On the existence of classical solutions for stationary extended mean field games

By Diogo A. Gomes, Stefania Patrizi, Vardan K. Voskanyan. 2013

157 citations recorded by the source corpus.

Abstract In this paper we consider extended stationary mean-field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean-field games in Gomes et al. (2012). In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish the existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in Lasry and Lions (2006), Gomes and Sanchez-Morgado (2011) or Gomes et al. (2012) this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where the existence of classical solutions can be proved.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/b952cc35327038dc142afe6345e32a7d7dec3ea1

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Citations in the corpus

Regularity for second order stationary mean-field games (2015, methodology): “In [11], the existence of smooth solutions for stationary mean-field games is established under quadratic type assumptions on H and polynomial or logarithmic assumptions on g.”
Deep Fictitious Play for Finding Markovian Nash Equilibrium in Multi-Agent Games (2019, background): “This is called the extended mean-field game and it has attracted lots of attention recently (see e.g., Gomes and Voskanyan (2013); Gomes et al. (2014); Gomes and Voskanyan (2016); Cardaliaguet and Lehalle (2018)).”
Time-Dependent Mean-Field Games in the Subquadratic Case (2013, background): “Time-Dependent Mean-Field Games 41 [32, 44] and environmental policy [38], finite state problems [19, 23, 24], explicit models [33, 48], obstacle-type problems [25], extended mean-field games [26, 30], probabilistic methods [13, 14], long-time behavior [10, 12], weak solutions to meanfield games [11”
Mean Field Games Models—A Brief Survey (2013, background): “This leads to the class of extended mean-ﬁeld games considered in [44] and [50].”
Some remarks on mean field games (2018, background): “Existence of solutions of such a system was proven for particular Hamiltonians in [22]; we refer to [23] for a more general result of existence for (2) and to [14, 24] for the 206 C.”
Time dependent mean-field games in the superquadratic case (2013, methodology): “Smooth solutions were studied in [22] (see also [26] for a related problem), [29, 31].”
Stationary focusing mean-field games (2016, background): “Existence of smooth solutions for (10) is still an open problem for general power-like nonlinearities, and known results require conditions on the exponents α, γ; see [14, 18, 19, 34].”
Existence of a solution to an equation arising from the theory of Mean Field Games (2015, unknown): “[7, 8, 15, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 39, 41, 42, 43, 49, 50, 51, 52]).”