  We obtain an existence and uniqueness theorem for fully coupled
forward-backward SDEs (FBSDEs) with jumps via the classical solution to the
associated quasilinear parabolic partial integro-differential equation (PIDE),
and provide the explicit form of the FBSDE solution. Moreover, we embed the
associated PIDE into a suitable class of non-local quasilinear parabolic PDEs
which allows us to extend the methodology of Ladyzhenskaya et al (1968) to
non-local PDEs of this class. Namely, we obtain the existence and uniqueness of
a classical solution to both the Cauchy problem and the initial-boundary value
problem for non-local quasilinear parabolic second-order PDEs.
