New numerical methods for mean field games with quadratic costs

Olivier Guéant

doi:10.3934/nhm.2012.7.315

Article Contents

2012, Volume 7, Issue 2: 315-336. Doi: 10.3934/nhm.2012.7.315

This issue Previous Article A-priori estimates for stationary mean-field games Next Article A modest proposal for MFG with density constraints

New numerical methods for mean field games with quadratic costs

Olivier Guéant^1,

1.
UFR de Math, Universit, 175, rue du Chevaleret, 75013 Paris

Received: November 2011

Revised: March 2012

Published: June 2012

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Abstract

Mean field games have been introduced by J.-M. Lasry and P.-L. Lions in [13, 14, 15] as the limit case of stochastic differential games when the number of players goes to $+\infty$. In the case of quadratic costs, we present two changes of variables that allow to transform the mean field games (MFG) equations into two simpler systems of equations. The first change of variables, introduced in [11], leads to two heat equations with nonlinear source terms. The second change of variables, which is introduced for the first time in this paper, leads to two Hamilton-Jacobi-Bellman equations. Numerical methods based on these equations are presented and numerical experiments are carried out.

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Mathematics Subject Classification: Primary: 35K91, 65M12; Secondary: 91A15.

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References

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