Stochastic approximations of set-valued dynamical systems: Convergence with positive probability to an attractor

Mathieu Faure (1) , Gregory Roth (1)

1 UNINE - Institut de Mathématiques

A succesful 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. Under an attainability condition, convergence to a given attractor of the flow induced by this dynamical system was proved to occur with positive probability (Benaïm, 1999) for a class of Robbins Monro algorithms. Benaïm et al. (2005) generalised this approach for stochastic approximation algorithms whose average behavior is related to a differential inclusion instead. We pursue the analogy by extending to this setting the result of convergence with positive probability to an attractor.

Mots clés: Stochastic approximations; set-valued dynamical systems; attractor; game theory; Markovian fictitious play

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Mathieu Faure, Gregory Roth. Stochastic approximations of set-valued dynamical systems: Convergence with positive probability to an attractor. Mathematics of Operations Research, 2010, 35 (3), pp.624-640. ⟨10.1287/moor.1100.0455⟩. ⟨hal-00383277v2⟩