  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.
