Linear‐Quadratic Mean‐Field Game of Mean‐Field Type Regime‐Switching Backward System

ABSTRACT This paper investigates linear‐quadratic mean‐field games for a weakly‐coupled backward large‐population system, where each individual's state is governed by a backward stochastic differential equation with Markov chain and a prescribed terminal condition. Using the Nash certainty equivalence principle and the stochastic maximum principle, we construct an auxiliary control problem and derive the decentralized strategies. The limiting process is governed by the consistency condition, represented as a fully coupled conditional mean‐field forward‐backward stochastic differential equation with Markov chain, whose well‐posedness is established by the method of continuation and the decoupling method, respectively. Furthermore, we demonstrate that the set of decentralized strategies constitutes an ‐Nash equilibrium. As an illustration, a network security problem is investigated and numerical results are presented.