Mean-field games of optimal stopping: A relaxed solution approach

SIAM Journal on Control and Optimization

Volumn 58, Issue 4, 2020, Pages 1795-1821

  Bouveret, Géraldine ^a,b   Dumitrescu, Roxana ^c   Tankov, Peter ^d  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

^c KING'S COLLEGE LONDON   (United Kingdom)

^d Ecole Nationale de la Statistique et de l'Administration Economique   (France)

Author keywords

Infinite dimensional linear programming; Mean field games; Optimal stopping; Relaxed solutions

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS;

MEAN FIELD GAMES; OCCUPATION MEASURE; OPTIMAL STOPPING; OPTIMAL STOPPING PROBLEM; OPTIMAL STOPPING TIME; OPTIMAL STRATEGIES; POTENTIAL GAMES; SOLUTION APPROACH;

COMPUTER GAMES;

EID: 85088965171     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/18M1233480     Document Type: Article

References (32)

1. L. Ambrosio, N. Fusco, and D. Pallara (2000), Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr. 254, Clarendon Press, Oxford.
2. J. Benamou and G. Carlier (2015), Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl., 167, pp. 1-26.
3. A. Bensoussan and J.-L. Lions (1982), Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam.
4. C. Bertucci (2017), Optimal stopping in mean field games, an obstacle problem approach, J. Math. Pures Appl., 120, pp. 165-194.
5. C. Bertucci (2020), A remark on Uzawa's algorithm and an application to mean-field games systems, ESAIM Math. Model. Numer. Anal., 54, pp. 1053-1071.
6. V. I. Bogachev (2007), Measure Theory, Springer, New York.
7. H. Brezis (2010), Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York.
8. R. Buckdahn, D. Goreac, and M. Quincampoix (2011), Stochastic optimal control and linear programming approach, Appl. Math. Optim., 63, pp. 257-276.
9. R. Carmona and F. Delarue (2013a), Mean field forward-backward stochastic differential equations, Electron. Commun. Probab., 18.
10. R. Carmona and F. Delarue (2013b), Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51, pp. 2705-2734.
11. R. Carmona and F. Delarue (2018), Probabilistic Theory of Mean Field Games with Applications I-II, Springer, New York.
12. R. Carmona, F. Delarue, and D. Lacker (2016), Mean field games with common noise, Ann. Probab., 44, pp. 3740-3803.
13. R. Carmona, F. Delarue, and D. Lacker (2017), Mean field games of timing and models for bank runs, Appl. Math. Optim., 76, pp. 217-260.
14. M. Cho and R. Stockbridge (2002), Linear programming formulation for optimal stopping problems, SIAM J. Control Optim., 40, pp. 1965-1982.
15. N. El Karoui, D. Huu Nguyen, and M. Jeanblanc-Picque (1987), Compactification methods in the control of degenerate diffusions: Existence of an optimal control, Stochastics, 20, pp. 169-219.
16. N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.-C. Quenez (1997), Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25, pp. 702-737.
17. L. Evans (1998), Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI.
18. A. Friedman (1975), Stochastic differential equations and applications, Vol. 1, Probab. Math. Stat. 28, Academic Press, New York, 1975.
19. A. Friedman (1983), Partial differential equations of parabolic type, Robert E. Krieger Publishing Company, Malabar, FL.
20. K. Helmes and R. H. Stockbridge (2007), Linear programming approach to the optimal stopping of singular stochastic processes, Stochastics, 79, pp. 309-335.
21. M. Huang, P. E. Caines, and R. P. Malhame (2006), Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6, pp. 221-252.
22. D. Lacker (2015), Mean field games via controlled martingale problems: Existence of Markovian equilibria, Stochastic Process. Appl., 125, pp. 2856-2894.
23. O. Ladyzhenskaya, V. Solonnikov, and N. Uraltseva (1968), Linear and quasilinear parabolic equations of parabolic type, AMS., Providence, RI.
24. J.-M. Lasry and P.-L. Lions (2006a), Jeux achamp moyen. i-le cas stationnaire, C. R. Math., 343, pp. 619-625.
25. J.-M. Lasry and P.-L. Lions (2006b), Jeux achamp moyen. ii-horizon fini et controle optimal, C. R. Math., 343, pp. 679-684.
26. J.-M. Lasry and P.-L. Lions (2007), Mean field games, Jpn. J. Math., 2, pp. 229-260.
27. A. McLennan (2018), Advanced Fixed Point Theory for Economics, Springer, New York.
28. M. Nutz (2018), A mean field game of optimal stopping, SIAM J. Control Optim., 56, pp. 1206-1221.
29. E. Pardoux (1998), Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, Springer, New York.
30. P. Protter (2004), Stochastic Integration and Differential Equations, 2nd ed., Springer, New York.
31. R. H. Stockbridge (1990), Time-average control of martingale problems: A linear programming formulation, Ann. Probab., 18, pp. 206-217.
32. V. Varadarajan (1965), Measures on topological spaces, Amer. Math. Soc. Ser. 2, 48 (1965), pp. 161-228.