Perturbations and projections of Kalman–Bucy semigroups

Bishop, Adrian N.; Moral, Pierre Del; Pathiraja, Sahani

doi:10.1016/j.spa.2017.10.006

Cited by 13 publications

(24 citation statements)

References 90 publications

(166 reference statements)

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“…theorems 1.1, 1.2, 1.3 and 1.4) are all given in this general setting. The broader impact of regularisation [26,13], and more sophisticated EnKF methods (e.g. [26,49]), on the sample covariance flow will be discussed elsewhere; e.g.…”

Section: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

A perturbation analysis of stochastic matrix Riccati diffusions

Bishop¹,

Moral²,

Niclas³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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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.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.

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Section: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

A perturbation analysis of stochastic matrix Riccati diffusions

Bishop¹,

Moral²,

Niclas³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…So-called variance inflation and localization are considered from a purely mathematical vantage in the linear-Gaussian setting in [14]. We refer further to [24,14,12] for some related and further background and references on the mathematical properties of the EnKF in the linear-Gaussian setting. See also [11] for a matrix-valued extension of the work considered herein, which seeks to remove the strong stability results on the signal required in prior work.…”

Section: Applications To Ensemble Kalman Filtersmentioning

confidence: 99%

On one-dimensional Riccati diffusions

Bishop¹,

Moral²,

Kamatani³

et al. 2019

Ann. Appl. Probab.

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This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration, and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.

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“…In this context, up to a change of probability space, the matrix-valued Riccati diffusion (1.2) describes the evolution of the sample covariance associated with these filters. With this application, the general form of (1.4) accommodates two popular EnKF filter models (determined by the binary switch κ ∈ {0, 1}) [24,41]; as well as accommodating a class of inflation-based regularization methods (determined by ̟ ≥ 0) [13]. The case ̟ = 0 in (1.4) corresponds to non-regularized models.…”

Section: Ensemble Kalman-bucy-type Filtersmentioning

confidence: 99%

“…This section is concerned with some applications of the results developed in the article to the analysis of the regularized EnKF filters discussed in [13]. We only consider an inflation-type regularisation often discussed in the EnKF literature [1,28,24].…”

Section: Regularized Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

On the stability of matrix-valued Riccati diffusions

Bishop¹,

Moral²

2019

Electron. J. Probab.

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The stability properties of matrix-valued Riccati diffusions are investigated. The matrixvalued 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. In addition, 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. 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. The analysis developed here applies to filtering problems with unstable signals. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation.1/2This special case (κ = 0) defines, in some sense, a minimal prototype of a forward-in-time matrixvalued Riccati diffusion in the space of symmetric positive (semi-)definite matrices. We let φ ǫ t (Q) := Q t be the stochastic flow of the matrix diffusion equation (1.2). Whenever it exists, the inverse stochastic flow of (1.2) is denoted by φ −ǫ t (Q) := Q −1 t . For any 0 ≤ s ≤ t, we let E ǫ s,t (Q) be the transition semigroup associated with the flow of random matrices [A − φ ǫ t (Q)S], i.e. the solution of the forward and backward equationsfor any 0 ≤ s ≤ t, with E ǫ t,t (Q) = I. When s = 0 we write E ǫ t (Q) instead of E ǫ 0,t (Q). We write φ t (Q) and E s,t (Q) instead of φ 0 t (Q), and E 0 s,t (Q), to denote the flow of the deterministic matrix Riccati differential equation when ǫ = 0, and the exponential semigroup defined via φ t (Q). 2

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Perturbations and projections of Kalman–Bucy semigroups

Cited by 13 publications

References 90 publications

A perturbation analysis of stochastic matrix Riccati diffusions

A perturbation analysis of stochastic matrix Riccati diffusions

On one-dimensional Riccati diffusions

On the stability of matrix-valued Riccati diffusions

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