Supermartingale shadow couplings: the decreasing case

DOI10.3150/23-BEJ1592arXiv2207.11732OpenAlexW4388514237MaRDI QIDQ6178554

Dominykas Norgilas, Shuoqing Deng, Erhan Bayraktar

Publication date: 16 January 2024

Published in: Bernoulli (Search for Journal in Brave)

Abstract: For two measures $mu$ and $u$ that are in convex-decreasing order, Nutz and Stebegg (Canonical supermartingale couplings, Ann. Probab., 46(6):3351--3398, 2018) studied the optimal transport problem with supermartingale constraints and introduced two canonical couplings, namely the increasing and decreasing transport plans, that are optimal for a large class of cost functions. In the present paper we provide an explicit construction of the decreasing coupling $p{i}^D$ by establishing a Brenier-type result: (a generalised version of) $p{i}^D$ concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale extit{shadow} measure and requires a suitable extension of the results by Juillet (Stability of the shadow projection and the left-curtain coupling, Ann. Inst. H. Poincar'e Probab. Statist., 52(4):1823--1843, November 2016) and Beiglb"ock and Juillet (Shadow couplings, Trans. Amer. Math. Soc., 374:4973--5002, 2021) established in the martingale setting. In particular, we prove the stability of the supermartingale shadow measure with respect to initial and target measures $mu,u$, introduce an infinite family of lifted supermartingale couplings that arise via shadow measure, and show how to explicitly determine the `martingale points' of each such coupling.

Full work available at URL: https://arxiv.org/abs/2207.11732

zbMATH Keywords

stabilitysupermartingalesoptimal transportBrenier's theorempeacocksconvex-decreasing order

Mathematics Subject Classification ID

Martingales with discrete parameter (60G42) Variational problems in a geometric measure-theoretic setting (49Q20) Stopping times; optimal stopping problems; gambling theory (60G40) Martingales with continuous parameter (60G44) Derivative securities (option pricing, hedging, etc.) (91G20)

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