Advertisement

Restoring uniqueness to mean-field games by randomizing the equilibria

  • François DelarueEmail author
Article

Abstract

We here address the question of restoration of uniqueness in mean-field games deriving from deterministic differential games with a large number of players. The general strategy for restoring uniqueness is inspired from earlier similar results on ordinary and stochastic differential equations. It consists in randomizing the equilibria through an external noise. As a main feature, we choose the external noise as an infinite dimensional Ornstein–Uhlenbeck process. We first investigate existence and uniqueness of a solution to the noisy system made of the mean-field game forced by the Ornstein–Uhlenbeck process. We also show how such a noisy system can be interpreted as the limit version of a stochastic differential game with a large number of players.

Keywords

Mean-field game Common noise Restoration of uniqueness SPDE Forward–backward system N-player game 

Mathematics Subject Classification

Primary 60H10 60H15 91A13 91A15 Secondary 60H30 

Notes

References

  1. 1.
    Ahuja, S.: Wellposedness of mean field games with common noise under a weak monotonicity condition. Technical report. http://arxiv.org/pdf/1406.7028v2.pdf (2015)
  2. 2.
    Bafico, R., Baldi, P.: Small random perturbations of Peano phenomena. Stochastics 6, 279–292 (1982)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springer Briefs in Mathematics. Springer, Berlin (2013)CrossRefGoogle Scholar
  4. 4.
    Bensoussan, A., Frehse, J., Yam, S.C.P.: The master equation in mean field theory. J. Math. Pures Appl. 103(9), 1441–1474 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bensoussan, A., Frehse, J., Yam, S.C.P.: On the interpretation of the Master Equation. Stochastic Process. Appl. 127(7), 2093–2137 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blackwell, D., Dubins, L.: An extension of Skorohod’s almost sure representation theorem. Proc. Am. Math. Soc. 89(4), 691–692 (1983).  https://doi.org/10.2307/2044607 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cardaliaguet, P.: Notes from P.L. Lions’ lectures at the Collège de France. Technical report. https://www.ceremade.dauphine.fr/~cardalia/MFG100629.pdf (2012)
  8. 8.
    Cardaliaguet, P.: Weak solutions for first order mean field games with local coupling. In: Bettiol, P., Colombo, G., Motta, M., Rampazzo, Z., et al. (eds.) Analysis and Geometry in Control Theory and its Applications. Springer INdAM Series, pp. 111–158. Rampazzo, Cham (2015)Google Scholar
  9. 9.
    Cardaliaguet, P., Delarue, F., Lasry, J.M., Lions, P.L.: The master equation and the convergence problem in mean field games. Technical report (2015)Google Scholar
  10. 10.
    Cardaliaguet, P., Graber, J.: Mean field games systems of first order. ESAIM Control Optim. Calc. Var. 21, 690–722 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cardaliaguet, P., Graber, J.: Second order mean field games with degenerate diffusion and local coupling. NoDEA 22, 1287–1317 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cardaliaguet, P., Mészáros, A.R., Santambrogio, F.: First order mean field games with density constraints: pressure equals price. Technical report. arXiv:1507.02019
  13. 13.
    Carmona, R., Delarue, F.: Mean field forward–backward stochastic differential equations. Electron. Commun. Probab. 68, 1–15 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Carmona, R., Delarue, F.: Probabilistic analysis of mean field games. SIAM J. Control Optim. 51, 2705–2734 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Carmona, R., Delarue, F.: The master equation for large population equilibriums. In: Crisan, D., Hambly, B., Zariphopoulou, T. (eds.) Stochastic Analysis and Applications, pp. 77–128. Springer, Berlin (2014)Google Scholar
  16. 16.
    Carmona, R., Delarue, F.: Probabilistic Theory of Mean Field Games: Vol. I, Mean Field FBSDEs, Control, and Games. Stochastic Analysis and Applications. Springer, Berlin (2018)zbMATHGoogle Scholar
  17. 17.
    Carmona, R., Delarue, F.: Probabilistic Theory of Mean Field Games: Vol. II, Mean Field Games with Common Noise and Master Equations. Stochastic Analysis and Applications. Springer, Berlin (2018)CrossRefGoogle Scholar
  18. 18.
    Carmona, R., Delarue, F., Lachapelle, A.: Control of McKean–Vlasov versus mean field games. Math. Financ. Econ. 7, 131–166 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Carmona, R., Delarue, F., Lacker, D.: Probabilistic analysis of mean field games with a common noise. Ann. Probab. 44, 3740–3803 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cerrai, S.: Second order PDE’s in finite and infinite dimension: a probabilistic approach. In: Lecture Notes in Mathematics, vol. 1762. Springer, Berlin (2001).  https://doi.org/10.1007/b80743 zbMATHGoogle Scholar
  21. 21.
    Chassagneux, J., Crisan, D., Delarue, F.: McKean–Vlasov FBSDEs and related master equation. Technical report (2015)Google Scholar
  22. 22.
    Crisan, D., Delarue, F.: Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations. J. Funct. Anal. 263(10), 3024–3101 (2012).  https://doi.org/10.1016/j.jfa.2012.07.015 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Davie, A.: Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN 24, Art. ID rnm124 (2007)Google Scholar
  24. 24.
    Delarue, F.: On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch. Processes Appl. 99, 209–286 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Delarue, F., Guatteri, G.: Weak existence and uniqueness for FBSDEs. Stoch. Processes Appl. 116, 1712–1742 (2006)CrossRefGoogle Scholar
  26. 26.
    Flandoli, F.: Random perturbation of PDEs and fluid dynamics: Ecole d’été de probabilités de Saint-Flour XL. In: Lecture Notes in Mathematics. Springer, Berlin (2011)Google Scholar
  27. 27.
    Flandoli, F., Gubinelli, M., Priola, E.: Well posedness of the transport equation by stochastic perturbation. Invent. Math. 180, 153 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Flandoli, F., Russo, F., Wolf, J.: Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40, 493–542 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Flandoli, F., Russo, F., Wolf, J.: Some SDEs with distributional drift. II. Lyons–Zheng structure, It’s formula and semimartingale characterization. Random Oper. Stoch. Equ. 2, 145–184 (2004)CrossRefGoogle Scholar
  30. 30.
    Fleming, W., Soner, M.: Controlled Markov Processes and Viscosity Solutions. Springer, Berlin (2010)zbMATHGoogle Scholar
  31. 31.
    Foguen Tchuendom, R.: Restoration of uniqueness of Nash equilibria for a class of linear–quadratic mean field games with common noise. Technical report (2015)Google Scholar
  32. 32.
    Gangbo, W., Świȩch, A.: Existence of a solution to an equation arising from the theory of mean field games. J. Differ. Eqs. 259(11), 6573–6643 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Giorgio, F., Fausto, G., Andrzej, S.: Stochastic Optimal Control in Infinite Dimension: Dynamic Programming and HJB Equations. Probability Theory and Stochastic Modelling, With a contribution by Marco Fuhrman and Gianmario Tessitore, vol. 82. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-53067-3 CrossRefzbMATHGoogle Scholar
  34. 34.
    Gomes, D., Pimentel, E.: Time-dependent mean-field games with logarithmic nonlinearities. SIAM J. Math. Anal. 47, 37983812 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Gomes, D., Pimentel, E., Sánchez-Morgado, H.: Time-dependent mean-field games in the sub-quadratic case. Commun. Partial Differ. Equ. 40, 40–76 (2015)CrossRefGoogle Scholar
  36. 36.
    Gomes, D., Pimentel, E., Sánchez-Morgado, H.: Time-dependent mean-field games in the superquadratic case. ESAIM Control Optim. Calc. Var. 22, 562580 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gomes, D., Saude, J.: Mean field games models—a brief survey. Technical report (2014)Google Scholar
  38. 38.
    Gomes, D., Voskanyan, V.: Extended mean field games-formulation, existence, uniqueness and examples. Technical report (2016)Google Scholar
  39. 39.
    Huang, M., Caines, P., Malhamé, R.: Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6, 221–252 (2006)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Huang, M., Caines, P., Malhamé, R.: Large population cost coupled LQG problems with nonuniform agents: individual mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans. Autom. Control 52, 1560–1571 (2007)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Kolokolstov, V., Troeva, M.: On the mean field games with common noise and the McKean–Vlasov SPDEs. Technical report (2015). arXiv:1506.04594
  42. 42.
    Krylov, N.V., Röckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131, 154–196 (2005)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Lacker, D.: A general characterization of the mean field limit for stochastic differential games. Probab. Theory Relat. Fields 165, 581648 (2016)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Lasry, J., Lions, P.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Lions, P.: Théorie des jeux à champs moyen et applications. Lectures at the Collège de France. http://www.college-de-france.fr/default/EN/all/equ_der/cours_et_seminaires.htm (2007–2008)
  48. 48.
    Lions, P.: Estimées nouvelles pour les équations quasilinéaires. Seminar in Applied Mathematics at the Collège de France. http://www.college-de-france.fr/site/pierre-louis-lions/seminar-2014-11-14-11h15.htm (2014)
  49. 49.
    Peng, S.: A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 2(4), 966–979 (1990)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Pham, H.: Continuous-Time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability. Springer, Berlin (2009)CrossRefGoogle Scholar
  51. 51.
    Veretennikov, A.Y.: Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. 111, 434–452 (1980)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  53. 53.
    Villani, C.: Optimal Transport, Old and New. Springer, Berlin (2009)CrossRefGoogle Scholar
  54. 54.
    Yong, J., Zhou, X.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, Berlin (1999)CrossRefGoogle Scholar
  55. 55.
    Zabczyk, J.: Parabolic equations on Hilbert spaces. In: Stochastic PDE’s and Kolmogorov equations in infinite dimensions (Cetraro, 1998).Lecture Notes in Mathematics, vol. 1715, pp. 117–213. Springer, Berlin (1999).  https://doi.org/10.1007/BFb0092419 Google Scholar
  56. 56.
    Zvonkin, A.K.: A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. 93, 129–149 (1974)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRS, LJADNice Cedex 02France

Personalised recommendations