Optimal Couplings of Levy Processes in the Class of Immersion Couplings
Abstract
We prove that any two Lévy processes on ℝd with finite second moments admit a coupling that is itself a Lévy process and is both locally and globally optimal for the quadratic cost among all immersion couplings. This strengthens the optimality result of Kang and Lim (2025), which was established among Feller couplings. Our approach uses a martingale criterion for local optimality that applies more generally to Markov processes. The argument rests on the existence of an optimal Lévy coupling and a dual formulation of the associated optimization problem, both previously established by Kang and Lim (2025). We give simpler, self-contained proofs of these two results using optimal transport between infinitely divisible measures and Lassonde's minimax principle.
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BibTeX
@misc{lim2026optimal,
title = {{Optimal Couplings of Levy Processes in the Class of Immersion Couplings}},
author = {Tau Shean Lim and Ray Shua Ooi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.24290},
eprint = {2606.24290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.24290},
}
RIS
TY - UNPB
TI - Optimal Couplings of Levy Processes in the Class of Immersion Couplings
AU - Tau Shean Lim
AU - Ray Shua Ooi
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2606.24290
UR - https://arxiv.org/abs/2606.24290
AB - We prove that any two Lévy processes on ℝ^d with finite second moments admit a coupling that is itself a Lévy process and is both locally and globally optimal for the quadratic cost among all immersion couplings. This strengthens the optimality result of Kang and Lim (2025), which was established among Feller couplings. Our approach uses a martingale criterion for local optimality that applies more generally to Markov processes. The argument rests on the existence of an optimal Lévy coupling and a dual formulation of the associated optimization problem, both previously established by Kang and Lim (2025). We give simpler, self-contained proofs of these two results using optimal transport between infinitely divisible measures and Lassonde's minimax principle.
ER -