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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: ergodicity, stability, Markov process, diffusion, stochastic algorithm, ODE method, Euler scheme, empirical measure
@article{PS_2001__5__141_0,
author = {Pag\`es, Gilles},
title = {Sur quelques algorithmes r\'ecursifs pour les probabilit\'es num\'eriques},
journal = {ESAIM: Probability and Statistics},
pages = {141--170},
publisher = {EDP-Sciences},
volume = {5},
year = {2001},
mrnumber = {1875668},
zbl = {0998.60073},
language = {fr},
url = {http://geodesic.mathdoc.fr/item/PS_2001__5__141_0/}
}
Pagès, Gilles. Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 141-170. http://geodesic.mathdoc.fr/item/PS_2001__5__141_0/