向量鞅最优运输与对偶性的几何
摘要
最优运输(Optimal Transport, OT)与鞅最优运输(Martingale Optimal Transport, MOT)理论受经济学与金融学问题启发，在过去几十年蓬勃发展，在理论与实践上取得重大进展。MOT考虑一种金融工具（称为期权）的定价与对冲问题，假设其收益依赖于单一资产价格。本文引入向量鞅最优运输(Vectorial Martingale Optimal Transport, VMOT)问题，考虑更普遍且现实的情境，即期权收益依赖于多种资产价格。我们处理这一给定市场信息（由标的资产价格的向量边际分布描述）的定价与对冲问题，这是稳健金融框架中密切相关的设置。我们确立VMOT问题作为无限维线性规划，其对偶规划存在最优解。这一对偶最优解的存在性结果意义重大：对偶最优解描述了一个负责期权收益支付的人如何构建最优对冲组合，更重要的是，它们能提供关于原始最优解（即VMOT）几何结构的关键信息。作为示例，我们表明给定边际的多个鞅在联合优化距离型成本函数期望时必然展现极端的条件下相关性结构。
引用
@article{arxiv.1611.01496,
title = {Geometry of vectorial martingale optimal transportations and duality},
author = {Tongseok Lim},
journal= {arXiv preprint arXiv:1611.01496},
year = {2023}
}
备注
The first version was titled "Multi-martingale optimal transport," which was later renamed "Geometry of multi-marginal martingale optimal transportations and duality," and the problem was dubbed "Multi-martingale optimal transport problem (MMOT)." In this version, we refer to the problem as "Vectorial martingale optimal transport problem (VMOT)". v4 will be published in Mathematical Programming