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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Interacting particle filtering with discrete-time observations : asymptotic behaviour in the Gaussian case

Auteur(s):

Code(s) de Classification MSC:

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