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Control of McKean–Vlasov dynamics versus mean field games

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Abstract

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 analysis of forward–backward stochastic differential equations, the coefficients of which depend on the marginal distributions of the solutions. We explain the difference between the nature and solutions to the two approaches by investigating the corresponding forward–backward systems. General results are stated and specific examples are treated, especially when cost functionals are of linear-quadratic type.

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References

  1. Andersson D., Djehiche B.: A maximum principle for sdes of mean-field type. Appl. Math. Optim. 63(3), 341–356 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bardi, M.: Explicit solutions of some linear quadratic mean field games. Technical report, Padova University, May 2011

  3. Cardaliaguet, P.: Notes on mean field games. Technical report (2010)

  4. Carmona, R., Delarue, F.: Forward backward stochastic differential equations of McKean–Vlasov type. Technical report (2012)

  5. Carmona, R., Delarue, F.: Optimal control of McKean–Vlasov stochastic dynamics. Technical report (2012)

  6. Carmona, R., Delarue, F.: Probabilistic analysis of mean field games. Technical report (2012)

  7. Carmona, R., Delarue, F., Espinosa, G-E., Touzi, N.: Singular forward–backward stochastic differential equations and emissions derivatives. Ann. Appl. Probab. (2012) (in press)

  8. Delarue F.: On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch. Process. Appl. 99, 209–286 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Delarue F.: Estimates of the Solutions of a System of Quasi-Linear PDEs. A Probabilistic Scheme. Séminaire de Probabilités XXXVII, pp. 290–332. Springer, New York (2003)

    Google Scholar 

  10. Fleming W., Soner M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (2010)

    Google Scholar 

  11. Guéant, O., Lasry, J.M., Lions, P.L.: Mean field games and applications. In: Carmona R. et~al. (eds.) Paris Princeton Lectures in Mathematical Finance IV, vol 2003 of Lecture Notes in Mathematics. Springer, New York (2010)

  12. Hamadène S.: Nonzero-sum linear quadratic stochastic differential games and backward forward equations. Stoch. Anal. Appl. 17, 117–130 (1999)

    Article  MATH  Google Scholar 

  13. Horst U.: Stationary equilibria in discounted stochastic games with weakly interacting players. Games Econ. Behav. 51, 83–108 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jovanovic B., Rosenthal R.W.: Anonymous sequential games. J. Math. Econ. 17, 77–87 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lasry J.M., Lions P.L.: Jeux à champ moyen I. Le cas stationnaire. Comptes Rendus de l’Académie des Sciences de Paris Ser. A 343, 619–625 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lasry J.M., Lions P.L.: Jeux à champ moyen II. Horizon fini et contrôle optimal. Comptes Rendus de l’Académie des Sciences de Paris Ser. A 343, 679–684 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lasry J.M., Lions P.L.: Mean field games. Japanese Journal of Mathematics 2, 229–260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lasry, J.M., Lions, P.L.: Mean field games. Cahiers de la Chaire Finance et Développement Durable 2 (2007)

  19. Lions, P.L.: Théorie des jeux à champs moyen et applications. Lectures of the Collége de France, Paris, France. http://www.college-de-france.fr/site/pierre-louis-lions/ (2007–2008)

  20. Peng S., Wu Z.: Fully coupled forward–backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37, 825–843 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sontag E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems (Second Edition). Springer, New-York (1998)

    MATH  Google Scholar 

  22. Sznitman, A.S.: Topics in propagation of chaos. In: Burkholder D.L. et al. (eds.) Ecole de Probabilités de Saint Flour, XIX-1989, vol 1464 of Lecture Notes in Mathematics, pp. 165–251. Springer, Berlin (1989)

  23. Yong J.: Linear forward backward stochastic differential equations. Appl. Math. Optim. 39, 93–119 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yong J.: Linear forward backward stochastic differential equations with random coefficients. Probab. Theory Relat. Fields 135, 53–83 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yong J., Zhou X.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    MATH  Google Scholar 

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Authors and Affiliations

  1. ORFE, Bendheim Center for Finance, Princeton University, Princeton, NJ, 08544, USA

    René Carmona & Aimé Lachapelle

  2. Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    François Delarue

Authors
  1. René Carmona
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  2. François Delarue
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  3. Aimé Lachapelle
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Correspondence to François Delarue.

Additional information

Paper presented at the conference “Stochastic Partial Differential Equations: Analysis, Numerics, Geometry and Modeling”, ETH Zürich, September 12–16, 2011 and the Third Humboldt-Princeton Conference, October 28–29, 2011.

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Carmona, R., Delarue, F. & Lachapelle, A. Control of McKean–Vlasov dynamics versus mean field games. Math Finan Econ 7, 131–166 (2013). https://doi.org/10.1007/s11579-012-0089-y

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  • DOI: https://doi.org/10.1007/s11579-012-0089-y

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