  This paper studies a new class of dynamic optimization problems of
large-population (LP) system which consists of a large number of negligible and
coupled agents. The most significant feature in our setup is the dynamics of
individual agents follow the forward-backward stochastic differential equations
(FBSDEs) in which the forward and backward states are coupled at the terminal
time. This current paper is hence different to most existing large-population
literature where the individual states are typically modeled by the SDEs
including the forward state only. The associated mean-field
linear-quadratic-Gaussian (LQG) 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 six
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 $\epsilon$-Nash equilibrium property of the derived
decentralized strategies. To this end, some estimates to backward stochastic
system are employed. In addition, due to the adaptiveness requirement to
forward-backward system, our arguments here are not parallel to those in its
forward case.
