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Sur quelques algorithmes récursifs pour les probabilitésnumériques

Published online by Cambridge University Press:  15 August 2002

Gilles Pagès
Gilles Pagès*
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, Université Paris 6, Case 188, 4 place Jussieu, 75252 Paris Cedex 05, France ; gpa@ccr.jussieu.fr.
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Abstract

The aim of this paper is to take an in-depth look at the long time behaviour of some continuous time Markovian dynamical systems and at its numerical analysis. We first propose a short overview of the main ergodicity properties of time continuous homogeneous Markov processes (stability, positive recurrence). The basic tool is a Lyapunov function. Then, we investigate if these properties still hold for the time discretization of these processes, either with constant or decreasing step (ODE method in stochastic approximation, Euler scheme for diffusions). We point out several advantages of the weighted empirical random measures associated to these procedures, especially with decreasing step, in terms of convergence and of rate of convergence. Several simulations illustrate these results.

Keywords

ErgodicitystabilityMarkov processdiffusionstochastic algorithmODE methodEuler schemeempirical measure.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 141 - 170
Copyright
© EDP Sciences, SMAI, 2001

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