings

Constructed a supermartingale coupling with infinite marginals.

Connected the coupling to local Lévy models in the limit.

Provided explicit computations for key financial models.

Abstract

We explicitly construct the supermartingale version of the Fr{\'e}chet-Hoeffding coupling in the setting with infinitely many marginal constraints. This extends the results of Henry-Labordere et al. obtained in the martingale setting. Our construction is based on the Markovian iteration of one-period optimal supermartingale couplings. In the limit, as the number of iterations goes to infinity, we obtain a pure jump process that belongs to a family of local L{\'e}vy models introduced by Carr et al. We show that the constructed processes solve the continuous-time supermartingale optimal transport problem for a particular family of path-dependent cost functions. The explicit computations are provided in the following three cases: the uniform case, the Bachelier model and the Geometric Brownian Motion case.