[Programmation dynamique pour les problèmes de contrôle à champs moyen]
For mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF ρ of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function
Pour les problèmes de contrôle stochastique à champs moyen, la programmation dynamique ne s'applique pas sans adaptation ; mais si l'on reformule le problème avec l'équation de Fokker–Planck, on peut le faire en utilisant une fonctionnelle valeur
Accepté le :
Publié le :
Mathieu Laurière 1 ; Olivier Pironneau 1
@article{CRMATH_2014__352_9_707_0, author = {Mathieu Lauri\`ere and Olivier Pironneau}, title = {Dynamic programming for mean-field type control}, journal = {Comptes Rendus. Math\'ematique}, pages = {707--713}, publisher = {Elsevier}, volume = {352}, number = {9}, year = {2014}, doi = {10.1016/j.crma.2014.07.008}, language = {en}, }
Mathieu Laurière; Olivier Pironneau. Dynamic programming for mean-field type control. Comptes Rendus. Mathématique, Volume 352 (2014) no. 9, pp. 707-713. doi : 10.1016/j.crma.2014.07.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.07.008/
[1] A maximum principle for SDEs of mean-field type, Appl. Math. Optim., Volume 63 (2011), pp. 341-356
[2] Control and Nash games with mean field effect, Chin. Ann. Math., Ser. B, Volume 34B (2013) no. 2, pp. 161-192
[3] Mean-Field Games and Mean-Field Type Control, Springer Briefs in Mathematics, 2014
[4] The master equation in mean-field theory, Asymptot. Anal. (2014) (in press)
[5] Controlled Markov Process and Viscosity Solutions, Springer, 2006
[6] Large deviations for a mean field model of systemic risk, SIAM J. Financ. Math., Volume 4 (2013) no. 1, pp. 151-184
[7] Mean field games and applications, Paris–Princeton Lectures on Mathematical Finance, Lecture Notes in Mathematics, Springer, 2011
[8] New development in freefem++, J. Numer. Math., Volume 20 (2012) no. 3–4, pp. 251-265
[9] Mean-field games, Jpn. J. Math., Volume 2 (2007), pp. 229-260
[10] Mean-field games, Cours au Collège de France (2007–2008) http://www.college-de-france.fr/site/pierre-louis-lions/course-2007-2008_1.htm
[11] Nonlinear Lévy processes and their characteristics | arXiv
[12] Applied Stochastic Control of Jump Diffusions, Springer, 2005
[13] Liquidity generated by heterogeneous beliefs and costly estimations, Netw. Heterog. Media, Volume 7 (2012) no. 2, pp. 349-361
[14] Optimal Stochastic Control, Stochastic Target Problems and Backard SDE, Field Inst. Monogr., vol. 29, Springer, 2013
[15] Stochastic Control, Applications of Mathematics Series, vol. 43, Springer, 1999
- Finite Approximations for Mean-Field Type Multi-agent Control and Their Near Optimality, Applied Mathematics Optimization, Volume 92 (2025) no. 1 | DOI:10.1007/s00245-025-10279-x
- Mean field type control problems, some Hilbert-space-valued FBSDES, and related equations, ESAIM: Control, Optimisation and Calculus of Variations, Volume 31 (2025), p. 33 | DOI:10.1051/cocv/2025022
- Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach, Journal of Differential Equations, Volume 430 (2025), p. 113205 | DOI:10.1016/j.jde.2025.02.076
- Stochastic McKean-Vlasov Control Problem with Regime-Switching and Its Applications, Journal of Systems Science and Complexity, Volume 38 (2025) no. 4, p. 1437 | DOI:10.1007/s11424-025-4283-4
- Generalized replicator dynamics based on mean-field pairwise comparison dynamic, Mathematics and Computers in Simulation, Volume 236 (2025), p. 200 | DOI:10.1016/j.matcom.2025.04.010
- Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem, Annals of Operations Research, Volume 336 (2024) no. 1-2, p. 1315 | DOI:10.1007/s10479-023-05293-7
- Stochastic recursive optimal control of McKean-Vlasov type: A viscosity solution approach, Journal of Differential Equations, Volume 409 (2024), p. 334 | DOI:10.1016/j.jde.2024.07.015
- Infinite Horizon Average Cost Optimality Criteria for Mean-Field Control, SIAM Journal on Control and Optimization, Volume 62 (2024) no. 5, p. 2776 | DOI:10.1137/23m1603649
- Deep learning for conditional McKean-Vlasov Jump diffusions, SSRN Electronic Journal (2024) | DOI:10.2139/ssrn.4760864
- Forward-Backward Sweep Method for the System of HJB-FP Equations in Memory-Limited Partially Observable Stochastic Control, Entropy, Volume 25 (2023) no. 2, p. 208 | DOI:10.3390/e25020208
- Time Efficient Offloading Optimization in Automotive Multi-Access Edge Computing Networks Using Mean-Field Games, IEEE Transactions on Vehicular Technology, Volume 72 (2023) no. 5, p. 6460 | DOI:10.1109/tvt.2022.3229888
- Dynamic Programming Principles for Mean-Field Controls with Learning, Operations Research, Volume 71 (2023) no. 4, p. 1040 | DOI:10.1287/opre.2022.2395
- Stochastic Fokker–Planck Equations for Conditional McKean–Vlasov Jump Diffusions and Applications to Optimal Control, SIAM Journal on Control and Optimization, Volume 61 (2023) no. 3, p. 1472 | DOI:10.1137/21m1461034
- Fully-coupled mean-field FBSDE and maximum principle for related optimal control problem, Systems Control Letters, Volume 177 (2023), p. 105550 | DOI:10.1016/j.sysconle.2023.105550
- McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations, Mathematics of Operations Research, Volume 47 (2022) no. 4, p. 2891 | DOI:10.1287/moor.2021.1232
- McKean–Vlasov optimal control: The dynamic programming principle, The Annals of Probability, Volume 50 (2022) no. 2 | DOI:10.1214/21-aop1548
- , 2021 IEEE International Conference on Systems, Man, and Cybernetics (SMC) (2021), p. 982 | DOI:10.1109/smc52423.2021.9658947
- Finite stateN-agent and mean field control problems, ESAIM: Control, Optimisation and Calculus of Variations, Volume 27 (2021), p. 31 | DOI:10.1051/cocv/2021032
- A stability property in mean field type differential games, Journal of Mathematical Analysis and Applications, Volume 498 (2021) no. 1, p. 124940 | DOI:10.1016/j.jmaa.2021.124940
- Lattice approximations of the first-order mean field type differential games, Nonlinear Differential Equations and Applications NoDEA, Volume 28 (2021) no. 6 | DOI:10.1007/s00030-021-00727-2
- , 2020 59th IEEE Conference on Decision and Control (CDC) (2020), p. 3860 | DOI:10.1109/cdc42340.2020.9303958
- A Tutorial On Mean-Field-Type Games and Risk-Aware Controllers, Annual Reviews in Control, Volume 50 (2020), p. 317 | DOI:10.1016/j.arcontrol.2020.05.003
- Berge equilibrium in linear-quadratic mean-field-type games, Journal of the Franklin Institute, Volume 357 (2020) no. 15, p. 10861 | DOI:10.1016/j.jfranklin.2020.08.019
- Nonexponential Sanov and Schilder theorems on Wiener space: BSDEs, Schrödinger problems and control, The Annals of Applied Probability, Volume 30 (2020) no. 3 | DOI:10.1214/19-aap1531
- Krasovskii–Subbotin Approach to Mean Field Type Differential Games, Dynamic Games and Applications, Volume 9 (2019) no. 3, p. 573 | DOI:10.1007/s13235-018-0282-6
- Mean-field optimal control as Gamma-limit of finite agent controls, European Journal of Applied Mathematics, Volume 30 (2019) no. 6, p. 1153 | DOI:10.1017/s0956792519000044
- A mean-field optimal control formulation of deep learning, Research in the Mathematical Sciences, Volume 6 (2019) no. 1 | DOI:10.1007/s40687-018-0172-y
- Target problem for mean field type differential game, IFAC-PapersOnLine, Volume 51 (2018) no. 32, p. 654 | DOI:10.1016/j.ifacol.2018.11.499
- Optimal Control of SDEs of McKean-Vlasov Type, Probabilistic Theory of Mean Field Games with Applications I, Volume 83 (2018), p. 513 | DOI:10.1007/978-3-319-58920-6_6
- MFGs with a Common Noise: Strong and Weak Solutions, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 107 | DOI:10.1007/978-3-319-56436-4_2
- Solving MFGs with a Common Noise, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 155 | DOI:10.1007/978-3-319-56436-4_3
- The Master Field and the Master Equation, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 239 | DOI:10.1007/978-3-319-56436-4_4
- Optimization in a Random Environment, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 3 | DOI:10.1007/978-3-319-56436-4_1
- Classical Solutions to the Master Equation, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 323 | DOI:10.1007/978-3-319-56436-4_5
- Convergence and Approximations, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 447 | DOI:10.1007/978-3-319-56436-4_6
- Extensions for Volume II, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 541 | DOI:10.1007/978-3-319-56436-4_7
- Viability Theorem for Deterministic Mean Field Type Control Systems, Set-Valued and Variational Analysis, Volume 26 (2018) no. 4, p. 993 | DOI:10.1007/s11228-018-0479-2
- , 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA) (2017), p. 1 | DOI:10.1109/cnsa.2017.7973931
- A risk analysis for a system stabilized by a central agent, Risk and Decision Analysis, Volume 6 (2017) no. 2, p. 97 | DOI:10.3233/rda-160117
- Dynamic Programming for Optimal Control of Stochastic McKean–Vlasov Dynamics, SIAM Journal on Control and Optimization, Volume 55 (2017) no. 2, p. 1069 | DOI:10.1137/16m1071390
- Limit Theory for Controlled McKean–Vlasov Dynamics, SIAM Journal on Control and Optimization, Volume 55 (2017) no. 3, p. 1641 | DOI:10.1137/16m1095895
- Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics, Transactions of the American Mathematical Society, Volume 370 (2017) no. 3, p. 2115 | DOI:10.1090/tran/7118
- Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource, Applied Mathematics Optimization, Volume 74 (2016) no. 3, p. 459 | DOI:10.1007/s00245-016-9385-x
- Dynamic Programming for Mean-Field Type Control, Journal of Optimization Theory and Applications, Volume 169 (2016) no. 3, p. 902 | DOI:10.1007/s10957-015-0785-x
- The Master equation in mean field theory, Journal de Mathématiques Pures et Appliquées, Volume 103 (2015) no. 6, p. 1441 | DOI:10.1016/j.matpur.2014.11.005
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