Mean Field Games and Mean Field Type Control Theory have brought major advances in decision making under uncertainty and control of dynamic systems. A significant step was taken when P.-L. Lions introduced a new partial differential equation known as the Master Equation, which conceptually synthesizes all potential approaches to the field. Unfortunately, this is an extraordinarily difficult equation. In this project, the investigators will develop new mathematical techniques to study the Master Equation. Therefore, the work is a contribution to the conceptual core of mean field theory, the master equation, focusing on two important applications: energy markets and stochastic control with partial information. The broader significance of this work is that it will produce new insights into economics, finance, machine learning, and other fields. Students at both graduate and undergraduate levels will be trained in a large variety of issues and concepts, with opportunities for research and applications.
The first problem of the project requires extended mean field control, where the extension lies in the fact that not only the probability distribution of the state, but also the probability distribution of the control enters in the model, and/or in the cost functional to be minimized. It is a new branch of the general theory, which the investigators will develop and apply to energy markets and related problems. The second problem regards nonlocal dependence, which the investigators will apply to the well-established domain of stochastic control with partial information, providing new insights from a mean field point of view. The mathematical techniques involved will include Hamilton-Jacobi equations on infinite dimensional spaces, backward stochastic partial differential equations, and forward-backward systems of nonlocal equations.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
