On the stability and the uniform propagation of chaos of a class of Extended Ensemble Kalman-Bucy filters

TL;DR

This paper investigates the exponential stability and uniform propagation of chaos in a class of Extended Ensemble Kalman-Bucy filters, providing the first non-asymptotic quantitative estimates for their stability and concentration properties.

Contribution

It introduces novel stability and propagation of chaos results for nonlinear ensemble Kalman-Bucy filters, with explicit quantitative bounds and concentration inequalities.

Findings

Exponential stability estimates for the nonlinear diffusions

Uniform propagation of chaos with non-asymptotic bounds

Exponential concentration inequalities for the 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. 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 non asymptotic estimates of the propagation of chaos properties of Extended Ensemble Kalman filters, including exponential concentration inequalities. To our knowledge these results seem to be the first results of this type for this class of nonlinear ensemble type Kalman-Bucy filters.