Indefinite Linear-Quadratic Mean-Field Game of Regime-Switching System

This paper studies an indefinite mean-field game with Markov jump parameters, where all agents' diffusion terms depend on control variables and both state and control average terms$(x^{(N)}, u^{(N)})$are considered. One notable aspect is the relaxation of the assumption regarding the positivity or non-negativity of weight matrices within costs, allowing for zero or even negative values. By virtue of mean-field methods and decomposition techniques, we have derived decentralized strategies presented by Hamiltonian systems and a new type of consistency condition system. These systems consist of fully coupled regime-switching forward-backward stochastic differential equations that do not conform to the Monotonicity condition. The well-posedness of these strategies is established by employing a relaxed compensator method with an easily verifiable Condition (RC) and the decomposition technique. Furthermore, we demonstrate that the resulting decentralized strategies achieve an$\epsilon$-Nash equilibrium in the indefinite case without any assumptions on admissible control sets using novel estimates of the disturbed state and cost function. Finally, our theoretical results are applied to resolve a class of mean-variance portfolio selection problems. We provide corresponding numerical simulation results and economic explanations.