McKean-Vlasov SDEs in nonlinear filtering

TL;DR

This paper introduces a unifying framework for deriving McKean-Vlasov representations of particle filters in nonlinear filtering, addressing challenges in high-dimensional systems and establishing conditions for well-posedness of associated Poisson equations.

Contribution

It systematically derives McKean-Vlasov SDE representations for several filters, providing a unified approach and analyzing conditions for the existence and uniqueness of solutions.

Findings

Derived Itô representations of filters as the approximation parameter tends to zero

Established conditions for the well-posedness of the weighted Poisson equation

Unified framework applicable to multiple particle filtering methods

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.