2023/12/27 by Mathias Beiglböck, Gudmund Pammer, Beiglböck, Mathias +5
Computer Science · Mathematics · #Data Management and Algorithms #FOS: Mathematics #Probability (math.PR) #Probability and Statistical Research #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2312.16725
openalex publication_date 2023/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Random variables Xi, i=1,2 are 'probabilistically equivalent' if they have the same law. Moreover, in any class of equivalent random variables it is easy to select canonical representatives. The corresponding questions are more involved for processes Xi on filtered stochastic bases (Ωi, \mathcal Fi, \mathbb Pi, (\mathcal Fit)t∈ [0,1]). Here equivalence in law does not capture relevant properties of processes such as the solutions to stochastic control or multistage decision problems. This motivates Aldous to introduce the stronger notion of synonymity based on prediction processes. Stronger still, Hoover--Keisler formalize what it means that Xi, i=1,2 have the same probabilistic properties. We establish that canonical representatives of the Hoover--Keisler equivalence classes are given precisely by the set of all Markov-martingale laws on a specific nested path space \mathsf M_∞. As a consequence we obtain that, modulo Hoover--Keisler equivalence, the class of stochastic processes forms a Polish space. On this space, processes are topologically close iff they model similar probabilistic phenomena. In particular this means that their laws as well as the information encoded in the respective filtrations are similar. Importantly, compact sets of processes admit a Prohorov-type characterization. We also obtain that for every stochastic process, defined on some abstract basis, there exists a process with identical probabilistic properties which is defined on a standard Borel space.