Control and optimal stopping Mean Field Games: a linear programming approach

Roxana Dumitrescu; Marcos Leutscher; Peter Tankov

doi:10.1214/21-EJP713

Control and optimal stopping Mean Field Games: a linear programming approach

Roxana Dumitrescu, Marcos Leutscher, Peter Tankov

Mathematics

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in earlier papers in the pure control case.

Original language	English
Article number	157
Journal	Electronic Journal Of Probability
Volume	26
DOIs	https://doi.org/10.1214/21-EJP713
Publication status	Published - 2021

Keywords

Continuous control
Controlled/stopped martingale problem
Infinite-dimensional linear programming
Mean-field games
Optimal stopping
Relaxed solutions

Access to Document

10.1214/21-EJP713

Cite this

@article{a99ff020df7c4f309512b976808d4375,

title = "Control and optimal stopping Mean Field Games: a linear programming approach",

abstract = "We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in earlier papers in the pure control case.",

keywords = "Continuous control, Controlled/stopped martingale problem, Infinite-dimensional linear programming, Mean-field games, Optimal stopping, Relaxed solutions",

author = "Roxana Dumitrescu and Marcos Leutscher and Peter Tankov",

note = "Funding Information: *Peter Tankov gratefully acknowledges financial support from the ANR (project EcoREES ANR-19-CE05-0042) and from the FIME Research Initiative. †Department of Mathematics, King{\textquoteright}s College London, Strand, London, WC2R 2LS, United Kingdom. E-mail: [email protected] ‡CREST, ENSAE, Institut Polytechnique de Paris, 5 avenue Henry Le Chatelier, 91120 Palaiseau, France. E-mail: [email protected] §CREST, ENSAE, Institut Polytechnique de Paris, 5 avenue Henry Le Chatelier, 91120 Palaiseau, France. E-mail: [email protected] Publisher Copyright: {\textcopyright} 2021, Institute of Mathematical Statistics. All rights reserved.",

year = "2021",

doi = "10.1214/21-EJP713",

language = "English",

volume = "26",

journal = "Electronic Journal Of Probability",

issn = "1083-6489",

publisher = "Institute of Mathematical Statistics",

}

TY - JOUR

T1 - Control and optimal stopping Mean Field Games

T2 - a linear programming approach

AU - Dumitrescu, Roxana

AU - Leutscher, Marcos

AU - Tankov, Peter

N1 - Funding Information: *Peter Tankov gratefully acknowledges financial support from the ANR (project EcoREES ANR-19-CE05-0042) and from the FIME Research Initiative. †Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom. E-mail: [email protected] ‡CREST, ENSAE, Institut Polytechnique de Paris, 5 avenue Henry Le Chatelier, 91120 Palaiseau, France. E-mail: [email protected] §CREST, ENSAE, Institut Polytechnique de Paris, 5 avenue Henry Le Chatelier, 91120 Palaiseau, France. E-mail: [email protected] Publisher Copyright: © 2021, Institute of Mathematical Statistics. All rights reserved.

PY - 2021

Y1 - 2021

N2 - We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in earlier papers in the pure control case.

AB - We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in earlier papers in the pure control case.

KW - Continuous control

KW - Controlled/stopped martingale problem

KW - Infinite-dimensional linear programming

KW - Mean-field games

KW - Optimal stopping

KW - Relaxed solutions

UR - http://www.scopus.com/inward/record.url?scp=85123090424&partnerID=8YFLogxK

U2 - 10.1214/21-EJP713

DO - 10.1214/21-EJP713

M3 - Article

AN - SCOPUS:85123090424

SN - 1083-6489

VL - 26

JO - Electronic Journal Of Probability

JF - Electronic Journal Of Probability

M1 - 157

ER -

Control and optimal stopping Mean Field Games: a linear programming approach

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this