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信号处理中滤波问题的经典降维理论，更广泛地应用于随机（偏）微分方程（S(P)DEs），该理论最初由Brigo等人（1998,1999）和Armstrong和Brigo （2016a, 2016b, 2016c）开发。该方法是基于在所选流形的切空间上的方程系数的投影或参数化，以一种对所选度量最优的方式，通常是均方。潜在的应用可能允许绕过许多工程领域的维数诅咒，包括SDEs的最优逼近，PDEs包括Fokker Planck Kolmogorov方程和热方程，以及SPDEs，如上述滤波问题。一个典型的例子是导航中的态度过滤，但信息几何文献中也出现了医学上的许多应用。该项目还旨在基于粗糙路径理论制定一个通用的降维路径框架。该项目是在三个数学领域的交叉点：S(P)DEs的几何，福克普朗克方程的变分方法与信息几何，粗糙路径理论。在项目的第二阶段，我们将寻求与工业界的积极合作，以获取实施过滤算法的数据。该项目属于EPSRC应用概率与统计增长领域。

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.