Fine properties of the optimal Skorokhod embedding problem
Fine properties of the optimal Skorokhod embedding problem
复制标题
DOI：
10.4171/jems/1122
复制
发表时间：
2019-03
影响因子：
2.6
通讯作者：
Mathias Beiglbock;Marcel Nutz;Florian Stebegg
中科院分区：
文献类型：
--
作者：
Mathias Beiglbock;Marcel Nutz;Florian Stebegg
我们研究了在给定的分布下停止布朗运动的问题，同时优化了一个依赖于(可能是随机的)停止时间和布朗运动的奖励函数。我们的第一个结果证明了嵌入$\nu$的停止时间集合$\mathcal{T}(\nu)$在随机嵌入的集合$\mathcal{R}(\nu)$中是弱稠密的。特别地，当报酬函数是半连续的时，$\mathcal{T}(\nu)$上的最优Skorokhod嵌入问题与$\mathcal{R}(\nu)$上的松弛Skorokhod嵌入问题具有相同的值，这与关于最优传输中Monge映射和Kantorovich耦合的一个基本结果是平行的.第二部分研究了线性规划意义下的对偶优化问题。虽然对偶解的存在性在以前的公式中是不成立的，但我们引入了对偶问题的一个松弛，它利用了一个新的紧性性质，得到了解的存在以及不存在对偶间隙，即使对于不规则的奖励函数也是如此。这导致了一个单调性原理，它补充了Beiglbock，Cox和Huesmann[最优传输和Skorokhod嵌入，发明数学，208：327-400,2017]的关键定理.我们证明了这些结果可以通过变分条件来刻画最优嵌入的几何.
We study the problem of stopping a Brownian motion at a given distribution $\nu$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $\mathcal{T}(\nu)$ of stopping times embedding $\nu$ is weakly dense in the set $\mathcal{R}(\nu)$ of randomized embeddings. In particular, the optimal Skorokhod embedding problem over $\mathcal{T}(\nu)$ has the same value as the relaxed one over $\mathcal{R}(\nu)$ when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem that exploits a novel compactness property and yields existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglb\"ock, Cox and Huesmann [Optimal transport and Skorokhod embedding, Invent. Math., 208:327-400, 2017]. We show that these results can be applied to characterize the geometry of optimal embeddings through a variational condition.