All adapted topologies are equal

Julio Backhoff-Veraguas; Daniel Bartl; Mathias Beiglböck; Manu Eder

doi:10.1007/s00440-020-00993-8

Titel

All adapted topologies are equal

Autor*in

Julio Backhoff-Veraguas

Institut für Mathematik, Fakultät für Mathematik, Universität Wien

Daniel Bartl

Institut für Mathematik, Fakultät für Mathematik, Universität Wien

Mathias Beiglböck

Institut für Mathematik, Fakultät für Mathematik, Universität Wien

... show all

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.

Stichwort

Aldous’ extended weak topologyHellwig’s information topologyNested distanceCausal optimal transportStability of optimal stoppingVershik’s iterated Kantorovich distance

Objekt-Typ

journal article

Sprache

Englisch [eng]

Persistent identifier

phaidra.univie.ac.at/o:1218519

DOI

10.1007/s00440-020-00993-8

Erschienen in

Titel

Probability Theory and Related Fields

Band