Topologies on unparametrised path space and correlation estimation

路的符号是由K.T.Chen在20世纪50年代引入的非交换指数，并在T.Lyons于20世纪90年代发展的粗糙路理论中成为一个中心对象。对于有界变差的连续路径，签名可以实现为在Young意义下定义的迭代积分序列。该签名为多模式、不规则抽样、按时间排序的数据提供了简明的摘要：它在重新参数化的一般概念(Hamble，Lyons 2010)及其项的因子衰减方面是独一无二的。此外，签名中的项类似于有限维数据的单项式：签名上的线性泛函可用于在非参数路径空间上一致逼近任何紧支撑的连续函数(Levin，Lyons，Ni 2013)。它还表明，在某些条件下，随机过程的期望签名是其定律的特征(Chevyrev，Lyons 2016和Chevyrev，Oberhauser 2022)。这些结果为签名在现代数据科学中的实际应用奠定了基础。这个项目旨在研究基本理论的各个方面，并开发用于相关性估计的路径方法。第一个项目涉及在非参数路径空间上选择合适的拓扑，这是充分理解近似理论的关键。由于拓扑没有典范选择，我们的目标是理解一大类自然选择的性质。这些性质包括：(完全)度量、可分、局部紧性、连续函数的可用性、紧集和Borel概率测度的性质。关于这些拓扑，文献中只有有限的结果。该项目将寻求将任何结果与重要的模型类别相关联，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新的资产相关性估计器。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(塞尔维等人)。2021年)。对于另一种加权内积，建立了布朗运动的预期签名和路径的双曲展开之间的联系(Cass，Lyons，Xu 2021)。后一项成果的应用、适应和推广的可能性将构成本项目的核心。通过最大化由CASS、Lyons和Xu提出的两个度量之间的一致性的概念，该项目试图为相关矩阵构造一个新的估计器。该项目将检查估计器的渐近性质，如一致性和收敛速度。它还将调查市场微观结构、异步观察和交易频率对估计者的影响。与EPSRC的战略和研究领域保持一致本项目属于EPSRC数学分析、统计学和应用概率的研究领域。合作者：Thomas Cass。

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.