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 obtained in the martingale setting. Our construction is based on the Markovian iteration of oneperiod 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.