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Supermartingale Brenier's Theorem with full-marginals constraint

Author

Listed:
  • Erhan Bayraktar
  • Shuoqing Deng
  • Dominykas Norgilas

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.

Suggested Citation

  • Erhan Bayraktar & Shuoqing Deng & Dominykas Norgilas, 2022. "Supermartingale Brenier's Theorem with full-marginals constraint," Papers 2212.14174, arXiv.org.
    • Handle: RePEc:arx:papers:2212.14174
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    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling

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