状态切换鞅的耦合与跟踪
Coupling and tracking of regime-switching martingales
摘要：
本文描述了标准布朗运动的两个显式耦合$B$和$V$，它们自然地推广了镜像耦合和同步耦合，并分别最大化和最小化（在所有时域上一致）两个状态切换鞅的耦合时间和跟踪误差.广义镜耦合最小化的耦合时间的两个鞅，同时最大化的跟踪误差的所有时间范围。广义同步耦合使耦合时间最大化，并使所有自适应耦合的跟踪误差最小化。证明是基于贝尔曼原则。我们给反例之间更广泛的随机积分类的两个耦合的约束最优性。
英文摘要：
This paper describes two explicit couplings of standard Brownian motions $B$ and $V$, which naturally extend the mirror coupling and the synchronous coupling and respectively maximise and minimise (uniformly over all time horizons) the coupling time and the tracking error of two regime-switching martingales. The generalised mirror coupling minimizes the coupling time of the two martingales while simultaneously maximising the tracking error for all time horizons. The generalised synchronous coupling maximises the coupling time and minimises the tracking error over all co-adapted couplings. The proofs are based on the Bellman principle. We give counterexamples to the conjectured optimality of the two couplings amongst a wider classes of stochastic integrals.
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