  We prove the well-posedness results, i.e. existence, uniqueness, and
stability, of the solutions to a class of nonlocal fully nonlinear parabolic
partial differential equations (PDEs), where there is an external time
parameter $t$ on top of the temporal and spatial variables $(s,y)$ and thus the
problem could be considered as a flow of equations. The nonlocality comes from
the dependence on the unknown function and its first- and second-order
derivatives evaluated at not only the local point $(t,s,y)$ but also at the
diagonal line of the time domain $(s,s,y)$. Such equations arise from
time-inconsistent problems in game theory or behavioural economics, where the
observations and preferences are (reference-)time-dependent. To address the
open problem of the well-posedness of the corresponding nonlocal PDEs (or the
time-inconsistent problems), we first study the linearized version of the
nonlocal PDEs with an innovative construction of appropriate norms and Banach
spaces and contraction mappings over which. With fixed-point arguments, we
obtain the well-posedness of nonlocal linear PDEs and establish a Schauder-type
prior estimate for the solutions. Then, by the linearization method, we
analogously establish the well-posedness under the fully nonlinear case.
Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic
PDE is an adapted solution to a flow of second-order forward-backward
stochastic differential equations.
