Supermartingale Brenier's Theorem with full-marginals constraint

Abstract

We explicitly construct the supermartingale version of the Fréchet-Hoeffding coupling in the setting with infinitely many marginal constraints. This extends the results of Henry-Labordère et al. [41] 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 Lévy models introduced by Carr et al. [22]. 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.