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

该项目旨在开发和推广经典降维理论，用于信号处理中的滤波问题，更一般地用于最初在Brigo et al（1998，1999）和Armstrong and Brigo（2016 a，2016 b，2016 c）中开发的随机（偏）微分方程（S（P）DE）。该方法是基于一个选定的流形或参数化的方式，是最佳的一个选定的度量，通常在均方的切线空间上的方程系数的投影。潜在的应用可能允许绕过许多工程领域中的维数灾难，并且包括SDES、PDE（包括Fokker Planck Kolmogorov和热方程）和SPDE（诸如上述滤波问题）的最佳近似。一个经典的例子是导航中的姿态滤波，但许多医学应用出现在信息几何文献中。该项目还旨在制定一个通用的路径降维框架的基础上粗糙路径理论。该项目是在数学的三个领域的交叉点：几何S（P）DE，变分方法的福克普朗克方程与信息几何，和粗糙路径理论。在该项目的第二阶段，我们将寻求与业界积极合作，以获得实施过滤算法所需的数据。该项目是在EPSRC增长领域的应用概率和统计。

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.