On the Stability of Kalman--Bucy Diffusion Processes

Abstract

The Kalman-Bucy filter is the optimal state estimator for an Ornstein-Uhlenbeck diffusion given that the system is partially observed via a linear diffusion-type (noisy) sensor. Under Gaussian assumptions, it provides a finite-dimensional exact implementation of the optimal Bayes filter. It is generally the only such finite-dimensional exact instance of the Bayes filter for continuous state-space models. Consequently, this filter has been studied extensively in the literature since the seminal 1961 paper of Kalman and Bucy. The purpose of this work is to review, re-prove and refine existing results concerning the dynamical properties of the KalmanBucy filter so far as they pertain to filter stability and convergence. The associated differential matrix Riccati equation is a focal point of this study with a number of bounds, convergence, and eigenvalue inequalities rigorously proven. New results are also given in the form of exponential and comparison inequalities for both the filter and the Riccati flow.

On the stability of matrix-valued Riccati diffusions

“…The proof of the r.h.s. uniform estimate is in [8,10]; e.g. it is easy to verify φ t (Q) ≤ c ( P ∞ ∨ Q ).…”

“…Let ψ ǫ t (Q, x) = X t denote the flow of the stochastic equation (1.6). Whenever ǫ = 0 = ̟, the diffusion process (1.6) resumes to the difference between the classical Kalman-Bucy filter [8] and the true signal state of an auxiliary linear-Gaussian process with drift matrix A and diffusion matrix R. In this case we have dX t = (A − P t S) X t dt + Σ 1/2 1,0 (P t ) dW t with the Riccati equation ∂ t P t = Θ(P t ) (1.8) We note that P t = E(X t X ′ t ) coincides with the covariance matrix of the state estimation error defined by the Ornstein-Uhlenbeck process in the l.h.s. of (1.8) and φ t (Q) = P t when P 0 = Q.…”

“…the time parameter. In particular, when the pair of matrices (A, R 1/2 ) is stabilisable, and (A, S 1/2 ) is detectable, then the error X t is a stable process in the sense that (A − P t S) delivers a uniformly exponentially stable linear (time-varying) system [8]. Moreover, there exists a unique matrix [35,40,37], and the convergence results in [36,17].…”

“…the time horizon. For further discussion on these stability properties we refer to [8,10] and the references therein. We point to [3,4,37] for precise definitions of controllable, stabilizable, and their duals, observable, detectable.…”

“…The stability of (A − P t S), and also of φ t (Q) → t→∞ P ∞ , is directly related [8,10] to the contractive properties of the deterministic semigroup E s,t (Q). The stochastic equation (1.6) implies that the stability properties of EnKF state estimators with ̟ = 0 depend on the stability properties of the stochastic exponential semigroup E ǫ s,t (Q).…”