Skip to main content
Erschienen in: Dynamic Games and Applications 3/2019

31.08.2018

Krasovskii–Subbotin Approach to Mean Field Type Differential Games

verfasst von: Yurii Averboukh

Erschienen in: Dynamic Games and Applications | Ausgabe 3/2019

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A mean field type differential game is a mathematical model of a large system of identical agents under mean field interaction controlled by two players with opposite purposes. We study the case when the dynamics of each agent is given by ODE and the players can observe the distribution of the agents. We construct suboptimal strategies and prove the existence of the value function.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
2.
Zurück zum Zitat Ambrosio L, Gigli N, Savaré G (2005) Gradient flows: in metric spaces and in the space of probability measures. Lectures in mathematics. ETH Zurich, BaselMATH Ambrosio L, Gigli N, Savaré G (2005) Gradient flows: in metric spaces and in the space of probability measures. Lectures in mathematics. ETH Zurich, BaselMATH
4.
Zurück zum Zitat Bardi M, Capuzzo-Dolcetta I (1996) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, BaselMATH Bardi M, Capuzzo-Dolcetta I (1996) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, BaselMATH
5.
Zurück zum Zitat Bayraktar E, Cosso A, Pham H (2018) Randomized dynamic programming principle and Feynman–Kac representation for optimal control of McKean–Vlasov dynamics. Trans Am Math Soc 370:2115–2160MathSciNetCrossRefMATH Bayraktar E, Cosso A, Pham H (2018) Randomized dynamic programming principle and Feynman–Kac representation for optimal control of McKean–Vlasov dynamics. Trans Am Math Soc 370:2115–2160MathSciNetCrossRefMATH
6.
Zurück zum Zitat Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. Springer, New YorkCrossRefMATH Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. Springer, New YorkCrossRefMATH
8.
Zurück zum Zitat Buckdahn R, Djehiche B, Li J (2011) A general stochastic maximum principle for SDEs of mean-field type. Appl Math Optim 64(2):197–216MathSciNetCrossRefMATH Buckdahn R, Djehiche B, Li J (2011) A general stochastic maximum principle for SDEs of mean-field type. Appl Math Optim 64(2):197–216MathSciNetCrossRefMATH
9.
Zurück zum Zitat Cardaliaguet P, Quincampoix M (2008) Deterministic differential games under probability knowledge of initial condition. Int Game Theory Rev 10(1):1–16MathSciNetCrossRefMATH Cardaliaguet P, Quincampoix M (2008) Deterministic differential games under probability knowledge of initial condition. Int Game Theory Rev 10(1):1–16MathSciNetCrossRefMATH
10.
Zurück zum Zitat Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Numerical methods for differential games. In: Bardi TPM, Raghavan TES (eds) Stochastic and differential games: theory and numerical methods, annals of the international society of dynamic games. Birkhäuser, Basel, pp 177–247CrossRef Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Numerical methods for differential games. In: Bardi TPM, Raghavan TES (eds) Stochastic and differential games: theory and numerical methods, annals of the international society of dynamic games. Birkhäuser, Basel, pp 177–247CrossRef
11.
Zurück zum Zitat Cardaliaguet P, Quincampoix M, Saint-Pierre P (2000) Numerical schemes for discontinuous value functions of optimal control. Set Valued Anal 8(1–2):111–126MathSciNetCrossRefMATH Cardaliaguet P, Quincampoix M, Saint-Pierre P (2000) Numerical schemes for discontinuous value functions of optimal control. Set Valued Anal 8(1–2):111–126MathSciNetCrossRefMATH
12.
13.
Zurück zum Zitat Carmona R, Delarue F (2013) Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Preprint arXiv:1303.5835 Carmona R, Delarue F (2013) Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Preprint arXiv:​1303.​5835
14.
Zurück zum Zitat Cavagnari G, Marigonda A (2015) Time-optimal control problem in the space of probability measures. In: Large-scale scientific computing, lecture notes in computer science, vol 9374, pp 109–116 Cavagnari G, Marigonda A (2015) Time-optimal control problem in the space of probability measures. In: Large-scale scientific computing, lecture notes in computer science, vol 9374, pp 109–116
16.
18.
19.
Zurück zum Zitat Dellacherie C, Meyer PAM (1978) Probabilities and potential, North-Holland mathematics studies, vol 29. North-Holland, Amsterdam Dellacherie C, Meyer PAM (1978) Probabilities and potential, North-Holland mathematics studies, vol 29. North-Holland, Amsterdam
20.
Zurück zum Zitat Djehiche B, Hamadène S (2016) Optimal control and zero-sum stochastic differential game problems of mean-field type. Preprint arXiv:1603.06071 Djehiche B, Hamadène S (2016) Optimal control and zero-sum stochastic differential game problems of mean-field type. Preprint arXiv:​1603.​06071
22.
Zurück zum Zitat Friedman A (1970) Existence of value and of saddle points for differential games of pursuit and evasion. J Differ Equ 7(1):92–110MathSciNetCrossRefMATH Friedman A (1970) Existence of value and of saddle points for differential games of pursuit and evasion. J Differ Equ 7(1):92–110MathSciNetCrossRefMATH
23.
Zurück zum Zitat Huang M, Malhamé R, Caines P (2005) Nash equilibria for large population linear stochastic systems with weakly coupled agents. In: Boukas E, Malhamé RP (eds) Analysis, control and optimization of complex dynamic systems. Springer, Berlin, pp 215–252CrossRef Huang M, Malhamé R, Caines P (2005) Nash equilibria for large population linear stochastic systems with weakly coupled agents. In: Boukas E, Malhamé RP (eds) Analysis, control and optimization of complex dynamic systems. Springer, Berlin, pp 215–252CrossRef
24.
25.
Zurück zum Zitat Khaled B, Meriem M, Brahim M (2014) Existence of optimal controls for systems governed by mean-field stochastic differential equations. Afr Stat 9(1):627–645MathSciNetMATH Khaled B, Meriem M, Brahim M (2014) Existence of optimal controls for systems governed by mean-field stochastic differential equations. Afr Stat 9(1):627–645MathSciNetMATH
26.
Zurück zum Zitat Kolokoltsov VN (2010) Nonlinear Markov process and kinetic equations. Cambridge University Press, CambridgeCrossRefMATH Kolokoltsov VN (2010) Nonlinear Markov process and kinetic equations. Cambridge University Press, CambridgeCrossRefMATH
27.
Zurück zum Zitat Kononenko A (1976) On equilibrium positional strategies in nonantagonistic differential games. Dokl Akad Nauk SSSR 231:285–288 (in Russian)MathSciNetMATH Kononenko A (1976) On equilibrium positional strategies in nonantagonistic differential games. Dokl Akad Nauk SSSR 231:285–288 (in Russian)MathSciNetMATH
28.
Zurück zum Zitat Krasovskii AN, Krasovskii NN (1995) Control under lack of information. Birkhäuser, BostonCrossRefMATH Krasovskii AN, Krasovskii NN (1995) Control under lack of information. Birkhäuser, BostonCrossRefMATH
29.
Zurück zum Zitat Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New YorkCrossRef Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New YorkCrossRef
30.
Zurück zum Zitat Lasry JM, Lions PL (2006) Jeux à champ moyen. I. Le cas stationnaire (French) [Mean field games. I. the stationary case]. C R Math Acad Sci Paris 343:619–625MathSciNetCrossRefMATH Lasry JM, Lions PL (2006) Jeux à champ moyen. I. Le cas stationnaire (French) [Mean field games. I. the stationary case]. C R Math Acad Sci Paris 343:619–625MathSciNetCrossRefMATH
31.
Zurück zum Zitat Lasry JM, Lions PL (2006) Jeux à champ moyen. II. Horizon fini et contrôle optimal (French) [Mean field games. II. finite horizon and optimal control]. C R Math Acad Sci Paris 343:679–684MathSciNetCrossRefMATH Lasry JM, Lions PL (2006) Jeux à champ moyen. II. Horizon fini et contrôle optimal (French) [Mean field games. II. finite horizon and optimal control]. C R Math Acad Sci Paris 343:679–684MathSciNetCrossRefMATH
32.
33.
Zurück zum Zitat Pham H, Wei X (2017) Dynamic programming for optimal control of stochastic McKean–Vlasov dynamics. SIAM J Control Optim 55:1069–1101MathSciNetCrossRefMATH Pham H, Wei X (2017) Dynamic programming for optimal control of stochastic McKean–Vlasov dynamics. SIAM J Control Optim 55:1069–1101MathSciNetCrossRefMATH
34.
Zurück zum Zitat Pham H, Wei X (2018) Bellman equation and viscosity solutions for mean-field stochastic control problem. ESAIM Control Optim Calc Var Pham H, Wei X (2018) Bellman equation and viscosity solutions for mean-field stochastic control problem. ESAIM Control Optim Calc Var
36.
Zurück zum Zitat Subbotin AI (1995) Generalized solutions of first-order PDEs. The dynamical perspective. Birkhäuser, BostonCrossRef Subbotin AI (1995) Generalized solutions of first-order PDEs. The dynamical perspective. Birkhäuser, BostonCrossRef
37.
Zurück zum Zitat Subbotin AI, Chentsov AG (1981) Optimization of guarantee in control problems. Nauka, Moscow (in Russian)MATH Subbotin AI, Chentsov AG (1981) Optimization of guarantee in control problems. Nauka, Moscow (in Russian)MATH
38.
Zurück zum Zitat Sznitman A (1991) Topics in propagation of chaos. Lecture notes in mathematics, vol 1464. Springer, Berlin, pp 165–251 Sznitman A (1991) Topics in propagation of chaos. Lecture notes in mathematics, vol 1464. Springer, Berlin, pp 165–251
40.
Zurück zum Zitat Varaya P, Lin J (1969) Existence of saddle points in differential games. SIAM J Control 7(1):142–157 Varaya P, Lin J (1969) Existence of saddle points in differential games. SIAM J Control 7(1):142–157
41.
Zurück zum Zitat Warga J (1972) Optimal control of differential and functional equations. Academic, New YorkMATH Warga J (1972) Optimal control of differential and functional equations. Academic, New YorkMATH
Metadaten
Titel
Krasovskii–Subbotin Approach to Mean Field Type Differential Games
verfasst von
Yurii Averboukh
Publikationsdatum
31.08.2018
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 3/2019
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-018-0282-6

Weitere Artikel der Ausgabe 3/2019

Dynamic Games and Applications 3/2019 Zur Ausgabe