A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g. Plug-Pichler - stochastic programming, Hellwig - game theory, Aldous - stability of optimal stopping, Hoover-Keisler - model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality. A natural compatible metric for the weak adapted topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug-Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.
翻译:一些研究人员独立地提出了一套扩展通常薄弱地形学的随机过程法的地形学,这取决于各自的科学背景,其动机是应用和与各个领域的联系(例如Plug-Pichler-Stochastic 编程、Hellwig-游戏理论、Aldous-最佳停车的稳定性、Hoover-Keisler-模型理论)。值得注意的是,所有这些看似独立的方法在有限的离散时间里界定了相同的适应性薄弱地形学。我们的第一个主要成果是建立一个经调整的经验措施的变异体,不断全面估计随机过程的规律。经调整的薄弱地形学的自然兼容度指标是经调整的Wasserstein距离的精细度,如Pflug-Pichler的精度作品所确立的那样。具体地说,经调整的瓦瑟斯坦距离可以控制在随机优化问题、定价和套期问题、最佳制止问题等方面出现的错误。在利普西茨维茨维茨模式上,这篇文章的第二个主要结果是为经调整的经验性测量度指标的趋近,我们所了解的史地获得的萨勒斯特列斯特列斯特里什内最佳浓度。