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Embedding measures into multi-dimensional stochastic processes and rough paths

将测量嵌入到多维随机过程和粗糙路径中

基本信息

批准号：
1941799
负责人：
金额：
--
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-1941799
关键词：
Embedding measures into multi dimensional

项目摘要

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.

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，使其匹配给定的分布。尽管是抽象的公式，这个问题已经发现了许多应用，包括数学金融，统计和功能极限定理。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的初始问题对于多维鞅有直接的意义（并且对于更一般的随机过程类也有一些重新表述）。虽然一些存在性结果是已知的，但当涉及到这种停时的具体构造时，文献变得非常稀疏。一个有吸引力的方法来多维Skorokhod嵌入是最近的鞅最优运输理论（“A。考克斯，M. Beiglbock和M. Huesmann“最佳运输和Skorokhod嵌入”，Inv. 2017年5月，2：327-400）。鞅最优运输导致证明存在性和最优的解决方案Skorokhod嵌入，即使底层过程是多维的，但证明是不建设性的，通常无助于实际构建一个给定的目标分布的停时。本研究提案的目标是产生新的方法来Skorokhod多维嵌入问题，可以导致这样的停时的具体建设;进一步的目的是扩展现有的理论，能够覆盖的例子中出现的粗糙路径理论，并探索与最优鞅运输和其他新的应用程序的连接。这个项目属于EPSRC统计和应用概率研究领域的福尔斯。作为第一步，我们将回顾Skorokhod嵌入的所谓根解：根表明，对于一维布朗运动，停止时间可以实现为时间-空间子集的击中时间。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.题名其余部分：考克斯.“根的障碍：建设，最优性和应用方差选项”。The Annals of Applied Probability，23（3）：859-894，2013; P. Gassiat，H. Oberhauser和G.多斯雷斯。“根的障碍，粘度解决方案的障碍问题和反映FBSDES。随机过程及其应用125.12（2015）：4601-4631。或者作为积分方程的解（参见P. Gassiat，A. Mijatovic和H Oberhauser。“根的障碍和布朗增量的生成的积分方程。”，应用概率年鉴25.4（2015）：2039-2065）。事实上，抽象的潜在的理论论据表明，根型解决方案持有的一大类多维马尔可夫过程，这些参数可以重新解释Skorokhod embeddings.This项目的最新进展属于EPSRC数学分析研究领域内福尔斯。