Geodesic
Controlled diffusion mean field games with common noise and McKean–Vlasov second order backward SDE
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 774-805 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a mean field game with common noise in which the diffusion coefficients may be controlled. We prove existence of a weak relaxed solution under some continuity conditions on the coefficients. We then show that, when there is no common noise, the solution of this mean field game is characterized by a McKean–Vlasov type second order backward SDE.
Keywords: stochastic control, Nash equilibrium, mean field game, exterior noise. 
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A. Barrasso; N. Touzi. Controlled diffusion mean field games with common noise and McKean–Vlasov second order backward SDE. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 774-805. http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a8/

[1] C. D. Aliprantis, K. C. Border, Infinite dimensional analysis, A Hitchhiker's guide, 2nd ed., Springer-Verlag, Berlin, 1999, xx+672 pp. | DOI | MR | Zbl

[2] A. Barrasso, F. Russo, “Path-dependent martingale problems and additive functionals”, Stoch. Dyn., 19:4 (2019), 1950027, 39 pp. | DOI | MR | Zbl

[3] P. Billingsley, Probability and measure, Wiley Ser. Probab. Math. Statist. Probab. Math. Statist., 2nd ed., John Wiley Sons, Inc., New York, 1986, xiv+622 pp. | MR | Zbl

[4] V. I. Bogachev, Osnovy teorii mery, v. 1, 2, NITs “Regulyarnaya i khaoticheskaya dinamika”, M.–Izhevsk, 2003, 544 s., 576 pp.; V. I. Bogachev, Measure theory, v. I, II, Springer-Verlag, Berlin, 2007, xviii+500 pp., xiv+575 pp. | DOI | MR | Zbl

[5] R. Carmona, F. Delarue, Probabilistic theory of mean field games with applications, v. 1, Probab. Theory Stoch. Model., 83, Mean field FBSDEs, control, and games, Springer, Cham, 2018, xxv+713 pp. | DOI | MR | Zbl

[6] R. Carmona, F. Delarue, Probabilistic theory of mean field games with applications, v. 2, Probab. Theory Stoch. Model., 84, Mean field games with common noise and master equations, Springer, Cham, 2018, xxiv+697 pp. | DOI | MR | Zbl

[7] R. Carmona, F. Delarue, D. Lacker, “Mean field games with common noise”, Ann. Probab., 44:6 (2016), 3740–3803 | DOI | MR | Zbl

[8] R. Carmona, D. Lacker, “A probabilistic weak formulation of mean field games and applications”, Ann. Appl. Probab., 25:3 (2015), 1189–1231 | DOI | MR | Zbl

[9] A. Cellina, “Approximation of set valued functions and fixed point theorems”, Ann. Mat. Pura Appl. (4), 82 (1969), 17–24 | DOI | MR | Zbl

[10] C. Dellacherie, P.-A. Meyer, Probabilités et potentiel, Ch. I–IV, Publ. Inst. Math. Univ. Strasbourg, XV, Actualités Sci. Indust., No. 1372, Hermann, Paris, 1975, x+291 pp. | MR | Zbl

[11] L. Q. Eifler, “Open mapping theorems for probability measures on metric spaces”, Pacific J. Math., 66:1 (1976), 89–97 | DOI | MR | Zbl

[12] I. Ekren, C. Keller, N. Touzi, Jianfeng Zhang, “On viscosity solutions of path dependent PDEs”, Ann. Probab., 42:1 (2014), 204–236 | DOI | MR | Zbl

[13] I. Ekren, N. Touzi, Jianfeng Zhang, “Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I”, Ann. Probab., 44:2 (2016), 1212–1253 | DOI | MR | Zbl

[14] N. El Karoui, S. Peng, M. C. Quenez, “Backward stochastic differential equations in finance”, Math. Finance, 7:1 (1997), 1–71 | DOI | MR | Zbl

[15] N. El Karoui, S. Méléard, “Martingale measures and stochastic calculus”, Probab. Theory Related Fields, 84:1 (1990), 83–101 | DOI | MR | Zbl

[16] U. G. Haussmann, “Existence of optimal Markovian controls for degenerate diffusions”, Stochastic differential systems (Bad Honnef, 1985), Lect. Notes Control Inf. Sci., 78, Springer, Berlin, 1986, 171–186 | DOI | MR | Zbl

[17] J.-B. Hiriart-Urruty, C. Lemaréchal, Fundamentals of convex analysis, Grundlehren Text Ed., Springer-Verlag, Berlin, 2001, x+259 pp. | DOI | MR | Zbl

[18] Ch. D. Horvath, “Extension and selection theorems in topological spaces with a generalized convexity structure”, Ann. Fac. Sci. Toulouse Math. (6), 2:2 (1993), 253–269 | DOI | MR | Zbl

[19] Minyi Huang, R. P. Malhamé, P. E. Caines, “Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle”, Commun. Inf. Syst., 6:3 (2006), 221–252 | DOI | MR | Zbl

[20] J. Jacod, Calcul stochastique et problèmes de martingales, Lecture Notes in Math., 714, Springer, Berlin, 1979, x+539 pp. | DOI | MR | Zbl

[21] O. Kallenberg, Random measures, theory and applications, Probab. Theory Stoch. Model., 77, Springer, Cham, 2017, xiii+694 pp. | DOI | MR | Zbl

[22] R. L. Karandikar, “On pathwise stochastic integration”, Stochastic Process. Appl., 57:1 (1995), 11–18 | DOI | MR | Zbl

[23] D. Lacker, “Mean field games via controlled martingale problems: existence of Markovian equilibria”, Stochastic Process. Appl., 125:7 (2015), 2856–2894 | DOI | MR | Zbl

[24] D. Lacker, Dense sets of joint distributions appearing in filtration enlargements, stochastic control, and causal optimal transport, 2018, 25 pp., arXiv: 1805.03185v1

[25] J.-M. Lasry, P.-L. Lions, “Mean field games”, Jpn. J. Math., 2:1 (2007), 229–260 | DOI | MR | Zbl

[26] Yiqing Lin, Zhenjie Ren, N. Touzi, Junjian Yang, “Second order backward SDE with random terminal time”, Electron. J. Probab., 25 (2020), 99, 43 pp. ; (2018), 36 pp., arXiv: 1802.02260 | DOI | MR | Zbl

[27] N. El Karoui, Nguyen Du'hŪŪ, M. Jeanblanc-Picqué, “Compactification methods in the control of degenerate diffusions: existence of an optimal control”, Stochastics, 20:3 (1987), 169–219 | DOI | MR | Zbl

[28] M. Nutz, “Pathwise construction of stochastic integrals”, Electron. Commun. Probab., 17 (2012), 24, 7 pp. | DOI | MR | Zbl

[29] D. Possamaï, Xiaolu Tan, Chao Zhou, “Stochastic control for a class of nonlinear kernels and applications”, Ann. Probab., 46:1 (2018), 551–603 | DOI | MR | Zbl

[30] H. M. Soner, N. Touzi, Jianfeng Zhang, “Wellposedness of second order backward SDEs”, Probab. Theory Related Fields, 153:1-2 (2012), 149–190 | DOI | MR | Zbl

[31] H. M. Soner, N. Touzi, Jianfeng Zhang, “Dual formulation of second order target problems”, Ann. Appl. Probab., 23:1 (2013), 308–347 | DOI | MR | Zbl

[32] D. W. Stroock, S. R. S. Varadhan, Multidimensional diffusion processes, Classics Math., Reprint of the 1997 ed., Springer-Verlag, Berlin, 2006, xii+338 pp. | DOI | MR | Zbl