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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
Approximation and Limit Results for Nonlinear Filters over an Infinite Time Interval: Part II, Random Sampling Algorithms
"... The paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval. Since the optimal filter can rarely be constructed, one needs to compute with numerically feasible approximations. The signal model can be a jumpdiffusion, reflected or no ..."
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Cited by 19 (8 self)
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The paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval. Since the optimal filter can rarely be constructed, one needs to compute with numerically feasible approximations. The signal model can be a jumpdiffusion, reflected or not. The observations can be taken either in discrete or continuous time. The cost of interest is the pathwise error per unit time over a long time interval. In a previous paper of the authors [2], it was shown, under quite reasonable conditions on the approximating filter and on the signal and noise processes that (as time, bandwidth, process and filter approximation, etc.) go to their limit in any way at all, the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used. When suitable approximating filters cannot be readily constructed due to excessive computational requirem...
Exponential Stability in Discrete Time Filtering for NonErgodic Signals
 System and Control Letters
, 1999
"... In this paper we prove exponential asymptotic stability for discrete time filters for signals arising as solutions of ddimensional stochastic difference equations. The observation process is the signal corrupted by an additive white noise of su#ciently small variance. The model for the signal admit ..."
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Cited by 13 (5 self)
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In this paper we prove exponential asymptotic stability for discrete time filters for signals arising as solutions of ddimensional stochastic difference equations. The observation process is the signal corrupted by an additive white noise of su#ciently small variance. The model for the signal admits nonergodic processes. We show that almost surely, the total variation distance between the optimal filter and an incorrectly initialized filter converges to 0 exponentially fast as time approaches #. Key Words: nonlinear filtering, asymptotic stability, measure valued processes. # Research suppored by the NSF grant DMI 9812857. 1 1 Introduction The central problem of nonlinear filtering is to study the conditional distribution of a signal process at any time instant given noisy observations on the signal available up until that time. If the signalobservation pair is Markov, the conditional distribution process, referred to hereafter as the optimal filter , is determined completely ...
Model robustness of finite state nonlinear filtering over the infinite time horizon
 Ann. Appl. Probab
"... Abstract. We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converge ..."
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Cited by 5 (5 self)
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Abstract. We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converges to zero uniformly over the infinite time interval as the misspecified model converges to the true model, provided the signal obeys a mixing condition. The filtering error is controlled through the exponential decay of the derivative of the nonlinear filter with respect to its initial condition. We allow simultaneously for misspecification of the initial condition, of the transition rates of the signal, and of the observation function. The first two cases are treated by relatively elementary means, while the latter case requires the use of Skorokhod integrals and tools of anticipative stochastic calculus. 1.
Monte Carlo algorithms and asymptotic problems in nonlinear filtering
 To Appear in Stochastics in Finite/Infinite Dimensions (Volume in honor of Gopinath Kallianpur
, 1999
"... This paper is an extension of [4], which dealt with a wide variety of approximations to optimal nonlinear filters over long time intervals, where ..."
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Cited by 5 (2 self)
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This paper is an extension of [4], which dealt with a wide variety of approximations to optimal nonlinear filters over long time intervals, where
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.
Ergodic properties of the Nonlinear Filter
 Stochastic Processes and their Applications, 95:1–24
, 2000
"... In a recent work [5] various Markov and ergodicity properties of the nonlinear filter, for the classical model of nonlinear filtering, were studied. It was shown that under quite general conditions, when the signal is a FellerMarkov process with values in a complete separable metric space E then th ..."
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Cited by 2 (1 self)
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In a recent work [5] various Markov and ergodicity properties of the nonlinear filter, for the classical model of nonlinear filtering, were studied. It was shown that under quite general conditions, when the signal is a FellerMarkov process with values in a complete separable metric space E then the pair process (signal, filter) is also a FellerMarkov process with state space E P(E), where P(E) is the space of probability measures on E. Furthermore, it was shown that if the signal has a unique invariant measure then, under appropriate conditions, uniqueness of the invariant measure for the above pair process holds within a certain restricted class of invariant measures. In many asymptotic problems concerning approximate filters [6, 7] it is desirable to have the uniqueness of the invariant measure to hold in the class of all invariant measures. In this paper we first show that for a rich class of filtering problems, when the signal has a unique invariant measure, the property of...
A Simple Asymptotically Optimal Filter Over An Infinite Horizon
"... . A filtering problem over an infinite horizon for a continuous time signal and discrete time observation in the presence of non Gaussian white noise is considered. Conditions are presented, under which a nonlinear Kalman type filter with limiter is asymptotically optimal in the mean square sense fo ..."
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Cited by 2 (0 self)
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. A filtering problem over an infinite horizon for a continuous time signal and discrete time observation in the presence of non Gaussian white noise is considered. Conditions are presented, under which a nonlinear Kalman type filter with limiter is asymptotically optimal in the mean square sense for long time intervals given provided the sampling frequency is su#ciently high. 1.
THE STABILITY OF QUANTUM MARKOV FILTERS
, 709
"... Abstract. When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be veri ..."
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Abstract. When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation. 1.
Stability and Approximation of Nonlinear Filters: an Information Theoretic Approach
, 2000
"... It has recently been proved by Clark, Ocone and Coumarbatch that the relative entropy (or Kullback Leibler information distance) between two nonlinear filters with different initial conditions is a supermartingale, hence its expectation can only decrease with time. This result was obtained for a v ..."
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Cited by 1 (0 self)
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It has recently been proved by Clark, Ocone and Coumarbatch that the relative entropy (or Kullback Leibler information distance) between two nonlinear filters with different initial conditions is a supermartingale, hence its expectation can only decrease with time. This result was obtained for a very general model, where the unknown state and observation processes form jointly a continuoustime Markov process. The purpose of this paper is (i) to extend this result to a large class of fdivergences, including the total variation distance, the Hellinger distance, and not only the KullbackLeibler information distance, and (ii) to consider not only robustness w.r.t. the initial condition of the filter, but also w.r.t. perturbation of the state generator. On the other hand, the model considered here is much less general, and consists of a diffusion process observed in discretetime. Keywords : nonlinear filtering, stability, relative entropy, KullbackLeibler information, Hellinger...