A nonlinear Bismut–Elworthy formula for HJB equations with quadratic Hamiltonian in Banach spaces

We consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes (Y, Z), with generator with quadratic growth with respect to Z. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut–Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to Z. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations in Banach spaces for the unknown v, with nonlinear term with quadratic growth with respect to ∇ v and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth.