Stochastic approximations and differential inclusions

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 Benaim and Hirsch is extended to this situation. Internally chain transitive sets and attractors are studied in...

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Bibliographic Details
Published inSIAM journal on control and optimization Vol. 44; no. 1; pp. 328 - 348
Main Authors BENAÏM, Michel, HOFBAUER, Josef, SORIN, Sylvain
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Subjects
Algorithms
Applied sciences
Approximation
Computer science; control theory; systems
Control theory. Systems
Dynamical systems
Exact sciences and technology
Game theory
Hypotheses
Mathematical functions
Mathematical programming
Operational research and scientific management
Operational research. Management science
Random variables
62P20
Differential equation
Stochastic approximation
Set computation
37B25
learning
Dynamical system
fictitious play 62L20
Attractor
Stochastic system
Game theory
34F05
34G25
Limit theorem
chain recurrence
Uncertain system
91A22
93E35
91A26
approachability
Differential inclusion
differential inclusions
set-valued dynamical systems
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Summary: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 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.
ISSN:0363-0129
1095-7138
DOI:10.1137/s0363012904439301