Geometric methods for dimensionality reductions of stochastic (partial) differential equations with applications to signal processing and finance

该项目旨在发展和推广经典的降维理论，用于信号处理中的滤波问题，以及更广泛的随机(偏微分方程)(S(P)DES)，最初由Brigo等人(1998年，1999年)和Armstrong和Brigo(2016a，2016b，2016c)发展。该方法基于方程系数在所选流形的切线空间上的投影，或以对所选度量最优的方式进行参数化，通常以均方为单位。潜在的应用可能允许绕过许多工程领域中的维度诅咒，包括对SDE、包括Fokker Planck Kolmogorov方程和热方程的PDE的最佳逼近，以及诸如上述过滤问题的SPDEs。一个经典的例子是导航中的姿态过滤，但许多来自医学的应用出现在信息几何文献中。该项目还旨在建立一个基于粗糙路径理论的通用路径降维框架。该项目是三个数学领域的交集：S(P)DES的几何，福克-普朗克方程的变分方法和信息几何，以及粗糙路径理论。在该项目的第二阶段，我们将寻求与业界的积极合作，以获得实施过滤算法所需的数据。该项目位于EPSRC应用概率和统计学的发展领域。

The project aims to develop and generalize the classic dimensionality reduction theory for the filtering problem in signal processing and more generally for Stochastic (Partial) Differential Equations (S(P)DEs) developed initially in Brigo et al (1998, 1999) and Armstrong and Brigo (2016a, 2016b, 2016c). The methodology is based on projection of the equation coefficients on the tangent space of a chosen manifold or parametrization in a way that is optimal for a chosen metric, typically in mean square. Potential applications might allow to bypass the curse of dimensionality in many fields of engineering and include optimal approximation of SDEs, PDEs including the Fokker Planck Kolmogorov and heat equations, and SPDEs such as the above mentioned filtering problem. A classic example is attitude filtering in navigation, but many applications from medicine are appearing in the information geometry literature. The project also aims at formulating a general pathwise framework for dimensionality reduction based on rough paths theory. The project is at the intersection of three areas of mathematics: geometry of S(P)DEs, the variational approach to the Fokker Planck equation with information geometry, and rough paths theory. In the second phase of the project we will seek active cooperation with industry to obtain data for the implementation of the filtering algorithms. The project is in the EPSRC grow area of applied probability and statistics.