Advertisement

The Master Equation for Large Population Equilibriums

  • René Carmona
  • François DelarueEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 100)

Abstract

We use a simple \(N\)-player stochastic game with idiosyncratic and common noises to introduce the concept of Master Equation originally proposed by Lions in his lectures at the Collège de France. Controlling the limit \(N\rightarrow \infty \) of the explicit solution of the \(N\)-player game, we highlight the stochastic nature of the limit distributions of the states of the players due to the fact that the random environment does not average out in the limit, and we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic Partial Differential Equations (SPDEs). The first one is a forward stochastic Kolmogorov equation giving the evolution of the conditional distributions of the states of the players given the common noise. The second is a form of stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the optimization problem when the flow of conditional distributions is given. Being highly coupled, the system reads as an infinite dimensional Forward Backward Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its Markov property lead to the representation of the solution of the backward equation (i.e. the value function of the stochastic HJB equation) as a deterministic function of the solution of the forward Kolmogorov equation, function which is usually called the decoupling field of the FBSDE. The (infinite dimensional) PDE satisfied by this decoupling field is identified with the master equation. We also show that this equation can be derived for other large populations equilibriums like those given by the optimal control of McKean-Vlasov stochastic differential equations. The paper is written more in the style of a review than a technical paper, and we spend more time motivating and explaining the probabilistic interpretation of the Master Equation, than identifying the most general set of assumptions under which our claims are true.

Keywords

Master Equation Larger Equilibrium Population Mean Field Games (MFG) Forward-backward Stochastic Differential Equations (FBSDE) McKean Vlasov Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    A. Bensoussan, J. Frehse, P. Yam, The master equation in mean-field theory. Technical report. http://arxiv.org/abs/1404.4150
  2. 2.
    P. Cardaliaguet, Notes on mean field games. Notes from P.L. Lions’ lectures at the Collège de France https://www.ceremade.dauphine.fr/cardalia/MFG100629.pdf (2012)
  3. 3.
    R. Carmona, F. Delarue, Forward-backward stochastic differential equations and controlled McKean Vlasov dynamics. in Annals of Probability To appearGoogle Scholar
  4. 4.
    R. Carmona, F. Delarue, Probabilistic analysis of mean field games. SIAM J. Control Optim. 51, 2705–2734 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    R. Carmona, F. Delarue, D. Lacker, Mean field games with a common noise. Technical report. http://arxiv.org/abs/1407.6181
  6. 6.
    R. Carmona, F. Delarue, A. Lachapelle, Control of McKean-Vlasov versus mean field games. Math. Financ. Econ. 7, 131–166 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    R. Carmona, J.P. Fouque, A. Sun, Mean field games and systemic risk. To appear in Communications in Mathematical SciencesGoogle Scholar
  8. 8.
    J.F. Chassagneux, D. Crisan, F. Delarue, McKean-Vlasov FBSDEs and related master equation. Work in progressGoogle Scholar
  9. 9.
    W. Fleming, M. Soner, Controlled Markov Processes and Viscosity Solutions (Springer, New York, 2010)Google Scholar
  10. 10.
    D.A. Gomes, J. Saude, Mean field games models—a brief survey. Technical report (2013)Google Scholar
  11. 11.
    O. Guéant, J.M. Lasry, P.L. Lions, Paris Princeton Lectures in Mathematical Finance IV, in Mean Field Games and Applications, Lecture Notes in Mathematics, ed. by R. Carmona, et al. (Springer, Berlin, 2010)Google Scholar
  12. 12.
    M. Huang, P.E. Caines, R.P. Malhamé, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6, 221–252 (2006)zbMATHMathSciNetGoogle Scholar
  13. 13.
    J.M. Lasry, P.L. Lions, Jeux à champ moyen I. Le cas stationnaire. Comptes Rendus de l’Académie des Sciences de Paris, ser. A 343(9), 619–625 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    J.M. Lasry, P.L. Lions, Jeux à champ moyen II. Horizon fini et contrôle optimal. Comptes Rendus de l’Académie des Sciences de Paris, ser. A 343(10), 679–684 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    J.M. Lasry, P.L. Lions, Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    P.L. Lions, Théorie des jeux à champs moyen et applications. Technical report, 2007–2008Google Scholar
  17. 17.
    J. Ma, H. Yin, J. Zhang, On non-Markovian forward-backward SDEs and backward stochastic PDEs. Stoch. Process. Appl. 122, 3980–4004 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications (Springer, New York, 1995)CrossRefzbMATHGoogle Scholar
  19. 19.
    S. Peng, Stochastic Hamilton Jacobi Bellman equations. SIAM J. Control Optim. 30, 284–304 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    A.S. Sznitman, Topics in propagation of chaos, in D.L. Burkholder et al., Ecole de Probabilités de Saint Flour, XIX-1989. Lecture Notes in Mathematics, vol. 1464 (Springer, Heidelberg, 1989), pp. 165–251Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.ORFEBendheim Center for Finance, Princeton UniversityPrincetonUSA
  2. 2.Laboratoire J.A. DieudonnéUniversité de Nice Sophia-AntipolisParc ValroseFrance

Personalised recommendations