Volume 27, 2021
|Number of page(s)||33|
|Published online||07 April 2021|
Finite state N-agent and mean field control problems*,**
Centre de Mathématiques Appliquées, École Polytechnique,
Route de Saclay,
Palaiseau Cedex, France.
*** Corresponding author: firstname.lastname@example.org
Accepted: 17 March 2021
We examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as N grows, of the value functions of the centralized N-agent optimal control problem to the limit mean field control problem value function, with a convergence rate of order . Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e. convergence of the N-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate.
Mathematics Subject Classification: 35B65 / 35F21 / 49L25 / 49M25 / 60F15 / 60J27 / 91A12
Key words: Mean field control problem / control of Markov chains / finite state space / cooperative games / social planner / Hamilton-Jacobi-Bellman equation / viscosity solution / finite difference approximation / classical solution / propagation of chaos
This research benefited from the support of the project ANR-16-CE40-0015-01 on “Mean Field Games”, LABEX Louis Bachelier Finance and Sustainable Growth - project ANR-11-LABX-0019, under the Investments for the Future program (in accordance with Article 8 of the Assignment Agreements for Communication Assistance), ECOREES ANR Project, FDD Chair and Joint Research Initiative FiME in partnership with Europlace Institute of Finance.
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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