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

Supermartingale Brenier's Theorem with full-marginals constraint