Optimal Coupling of Jumpy Brownian Motion on the Circle

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.