A variational approach to second order mean field games with density constraints: the stationary case

By Alpár R. Mészáros, Francisco J. Silva. 2015

66 citations recorded by the source corpus.

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain $\Omega \subset \mathbb{R}^d$. We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order $q' \in ]1, d/(d-1)[$, the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou-Brenier, we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order $q'\geq d/(d-1)$, the previous arguments do not apply and we prove the existence by means of an approximation argument.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/fb4a5cc8c8b85da37d522597f565f23b1c7cbf99

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Filippo Santambrogio
  2. P. J. Graber
  3. Pratik Chaudhari
  4. Stefano Soatto

In Finance

  1. Filippo Santambrogio
  2. P. J. Graber
  3. G. Peyr'e
  4. Marco Cuturi
  5. Luigi Ambrosio
  6. Tudor Manole
  7. Jonathan Niles-Weed
  8. Michael S. Albergo

Citations in the corpus