Supermartingale Brenier's Theorem with full-marginals constraint

Erhan Bayraktar; Shuoqing Deng; Dominykas Norgilas

doi:10.3934/fmf.2023010

Article Contents

2023, Volume 2, Issue 2: 202-243. Doi: 10.3934/fmf.2023010 open access

This issue Previous Article Representation for martingales living after a random time with applications Next Article Option returns

Supermartingale Brenier's Theorem with full-marginals constraint

1.
Department of Mathematics, University of Michigan, USA
2.
Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong

^*Corresponding author Erhan Bayraktar
^*Corresponding author Erhan Bayraktar

Received: September 2022

Revised: November 2022

Early access: March 2023

Published: June 2023

The first author is supported by the National Science Foundation under grant DMS-2106556 and by the Susan M. Smith chair. The second author is supported by the HKUST Start-up grant no. R9826

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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.

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Mathematics Subject Classification: Primary: 60G40, 60G05; Secondary: 49M29.

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