  We consider a backward stochastic differential equation in a Markovian
framework for the pair of processes $(Y,Z)$, with generator with quadratic
growth with respect to $Z$. Under non-degeneracy assumptions, we prove an
analogue of the well-known Bismut-Elworty formula when the generator has
quadratic growth with respect to $Z$. Applications to the solution of a
semilinear Kolmogorov equation for the unknown $v$ with nonlinear term with
quadratic growth with respect to $\nabla v$ and final condition only bounded
and continuous are given, as well as applications to stochastic optimal control
problems with quadratic growth.
