  We discuss BSDE with drivers containing nonlinearities of the type $p(y)|z|$
and $p(y)|z|^2$ with $p$ a polynomial of any degree. Sufficient conditions are
given for existence and uniqueness of solutions as well as comparison results.
We then connect the results to the Markovian FBSDE setting, discussing
applications in the theory of PDE perturbation and stating a result concerning
a large deviations principle for the first component of the solution to the
BSDE.
