Linear–Quadratic Mean-Field Game for Stochastic Delayed Systems

This paper studies the linear–quadratic mean-field game (MFG) for a class of stochastic delayed systems. We consider a large-population system, where the dynamics of each agent is modeled by a stochastic differential delayed equation. The consistency condition is derived through an auxiliary system, which is an anticipated forward–backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such an AFBSDDE system can be obtained using a continuation method. Thus, the MFG strategies can be defined on an arbitrary time horizon, not necessary on a small time horizon by a commonly used contraction mapping method. Moreover, the decentralized strategies are verified to satisfy the $\epsilon$ -Nash equilibrium property. For illustration, three special cases of delayed systems are further explored, for which the closed-loop and open-loop MFG strategies are derived, respectively.