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48
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 ...
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.
Robustness of Zakai's equation via FeynmanKac representations
 Stochastic Analysis, Control, Optimization and Applications : A Volume in Honor of W.H. Fleming, Systems & Control : Foundations & Applications
, 1997
"... We propose to study the sensitivity of the optimal filter to its initialization, by looking at the distance between two differently initialized filtering processes in terms of the ratio between two simple FeynmanKac integrals in the product space. We illustrate, by considering two simple examples, ..."
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Cited by 7 (1 self)
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We propose to study the sensitivity of the optimal filter to its initialization, by looking at the distance between two differently initialized filtering processes in terms of the ratio between two simple FeynmanKac integrals in the product space. We illustrate, by considering two simple examples, how this approach may be employed to study the asymptotic decay rate, as the difference between the growth rates of the two integrals. We apply asymptotic methods, such as large deviations, to estimate these growth rates. The examples we consider are the linear case, where we recover known results, and a case where the drift term in the state process is nonlinear. In both cases, only the small noise regime and only onedimensional diffusions are studied.
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
Uniform time average consistency of Monte Carlo particle filters
 BP 101  54602 VillerslèsNancy Cedex Centre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau  Rocquencourt  BP 105  78153 Le Chesnay Cedex Centre de recherche INRIA Rennes – Bretagne Atlantique : IRISA, Campus universitaire de Beaulieu 
, 2009
"... Abstract. We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further a ..."
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Cited by 5 (0 self)
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Abstract. We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further assumptions when the signal state space is compact, as well as in the noncompact setting when the signal is geometrically ergodic and the observations satisfy additional regularity assumptions. 1.
Stability of nonlinear filters in nonmixing case
, 2004
"... 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 an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing conditi ..."
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Cited by 5 (1 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 an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, 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 the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure. 1. Introduction and the main result. This paper addresses the stability problem of the nonlinear filtering equation with respect to its initial condition. We consider a homogeneous ergodic Markov chain (Xn)n≥0 with values in S ⊆ Rd regarded as a signal to be filtered from observation of (Yn)n≥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.
L 1 convergence of smoothing densities in non parametric state space models submitted to Stat. Inf. for Stoch
 Proc
, 2005
"... This paper addresses the problem of reconstructing partially observed stochastic processes. The L 1 convergence of the filtering and smoothing densities in state space models is studied, when the transition and emission densities are estimated using non parametric kernel estimates. An application to ..."
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Cited by 4 (3 self)
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This paper addresses the problem of reconstructing partially observed stochastic processes. The L 1 convergence of the filtering and smoothing densities in state space models is studied, when the transition and emission densities are estimated using non parametric kernel estimates. An application to real data is proposed, in which a wave time series is forecasted given a wind time series.
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.
Exponential Forgetting and Geometric Ergodicity in StateSpace Models
 in Proc. 41th IEEE Conf. Decision and Control, 2002
, 2002
"... In this paper, the problem of exponential forgetting and geometric ergodicity for optimal filtering in gen eral state spase models is considered. We consider here statespace models where the latent process is modeled by a Markov chain taking its values in a continuous space and the observation at ..."
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Cited by 3 (1 self)
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In this paper, the problem of exponential forgetting and geometric ergodicity for optimal filtering in gen eral state spase models is considered. We consider here statespace models where the latent process is modeled by a Markov chain taking its values in a continuous space and the observation at each point admits a distribution dependent of both the current state of the Markov chain and the past observation. Under given regularity assumptions, we establish that 1) the filter, and its derivatives with respect to some parameters in the model, have exponential forgetting properties and 2) the extended Markov chain, whose components are the latent process, the observation sequence, the filter and its derivatives is geometrically ergodic.