Electronic Journal of Probability

On the stability of matrix-valued Riccati diffusions

Adrian N. Bishop and Pierre Del Moral

Full-text: Open access

Abstract

The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation. This class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. The analysis developed here applies to filtering problems with unstable signals.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 84, 40 pp.

Dates
Received: 29 January 2019
Accepted: 13 July 2019
First available in Project Euclid: 10 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1568080863

Digital Object Identifier
doi:10.1214/19-EJP342

Mathematical Reviews number (MathSciNet)
MR4003137

Zentralblatt MATH identifier
07107391

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G52: Stable processes 93E11: Filtering [See also 60G35] 60G99: None of the above, but in this section

Keywords
ensemble filtering methods ensemble Kalman-Bucy filters matrix quadratic stochastic differential equations matrix Riccati diffusion equations matrix Riccati equations uniform fluctuation estimates uniform stability estimates

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bishop, Adrian N.; Del Moral, Pierre. On the stability of matrix-valued Riccati diffusions. Electron. J. Probab. 24 (2019), paper no. 84, 40 pp. doi:10.1214/19-EJP342. https://projecteuclid.org/euclid.ejp/1568080863


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