We study the existence, optimality, and construction of non-randomised stopping times that solve the Skorokhod embedding problem (SEP) for Markov processes which satisfy a duality assumption. These stopping times are hitting times of space-time subsets, so-called Root barriers. Our main result is, besides the existence and optimality, a potential-theoretic characterisation of this Root barrier as a free boundary. If the generator of the Markov process is sufficiently regular, this reduces to an obstacle PDE that has the Root barrier as free boundary and thereby generalises previous results from one-dimensional diffusions to Markov processes. However, our characterisation always applies and allows, at least in principle, to compute the Root barrier by dynamic programming, even when the well-posedness of the informally associated obstacle PDE is not clear. Finally, we demonstrate the flexibility of our method by replacing time by an additive functional in Root’s construction. Already for multi-dimensional Brownian motion this leads to new class of constructive solutions of (SEP).

我们研究了满足二元性假设的马尔可夫过程的解决Skorokhod嵌入问题（SEP）的非随机停止时间的存在，最优性和构造。这些停止时间是时空子集（所谓的根障碍）的命中时间。我们的主要结果是，除了存在性和最优性之外，该根垒作为自由边界的潜在理论表征。如果马尔可夫过程的生成器足够规则，则这将减少为具有根边界作为自由边界的障碍物PDE，从而概括从一维扩散到马尔可夫过程的先前结果。但是，我们的表征始终适用，并且至少在原理上允许通过动态编程来计算Root障碍，即使不清楚与非正式关联的障碍物PDE的正确摆放位置。最后，我们通过用Root构造中的加法函数代替时间来证明我们方法的灵活性。对于多维布朗运动，这已经导致了（SEP）的一类新的构造解。