  We prove the global-in-time well-posedness for a broad class of mean field
game problems, which is beyond the special linear-quadratic setting, as long as
the mean field sensitivity is not too large. Through the stochastic maximum
principle, we adopt the FBSDE approach to investigate the unique existence of
the corresponding equilibrium strategies. The corresponding FBSDEs are first
solved locally in time, then by controlling the sensitivity of the backward
solutions with respect to the initial condition via some suitable apriori
estimates for the corresponding Jacobian flows, the global-in-time solution is
warranted. Further analysis on these Jacobian flows will be discussed to
establish the regularities, such as linear functional differentiability, of the
respective value functions that leads to the ultimate classical well-posedness
of the master equation on $\mathbb{R}^d$. To the best of our knowledge, it is
the first article to deal with the mean field game problem, as well as its
associated master equation, with general cost functionals having quadratic
growth under the small mean field effect. In this current approach, we directly
impose the structural conditions on the cost functionals, rather than
conditions on the Hamiltonian. The advantages of this are threefold: (i)
compared with imposing conditions on Hamiltonian, the structural conditions
imposed in this work are easily verified, and less demanding on the regularity
requirements of the cost functionals while solving the master equation; (ii)
the displacement monotonicity is basically just a direct consequence of small
mean field effect in the structural conditions; and (iii) when the mean field
effect is not that small, we can still provide an accurate lifespan for the
local existence. The method in this work can be readily extended to the case
with nonlinear drift and non-separable cost functionals.
