Abstract
Mean field games models describing the limit case of a large class of stochastic differential games, as the number of players goes to +∞, were introduced by Lasry and Lions [C R Acad Sci Paris 343(9/10) (2006); Jpn. J. Math. 2(1) (2007)]. We use a change of variables to transform the mean field games equations into a system of simpler coupled partial differential equations in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the mean field games equations.
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Notes
- 1.
In our case, this assumption consists only in assuming that the initial datum is a probability distribution function m 0.
- 2.
In terms of the initial MFG problem, the optimal control ∇ u and the subsequent distribution m are not changed if we subtract \(\|{f\|}_{\infty }\) to f.
References
Evans, L.C.: Partial Differential Equations (Graduate Studies in Mathematics, vol. 19). American Mathematical Society, Providence, RI (2009)
Guéant, O.: Mean field games equations with quadratic Hamiltonian: a specific approach. Math. Models Methods Appl. Sci., 22, (2012)
Guéant, O., Lasry, J.M., Lions, P.L.: Mean field games and applications. In: Paris Princeton Lectures on Mathematical Finance (2010)
Lasry, J.-M., Lions, P.-L.: Jeux champ moyen. I. Le cas stationnaire. C. R. Acad. Sci. Paris 343(9), 619–625
Lasry, J.-M., Lions, P.-L.: Jeux champ moyen. II. Horizon fini et contrôle optimal. C. R. Acad. Sci. Paris 343(10), 679–684 (2006)
Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Lasry, J.-M., Lions, P.-L.: Cours au collège de france: théorie des jeux champs moyens. http://www.college-de-france.fr/default/EN/all/equ\_der/audio\_video.jsp. (2008)
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Guéant, O. (2013). Mean Field Games with a Quadratic Hamiltonian: A Constructive Scheme. In: Cardaliaguet, P., Cressman, R. (eds) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol 12. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8355-9_12
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DOI: https://doi.org/10.1007/978-0-8176-8355-9_12
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