Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersson, D., Djehiche, B.: A maximum principle for sdes of mean-field type. Appl. Math. Optim. 63(3), 341–356 (2011)
Bardi, M.: Explicit solutions of some linear-quadratic mean field games. Netw. Heterog. Media 7(2), 243–261 (2012)
Bensoussan, A.: Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (2004)
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)
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)
Bensoussan, A., Frehse, J., Yam, S.C.P.: Mean Field Games and Mean Field Type Control Theory. Springer, Berlin (2013)
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)
Bollobás, B.: Linear Analysis: An Introductory Course. Cambridge University Press, Cambridge (1999)
Buckdahn, R., Djehiche, B., Li, J., Peng, S.: Mean-field backward stochastic differential equations: a limit approach. Ann. Probab. 37(4), 1524–1565 (2009)
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)
Cardaliaguet, P.: Notes on Mean Field Games. Technical Report (2010)
Carmona, R., Delarue, F.: Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51(4), 2705–2734 (2013)
Carmona, R., Fouque, J.-P., Sun, Li-Hsien: Mean field games and systemic risk. Commun. Math. Sci. 13(4), 911–933 (2015)
Carmona, R., Delarue, F., Lacker, D.: Mean field games with common noise. Ann. Probab. (2016) (Forthcoming)
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)
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)
Huang, J., Wang, S.: A class of mean-field LQG games with partial information. arXiv:1403.5859 (2014) (preprint)
Huang, M.: Large-population lqg games involving a major player: the nash certainty equivalence principle. SIAM J. Control Optim. 48(5), 3318–3353 (2010)
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)
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)
Hunter, J.K.: Measure theory. University Lecture Notes, Department of Mathematics, University of California at Davis. http://www.math.ucdavis.edu/~hunter/measure_theory (2011)
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)
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)
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)
Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Meyer-Brandis, T., Øksendal, B., Zhou, X.: A mean-field stochastic maximum principle via Malliavin calculus. Stochastics 84(5–6), 643–666 (2012)
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)
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)
Stein, E.M., Shakarchi, R.: Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, Princeton (2005)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chau, M.H.M., Lai, Y. & Yam, S.C.P. Discrete-Time Mean Field Partially Observable Controlled Systems Subject to Common Noise. Appl Math Optim 76, 59–91 (2017). https://doi.org/10.1007/s00245-017-9437-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-017-9437-x