×

Mean field games with singular controls. (English) Zbl 1386.93306

Summary: This paper establishes the existence of relaxed solutions to mean field games (MFGs for short) with singular controls. We also prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls. Our existence and approximation results strongly hinge on the use of the Skorokhod \(M_1\) topology on the space of càdlàg functions.

MSC:

93E20 Optimal stochastic control
91B70 Stochastic models in economics
60H30 Applications of stochastic analysis (to PDEs, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] S. Ahuja, {\it Wellposedness of mean field games with common noise under a weak monotonicity condition}, SIAM J. Control Optim., 51 (2016), pp. 30-48, . · Zbl 1327.93403
[2] C. Aliprantis and K. Border, {\it Infinite Dimensional Analysis: A Hitchhiker’s Guide}, Springer-Verlag, Berlin, Heidelberg, 2006. · Zbl 1156.46001
[3] A. Bensoussan, K. Sung, P. Yam, and S. Yung, {\it Linear quadratic mean field games}, J. Optim. Theory Appl., 169 (2016), pp. 496-529. · Zbl 1343.91010
[4] R. Carmona and F. Delarue, {\it Probabilistic analysis of mean-field games}, SIAM J. Control Optim., 51 (2013), pp. 2705-2734, . · Zbl 1275.93065
[5] R. Carmona, F. Delarue, and A. Lachapelle, {\it Control of McKean-Vlasov dynamics versus mean field games}, Math. Financ. Econ., 7 (2013), pp. 131-166. · Zbl 1269.91012
[6] R. Carmona, F. Delarue, and D. Lacker, {\it Mean field games with common noise}, Ann. Probab., 44 (2016), pp. 3740-3803. · Zbl 1422.91083
[7] R. Carmona, J. Fouque, and L. Sun, {\it Mean field games and systemic risk}, Commun. Math. Sci., 13 (2015), pp. 911-933. · Zbl 1337.91031
[8] R. Carmona and D. Lacker, {\it A probabilistic weak formulation of mean field games and applications}, Ann. Appl. Probab., 25 (2015), pp. 1189-1231. · Zbl 1332.60100
[9] R. Carmona and X. Zhu, {\it A probabilistic approach to mean field games with major and minor players}, Ann. Appl. Probab., 26 (2016), pp. 1535-1580. · Zbl 1342.93121
[10] P. Chan and R. Sircar, {\it Fracking, renewables and mean field games}, SIAM Rev., 59 (2016), pp. 588-615, . · Zbl 1369.91146
[11] R. Elie, T. Mastrolia, and D Possamaï, {\it A Tale of a Principal and Many Many Agents}, preprint, , 2016. · Zbl 1443.91198
[12] O. Gueant, J. Lasry, and P. Lions, {\it Mean field games and applications}, Paris-Princeton Lectures on Mathematical Finance, Lecture Notes in Math. 2003, Springer, Berlin, 2011, pp. 205-266. · Zbl 1205.91027
[13] X. Guo and J. Lee, {\it Mean Field Games with Singular Controls of Bounded Velocity}, preprint, , 2017.
[14] U. Haussmann and J. Lepeltier, {\it On the existence of optimal controls}, SIAM. J. Control Optim., 28 (1990), pp. 851-902, . · Zbl 0712.49013
[15] U. Haussmann and W. Suo, {\it Singular optimal stochastic controls I: Existence}, SIAM. J. Control Optim., 33 (1995), pp. 916-936, . · Zbl 0925.93958
[16] U. Horst and F. Naujokat, {\it When to cross the spread: Trading in two-sided limit order books}, SIAM J. Financial Math., 5 (2014), pp. 278-315, . · Zbl 1308.93224
[17] U. Horst and J. Scheinkman, {\it Equilibria in systems of social interactions}, J. Econom. Theory, 130 (2006), pp. 44-77. · Zbl 1141.91343
[18] U. Horst and J. Scheinkman, {\it A limit theorem for systems of social interactions}, J. Math. Econom., 45 (2009), pp. 609-623. · Zbl 1195.91089
[19] M. Huang, R. Malhamé, and P. Caines, {\it Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle}, Commun. Inf. Syst., 6 (2006), pp. 221-252. · Zbl 1136.91349
[20] J. Jacod and J. Mémin, {\it Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité}, Séminaire de probabilités XV, University of Strasbourg, Lecture Notes in Math. 850, Springer, Berlin, New York, 1981, pp. 529-546. · Zbl 0458.60016
[21] S. Jaimungal and M. Nourian, {\it Mean-Field Game Strategies for Optimal Execution}, 2015, . · Zbl 1410.91498
[22] I. Karatzas and S.E. Shreve, {\it Brownian Motion and Stochastic Calculus}, Grad. Texts in Math. 113, Springer-Verlag, New York, 1991. · Zbl 0734.60060
[23] N. El Karoui and S. Meleard, {\it Martingale measures and stochastic calculus}, Probab. Theory Related Fields, 84 (1990), pp. 83-101. · Zbl 0694.60041
[24] N. El Karoui, D. Huu Nguyen, and M. Jeanblanc-Picqué, {\it Compactification methods in the control of degenerate diffusions: Existence of an optimal control}, Stochastics, 20 (1987), pp. 196-219. · Zbl 0613.60051
[25] D. Lacker, {\it Mean field games via controlled martingale problems: Existence of Markovian equilibria}, Stochastic Process. Appl., 125 (2015), pp. 2856-2894. · Zbl 1346.60083
[26] D. Lacker, {\it Limit theory for controlled McKean-Vlasov dynamics}, SIAM. J. Control Optim., 2 (2017), pp. 1641-1672, . · Zbl 1362.93167
[27] J. Lasry and P. Lions, {\it Mean field games}, Jpn. J. Math., 2 (2007), pp. 229-260. · Zbl 1156.91321
[28] G. Pang and W. Whitt, {\it Continuity of a queueing integral representation in the \(M_1\) topology}, Ann. Appl. Probab., 20 (2010), pp. 214-237. · Zbl 1186.60098
[29] C. Villani, {\it Optimal Transport: Old and New}, Grundlehren der Mathematischen Wissenschaften 338, Springer-Verlag, Berlin, Heidelberg, 2009. · Zbl 1156.53003
[30] W. Whitt, {\it Stochastic Process Limits: an Introduction to Stochastic Process Limits and Their Application to Queues}, Springer-Verlag, New York, 2002. · Zbl 0993.60001
[31] L. Young, {\it Generalized curves and the existence of an attained absolute minimum in the calculus of variations}, C. R. Soc. Sci. Letters Varsovie, III, 30 (1937), pp. 212-234 · Zbl 0019.21901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.