Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60G35 Applications (signal detection, filtering, etc.), See Also {62M20, 93E10, 93E11, 94Axx}
- 60F17 Functional limit theorems; invariance principles
- 93E11 Filtering, See also {60G35}

**Résumé:** The aim of this paper is to investigate the behaviour of some
Monte-Carlo approximation schemes for the filter in the case of
discrete-time observations, as time goes to infinity. Two
approximation schemes are considered: one is based on a ``na\"\i
ve'' Monte-Carlo simulation, the other one, introduced in a previous
paper [3], is based on an interacting particle scheme. An associated
central limit theorem shows that the normalized difference between the
approximate filter and the true filter on a given test function $f$
and at time $n$ is
asymptotically centered Gaussian (as
the number of simulated variables increases) with variance, say,
$\Ga_n(f)$: this $\Ga_n(f)$ depends on the observations, and so is
itself a random variable.
In order to determine the precise behaviour of $\Ga_n(f)$ as $n$
increases, we consider a very particular
(but hopefully representative of the general situation) case, namely the
state process is an Ornstein-Uhlenbeck process, while the observations, taking
place at integer times, equal the state process plus a Gaussian error. We
consider only the case when the state process is ergodic.
The result is that the random variables $\Ga_n(f)$ stay bounded
in probability for the interacting particle scheme, while they grow
exponentially fast for the na\"\i ve scheme.

**Mots Clés:** *Filtering ; Monte-Carlo methods ; Diffusion processes ; Interacting particle systems*

**Date:** 2000-02-16

**Prépublication numéro:** *PMA-565*