Information is an inherent component of stochastic processes and to measure the distance between different stochastic processes it is not sufficient to consider the distance between their laws. Instead, the information which accumulates over time and which is mathematically encoded by filtrations has to be accounted for as well. The nested distance/bicausal Wasserstein distance addresses this challenge by incorporating the filtration. It is of emerging importance due to its applications in stochastic analysis, stochastic programming, mathematical economics and other disciplines. This article establishes a number of fundamental properties of the nested distance. In particular we prove that the nested distance of processes generates a Polish topology but is itself not a complete metric. We identify its completion to be the set of nested distributions, which are a form of generalized stochastic processes. We also characterize the extreme points of the set of couplings which participate in the definition of the nested distance, proving that they can be identified with adapted deterministic maps. Finally, we compare the nested distance to an alternative metric, which could possibly be easier to compute in practical situations.

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过程距离的基本属性

信息是随机过程的固有组成部分，要衡量不同随机过程之间的距离，仅考虑它们的规律之间的距离是不够的。取而代之的是，还必须考虑随时间累积并通过过滤进行数学编码的信息。嵌套距离/双因果 Wasserstein 距离通过合并过滤解决了这一挑战。由于其在随机分析、随机规划、数理经济学和其他学科中的应用，它具有新兴的重要性。本文建立了嵌套距离的许多基本属性。我们特别证明了过程的嵌套距离生成了波兰拓扑，但它本身并不是一个完整的度量。我们将其完成确定为一组嵌套分布，这是一种广义随机过程。我们还描述了参与嵌套距离定义的耦合集的极值点，证明它们可以用适应的确定性映射来识别。最后，我们将嵌套距离与替代度量进行比较，这在实际情况下可能更容易计算。