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

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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.