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

• Published in: Applied mathematics & optimization Vol. 76; no. 1; pp. 59 - 91
• Main Authors: Chau, M. H. M., Lai, Y., Yam, S. C. P.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2017; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Control; Discrete time systems; Games; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Maximum principle; Numerical and Computational Physics; Optimal control; Probability theory; Production planning; Randomness; Separation; Simulation; Stochastic processes; Systems Theory; Theoretical; Common noise; Discrete-time; Partial observation; Mean field type stochastic control; Mean field game
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-017-9437-x

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.