Journal
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Volume 130, Issue 9, Pages 5575-5591Publisher
ELSEVIER
DOI: 10.1016/j.spa.2020.03.017
Keywords
Optimal transport; Nested distance; Martingales; Causal Wasserstein distance; Information topology
Categories
Funding
- Austrian Science Fund (FWF) [Y782-N25]
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To quantify the difference of distinct stochastic processes it is not sufficient to consider the distance of their states and corresponding probabilities. Instead, the information, which evolves and accumulates over time and which is mathematically encoded by filtrations, has to be accounted for as well. The nested distance, also known as bicausal Wasserstein distance, recognizes this component and involves the filtration properly. This distance is of emerging importance due to its applications in stochastic analysis, stochastic programming, mathematical economics and other disciplines. This paper investigates the basic metric and topological properties of the nested distance on the space of discrete-time processes. In particular we prove that the nested distance generates a Polish topology, although the genuine space is not complete. Moreover we identify its completion to be the space of nested distributions, a space of generalized stochastic processes. (C) 2020 Published by Elsevier B.V.
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