Applied Mathematics & Optimization

, Volume 76, Issue 1, pp 59–91 | Cite as

Discrete-Time Mean Field Partially Observable Controlled Systems Subject to Common Noise

  • M. H. M. Chau
  • Y. Lai
  • S. C. P. YamEmail author


In this article, we provide the first systemic study on discrete time partially observable mean field systems in the presence of a common noise. Each player makes decision solely based on the observable process. Both the mean field games and the related tractable mean field type stochastic control problem are studied. We first solve the mean field type control problem using classical discrete time Kalman filter with notable modifications. The unique existence of the resulted forward backward stochastic difference system is then established by separation principle. The mean field game problem is also solved via an application of stochastic maximum principle, while the existence of the mean field equilibrium is shown by the Schauder’s fixed point theorem.


Discrete-time Mean field game Mean field type stochastic control Partial observation Common noise 



The first author-Michael Chau acknowledges the financial support from Imperial College London and The University of Hong Kong, and the present work constitutes a part of his work for his postgraduate dissertation. The third author-Phillip Yam acknowledges the financial support from The Hong Kong RGC GRF 14301015 with the project title: Advance in Mean Field Theory and GRF 11303316 with the project title: Mean Field Control with Partial Information. Phillip Yam also acknowledges the financial support from Department of Statistics of Columbia University in the City of New York during the period he was a visiting faculty member.


  1. 1.
    Andersson, D., Djehiche, B.: A maximum principle for sdes of mean-field type. Appl. Math. Optim. 63(3), 341–356 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bardi, M.: Explicit solutions of some linear-quadratic mean field games. Netw. Heterog. Media 7(2), 243–261 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bensoussan, A.: Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  4. 4.
    Bensoussan, A., Chau, M.H.M., Yam, S.C.P.: Mean field stackelberg games: aggregation of delayed instructions. SIAM J. Control Optim. 53(4), 2237–2266 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bensoussan, A., Chau, M.H.M., Yam, S.C.P.: Mean field games with a dominating player. Appl. Math. Optim. 74(1), 91–128 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bensoussan, A., Frehse, J., Yam, S.C.P.: Mean Field Games and Mean Field Type Control Theory. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bensoussan, A., Sung, J., Yam, S.C.P., Yung, S.P.: Linear-quadratic mean field games. J. Optim. Theory Appl. 169(2), 496–529 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bollobás, B.: Linear Analysis: An Introductory Course. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Buckdahn, R., Djehiche, B., Li, J., Peng, S.: Mean-field backward stochastic differential equations: a limit approach. Ann. Probab. 37(4), 1524–1565 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Buckdahn, R., Djehiche, B., Li, J.: A general stochastic maximum principle for sdes of mean-field type. Appl. Math. Optim. 64(2), 197–216 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cardaliaguet, P.: Notes on Mean Field Games. Technical Report (2010)Google Scholar
  12. 12.
    Carmona, R., Delarue, F.: Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51(4), 2705–2734 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Carmona, R., Fouque, J.-P., Sun, Li-Hsien: Mean field games and systemic risk. Commun. Math. Sci. 13(4), 911–933 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Carmona, R., Delarue, F., Lacker, D.: Mean field games with common noise. Ann. Probab. (2016) (Forthcoming)Google Scholar
  15. 15.
    Garnier, J., Papanicolaou, G., Yang, T.W.: Large deviations for a mean field model of systemic risk. SIAM J. Financ. Math. 4(1), 151–184 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guéant, O., Lasry, J.M., Lions, P.L.: Mean Field Games and Applications. Paris-Princeton Lectures on Mathematical Finance 2010, pp. 205–266. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    Huang, J., Wang, S.: A class of mean-field LQG games with partial information. arXiv:1403.5859 (2014) (preprint)
  18. 18.
    Huang, M.: Large-population lqg games involving a major player: the nash certainty equivalence principle. SIAM J. Control Optim. 48(5), 3318–3353 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Huang, M., Caines, P.E., Malhamé, R.P.: Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, vol. 1, pp. 98–103. IEEE, New York (2003)Google Scholar
  20. 20.
    Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hunter, J.K.: Measure theory. University Lecture Notes, Department of Mathematics, University of California at Davis. (2011)
  22. 22.
    Kolokoltsov, V.N., Troeva, M., Yang, W.: On the rate of convergence for the mean-field approximation of controlled diffusions with large number of players. Dyn. Games Appl. 4(2), 208–230 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lasry, J.M., Lions, P.L.: Jeux à champ moyen. i–le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–925 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Meyer-Brandis, T., Øksendal, B., Zhou, X.: A mean-field stochastic maximum principle via Malliavin calculus. Stochastics 84(5–6), 643–666 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Nourian, M., Caines, P.E.: \(\epsilon \)-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim. 51(4), 3302–3331 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Şen, N., Caines, P.E.: Mean field games with partially observed major player and stochastic mean field. In: 53rd IEEE Conference on Decision and Control, pp. 2709–2715. IEEE, New York (2014)Google Scholar
  29. 29.
    Stein, E.M., Shakarchi, R.: Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, Princeton (2005)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Department of Statistics and Actuarial ScienceThe University of Hong KongPokfulamHong Kong
  3. 3.Department of StatisticsThe Chinese University of Hong KongSha TinHong Kong
  4. 4.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Personalised recommendations