Electronic Journal of Probability

On the stability and the concentration of extended Kalman-Bucy filters

Pierre Del Moral, Aline Kurtzmann, and Julian Tugaut

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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 and fully observable 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.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 91, 30 pp.

Dates
Received: 22 August 2017
Accepted: 12 June 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717750

Digital Object Identifier
doi:10.1214/18-EJP188

Mathematical Reviews number (MathSciNet)
MR3858919

Zentralblatt MATH identifier
1401.93199

Subjects
Primary: 93C55: Discrete-time systems 93D20: Asymptotic stability 93E11: Filtering [See also 60G35] 60M20 60G25: Prediction theory [See also 62M20]

Keywords
Concentration inequalities non asymptotic exponential stability Lyapunov exponents extended Kalman-Bucy filter Riccati equation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Del Moral, Pierre; Kurtzmann, Aline; Tugaut, Julian. On the stability and the concentration of extended Kalman-Bucy filters. Electron. J. Probab. 23 (2018), paper no. 91, 30 pp. doi:10.1214/18-EJP188. https://projecteuclid.org/euclid.ejp/1536717750


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