BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to the study of the differentiability of solutions to real-valued backward stochastic differential equations (BSDEs for short) with quadratic generators driven by a cylindrical Wiener process. The main novelty of this problem consists in the fact that the gradient equation of a quadratic BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO martingales. We show some applications to the nonlinear Kolmogorov equations.