Abstract
We study optimal Markovian couplings of Markov processes, where the optimality is understood in terms of minimization of concave transport costs between evaluations of the coupled processes at corresponding times. We provide explicit constructions of such optimal couplings for one-dimensional finite-activity Lévy processes (continuous-time random walks) whose jump distributions are unimodal but not necessarily symmetric. Remarkably, the optimal Markovian coupling does not depend on the specific concave transport cost. To this end, we combine McCann’s results on optimal transport and Rogers’ results on random walks with a novel uniformization construction that allows us to characterize all Markovian couplings of finite-activity Lévy processes. In particular, we show that the optimal Markovian coupling for finite-activity Lévy processes with non-symmetric unimodal Lévy measures has to allow for non-simultaneous jumps of the two coupled processes.
Acknowledgements
This is a theoretical research paper and, as such, no new data were created during this study. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising. The first author acknowledges support by the EPSRC grant EP/K013939. A substantial part of this work was completed while the second author was affiliated to the University of Warwick and supported by the EPSRC grant EP/P003818/1. The third author was supported by the EPSRC grants EP/P003818/1, EP/V009478/1 & EP/W006227/1. The first and third authors were also supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1 and The Turing Institute Programme on Data-Centric Engineering funded by the Lloyd’s Register Foundation.
Citation
Wilfrid S. Kendall. Mateusz B. Majka. Aleksandar Mijatović. "Optimal Markovian coupling for finite activity Lévy processes." Bernoulli 30 (4) 2821 - 2845, November 2024. https://doi.org/10.3150/23-BEJ1696
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