We study the problem of bounding path-dependent expectations (within any finite time horizon $d$) over the class of discrete-time martingales whose marginal distributions lie within a prescribed tolerance of a given collection of benchmark marginal distributions. This problem is a relaxation of the martingale optimal transport (MOT) problem and is motivated by applications to super-hedging in financial markets. We show that the empirical version of our relaxed MOT problem can be approximated within $O\left( n^{-1/2}\right)$ error where $n$ is the number of samples of each of the individual marginal distributions (generated independently) and using a suitably constructed finite-dimensional linear programming problem.
翻译:我们研究了将依赖路径的预期(在任何有限的时间范围内以美元计算)与离散时间马丁堡等级挂钩的问题,这些马丁堡的边际分布是在对某一集的基准边际分布的一定容忍度之内的,这是马丁加勒最佳运输(MOT)问题的缓解,其动机是金融市场的超载应用。我们表明,我们放松的MOT问题的经验性版本可以与美元左翼(n ⁇ -1/2 ⁇ right)差相近,因为美元是每个个别边际分布(独立产生)的样本数量,并使用适当构建的有限线性编程问题。