Mean Field Control with Partial Information

Project: Research

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Mean field games have been introduced in 2006, independently by Lasry-Lions andHuang-Caines-Malhame. Since then, this domain has attracted a considerable interestworldwide. This interest keeps growing, after almost a decade. The mean fieldcommunity meets every other year and has become quite large. The domain is nowconstantly represented in all large conferences on control or applied mathematics. Onereason is certainly due to the mathematical new challenges which appear continuously inthe domain. Another one concerns the attractiveness of the concepts for applications.The idea that decision makers cannot act independently of large crowds of otherindividuals in which they are embedded is now obvious. At the same time, traditionaltools in optimization, control theory are not adapted. Mean field theories, whichincorporate ideas from physics, making an analogy between individuals in a largecommunity and particles in a medium, provide a clear progress. At the same time, thenew mathematical problems at stake are huge.The P.I. has benefited in 2013 of an RGC grant on mean field control, in which theobjectives were first to provide a comprehensive approach to mean field games andmean field type control, and second to waive the limitation that players are identical. Wenow address a completely new topic. The decision makers do not have full informationabout the past and the present. They observe a signal, which is affected by measurementerrors.This problem corresponds to concrete motivations, well known in the standard stochasticcontrol framework. It has very little been addressed up to now, except in particularsituations. Only recently ?en and Caines have studied nonlinear filtering for Mc Kean-Vlasov diffusions.An important aspect is to describe the uncertainties on the system itself. We considertwo types of noise, the local noise and the common noise. This refers to the origin ofmean field modeling. There is a large community of identical agents. The state evolutionof each of them is affected individually by a local noise, and a common noise, whichaffects all of them. Think of the state of the economy, as an example.Our main effort will be to derive a stochastic maximum principle to characterize optimaldecisions. Basically, we will try to extend to Mac Kean diffusions, what is known forordinary diffusions. As described in the proposal, we are confident it can be done. Thereare, however, daunting mathematical challenges, which motivates our research.


Project number9042386
Grant typeGRF
Effective start/end date1/01/171/12/20

    Research areas

  • Mean Field Theory , Hamilton Jacobi Bellman , Fokker Planck , Partial Observation , Nonlinear Filtering