Time dependent mean-field games in the superquadratic case

By Diogo A. Gomes, Edgard A. Pimentel, H'ector S'anchez-Morgado. 2013

108 citations recorded by the source corpus.

We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as the key techniques used in the subquadratic case do not extend to the superquadratic setting. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton-Jacobi equation are obtained through a novel set of techniques. These explore the parabolic nature of the problem through the non-linear adjoint method. Well-posedness is proved by combining Lipschitz regularity for the Hamilton-Jacobi equation with polynomial estimates for solutions of the Fokker-Planck equation. Existence of classical solutions can then be proved under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to the current understanding of superquadratic Hamilton-Jacobi equations.

Subfields: Applied Mathematics, Finance, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a6daa11a1cf610f38e6a157c18a4cbba4ba3b18d

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. Edgard A. Pimentel
2. Diogo A. Gomes
3. P. J. Graber
4. Alessio Porretta
5. H'ector S'anchez-Morgado
6. Marco Cirant
7. Vardan K. Voskanyan
8. Martino Bardi

In Finance

1. Edgard A. Pimentel
2. Diogo A. Gomes
3. Alessio Porretta
4. P. J. Graber
5. H'ector S'anchez-Morgado
6. Marco Cirant
7. Martino Bardi
8. Pierre Cardaliaguet

In Numerical Analysis

1. Diogo A. Gomes
2. Edgard A. Pimentel
3. H'ector S'anchez-Morgado
4. Martino Bardi
5. P. J. Graber
6. Guy Barles

Citations in the corpus

• Regularity for second order stationary mean-field games (2015, background): “Classical solutions have been considered in [14], [13] and [12].”
• Time-Dependent Mean-Field Games in the Subquadratic Case (2013, methodology): “In a companion paper [GPSM13] we address superquadratic Hamiltonians which require different techniques.”
• Second order mean field games with degenerate diffusion and local coupling (2014, background): “, [8, 17, 18, 19, 21, 12, 13, 22]) or by stochastic techniques (see, e.”
• Mean field games models of segregation (2016, background): “[31] in the superquadratic case (see also Ref.”