A Bismut Elworthy formula for quadratic BSDEs

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.