A perturbation analysis of stochastic matrix Riccati diffusions

Abstract

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.Résumé: 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.

On the stability of matrix-valued Riccati diffusions

“…Now we turn to quantifying the fluctuations of the matrix Riccati diffusions around their limiting values when the diffusion parameter ǫ tends to 0. The next theorem extends (in some directions) the uniform fluctuation estimates obtained in [12]. In some results in [12] time-uniform estimates were obtained only with A stable, whereas here we accommodate more general matrix models with possibly unstable modes.…”

“…The present article thus complements the recent article [11] dedicated to the stability and fluctuation properties of general one-dimensional Riccati diffusions. And this article extends [12,20] in a number of directions, and corrects some claims in [20].…”

“…The l.h.s. under bias estimate is a consequence of the inequality (4.5); see also Theorem 1.3 in [12]. The proof of the r.h.s.…”

“…Later we also consider applications of this work to EnKF theory, and we consider the stability of the associated Ornstein-Uhlenbeck-type vector-valued flow ψ ǫ t (Q, x). The analysis of Riccati diffusions of the form (1.2) with the parameters (κ, ̟) = (1, 0) has been started in [12,20]. In these articles, the authors provide several uniform convergence results when S is proportional to the identity and when A is a stable matrix.…”

“…In these articles, the authors provide several uniform convergence results when S is proportional to the identity and when A is a stable matrix. In [12] we also provide a complete Taylor-type stochastic expansion of the Riccati flow with estimates given at any order with bounded remainder terms, and with a fluctuation analysis considered over the entire path space of the matrix-valued stochastic Riccati flow. We remark that in the scalar case we also have a complete time-uniform fluctuation and stability analysis of one-dimensional Riccati diffusions in [11].…”