McKean-Vlasov SDEs in nonlinear filtering
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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 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 Crisan Xiong (2010) and Clark Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive Itô representations of their limiting forms as the approximation parameter δ→ 0. All filters require the solution of a Poisson equation defined on ℝ^d, for which existence and uniqueness of solutions can be a non-trivial 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.
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