# All Adapted Topologies are Equal

## TL;DR

This paper demonstrates that various topologies on the set of stochastic process laws, developed for different purposes, are actually equivalent in finite discrete time, unifying their theoretical framework.

## Contribution

It proves that all these different adapted topologies coincide in finite discrete time, providing a unified understanding of their structure and properties.

## Key findings

- All adapted topologies are equivalent in finite discrete time.
- The weak adapted topology is characterized by continuity of optimal stopping problems.
- Different approaches to defining topologies on stochastic laws unify under this framework.

## Abstract

A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion-Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle's causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this 'weak adapted topology' is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.