论文标题
Supermartingale影子耦合：减少案例
Supermartingale shadow couplings: the decreasing case
论文作者
论文摘要
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大量的成本功能。在本文中，我们通过建立brenier型结果来提供减少耦合$π^d $的明确结构：（）$π^d $ compentes of两个函数的图表。 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. Poincaré Probab. Statist., 52(4):1823--1843, November 2016) and Beiglböck and Juillet (Shadow couplings, Trans. Amer。特别是，我们证明了超级智能阴影量度相对于初始和目标度量$μ，ν$的稳定性，引入了通过阴影措施产生的无限升起的超级马丁盖尔耦合家族，并显示如何明确确定每种此类耦合的``martingale points''。
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 $π^D$ by establishing a Brenier-type result: (a generalised version of) $π^D$ 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. 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 $μ,ν$, 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.
