Optimal Coupling of Jumpy Brownian Motion on the Circle

TL;DR

This paper investigates optimal coupling strategies for jumpy Brownian motion on a circle, revealing a phase transition at a critical jump rate where different couplings become optimal for minimizing or maximizing coupling times.

Contribution

It introduces a stochastic control framework to determine optimal co-adapted couplings for jumpy Brownian motion, identifying a critical jump rate that switches the optimal strategy.

Findings

Existence of a critical jump rate $oxed{0.083}$ where coupling strategies switch.

Mirror coupling minimizes mean coupling time for low jump rates.

Synchronous coupling maximizes the Laplace transform of coupling time for high jump rates.

Abstract

Consider a Brownian motion on the circumference of the unit circle, which jumps to the opposite point of the circumference at incident times of an independent Poisson process of rate $\lambda$. We examine the problem of coupling two copies of this `jumpy Brownian motion' started from different locations, so as to optimise certain functions of the coupling time. We describe two intuitive co-adapted couplings (`Mirror' and `Synchronous') which differ only when the two processes are directly opposite one another, and show that the question of which strategy is best depends upon the jump rate $\lambda$ in a non-trivial way. More precisely, we use the theory of stochastic control to show that there exists a critical value $\lambda^\star = 0.083\dots$ such that the Mirror coupling minimises the mean coupling time within the class of all co-adapted couplings when $\lambda<\lambda^\star$, but for $\lambda\ge \lambda^\star$ the Synchronous coupling uniquely maximises the Laplace transform $\mathbb{E}[e^{-\gamma T}]$ of all coupling times $T$ within this class. We also provide an explicit description of a (non co-adapted) maximal coupling for any jump rate in the case that the two jumpy Brownian motions begin at antipodal points of the circle.