On the Stability and the Uniform Propagation of Chaos of a Class of Extended Ensemble Kalman--Bucy Filters

Abstract

This article is concerned with the exponential stability and the uniform propagation of chaos properties of a class of extended ensemble Kalman--Bucy filters with respect to the time horizon. This class of nonlinear filters can be interpreted as the conditional expectations of nonlinear McKean--Vlasov-type diffusions with respect to the observation process. We consider filtering problems with Langevin-type signal processes observed by some noisy linear and Gaussian-type sensors. In contrast with more conventional Langevin nonlinear drift type processes, the mean field interaction is encapsulated in the covariance matrix of the diffusion. The main results discussed in the article are quantitative estimates of the exponential stability properties of these nonlinear diffusions. These stability properties are used to derive uniform and nonasymptotic estimates of the propagation of chaos properties of extended ensemble Kalman filters, including exponential concentration inequalities. To our knowledge these res...