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Pagès, Gilles
Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM: Probability and Statistics, 5 (2001), p. 141-170
Texte intégral djvu | pdf | Analyses MR 1875668 | Zbl 0998.60073 | 2 citations dans Numdam
Class. Math.: 65C30, 62L20
Mots clés: ergodicity, stability, Markov process, diffusion, stochastic algorithm, ODE method, Euler scheme, empirical measure

URL stable: http://www.numdam.org/item?id=PS_2001__5__141_0

Résumé

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.

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