Optimal Markovian coupling for finite activity L\'evy processes

TL;DR

This paper constructs explicit optimal Markovian couplings for one-dimensional finite-activity Lévy processes with unimodal jump distributions, showing that optimality is independent of the specific concave transport cost.

Contribution

It provides a novel uniformization method to characterize all Markovian couplings for these processes and identifies the structure of the optimal coupling, including non-simultaneous jumps.

Findings

Optimal Markovian couplings are explicitly constructed.

Optimality does not depend on the specific concave transport cost.

Non-simultaneous jumps are necessary for optimal coupling with non-symmetric measures.

Abstract

We study optimal Markovian couplings of Markov processes, where the optimality is understood in terms of minimization of concave transport costs between the time-marginal distributions of the coupled processes. We provide explicit constructions of such optimal couplings for one-dimensional finite-activity L\'evy 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 L\'evy processes. In particular, we show that the optimal Markovian coupling for finite-activity L\'evy processes with non-symmetric unimodal L\'evy measures has to allow for non-simultaneous jumps of the two coupled processes.