Results 1  10
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14
Exponential Stability for Nonlinear Filtering
, 1996
"... 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. C ..."
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Cited by 53 (2 self)
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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 discretecountable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...
Asymptotic stability of the Wonham filter: ergodic and nonergodic signals
 SIAM J. Control Optim
"... Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the ..."
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Cited by 25 (13 self)
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Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure. Key words. Nonlinear filtering, stability, Wonham filter
Robustness of Nonlinear Filters over the Infinite Time Interval
 SIAM J. on Control and Optimization
, 1997
"... Nonlinear filtering is one of the classical areas of stochastic control. From the point of view of practical usefulness, it is important that the filter not be too sensitive to the assumptions made on the initial distribution, the transition function of the underlying signal process and the model fo ..."
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Cited by 18 (7 self)
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Nonlinear filtering is one of the classical areas of stochastic control. From the point of view of practical usefulness, it is important that the filter not be too sensitive to the assumptions made on the initial distribution, the transition function of the underlying signal process and the model for the observation. This is particularly acute if the filter is of interest over a very long or potentially infinite time interval. Then the effects of small errors in the model which is used to construct the filter might accumulate to make the output useless for large time. The problem of asymptotic sensitivity to the initial condition has been treated in several papers. We are concerned with this as well as with the sensitivity to the signal model, uniformly over the infinite time interval. It is conceivable that the effects of even small errors in the model will accumulate so that the filter will eventually be useless. The robustness is shown for three classes of problems. For the first tw...
R.Liptser, Stability of nonlinear filters in nonmixing
"... The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The ..."
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Cited by 14 (6 self)
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The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is a uniform positiveness of this density. We show that this requirement might be weakened regardless of an observation process structure.
On a role of predictor in the filtering stability
 Electron. Comm. Probab
"... Abstract. When is a nonlinear filter stable with respect to its initial condition? In spite of the recent progress, this question still lacks a complete answer in general. Currently available results indicate that stability of the filter depends on the signal ergodic properties and the observation p ..."
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Cited by 8 (1 self)
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Abstract. When is a nonlinear filter stable with respect to its initial condition? In spite of the recent progress, this question still lacks a complete answer in general. Currently available results indicate that stability of the filter depends on the signal ergodic properties and the observation process regularity and may fail if either of the ingredients is ignored. In this note we address the question of stability in a particular weak sense and show that the estimates of certain functions are always stable. This is verified without dealing directly with the filtering equation and turns to be inherited from certain onestep predictor estimates. 1.
Asymptotic Stability, Ergodicity and Other Asymptotic Properties of the Nonlinear Filter
, 2002
"... In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation # field and the tail # field of the ..."
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Cited by 4 (0 self)
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In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation # field and the tail # field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain # fields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as (a) uniqueness of invariant measure for the signal, (b) uniqueness of invariant measure for the pair (signal, filter), (c) a finite memory property of the filter , (d) a property of finite time dependence between the signal and observation # fields and (e) asymptotic stability of the filter. Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold.
Stability of the nonlinear filter for slowly switching Markov chains. Stochastic Process
 Appl
"... Dedicated to Robert Liptser on the occasion of his 70th birthday Abstract. Exponential stability of the nonlinear filtering equation is revisited, when the signal is a finite state Markov chain. An asymptotic upper bound for the filtering error due to incorrect initial condition is derived in the ca ..."
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Cited by 3 (1 self)
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Dedicated to Robert Liptser on the occasion of his 70th birthday Abstract. Exponential stability of the nonlinear filtering equation is revisited, when the signal is a finite state Markov chain. An asymptotic upper bound for the filtering error due to incorrect initial condition is derived in the case of slowly switching signal.
INTRINSIC METHODS IN FILTER STABILITY
"... Abstract. The purpose of this article is to survey some intrinsic methods for studying the stability of the nonlinear filter. By ‘intrinsic ’ we mean methods which directly exploit the fundamental representation of the filter as a conditional expectation through classical probabilistic techniques su ..."
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Cited by 3 (1 self)
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Abstract. The purpose of this article is to survey some intrinsic methods for studying the stability of the nonlinear filter. By ‘intrinsic ’ we mean methods which directly exploit the fundamental representation of the filter as a conditional expectation through classical probabilistic techniques such as change of measure, martingale convergence, coupling, etc. Beside their conceptual appeal and the additional insight gained into the filter stability problem, these methods allow one to establish stability of the filter under weaker conditions compared to other methods, e.g., to go beyond strong mixing signals, to reveal connections between filter stability and classical notions of observability, and to discover links to martingale convergence and information theory. 1. Inroduction Consider a pair of random sequences (X, Y) = (Xn, Yn)n∈Z+, where the signal component Xn takes values in a Polish space 1 S and the observation component Yn takes values in R p for some p ≥ 1. The classical filtering problem is to compute the conditional distribution πn(·) = P(Xn ∈ · F Y 0,n), (1.1) where F Y k,n stands for the σalgebra of events generated by Ym, k ≤ m ≤ n (similarly, we will use below the σalgebra F X k,n generated by Xm, k ≤ m ≤ n). Once πn is found, the optimal mean square estimate of f(Xn) can be calculated as E(f(Xn)F Y ∫ 0,n) = f(x) πn(dx) for any function f with Ef(Xn)  2 < ∞. If both X and (X, Y) are Markov processes, πn satisfies a recursive filtering equation. Specifically, let Λ and ν denote the transition probability and the initial distribution of X, i.e., for A ∈ B(S) ν(A) = P(X0 ∈ A), Λ(Xn−1, A) = P(Xn ∈ AF X 0,n−1)
AN ERGODIC THEOREM FOR FILTERING WITH APPLICATIONS TO STABILITY
, 2006
"... Abstract. Ergodic properties of the signalfiltering pair are studied for continuous time finite Markov chains, observed in white noise. The obtained law of large numbers is applied to the stability problem of the nonlinear filter with respect to initial conditions. The FurstenbergKhasminskii formu ..."
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Cited by 2 (2 self)
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Abstract. Ergodic properties of the signalfiltering pair are studied for continuous time finite Markov chains, observed in white noise. The obtained law of large numbers is applied to the stability problem of the nonlinear filter with respect to initial conditions. The FurstenbergKhasminskii formula is derived for the top Lyapunov exponent of the Zakai equation and is used to estimate the stability index of the filter. 1.
ON EXPONENTIAL STABILITY OF WONHAM FILTER
, 2002
"... Abstract. We give elementary proof of a stability result from Atar and Zeitouni, [1], concerning an exponential asymptotic (t → ∞) for filtering estimates generated by wrongly initialized Wonham filter. This proof is based on new exponential bound having independent interest. 1. ..."
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Cited by 1 (1 self)
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Abstract. We give elementary proof of a stability result from Atar and Zeitouni, [1], concerning an exponential asymptotic (t → ∞) for filtering estimates generated by wrongly initialized Wonham filter. This proof is based on new exponential bound having independent interest. 1.