Topologies on unparametrised path space and correlation estimation

非参数化路径空间拓扑和相关性估计

• 批准号: 2601859
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2601859
• 关键词: Topologies; unparametrised; path; space; correlation

The signature of a path is a non-commutative exponential introduced by K.T. Chen in the 1950s, and appears as a central object in the theory of rough paths developed by T. Lyons in the 1990s. For continuous paths of bounded variation, the signature may be realised as a sequence of iterated integrals defined in the Young sense. The signature provides a succinct summary for multimodal, irregularly sampled, time-ordered data: it is unique up to a general notion of reparameterisation (Hambly, Lyons 2010) and its terms decay factorially. Furthermore, the terms in the signature act as an analogue to monomials for finite dimensional data: linear functionals on the signature can be used to uniformly approximate any compactly supported continuous function on unparameterised path space (Levin, Lyons, Ni 2013). It has also been shown, under certain conditions, that the expected signature of a stochastic process characterises its law (Chevyrev, Lyons 2016 and Chevyrev, Oberhauser 2022). These results underpin the practical use of the signature in modern data science.This project aims to investigate aspects of the underlying theory, and develop pathwise methods for correlation estimation.A first project relates to the selection of a suitable topology on the space of unparameterised paths, key to a full understanding of the approximation theory. Since there is no canonical choice for the topology, our objective is to understand the properties of a broad class of natural choices. Such properties include: (complete) metrisability, separability, local compactness, and the availability of continuous functions, compact sets, and the properties of Borel probability measures. Only limited results concerning these topologies exist in the literature. The project will look to relate any results to important classes of models such as the fixed time solution of a controlled differential equation.A second project involves constructing novel estimators for asset correlation using signature kernels. A signature kernel is the inner product between two signatures. For a certain choice of inner product, the signature kernel solves a Goursat PDE (Salvi et al. 2021). For an alternative, weighted, inner product, a connection between the expected signature of Brownian motion and the hyperbolic development of a path has been established (Cass, Lyons, Xu 2021). The application, adaptation and possible of extension of this latter result will form the core of this project. By maximising a notion of alignment between two measures, introduced by Cass, Lyons and Xu, this project seeks to construct a new estimator for the correlation matrix. The project will examine the asymptotic properties of the estimator, such as consistency and rates of convergence. It will also investigate the impact of market microstructure, asynchronous observations and frequency of trading on the estimator.Alignment to EPSRC's strategies and research areasThis project falls within the EPSRC research areas of Mathematical analysis, and Statistics and applied probability.CollaboratorsMy supervisor: Thomas Cass.

路径的签名是由K. T. Chen在20世纪50年代提出了粗糙路径理论，并在T. 90年代的里昂。对于有界变化的连续路径，签名可以实现为定义在Young意义上的迭代积分序列。该特征为多模态、不规则采样、时间排序的数据提供了简洁的总结：它在重新参数化的一般概念下是唯一的（Hambly，里昂2010），并且其项按因子衰减。此外，签名中的项充当有限维数据的单项式的类似物：签名上的线性泛函可以用于一致地近似非参数化路径空间上的任何紧支撑连续函数（Levin，里昂，Ni 2013）。在某些条件下，还表明随机过程的预期特征表征其规律（Chevyrev，里昂2016和Chevyrev，Oberhauser 2022）。这些结果支持了签名在现代数据科学中的实际应用。本项目旨在研究基础理论的各个方面，并开发相关估计的路径方法。第一个项目涉及在非参数化路径空间上选择合适的拓扑结构，这是充分理解近似理论的关键。由于没有规范的选择拓扑，我们的目标是了解性质的一个广泛的类的自然选择。这些特性包括：（完全）可度量性，可分性，局部紧性，连续函数的可用性，紧集，以及Borel概率测度的性质。只有有限的结果，这些拓扑结构存在于文献中。该项目将寻求将任何结果与重要类别的模型相关联，例如受控微分方程的固定时间解。第二个项目涉及使用签名核构建资产相关性的新估计。签名核是两个签名之间的内积。对于特定的内积选择，签名内核解决了Goursat PDE（Salvi et al. 2021）。对于另一种加权内积，已经建立了布朗运动的预期签名与路径的双曲发展之间的联系（卡斯，里昂，徐2021）。后一项成果的应用、调整和可能的推广将构成本项目的核心。通过最大化两个措施之间的对齐的概念，介绍了卡斯，里昂和徐，该项目旨在构建一个新的估计的相关矩阵。该项目将检查估计量的渐近性质，如一致性和收敛速度。它还将调查市场微观结构的影响，异步观测和交易频率的估计。调整EPSRC的战略和研究领域这个项目福尔斯属于EPSRC的数学分析，统计和应用概率的研究领域。合作者我的导师：托马斯卡斯。