AI 中文总结
研究半离散情形下q - 巴斯鞅的存在性，采用几何方法，通过分析凸多边形链的合适参数化，证明了当初始边际支撑在有限多个原子上时q - 巴斯鞅存在且相关巴斯测度在加法平移常数意义下唯一。
AI 中文摘要
q - 巴斯鞅类为鞅最优传输中的核心问题提供了自然答案：如何构造具有规定初始和终端边际且转移核尽可能接近给定参考测度q的鞅。我们证明了当初始边际支撑在有限多个原子上时q - 巴斯鞅的存在性，并建立了相关巴斯测度在加法平移常数意义下的唯一性。我们的方法是几何的，依赖于对凸多边形链的合适参数化分析。
英文摘要
The class of $q$-Bass martingales provides a natural answer to a central question in martingale optimal transport: how to construct martingales with prescribed initial and terminal marginals whose transition kernel remains as close as possible to a given reference measure $q$. We prove the existence of $q$-Bass martingales when the initial marginal is supported on finitely many atoms, and establish uniqueness, up to an additive translation constant, of the associated Bass measure. Our approach is geometric and relies on the analysis of a suitable parametrization of convex polygonal chains.
