Stochastic PDE projection on manifolds Assumed-Density and Galerkin Filters - Damiano Brigo, John Armstrong

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Authors : Damiano Brigo, John Armstrong
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_76
Video : http://www.youtube.com/watch?v=BPkO4sDSeE4
Slides: Brigo_Stochastic PDE projection.pdf
Presentation : https://www.see.asso.fr/node/14269
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finite-dimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (Kushner-Stratonovich) or a specific unnormalized (Zakai) density of the optimal filter.