MCRF
A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon
Jianhui Huang Xun Li Jiongmin Yong
Mathematical Control & Related Fields 2015, 5(1): 97-139 doi: 10.3934/mcrf.2015.5.97
A linear-quadratic (LQ, for short) optimal control problem is considered for mean-field stochastic differential equations with constant coefficients in an infinite horizon. The stabilizability of the control system is studied followed by the discussion of the well-posedness of the LQ problem. The optimal control can be expressed as a linear state feedback involving the state and its mean, through the solutions of two algebraic Riccati equations. The solvability of such kind of Riccati equations is investigated by means of semi-definite programming method.
keywords: MF-stabilizability Riccati equation. Mean-field stochastic differential equation linear-quadratic optimal control
MCRF
Linear quadratic mean-field-game of backward stochastic differential systems
Kai Du Jianhui Huang Zhen Wu
Mathematical Control & Related Fields 2018, 8(3&4): 653-678 doi: 10.3934/mcrf.2018028

This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

keywords: Backward mean-field game (BMFG) \begin{document}$\epsilon$\end{document}-Nash equilibrium Hamiltonian-type consistency condition (HCC) Riccati-type consistency condition (RCC) backward stochastic differential equation (BSDE)

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