Volume 28, 2022
|Number of page(s)||25|
|Published online||07 July 2022|
Dynamic optimization problems for mean-field stochastic large-population systems*
School of Mathematics, Shandong University, Jinan, Shandong 250100, PR China
2 School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan, Shandong 250014, PR China
** Corresponding author: firstname.lastname@example.org
Accepted: 10 June 2022
This paper considers dynamic optimization problems for a class of control average meanfield stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation with individual noise and common noise, and the weight coefficients in the corresponding cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation. Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence equation systems are discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.
Mathematics Subject Classification: 60H10 / 60H30 / 91A10 / 91A25 / 93E20
Key words: Mean-field stochastic differential equation / mean-field game / stochastic Hamiltonian system / Riccati equation / ϵ-Nash equilibrium
N. Li acknowledges the National Natural Science Foundation of China (12171279, 11801317), the Natural Science Foundation of Shandong Province (ZR2019MA013), and the Colleges and Universities Youth Innovation Technology Program of Shandong Province (2019KJI011). Z. Wu acknowledges the National Natural Science Foundation of China (11831010, 61961160732), the Natural Science Foundation of Shandong Province (ZR2019ZD42), and the Taishan Scholars Climbing Program of Shandong (TSPD20210302).
© The authors. Published by EDP Sciences, SMAI 2022
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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