scholarly journals Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective

2019 ◽  
Vol 57 (6) ◽  
pp. 3666-3693 ◽  
Author(s):  
Beatrice Acciaio ◽  
Julio Backhoff-Veraguas ◽  
René Carmona
Keyword(s):  
Maximum Principle ◽  
Mean Field ◽  
Stochastic Maximum Principle ◽  
Control Problems ◽  
Field Control

scholarly journals Finite state N-agent and mean field control problems

2021 ◽  
Author(s):  
Alekos Cecchin
Keyword(s):  
Control Problem ◽  
Value Function ◽  
Mean Field ◽  
Control Problems ◽  
Value Functions ◽  
Limit Value ◽  
Field Control ◽  
Finite State ◽  
Finite Time Horizon ◽  
Hamilton Jacobi Bellman

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 $\frac{1}{\sqrt{N}}$. 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.

scholarly journals A Stochastic Maximum Principle for Markov Chains of Mean-Field Type

Games ◽  
2018 ◽  
Vol 9 (4) ◽  
pp. 84 ◽  
Author(s):  
Salah Choutri ◽  
Tembine Hamidou
Keyword(s):  
Maximum Principle ◽  
Markov Chains ◽  
Martingale Problem ◽  
Mean Field ◽  
Necessary Optimality Conditions ◽  
Stochastic Maximum Principle ◽  
Jump Processes ◽  
Field Type ◽  
Cost Functionals ◽  
Well Posed

We derive sufficient and necessary optimality conditions in terms of a stochastic maximum principle (SMP) for controls associated with cost functionals of mean-field type, under dynamics driven by a class of Markov chains of mean-field type which are pure jump processes obtained as solutions of a well-posed martingale problem. As an illustration, we apply the result to generic examples of control problems as well as some applications.

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