# American Institute of Mathematical Sciences

June  2012, 7(2): 315-336. doi: 10.3934/nhm.2012.7.315

## New numerical methods for mean field games with quadratic costs

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

Received  November 2011 Revised  March 2012 Published  June 2012

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.
Citation: Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315
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