M. Benaim, J. Hofbauer and S. Sorin:

Stochastic approximation and differential inclusions

SIAM J. Control and Optimization 44 (1) (2005), 328348.

Abstract. The dynamical systems approach to stochastic approximation is generalized to the case where the mean differential equation is replaced b a differential inclusion. The limit set theorem of Benaim 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.

reprint from SIAM