McKean--Vlasov SDEs in Nonlinear Filtering

Abstract

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 It\^{o} representations of their limiting forms as the approximation parameter $\delta \rightarrow 0$. All filters require the solution of a Poisson equation defined on $\mathbb{R}^{d}$, 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.