Abstract
The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued 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. 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. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation. 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. The analysis developed here applies to filtering problems with unstable signals.
Highlights
We introduce some matrix notation needed from the onset
The matrix-valued Riccati diffusions discussed in this article are defined by the stochastic model dQt = Θ(Qt) dt + dMt with t ∈ [0, ∞[, Q0 = Q ∈ Sr0, and some noise parameter ≥ 0
With ≤ ε0, we prove that the matrix Riccati diffusion (1.3) has a unique strong solution in Sr+ and that it never hits the boundary ∂Sr+ on any positive time horizon
Summary
Let Mr be the set of (r × r) real matrices with r ≥ 1. Let Sr ⊂ Mr be the subset of symmetric matrices, and Sr0, and Sr+ the subsets of positive semi-definite and definite matrices respectively. We denote by 0 and I the null and identity matrices, for any r ≥ 1. Given R ∈ ∂Sr+ := Sr0 − Sr+ we denote by R1/2 a (non-unique) symmetric square root of R. When R ∈ Sr+ we choose the unique symmetric square root. We write A the transpose of A, and Asym = (A + A )/2 its symmetric part. We denote by Absc(A) := max {Re(λ) : λ ∈ Spec(A)} its spectral abscissa. Let μ(A) = λ1(Asym) denote the (2-)logarithmic “norm” (which can be < 0).
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