McKean-Vlasov SDEs in nonlinear filtering
- Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be aVarious particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.…
Author details: | Sahani Darschika PathirajaORCiD, Sebastian ReichORCiDGND, Wilhelm StannatORCiD |
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DOI: | https://doi.org/10.1137/20M1355197 |
ISSN: | 0363-0129 |
ISSN: | 1095-7138 |
Title of parent work (English): | SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics |
Publisher: | Society for Industrial and Applied Mathematics |
Place of publishing: | Philadelphia |
Publication type: | Article |
Language: | English |
Date of first publication: | 2022/11/04 |
Publication year: | 2021 |
Release date: | 2023/11/13 |
Tag: | McKean-Vlasov; Poincare inequality; data assimilation; feedback particle filter; mean-field equations; nonlinear filtering; well-posedness |
Volume: | 59 |
Issue: | 6 |
Number of pages: | 28 |
First page: | 4188 |
Last Page: | 4215 |
Funding institution: | Deutsche Forschungsgemeinschaft (DFG) German Research Foundation (DFG) [SFB1294/1-318763901] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access |
License (German): | CC-BY - Namensnennung 4.0 International |