On the stability of matrix-valued Riccati diffusions

Abstract

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

An explicit Floquet-type representation of Riccati aperiodic exponential semigroups

“…more particles). The stability analysis of these matrix diffusion processes rely on the stability properties of (1.1); see for instance [7,8,6,26].…”

“…For example, in ensemble Kalman filtering theory, the former often do not exist while the later do; see e.g. [7,8,6].…”

On the incorporation of box-constraints for ensemble Kalman inversion

“…The condition A Γ −1 A > 0 is well-known in the context of data assimilation related to as the strong observability condition. As a result this condition for the case of ensemble Kalman filter has been well-documented, see e,g, [8,25].…”

Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients

“…by studying the stability properties of ensemble Kalman-Bucy-type filters [21]. See [9,8,7] for dedicated technical discussions on related ensemble filters, and associated literature review (which is beyond the focus of this article). Related applications of our type of stability result are in [25].…”

“…The matrix S := C ′ R −1 o C is defined in terms of the sensor observation matrix C, and the covariance diffusion matrix R o of the observation noise. The dominating matrices (U , V) depend on the different variants of ensemble Kalman-Bucy filters one may consider [7]. Now, the difference between the ensemble Kalman-Bucy filter sample mean and the true signal state (i.e.…”