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-Elworthy 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 ▿v and final condition only bounded and continuous are given, as well as applications to stochastic optimal control problems with quadratic growth.