A perturbation analysis of stochastic matrix Riccati diffusions. (English. French summary) Zbl 1434.60020

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

Ann. Inst. Henri Poincaré, Probab. Stat. 56, No. 2, 884-916 (2020).

Summary: 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.

MSC:

60B20	Random matrices (probabilistic aspects)
15B52	Random matrices (algebraic aspects)
62M20	Inference from stochastic processes and prediction
60G35	Signal detection and filtering (aspects of stochastic processes)

Keywords:

Riccati matrix differential equation; covariance matrices; filtering; ensemble Kalman filters; interacting particle systems; perturbation theory

Software:

EnKF; mftoolbox

References:

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