论文标题
在圆圈上跳跃布朗运动的最佳耦合
Optimal Coupling of Jumpy Brownian Motion on the Circle
论文作者
论文摘要
考虑在单位圆的圆周上进行布朗尼运动，该运动在汇率$λ$的独立泊松过程的事件时跳到周长的相对点。我们研究了从不同位置开始耦合此“跳跃布朗运动”的两个副本的问题，以优化耦合时间的某些功能。我们描述了两个直观的共同适应耦合（``镜像''和`同步'），仅当两个过程彼此直接相反时，它们只有在彼此对立时有所不同，并表明哪种策略最佳取决于以非平凡的方式跳跃率$λ$。 More precisely, we use the theory of stochastic control to show that there exists a critical value $λ^\star = 0.083\dots$ such that the Mirror coupling minimises the mean coupling time within the class of all co-adapted couplings when $λ<λ^\star$, but for $λ\ge λ^\star$ the Synchronous coupling uniquely maximises the laplace变换$ \ mathbb {e} [e^{ - γt}] $的所有耦合时间$ t $在此类中。我们还提供了（非共同适应的）最大耦合的明确描述，即在两个跳跃的布朗尼运动开始于圆的抗虫点开始的情况下，任何跳跃速率的最大耦合。
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 $λ$. 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 $λ$ in a non-trivial way. More precisely, we use the theory of stochastic control to show that there exists a critical value $λ^\star = 0.083\dots$ such that the Mirror coupling minimises the mean coupling time within the class of all co-adapted couplings when $λ<λ^\star$, but for $λ\ge λ^\star$ the Synchronous coupling uniquely maximises the Laplace transform $\mathbb{E}[e^{-γT}]$ 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.
