Supermartingale Brenier's Theorem with full-marginals constraint

Erhan Bayraktar, Shuoqing Deng and Dominykas Norgilas

Papers from arXiv.org

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.

Date: 2022-12
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Chapter: Supermartingale Brenier’s Theorem with Full-Marginal Constraint (2023)
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