Individual and mass behavior in large population forward–backward stochastic control problems: Centralized and Nash equilibrium solutions

Abstract

AbstractThis article studies a class of dynamic optimization of large‐population system in which a large number of negligible agents are coupled via state‐average in their cost functional and state dynamics. The most significant feature in our setup is the dynamics of individual agents are modeled by forward–backward stochastic differential equations. The associated mean‐field game, in its forward–backward sense, is also formulated to seek the decentralized strategies. Unlike the forward case, the consistency conditions of our forward–backward mean‐field games involve five Riccati and force rate equations. Moreover, their initial and terminal conditions are mixed thus some special decoupling technique is applied here. We also verify the ‐Nash equilibrium property of the derived decentralized strategies. To this end, some estimates to backward stochastic system are employed.