Mean Field Control with Partial Information
DescriptionMean field games have been introduced in 2006, independently by Lasry-Lions and Huang-Caines-Malhame. Since then, this domain has attracted a considerable interest worldwide. This interest keeps growing, after almost a decade. The mean field community meets every other year and has become quite large. The domain is now constantly represented in all large conferences on control or applied mathematics. One reason is certainly due to the mathematical new challenges which appear continuously in the domain. Another one concerns the attractiveness of the concepts for applications. The idea that decision makers cannot act independently of large crowds of other individuals in which they are embedded is now obvious. At the same time, traditional tools in optimization, control theory are not adapted. Mean field theories, which incorporate ideas from physics, making an analogy between individuals in a large community and particles in a medium, provide a clear progress. At the same time, the new mathematical problems at stake are huge. The P.I. has benefited in 2013 of an RGC grant on mean field control, in which the objectives were first to provide a comprehensive approach to mean field games and mean field type control, and second to waive the limitation that players are identical. We now address a completely new topic. The decision makers do not have full information about the past and the present. They observe a signal, which is affected by measurement errors.This problem corresponds to concrete motivations, well known in the standard stochastic control framework. It has very little been addressed up to now, except in particular situations. 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 consider two types of noise, the local noise and the common noise. This refers to the origin of mean field modeling. There is a large community of identical agents. The state evolution of each of them is affected individually by a local noise, and a common noise, which affects 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 optimal decisions. Basically, we will try to extend to Mac Kean diffusions, what is known for ordinary diffusions. As described in the proposal, we are confident it can be done. There are, however, daunting mathematical challenges, which motivates our research.
|Effective start/end date||1/01/17 → …|
- Mean Field Theory , Hamilton Jacobi Bellman , Fokker Planck , Partial Observation , Nonlinear Filtering