2019/05/31 by Julio Backhoff‐Veraguas, Julio Backhoff-Veraguas, Daniel Bartl +2 · 54 citations
Economics, Econometrics and Finance · Mathematics · #Bounded function #Combinatorics #Comparison of topologies #Computer science #Convergence (economics) #Discrete mathematics #Extension topology #General topology #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Network topology #Random Matrices and Applications #Stochastic processes and financial applications #Symmetrization #Topological space #Topology (electrical circuits) #Weak topology (polar topology) #math.PR #msc:60B10 #msc:60G07
paper · pdf · doi:10.1007/s00440-020-00993-8
published in Probability Theory and Related Fields 178(3-4), 1125-1172 (Springer Science+Business Media) · Minor clarifying changes; 37 pages
crossref issued 2020/09/14 · crossref published 2020/09/14 · crossref published-online 2020/09/14 · openalex publication_date 2020/09/14 · crossref created 2020/09/14 · arxiv created 2020/09/29 · arxiv updated 2020/09/30 · crossref published-print 2020/12/01 · crossref deposited 2021/09/13 · openalex created_date 2025/10/10 · crossref indexed 2026/07/31 · openalex updated_date 2026/08/05
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.