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},
doi = {10.1051/cocv/2021085},
language = {en},
url = {https://www.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 PB - EDP-Sciences UR - https://www.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 %I EDP-Sciences %U https://www.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), article no. 89. doi: 10.1051/cocv/2021085
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