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Mean Field Games with a Quadratic Hamiltonian: A Constructive Scheme

Part of the Annals of the International Society of Dynamic Games book series (AISDG,volume 12)

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.

Keywords

  • Mean field games
  • Forward–backward equations
  • Monotonic schemes

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Notes

  1. 1.

    In our case, this assumption consists only in assuming that the initial datum is a probability distribution function m 0.

  2. 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

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Correspondence to Olivier Guéant .

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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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