英文标题：
《Fine Properties of the Optimal Skorokhod Embedding Problem》
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作者：
Mathias Beiglb\\\"ock, Marcel Nutz, Florian Stebegg
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最新提交年份：
2020
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英文摘要：
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.
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中文摘要：
我们研究了在给定分布$\\nu$下停止布朗运动，同时优化依赖于（可能随机的）停止时间和布朗运动的奖励函数的问题。我们的第一个结果确定，在随机嵌入集$\\mathcal{R}（\\nu）$中，停止时间嵌入集$\\mathcal{T}（\\nu）$是弱稠密的。特别地，当奖励函数为半连续时，$\\数学{T}（\\nu）$上的最优Skorokhod嵌入问题与$\\数学{R}（\\nu）$上的松弛问题具有相同的值，这与最优传输中Monge映射和Kantorovich耦合的基本结果相似。第二部分研究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失败了，但我们引入了对偶问题的松弛，该问题利用了一个新的紧性性质，并产生了解的存在性以及对偶间隙的存在性，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，该原理补充了Beiglb“ock、Cox和Huesmann的关键定理【最优传输和Skorokhod嵌入，发明数学，208:327-4002017】。我们表明，这些结果可以通过变分条件来表征最优嵌入的几何特征。
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分类信息：

一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Mathematics 数学
二级分类：Optimization and Control 优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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