Algorithmes d'approximation stochastiques et inclusions différentielles

A successful method to describe the asymptotic behavior of a discrete time stochastic process governed by some recursive formula is to relate it to the limit sets of a well chosen mean differential equation. Benaïm, Hofbauer and Sorin generalised this approach to stochastic approximation algorithms whose average behavior is related to a differential inclusion instead. The aim of this thesis is to pursue this analogy by extending to this setting the following results. First, under an attainability condition, we prove that convergence to a given attractor of the dynamical system induced by this differential inclusion occurs with positive probability, for a class of Robbins Monro algorithms. Next we generalize a result of Benaïm and Schreiber which characterizes the ergodic behavior of algorithms. In particular, we prove that the weak* limit points of the empirical measures associated to such processes are almost surely invariant for the associated deterministic dynamics. To do this, we give two equivalent definitions of the invariance of a measure for a set-valued dynamical system continuous in time. Secondly, we consider approximation algorithms with constant step size associated to a differential inclusion. We prove that over any finite time span, the sample paths of the stochastic process are closely approximated by a solution of the differential inclusion with high probability. We then analyze infinite horizon behavior, showing that stationary measures of the stochastic process must become concentrated on the Birkhoff center of the deterministic system.

Thèse de doctorat : Université de Neuchâtel, 2011