Control of McKean–Vlasov dynamics versus mean field games

@article{Carmona2012ControlOM,
  title={Control of McKean–Vlasov dynamics versus mean field games},
  author={Ren{\'e} A. Carmona and François Delarue and Aim{\'e} Lachapelle},
  journal={Mathematics and Financial Economics},
  year={2012},
  volume={7},
  pages={131-166}
}
We discuss and compare two investigation methods for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which optimization and passage to the limit are performed. When optimizing first, the asymptotic problem is usually referred to as a mean-field game. Otherwise, it reads as an optimization problem over controlled dynamics of McKean–Vlasov type. Both problems lead to the… 
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References

SHOWING 1-10 OF 33 REFERENCES
Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics
The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean–Vlasov type. Motivated by the recent interest in
Probabilistic Analysis of Mean-Field Games
TLDR
It is proved that a solution of the Mean-Field Game problem as formulated by Lasry and Lions, does indeed provide a solution and existence and regularity of the corresponding value function are proved.
Explicit solutions of some linear-quadratic mean field games
  • M. Bardi
  • Mathematics
    Networks Heterog. Media
  • 2012
TLDR
The quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games is solved and the L-Q model is compared with other Mean Field models of population distribution.
Mean field games
Abstract.We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field
A Maximum Principle for SDEs of Mean-Field Type
We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the
Controlled Markov processes and viscosity solutions
This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems
Linear Forward—Backward Stochastic Differential Equations
Abstract. The problem of finding adapted solutions to systems of coupled linear forward—backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of
Linear forward-backward stochastic differential equations with random coefficients
Abstract.Solvability of linear forward-backward stochastic differential equations (FBSDEs, for short) with random coefficients is studied. A decoupling reduction method is introduced via which a
Stationary equilibria in discounted stochastic games with weakly interacting players
  • U. Horst
  • Mathematics, Economics
    Games Econ. Behav.
  • 2005
...
...