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。尽管这个问题是抽象的，但它已经得到了许多应用，包括数学金融、统计学和函数极限定理。最近，它在将最优运输和鞅理论的思想结合起来方面发挥了核心作用。Skorokhod的初始问题对于多维鞅立即有意义(并且经过一些重新表述也适用于更一般的随机过程类)。虽然一些存在的结果是已知的，但当涉及到这种停止时间的具体构造时，文献很快就变得稀疏了。多维Skorokhod嵌入的一个吸引人的方法是最近的最新的鞅最优传输理论(“A.Cox，M.Beiglbock和M.Huesmann。“最佳运输和Skorokhod嵌入”，Inv.数学课。2017年5月2：327-400)。即使在基本过程是多维的情况下，鞅最优传输也得到了Skorokhod嵌入问题解的存在性和最优性的证明，但这些证明是非构造性的，并且通常无助于实际构造给定目标分布的停止时间。这个研究方案的目的是给出Skorokhod多维嵌入问题的新方法，从而得到这种停止时间的具体构造；另一个目的是扩展现有的理论，使其能够涵盖粗糙路理论中出现的例子，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。巴黎多芬大学的Paul Gassiat将作为合作者参与其中。作为第一步，我们将回顾Skorokhod嵌入的所谓根解：Root表明，对于一维布朗运动，停止时间可以实现为时空子集的命中时间。在这种情况下，最近的工作表明，这一时空子集可以计算为抛物型偏微分方程的自由边界(参见A.M.G.Cox，J.Wang。根的障碍：构造、最优性和方差期权的应用应用概率年鉴，23(3)：859-894,2013；P.Gassiat，H.Oberhauser和G.dos Reis。根的障碍，障碍问题的粘性解决方案和反映的FBSDE。随机过程及其应用125.12(2015年)：4601-4631.或者作为积分方程解的另一种选择，(见P.Gassiat，A.Mijatovic和H Oberhauser。《根势垒积分方程与布朗增量的生成》，《应用概率年鉴》25.4(2015)：2039-2065)。事实上，抽象势理论论证表明，根型解适用于一大类多维马尔可夫过程，这些论点可以用Skorokhod嵌入的最新进展来重新解释。本项目属于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.