Time-Dependent Mean-Field Games in the Subquadratic Case

Communications in Partial Differential Equations

Volumn 40, Issue 1, 2015, Pages 40-76

  Gomes, Diogo A ^a   Pimentel, Edgard A ^b   Sánchez Morgado, Héctor ^c  

^a KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Saudi Arabia)

^b IMPA   (Brazil)

^c UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO   (Mexico)

Author keywords

A priori estimates; Classical solutions; Mean field games

Indexed keywords

EID: 84911982713     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605302.2014.903574     Document Type: Article

References (51)

1. Achdou , Y. (2013). Finite difference methods for mean field games . In: Loreti, P., Tchou, N.A., eds., Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications . New York : Springer , pp. 1 – 47 .
2. Achdou , Y. , Camilli , F. , Capuzzo-Dolcetta , I. (2012). Mean field games: numerical methods for the planning problem . SIAM J. Control Optim. 50 : 77 – 109 .
3. Achdou , Y. , Capuzzo-Dolcetta , I. (2010). Mean field games: numerical methods . SIAM J. Numer. Anal. 48 : 1136 – 1162 .
4. Amann , H. , Crandall , M.G. (1978). On some existence theorems for semi-linear elliptic equations . Indiana Univ. Math. J. 27 : 779 – 790 .
5. Bardi , M. , Feleqi , E. (2013). Nonlinear elliptic systems and mean field games. Preprint .
6. Cagnetti , F. , Gomes , D. , Mitake , H. , Tran , H. (2013). A new method for large time behavior of convex hamilton-jacobi equations: degenerate equations and weakly coupled systems. Preprint .
7. Cagnetti , F. , Gomes , D. , Tran , H.V. (2013). Adjoint methods for obstacle problems and weakly coupled systems of PDE . ESAIM: Cont. Opt. Calc. Var. 19 : 754 – 779 .
8. Cagnetti , F., Gomes , D., Tran , H.V. Aubry-Mather measures in the non convex setting. Submitted.
9. Cardaliaguet , P. (2011). Notes on mean-field games .
10. Cardaliaguet , P. (2013). Long time average of first order mean-field games and weak kam theory . Dyn. Games Appl. 19 : 473 – 488 .
11. Cardaliaguet , P. Weak solutions for first order mean-field games with local coupling. Preprint .
12. Cardaliaguet , P. , Lasry , J.-M. , Lions , P.-L. , Porretta , A. (2012). Long time average of mean field games . Netw. Heterog. Media 7 : 279 – 301 .
13. Carmona , R. , Delarue , F. (2013). Mean field forward-backward stochastic differential equations . Electron. Commun. Probab. 18 (68): 15 .
14. Carmona , R. , Delarue , F. (2013). Probabilistic analysis of mean-field games . SIAM J. Control Optim. 51 : 2705 – 2734 .
15. Evans , L.C. (2003). Some new pde methods for weak kam theory . Calc. Var. Part. Diff. Eqs. 17 : 159 – 177 .
16. Evans , L.C. (2009). Further PDE methods for weak KAM theory . Calc. Var. Partial Diff. Eqs. 35 : 435 – 462 .
17. Evans , L.C. (2010). Adjoint and compensated compactness methods for Hamilton-Jacobi PDE . Arch. Ration. Mech. Anal. 197 : 1053 – 1088 .
18. Evans , L.C. , Smart , C.K. (2011). Adjoint methods for the infinity Laplacian partial differential equation . Arch. Ration. Mech. Anal. 201 : 87 – 113 .
19. Ferreira , R. , Gomes , D. (2014). On the convergence of finite state mean-field games through Γ-convergence . J. Math. Anal. Appl. 418 : 211 – 230 .
20. Fleming , W.H. , Soner , H.M. (2006). Controlled Markov Processes and Viscosity Solutions . Stochastic Modelling and Applied Probability , vol. 25. New York : Springer-Verlag .
21. Friedman , A. (1969). Partial Differential Equations . New York : Holt, Rinehart & Winston .
22. Gomes , D. , Iturriaga , R. , Sánchez-Morgado , H. , Yu , Y. (2010). Mather measures selected by an approximation scheme . Proc. Amer. Math. Soc. 138 : 3591 – 3601 .
23. Gomes , D. , Mohr , J. , Souza , R.R. (2010). Discrete time, finite state space mean field games . J. Math. Pure. Appl. 93 : 308 – 328 .
24. Gomes , D. , Mohr , J. , Souza , R.R. (2013). Continuous time finite state mean-field games . Appl. Math. and Opt. 68 : 99 – 143 .
25. Gomes , D. , Patrizi , S. Obstacle and weakly coupled systems problem in mean field games. Preprint .
26. Gomes , D. , Patrizi , S. , Voskanyan , V. (2014). On the existence of classical solutions for stationary extended mean field games . Nonlinear Anal. 99 : 49 – 79 .
27. Gomes , D. , Pimentel , E. , Sanchez-Morgado , H. Time dependent mean-field games in the superquadratic case. Preprint .
28. Gomes , D. , Sánchez-Morgado , H. (2014). On the stochastic Evans-Aronsson problem . Trans. Amer. Math. Soc. 366 : 903 – 929 .
29. Gomes , D. , Saúde , J. (2014). Mean-field games: A brief survey . Dyn. Games Appl. 4 : 110 – 154 .
30. Gomes , D. , Voskanyan , V. Extended mean-field games - formulation, existence, uniqueness and examples. Preprint .
31. Gomes , D.A. , Pires , G.E. , Sánchez-Morgado , H. (2012). A-priori estimates for stationary mean-field games . Netw. Heterog. Media 7 : 303 – 314 .
32. Guéant , O. (2009). Mean field games and applications to economics. Ph.D. Thesis, Université Paris Dauphine, Paris, France.
33. Guéant , O. (2009). A reference case for mean field games models . J. Math. Pures Appl. (9) 92 : 276 – 294 .
34. Guéant , O. , Lasry , J.-M. , Lions , P.-L. (2011). Mean field games and applications. In: Paris-Princeton lectures on Mathematical Finance. Lecture Notes in Mathematics, vol. 2003. Berlin: Springer, pp. 206–266 .
35. Huang , M. , Caines , P.E. , Malhamé , R.P. (2007). Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralizε-Nash equilibria . IEEE Trans. Automat. Control 52 : 1560 – 1571 .
36. Huang , M. , Malhamé , R.P. , Caines , P.E. (2006). Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle . Commun. Inf. Syst. 6 : 221 – 251 .
37. Kolokoltsov , V.N. , Li , J. , Yang , W. (2011). Mean field games and nonlinear markov processes. Preprint .
38. Lachapelle , A. , Salomon , J. , Turinici , G. (2010). Computation of mean field equilibria in economics . Math. Model. Meth. Appl. Sci. 20 : 567 – 588 .
39. Ladyzenskaja , O.A. , Solonnikov , V.A. , Ural'ceva , N.N. (1967). Linear and Quasilinear Equations of Parabolic Type . Translations of Mathematical Monographs , vol. 27. Providence, RI : American Mathematical Society .
40. Lasry , J.-M. , Lions , P.-L. (2006). Jeux à champ moyen. I. Le cas stationnaire [Mean field games. I. The stationary case.] . C. R. Math. Acad. Sci. Paris 343 : 619 – 625 .
41. Lasry , J.-M. , Lions , P.-L. (2006). Jeux à champ moyen. II. Horizon fini et contr\^ole optimal [Mean field games. II. Finite horizon and optimal control.] . C. R. Math. Acad. Sci. Paris 343 : 679 – 684 .
42. Lasry , J.-M. , Lions , P.-L. (2007). Mean field games . Jpn. J. Math. 2 : 229 – 260 .
43. Lasry , J.-M. , Lions , P.-L. (2007). Mean field games. Cahiers de la Chaire Finance et Développement Durable.
44. Lasry , J.-M. , Lions , P.-L. , Guéant , O. Application of mean field games to growth theory. Preprint .
45. Lions , P.-L. (1995). Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models. Oxford Lecture Series in Mathematics and its Applications, vol. 3. Oxford, Oxford University Press .
46. Lions , P.-L. (2011). Mean-field games. Video lecture series. Paris: Collége de France .
47. Lions , P.-L. (2012). IMA, University of Minessota. Course on mean-field games. Available at: http://www.ima.umn.edu/2012-2013/sw11.12-13.12/ (accessed 3 September 2014) .
48. Nguyen , S.L. , Huang , M. (2012). Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players . SIAM J. Control Optim. 50 : 2907 – 2937 .
49. Pimentel , E. Time Dependent Mean-Field Games. IST-UTL. Doctoral thesis, Lisbon, Portugal .
50. Porretta , A. (2014). On the planning problem for the mean-field games system . Dyn. Games Appl. 4 : 231 – 256 .
51. Tran , H.V. (2011). Adjoint methods for static Hamilton-Jacobi equations . Calc. Var. PDE. 41 : 301 – 319 .