Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussan et al., Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in Bandini et al. [Stochastic Process. Appl. 129 (2019) 674–711], which is fundamentally different from our present proposed framework.
Keywords: Duncan-Mortensen-Zakai equations, mean field type control problem, Bellman and master equations, filtering formulae with non-Gaussian initial conditions, linear dynamics and quadratic payoff, settings with Gaussian or non-Gaussian initial distributions, Riccati equations
@article{COCV_2021__27_1_A91_0,
author = {Bensoussan, Alain and Yam, Sheung Chi Phillip},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Mean field approach to stochastic control with partial information},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
number = {1},
doi = {10.1051/cocv/2021085},
language = {en},
url = {https://numdam.org/articles/10.1051/cocv/2021085/}
}
TY - JOUR AU - Bensoussan, Alain AU - Yam, Sheung Chi Phillip ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Mean field approach to stochastic control with partial information JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 IS - 1 PB - EDP-Sciences UR - https://numdam.org/articles/10.1051/cocv/2021085/ DO - 10.1051/cocv/2021085 LA - en ID - COCV_2021__27_1_A91_0 ER -
%0 Journal Article %A Bensoussan, Alain %A Yam, Sheung Chi Phillip %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Mean field approach to stochastic control with partial information %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %N 1 %I EDP-Sciences %U https://numdam.org/articles/10.1051/cocv/2021085/ %R 10.1051/cocv/2021085 %G en %F COCV_2021__27_1_A91_0
Bensoussan, Alain; Yam, Sheung Chi Phillip. Mean field approach to stochastic control with partial information. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021) no. 1, article no. 89. doi: 10.1051/cocv/2021085
[1] , and , Stochastic filtering of a pure jump process with predictable jumps and path-dependent local characteristics. Preprint (2020). | arXiv
[2] , , and , Randomized filtering and bellman equation in Wasserstein space for partial observation control problem. Stoch. Process. Appl. 129 (2019) 674–711.
[3] , Stochastic Control of Partially Observable Systems. Cambridge University Press (1992).
[4] , and , Vol. 101 of Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013).
[5] , and , A mean-field stochastic control problem with partial observations. Ann. Appl. Probab. 27 (2017) 3201–3245.
[6] , Stochastic filtering and optimal control of pure jump Markov processes with noise-free partial observation. ESAIM: COCV 26 (2020) 25.
[7] , , and , The Master Equation and the Convergence Problem in Mean Field Games. Preprint (2015). | arXiv
[8] , and , Discrete-time mean field partially observable controlled systems subject to common noise. Appl. Math. Optim. 76 (2017) 59–91.
[9] and , ϵ-Nash Equilibria for major minor LQG mean field games with partial observations of all agents. IEEE Trans. Autom. Control 66 (2021) 2778–2786.
[10] and , Hamilton-Jacobi-Bellman equations for the optimal control of the Duncan-Mortensen-Zakai equation. J. Funct. Anal. 172 (2000) 466–510.
[11] , Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolutions. Acta Math. 161 (1988) 243–278.
[12] , Filtering formulae for partially observed linear systems with non-Gaussian initial conditions. Stochastics 16 (1986) 1–24.
[13] , Vol. 72 of Stochastic control theory: Dynamic programming principle. Springer (2014).
[14] , and , Partially-observed discrete-time risk-sensitive mean-field games. In 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE (2019) 317–322.
[15] and , Mean field game theory for agents with individual-state partial observations. In 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE (2016) 6105–6110.
[16] , , and , Partially Observable Mean Field Reinforcement Learning. Preprint (2020). | arXiv
[17] , The maximum principle for partially observed optimal control of stochastic differential equations. SIAM J. Control Optim. 36 (1998) 1596–1617.
Cité par Sources :
