Exponential stability for nonlinear filtering

doi:10.1016/S0246-0203(97)80110-0

Annales de l'Institut Henri Poincare (B) Probability and Statistics

Volume 33, Issue 6, 1997, Pages 697-725

https://doi.org/10.1016/S0246-0203(97)80110-0 Get rights and content

Abstract

We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discrete-countable state spaces. A similar question regarding the optimal smoother is investigated and a stability criterion is provided.

Résumé

Nous étudions la stabilité du filtre optimal par rapport à ses conditions initiales. Le taux de décroissance exponentielle est calculé dans un cadre général, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des critères de stabilité exponentielle et des bornes explicites sur le taux sont calculés pour les cas particuliers d'une diffusion sur une variété compacte, ainsi que pour des chaînes de Markov sur un espace discret ou continu.

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^∗: This work was partially supported by the Israel Science Foundation, administered by the Israeli Academy of Sciences and Humanities.

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Exponential stability for nonlinear filtering

Abstract

Résumé

J. Multivariate Anal.

Systems & Control Letters

Lyapunov Exponents for Finite State Nonlinear Filtering

Siam J. Contr. Optim.

Extensions of Jentzsch's Theorem

Trans. Am. Math. Soc.

Lattice Theory

Théorèmes limites pour les systèmes linéaires à coefficients markoviens

Prob. Theory Rel. Fields

Products of Random Matrices with Applications to Schrödinger Operators

Heat Kernels and Spectral Theory

Lyapunov exponents for filtering problems

Stochastic Parabolic Equations in Bounded Domains: Random Evolution Operator and Lyapunov Exponents

Stochastics and Stochastic Reports