Indefinite Linear-Quadratic Partially Observed Mean-Field Game

This paper investigates an indefinite linear-quadratic partially observed mean-field game with common noise, incorporating both state-average and control-average effects. In our model, each agent’s state is observed through both individual and public observations, which are modeled as general stochastic processes rather than Brownian motions. It is noteworthy that the weighting matrices in the cost functional are allowed to be indefinite. We derive the optimal decentralized strategies using the Hamiltonian approach and establish the well-posedness of the resulting Hamiltonian system by employing a relaxed compensator. The associated consistency condition and the feedback representation of decentralized strategies are also established. Furthermore, we demonstrate that the set of decentralized strategies forms an [Formula: see text]-Nash equilibrium. As an application, we solve a mean-variance portfolio selection problem. Funding: This work was supported by National Key Research and Development Program of China [Grant 2023YFA1009200], the National Natural Science Foundation of China [Grants 12371447, 12022108, 12521001, 11971267, 62561160159, 12401583, and 12571587], the Shandong Provincial Natural Science Foundation [Grants ZR2022JQ01 and ZR2024MA095], the Basic Research Program of Jiangsu [Grant BK20240416], the Taishan Scholars Climbing Program of Shandong [Grant TSPD20210302], the Distinguished Young Scholars Program of Shandong University, and the Shandong Provincial Key Laboratory of Stochastic System Control and Scientific Computing.