Stochastic Approximations and Differential Inclusions

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Stochastic Approximations and Differential Inclusions

Michel Benaïm, Josef Hofbauer, Sylvain Sorin

To cite this version:

Michel Benaïm, Josef Hofbauer, Sylvain Sorin. Stochastic Approximations and Differential Inclusions.

2003. �hal-00242990�

Stochastic Approximations and Differential Inclusions

Michel Benaim

Josef Hofbauer

Sylvain Sorin

December 2003

Cahier n° 2003-029

ECOLE POLYTECHNIQUE

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

LABORATOIRE D'ECONOMETRIE

1rue Descartes F-75005 Paris (33) 1 55558215 http://ceco.polytechnique.fr/

mailto:labecox@poly.polytechnique.fr

Stochastic Approximations and Differential Inclusions

Michel Benaim

Josef Hofbauer

Sylvain Sorin

December 2003 Cahier n° 2003-029

Résumé:

L'approche en termes de systèmes dynamiques de l'approximation stochastique est étendue au cas ou l'équation différentielle moyenne est remplacée par une inclusion différentielle.

Le théorème de Benaim et Hirsch sur l'ensemble limite est étendu a ce cas. On étudie en détail les ensembles ICT et les attracteurs. On donne des applications a des questions de théorie des jeux, en particulier concernant le théorème d'approchabilite de Blackwell et la convergence de "fictitious play".

Abstract:

The dynamical systems approach to stochastic approximation is generalized to the case where the mean differential equation is replaced by a differential inclusion.

The limit set theorem of Bena\"{\i}m and Hirsch is extended to this situation. Internally chain transitive sets and attractors are studied in detail for set-valued dynamical systems.

Applications to game theory are given, in particular to Blackwell's approachability theorem and the convergence of fictitious play.

Mots clés :

Approximation stochastique, système dynamique multivalue

Key Words :

Stochastic approximation, multivalued dynamical systems

Classification AMS:

91A05, 37C

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