this article, we use the approach based on the Hilbert metric to study the asymptotic behavior of the optimal filter, and to prove as in [9] the uniform convergence of several particle filters, such as the interacting particle filter (IPF) and some other original particle filters. A common assumption to prove stability results, see e.g. in [9, Theorem 2.4], is that the Markov transition kernels are mixing, which implies that the hidden state sequence is ergodic. Our results are obtained under the assumption that the nonnegative kernels describing the evolution of the unnormalized optimal filter, and incorporating simultaneously the Markov transition kernels and the likelihood functions, are mixing. This is a weaker assumption, see Proposition 3.9, which allows to consider some cases, similar to the case studied in [6], where the hidden state sequence is not ergodic, see Example 3.10. This point of view is further developped by Le Gland and Oudjane in [22] and by Oudjane and Rubenthaler in [28]. Our main contribution is to study also the stability of the optimal filter w.r.t. the model, when the local error is propagated by mixing kernels, and can be estimated in the Hilbert metric, in the total variation norm, or in a weaker distance suitable for random probability distributions. AMS 1991 subject classifications. Primary 93E11, 93E15, 62E25; secondary 60B10, 60J27, 62G07, 62G09, 62L10

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 non-ergodic 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

In this paper a new generation of particle filters for nonlinear discrete time processes is proposed, based on convolution kernel probability density estimation. The main advantage of this approach is to be free of the limitations encountered by the current particle filters when the likelihood of the observation variable is analytically unknown or when the observation noise is null or too small. To illustrate this convolution kernel approach the counterparts of the well-known sequential importance sampling (SIS) and sequential importance sampling-resampling (SIS-R) filters are considered and their stochastic convergence to the optimal filter under different modes are proved. Their good behaviour with respect to that of these filters is shown on several simulated case studies.

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.

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 one-step predictor estimates. 1.

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 σ-algebra of events generated by Ym, k ≤ m ≤ n (similarly, we will use below the σ-algebra 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 E|f(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 ∈ A|F X 0,n−1)

Abstract. Ergodic properties of the signal-filtering 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 Furstenberg-Khasminskii formula is derived for the top Lyapunov exponent of the Zakai equation and is used to estimate the stability index of the filter. 1.

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.

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apport de recherche Convolution particle filters for parameter estimation in general state-space models