发表机构
Department of Mathematics, ETH Zürich（苏黎世联邦理工学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一维 $q$-Bass 鞅的不动点问题，在一般凸序边际下证明存在性、唯一性、稳定性及 $\mathcal{W}_\infty$-收敛性，并推广 Benamou-Brenier 公式至任意加性参考过程。
AI 中文摘要
我们建立了一维 $q$-Bass 鞅的存在性、唯一性、稳定性与收敛性结果，这类鞅的特征在于其转移核最接近参考测度 $q$，且具有给定的初始与终端边际分布。其存在性等价于一个关于概率分布的不动点问题的可解性。基于 Acciaio 和 Marini (2026) 的工作（该工作要求第一边际分布支撑在有限多个点上），我们研究凸序下一般边际分布的情形。在假设 $q\ll\lambda$ 下，我们证明了不动点分布的存在性、唯一性与稳定性，不动点迭代的 $\mathcal{W}_\infty$-收敛性，以及支撑直径估计。我们还把鞅 Benamou-Brenier 公式从布朗运动推广到任意加性参考过程 $X$，并证明相应的 $X$-Bass 鞅在存在时是最优的，且可解释为 $X$ 的适应 Wasserstein 投影。
英文摘要
We establish existence, uniqueness, stability, and convergence results for one-dimensional $q$-Bass martingales, characterized as the martingales with prescribed initial and terminal marginals whose transition kernels are closest to a reference measure $q$. Their existence is equivalent to the solvability of a fixed-point problem for probability distributions. Building on Acciaio and Marini (2026), that requires the first marginal to be supported on finitely many points, we study the case of general marginals in convex order. Under the assumption that $q\llλ$, we prove existence, uniqueness and stability of fixed-point distributions, $\mathcal{W}_\infty$-convergence of the fixed-point iteration, and support-diameter estimates. We also extend the martingale Benamou-Brenier formula from Brownian motion to any additive reference process $X$ and show that the corresponding $X$-Bass martingale is optimal whenever it exists, with an interpretation as an adapted Wasserstein projection of $X$.
