On the Stability and the Exponential Concentration of Extended Kalman-Bucy filters

TL;DR

This paper establishes new exponential stability and concentration inequalities for extended Kalman-Bucy filters, providing non-asymptotic confidence bounds and stability estimates for nonlinear filtering under partial observations.

Contribution

It introduces the first non-asymptotic exponential concentration and stability results for extended Kalman-Bucy filters, combining advanced probabilistic and spectral analysis techniques.

Findings

Derived new concentration inequalities for partially observed signals.

Provided explicit non-asymptotic estimates for filter forgetting rates.

Established stability bounds for stochastic Riccati equations.

Abstract

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estimates in terms of the filter forgetting properties with respect to erroneous initial conditions. For uniformly stable signals, we also provide explicit non-asymptotic estimates for the exponential forgetting rate of the filters and the associated stochastic Riccati equations w.r.t. Frobenius norms. These non asymptotic exponential concentration and quantitative stability estimates seem to be the first results of this type for this class of nonlinear filters. Our techniques combine $\chi$-square concentration inequalities and Laplace estimates with spectral and random matrices theory, and the non asymptotic stability theory of quadratic type stochastic processes.