Abstract
The goal of this chapter is to propose solutions to asymptotic forms of the search for Nash equilibria for large stochastic differential games with mean field interactions. We implement the Mean Field Game strategy, initially developed by Lasry and Lions in an analytic set-up, in a purely probabilistic framework. The roads to solutions go through a class of standard stochastic control problems followed by fixed point problems for flows of probability measures. We tackle the inherent stochastic optimization problems in two different ways. Once by representing the value function as the solution of a backward stochastic differential equation (reminiscent of the so-called weak formulation approach), and a second time using the Pontryagin stochastic maximum principle. In both cases, the optimization problem reduces to the solutions of a Forward-Backward Stochastic Differential Equation (FBSDE for short). The search for a fixed flow of probability measures turns the FBSDE into a system of equations of the McKean-Vlasov type where the distribution of the solution appears in the coefficients. In this way, both the optimization and interaction components of the problem are captured by a single FBSDE, avoiding the twofold reference to Hamilton-Jacobi-Bellman equations on the one hand, and to Kolmogorov equations on the other hand.
References
S. Ahuja. Wellposedness of mean field games with common noise under a weak monotonicity condition. SIAM Journal on Control and Optimization, 54:30–48, 2016.
R. Aumann. Markets with a continuum of traders. Econometrica, 32:39–50, 1964.
M. Bardi. Explicit solutions of some linear quadratic mean field games. Networks and Heterogeneous Media, 7:243–261, 2012.
A. Bensoussan, J. Frehse, and P. Yam. Mean Field Games and Mean Field Type Control Theory. SpringerBriefs in Mathematics. Springer-Verlag New York, 2013.
A. Bensoussan, K.C.J. Sung, S.C.P. Yam, and S.P. Yung. Linear quadratic mean field games. Journal of Optimization Theory and Applications, 169:469–529, 2016.
D.P. Bertsekas. Nonlinear Programming. Athena Scientific, 1995.
J.-M. Bismut. Théorie probabiliste du contrôle des diffusions, Memoirs of the American Mathematical Society, 167(4), 1976.
J.M. Bismut. An introductory approach to duality in optimal stochastic control. SIAM Review, 20:62–78, 1978.
V.S. Borkar. Controlled diffusion processes. Probability Surveys, 2:213–244, 2005.
R. Buckdahn, B. Djehiche, and J. Li. Mean field backward stochastic differential equations and related partial differential equations. Stochastic Processes and their Applications, 119:3133–3154, 2007.
R. Buckdahn, B. Djehiche, J. Li, and S. Peng. Mean field backward stochastic differential equations: A limit approach. Annals of Probability, 37:1524–1565, 2009.
P. Cardaliaguet. Notes from P.L. Lions’ lectures at the Collège de France. Technical report, https://www.ceremade.dauphine.fr/$sim$cardalia/MFG100629.pdf, 2012.
P. Cardaliaguet. Weak solutions for first order mean field games with local coupling. In P. Bettiol et al., editors, Analysis and Geometry in Control Theory and its Applications. Springer INdAM Series, pages 111–158. Springer International Publishing, 2015.
P. Cardaliaguet and J. Graber. Mean field games systems of first order. ESAIM: Control, Optimisation and Calculus of Variations, 21:690–722, 2015.
P. Cardaliaguet, J. Graber, A. Porretta, and D. Tonon. Second order mean field games with degenerate diffusion and local coupling. Nonlinear Differential Equations and Applications NoDEA, 22:1287–1317, 2015.
P. Cardaliaguet, A. R. Mészáros, and F. Santambrogio. First order mean field games with density constraints: Pressure equals price. SIAM Journal on Control and Optimization, 54:2672–2709, 2016.
R. Carmona. Lectures on BSDEs, Stochastic Control and Stochastic Differential Games. SIAM, 2015.
R. Carmona and F. Delarue. Mean field forward-backward stochastic differential equations. Electronic Communications in Probability, 2013.
R. Carmona, F. Delarue, and A. Lachapelle. Control of McKean-Vlasov versus mean field games. Mathematics and Financial Economics, 7:131–166, 2013.
R. Carmona and D. Lacker. A probabilistic weak formulation of mean field games and applications. Annals of Applied Probability, 25:1189–1231, 2015.
P. G. Ciarlet. Introduction to Numerical Linear Algebra and Optimisation. Cambridge Texts in Applied Mathematics. Cambridge University Press, 1989.
D. Duffie and Y. Sun. Existence of independent random matching. Annals of Applied Probability, 17:385–419, 2007.
W.H. Fleming and M. Soner. Controlled Markov Processes and Viscosity Solutions. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2010.
W.H. Fleming. Generalized solutions in optimal stochastic control. In Proceedings of the Second Kingston Conference on Differential Games, pages 147–165. Marcel Dekker, 1977.
D. Fudenberg and D. Levine. Open-loop and closed-loop equilibria in dynamic games with many players. Journal of Economic Theory, 44:1–18, 1988.
D. Fudenberg and J. Tirole. Game Theory. MIT Press, 1991.
J. Gatheral, A. Schied, and A. Slynko. Transient linear price impact and Fredholm integral equations. Mathematical Finance, 22:445–474, 2012.
D.A. Gomes, L. Nurbekyan, and E. Pimentel. Economic Models and Mean-field Games Theory. Publicaões Matemáticas, IMPA, Rio, Brazil, 2015.
D.A. Gomes and E. Pimentel. Time-dependent mean-field games with logarithmic nonlinearities. SIAM Journal of Mathematical Analysis, 47:3798–3812, 2015.
D.A. Gomes and E. Pimentel. Local regularity for mean-field games in the whole space. Minimax Theory and its Applications, 1:65–82, 2016.
D.A. Gomes, E. Pimentel, and H. Sánchez-Morgado. Time-dependent mean-field games in the sub- quadratic case. Communications in Partial Differential Equations, 40:40–76, 2015.
D.A. Gomes, E. Pimentel, and H. Sánchez-Morgado. Time-dependent mean-field games in the superquadratic case. ESAIM: Control, Optimisation and Calculus of Variations, 22:562–580, 2016.
D.A. Gomes, E. Pimentel, and V. Voskanyan. Regularity Theory for Mean-Field Game Systems. SpringerBriefs in Mathematics Springer International Publishing, 2016.
O. Guéant. Mean field games equations with quadratic Hamiltonian: A specific approach. Mathematical Models and Methods in Applied Sciences, 22:291–303, 2012.
O. Guéant, J.M. Lasry, and P.L. Lions. Mean field games and applications. In R. Carmona et al., editors, Paris Princeton Lectures on Mathematical Finance 2010. Volume 2003 of Lecture Notes in Mathematics. Springer-Verlag Berlin Heidelberg, 2010.
S. Hamadène and J.P. Lepeltier. Backward equations, stochastic control and zero-sum stochastic differential games. Stochastics and Stochastic Reports, 54:221–231, 1995.
Y. Hu. Stochastic maximum principle. In John Baillieul, Tariq Samad, editors, Encyclopedia of Systems and Control, pages 1347–1350. Springer-Verlag London, 2015.
M. Huang, P.E. Caines, and R.P. Malhamé. Individual and mass behavior in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In Proceedings of the 42nd IEEE International Conference on Decision and Control, pages 98–103. 2003.
M. Huang, P.E. Caines, and R.P. Malhamé. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communications in Information and Systems, 6:221–252, 2006.
M. Huang, P.E. Caines, and R.P. Malhamé. Large population cost coupled LQG problems with nonuniform agents: individual mass behavior and decentralized ε-Nash equilibria. IEEE Transactions on Automatic Control, 52:1560–1571, 2007.
M. Huang, R.P. Malhamé, and P.E. Caines. Nash equilibria for large population linear stochastic systems with weakly coupled agents. In R.P. Malhamé, E.K. Boukas, editors, Analysis, Control and Optimization of Complex Dynamic Systems, pages 215–252. Springer-US, 2005.
N. El Karoui, S. Peng, and M.C. Quenez. Backward stochastic differential equations in finance. Mathematical Finance, 7:1–71, 1997.
D. Lacker. Mean field games via controlled martingale problems: Existence of markovian equilibria. Stochastic Processes and their Applications, 125:2856–2894, 2015.
J.M. Lasry and P.L. Lions. Jeux à champ moyen I. Le cas stationnaire. Comptes Rendus de l’Académie des Sciences de Paris, ser. I, 343:619–625, 2006.
J.M. Lasry and P.L. Lions. Jeux à champ moyen II. Horizon fini et contrôle optimal. Comptes Rendus de l’Académie des Sciences de Paris, ser. I, 343:679–684, 2006.
J.M. Lasry and P.L. Lions. Mean field games. Japanese Journal of Mathematics, 2:229–260, 2007.
P.L. Lions. Théorie des jeux à champs moyen et applications. Lectures at the Collège de France. http://www.college-de-france.fr/default/EN/all/equ_der/cours_et_seminaires.htm, 2007–2008.
J. Ma and J. Yong. Forward-Backward Stochastic Differential Equations and their Applications. Volume 1702 of Lecture Notes in Mathematics. Springer-Verlag Berlin Heidelberg, 2007.
E. Pardoux and S. Peng. Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 14:55–61, 1990.
E. Pardoux and S. Peng. Backward SDEs and quasilinear PDEs. In B. L. Rozovskii and R. B. Sowers, editors, Stochastic Partial Differential Equations and Their Applications. Volume 176 of Lecture Notes in Control and Information Sciences. Springer-Verlag Berlin Heidelberg, 1992.
E. Pardoux and A. Rǎşcanu. Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic Modelling and Applied Probability. Springer International Publishing, 2014.
S. Peng. A general stochastic maximum principle for optimal control problems. SIAM Journal on Control and Optimization, 2:966–979, 1990.
S. Peng. A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics and Stochastics Reports, 38:119–134, 1992.
H. Pham. On some recent aspects of stochastic control and their applications. Probability Surveys, 2:506–549, 2005.
H. Pham. Continuous-time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability. Springer-Verlag Berlin Heidelberg, 2009.
Y. Sun. The exact law of large numbers via Fubini extension and characterization of insurable risks. Journal of Economic Theory, 126:31–69, 2006.
H. Tembine, Q. Zhu, and T. Basar. Risk-sensitive mean-field stochastic differential games. IEEE Transactions on Automatic Control, 59:835–850, 2014.
N. Touzi. Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs. Springer-Verlag New York, 2012.
J. Yong. Linear forward backward stochastic differential equations. Applied Mathematics & Optimization, 39:93–119, 1999.
J. Yong. Linear forward backward stochastic differential equations with random coefficients. Probability Theory and Related Fields, 135:53–83, 2006.
J. Yong and X. Zhou. Stochastic Controls: Hamiltonian Systems and HJB Equations. Stochastic Modelling and Applied Probability. Springer-Verlag New York, 1999.
L.C. Young. Calculus of variations and control theory. W.B. Saunders, Philadelphia, 1969.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Carmona, R., Delarue, F. (2018). Stochastic Differential Mean Field Games. In: Probabilistic Theory of Mean Field Games with Applications I. Probability Theory and Stochastic Modelling, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-58920-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-58920-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-56437-1
Online ISBN: 978-3-319-58920-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)