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 Itô representations of their limiting forms as the approximation parameter $\delta \to 0$. 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.

References in corpus (3)

• L2 convergence of smooth approximations of Stochastic Differential Equations with unbounded coefficients
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
• Feedback Particle Filter on Matrix Lie Groups

Cited by in corpus (7)

• On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering
• L2 convergence of smooth approximations of Stochastic Differential Equations with unbounded coefficients
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
• Analysis of the Ensemble Kalman--Bucy Filter for correlated observation noise
• A Unification of Weighted and Unweighted Particle Filters
• Analysis of the feedback particle filter with diffusion map based approximation of the gain
• Optimal Transportation Methods in Nonlinear Filtering: The feedback particle filter