Résultat de la recherche
Asymptotic Pseudo-Trajectories and Chain-recurrent flows, with Applications
Learning Processes, Mixed Equilibria and Dynamical Systems Arising from Fictitious Play in Perturbed Games
Dynamics of Morse-Smale urn processes
We consider stochastic processes {x(n)}(n greater than or equal to 0) of the form x(n+1)-x(n)=gamma(n+1)(F(x(n))+U-n+1) where F : R(m) --> R(m) is C-2, {gamma(i)}(i greater than or equal to 1) is a Sequence of positive numbers decreasing to 0 and {U-i}(i greater than or equal to 1) is a sequence of uniformly bounded R(m)-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Gamma, the probability of sample paths approaching Gamma is positive. This gives the generic behavior of three-color urn models.
Stochastic approximation algorithms with constant step size whose average is cooperative
We consider stochastic approximation algorithms with constant step size whose average ordinary differential equation (ODE) is cooperative and irreducible. We show that, under mild conditions on the noise process, invariant measures and empirical occupations measures of the process weakly converge (as the time goes to infinity and the step size goes to zero) toward measures which are supported by stable equilibria of the ODE. These results are applied to analyzing the long-term behavior of a class of learning processes arising in game theory.