Embedding measures into multi-dimensional stochastic processes and rough paths

随机分析中的一个经典问题是Skorokhod嵌入问题：给定一个布朗运动和实数上的分布，任务是停止布朗轨迹，使其与给定的分布相匹配。尽管这个问题的表述是抽象的，但它已经找到了许多应用，包括数学、金融、统计学和函数极限定理。最近，它在结合最优运输和鞅理论的思想方面发挥了核心作用。Skorokhod最初的问题对多维鞅（以及对更一般的随机过程的一些重新表述）立即有意义。虽然一些存在结果是已知的，但当涉及到这种停止时间的具体结构时，文献很快就会变得稀疏。最近的鞅最优输运理论（A. Cox， M. Beiglbock and M. Huesmann）是研究多维Skorokhod嵌入的一个有吸引力的方法。“最优运输和Skorokhod嵌入”，《数学》。2017年5月，2:327-400)。鞅最优传输即使在底层过程是多维的情况下也能证明Skorokhod嵌入的存在性和最优性，但这些证明是非建设性的，通常无助于实际构建给定目标分布的停止时间。这项研究计划的目标是为Skorokhod多维嵌入问题提供新的方法，可以导致这种停止时间的具体构建；进一步的目标是扩展现有的理论，使其能够涵盖粗糙路径理论中出现的例子，并探索与最优鞅传输和其他新应用的联系。该项目属于EPSRC统计和应用概率研究领域。来自巴黎多芬大学的Paul Gassiat将作为合作者参与其中。作为第一步，我们将重温所谓的Skorokhod嵌入的根解：根表明，对于一维布朗运动，停止时间可以实现为一个时空子集的命中时间。在这种情况下，最近的工作表明，这个时空子集可以计算为抛物型偏微分方程的自由边界（见a . M. G. Cox， J.Wang）。“根的障碍：构造、最优性和方差选择的应用”。应用概率学报，23(3):859-894,2013；P. Gassiat， H. Oberhauser和G. dos Reis。“根的屏障，粘度解决障碍问题和反映FBSDEs。”随机过程及其应用，125.12(2015):4601-4631.)或作为积分方程的解(见P. Gassiat， A. Mijatovic和H . Oberhauser。“根势垒的积分方程和布朗增量的生成”，应用概率年鉴（2015）：2039-2065。事实上，抽象的潜在理论论证表明，根型解适用于一大类多维马尔可夫过程，这些论证可以用Skorokhod嵌入的最新进展来重新解释。该项目属于EPSRC数学分析研究领域。

A classic problem in stochastic analysis is the Skorokhod embedding problem: given a Brownian motion and a distribution on the reals, the task is to stop the Brownian trajectories such that it matches the given distribution. Despite the abstract formulation, this problem has found many applications including mathematical finance, statistics, and functional limit theorems. More recently, it played a central part in combining ideas from optimal transport and martingale theory. Skorokhod's initial question makes immediately sense for multi-dimensional martingales (and with some reformulations also for classes of more general stochastic processes). While some existence results are known, the literature gets very quickly sparse when it comes to concrete constructions of such stopping times. An attractive approach to the multidimensional Skorokhod embedding is the recent martingale optimal transport theory ("A. Cox, M. Beiglbock and M. Huesmann. "Optimal transport and Skorokhod embedding", Inv. Math. May 2017,2: 327-400). Martingale optimal transport leads to proofs of existence and optimality of solutions to Skorokhod embeddings even when the underlying process is multidimensional, but the proofs are not-constructive and typically do not help to actually construct the stopping time for a given target distribution. The goal of this research proposal is to produce new approaches to the Skorokhod multidimensional embedding problem that can lead to a concrete construction of such stopping times; a further aim is to extend the existing theory to be able to cover examples that arise in rough path theory and to explore connections with optimal martingale transport and other new applications. This project falls within the EPSRC Statistics and applied probability research area. Paul Gassiat from University Paris-Dauphine will be involved as a collaborator.As a first step, we will revisit the so-called Root solution of the Skorokhod embedding: Root showed that for one-dimensional Brownian motion, the stopping time can be realized as the hitting time of a subset of time-space. In this case, recent work has shown that this subset of time-space can be computed as the free boundary of a parabolic partial differential equation (see, A. M. G. Cox, J.Wang. "Root's barrier: Construction, optimality, and applications to variance options". The Annals of Applied Probability, 23(3):859-894, 2013; P. Gassiat, H. Oberhauser, and G. dos Reis. "Root's barrier, viscosity solutions of obstacle problems and reflected FBSDEs." Stochastic Processes and their Applications 125.12 (2015): 4601-4631.) or alternatively as the solution of an integral equation, (see P. Gassiat, A. Mijatovic and H Oberhauser. "An integral equation for Root's barrier and the generation of Brownian increments.", The Annals of Applied Probability 25.4 (2015): 2039-2065). In fact, abstract potential theoretic arguments show that Root type solutions hold for a large class of multidimensional Markov processes and these arguments can be reinterpreted in terms of recent advances in Skorokhod embeddings.This project falls within the EPSRC Mathematical Analysis research area.