Supermartingale shadow couplings: the decreasing case

For two measures $\mu$ and $\nu$ 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 $\pi^D$ by establishing a Brenier-type result: (a generalised version of) $\pi^D$ concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale \textit{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 $\mu,\nu$, 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.