A one-sided constrained martingale transport between two uniform laws
Abstract
We minimize 𝔼[h(Y−X)] over martingale couplings of Unif[−1,1] and Unif[−2,2] satisfying Y≥ X−k, where h∈ C1([−3,3]) has convex derivative. Feasibility holds exactly for k≥1. For each such k, we construct a coupling that minimizes all costs in this class. For 1<k<3, its support consists of two graphs, with D(x)=x−k on [k−2,1]. The maps admit an explicit parametrization for 2≤ k<3 and are determined by scalar equations with unique admissible roots for 1<k<2. We prove optimality by a dual inequality, using an analytic estimate in the latter range. At k=1 the optimizer is Y=X±1 with equal probabilities; for k≥3 it is the ordinary left-curtain coupling. In the unconstrained problem, left-monotonicity identifies the left-curtain coupling, which is optimal for this cost class. Under the constraint, a discrete example shows that the corresponding support condition, even together with every two-source comparison, does not imply optimality.
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BibTeX
@misc{bayraktar2026onesided,
title = {{A one-sided constrained martingale transport between two uniform laws}},
author = {Erhan Bayraktar and Xin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29997},
eprint = {2609.29997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29997},
}
RIS
TY - UNPB
TI - A one-sided constrained martingale transport between two uniform laws
AU - Erhan Bayraktar
AU - Xin Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29997
UR - https://arxiv.org/abs/2609.29997
AB - We minimize 𝔼[h(Y−X)] over martingale couplings of Unif[−1,1] and Unif[−2,2] satisfying Y≥ X−k, where h∈ C^1([−3,3]) has convex derivative. Feasibility holds exactly for k≥1. For each such k, we construct a coupling that minimizes all costs in this class. For 1<k<3, its support consists of two graphs, with D(x)=x−k on [k−2,1]. The maps admit an explicit parametrization for 2≤ k<3 and are determined by scalar equations with unique admissible roots for 1<k<2. We prove optimality by a dual inequality, using an analytic estimate in the latter range. At k=1 the optimizer is Y=X±1 with equal probabilities; for k≥3 it is the ordinary left-curtain coupling. In the unconstrained problem, left-monotonicity identifies the left-curtain coupling, which is optimal for this cost class. Under the constraint, a discrete example shows that the corresponding support condition, even together with every two-source comparison, does not imply optimality.
ER -