Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A perturbation analysis of stochastic matrix Riccati diffusions

Adrian N. Bishop, Pierre Del Moral, and Angèle Niclas

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Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman–Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.


Les équations de Riccati matricielles jouent un rôle important dans la théorie du filtrage et du contrôle optimal. Cet article présente une théorie des perturbations d’une classe d’équations de Riccati matricielles stochastiques. Ces modèles probabilistes sont d’un usage courant dans la théorie des filtres de Kalman d’Ensemble. Ils représentent dans ce contexte l’évolution des matrices de covariance empiriques associées à un ensemble de diffusions en interaction. Les perturbations aléatoires résultent des fluctuations stochastiques d’un système de particules de type champ moyen interagissant avec la mesure empirique du système. Nous présentons dans cet article une formule de Taylor non asymptotique pour des flots stochastiques de diffusion de Riccati matricelles par rapport à un paramètre de fluctuation. Ces développements sont fondés sur un nouveau calcul différentiel stochastique et une analyse fine de semigroupes non linéaires dans des espaces de matrices. Ces résultats permettent de quantifier avec précision les fluctuations des flots de matrices stochastiques autour des systèmes limites à tout ordre. Nous illustrons ces résultats avec une preuve de la convergence des matrices empiriques de filtres de Kalman d’Ensemble vers la solution d’équations de Riccati déterministes lorsque le nombre de particules tends vers l’infini. Nous présentons dans ce cadre des estimations fines des biais et des variances, ainsi qu’un theorème de la limite centrale fonctionnel au niveau du processus matriciel.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 884-916.

Received: 29 March 2018
Accepted: 25 March 2019
First available in Project Euclid: 16 March 2020

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Mathematical Reviews number (MathSciNet)

Primary: 11M50: Relations with random matrices 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Riccati matrix differential equation Covariance matrices Filtering Ensemble Kalman filters Interacting particle systems Perturbation theory


Bishop, Adrian N.; Del Moral, Pierre; Niclas, Angèle. A perturbation analysis of stochastic matrix Riccati diffusions. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 884--916. doi:10.1214/19-AIHP987.

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