All adapted topologies are equal

By Julio Backhoff-Veraguas, Daniel Bartl, Mathias Beiglbock, Manu Eder. Probability theory and related fields, 2019

78 citations recorded by the source corpus.

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.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/edf9fd0979ef6e64c729ca11db080e2e92d6e935

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Citations in the corpus

The Wasserstein space of stochastic processes (2021, methodology): “Its proof is similar to [11] and is included here for the convenience of the”
Estimating processes in adapted Wasserstein distance (2020, background): “Adapted distances / topologies on laws of processes are of fundamental importance in questions of stability in mathematical finance and stochastic control, see [13, 5, 18, 1, 9, 3, 7].”
Adapted Wasserstein distances and stability in mathematical finance (2019, background): “The articles [8, 6, 23] investigate basic properties of this topology; e.”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “[33, 5, 4]) and will allow us to carry out proofs that closely resemble familiar arguments from classical optimal transport.”