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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 aregiven for existence and uniqueness of solutions as well as comparison results.We then connect the results to the Markovian FBSDE setting, discussingapplications in the theory of PDE perturbation and stating a result concerninga large deviations principle for the first component of the solution to theBSDE.

Quadratic FBSDE with generalized Burgers' type nonlinearities, perturbations and large deviations