# Fundamental Properties of Process Distances

## TL;DR

This paper explores the fundamental mathematical properties of the nested distance, a metric for comparing stochastic processes that accounts for information flow over time, revealing its topological and geometric structure.

## Contribution

It establishes that the nested distance generates a Polish topology, identifies its completion with nested distributions, and characterizes its extreme points, advancing the theoretical understanding of process distances.

## Key findings

- Nested distance generates a Polish topology.
- Its completion consists of nested distributions.
- Extreme points are characterized by adapted deterministic maps.

## Abstract

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.