A probabilistic approach to extended finite state mean field games

René Carmona, Peiqi Wang

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We develop a probabilistic approach to continuous-time finite state mean field games. Based on an alternative description of continuous-time Markov chains by means of semimartingales and the weak formulation of stochastic optimal control, our approach not only allows us to tackle the mean field of states and the mean field of control at the same time, but also extends the strategy set of players from Markov strategies to closed-loop strategies. We show the existence and uniqueness of Nash equilibrium for the mean field game as well as how the equilibrium of a mean field game consists of an approximative Nash equilibrium for the game with a finite number of players under different assumptions of structure and regularity on the cost functions and transition rate between states.

Original languageEnglish (US)
Pages (from-to)471-502
Number of pages32
JournalMathematics of Operations Research
Issue number2
StatePublished - May 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Computer Science Applications
  • Management Science and Operations Research


  • Approximative Nash equilibrium
  • Finite state space
  • McKean–Vlasov BSDE
  • Mean field games
  • Weak formulation of optimal control


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