论文标题
最佳的马尔可夫耦合，用于有限活动莱维过程
Optimal Markovian coupling for finite activity Lévy processes
论文作者
论文摘要
我们研究了马尔可夫过程的最佳马尔可夫耦合，在这些过程中，从最小化耦合过程的时间 - 边界分布之间的凹形运输成本来理解了最优性。我们为一维有限活性过程（连续时间随机步行）提供了此类最佳耦合的明确结构，其跳跃分布是单峰的，但不一定是对称的。值得注意的是，最佳马尔可夫耦合不取决于特定的凹入运输成本。为此，我们将McCann在最佳运输方面的结果和Rogers的结果结合在一起，并在随机步行中的结果与一种新颖的均匀化结构，使我们能够表征有限活性莱维过程的所有马尔可夫耦合。特别是，我们表明，具有非对称单峰莱维措施的有限活性过程的最佳马尔可夫耦合必须允许两种耦合过程的非同样跳跃。
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 Lévy 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 Lévy processes. In particular, we show that the optimal Markovian coupling for finite-activity Lévy processes with non-symmetric unimodal Lévy measures has to allow for non-simultaneous jumps of the two coupled processes.
