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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
Article
  • 248 Downloads

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.

Keywords

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

Notes

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.

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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

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