Supermartingale shadow couplings: The decreasing case

Abstract

For two measures μ and ν 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 ${\mathrm{\pi }}^D$ by establishing a Brenier-type result: (a generalised version of) ${\mathrm{\pi }}^D$ concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale 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. Poincaré Probab. Statist. 52(4) 1823–1843, November 2016) and Beiglböck 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 $\mathrm{\mu },\mathrm{\nu }$, 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.