Stochastic optimal control problems and parabolic equations in Banach spaces

We consider stochastic optimal control problems in Banach spaces. These problems are related to nonlinear controlled equations with dissipative nonlinearity and are treated via the backward stochastic differential equation approach, which also allows us to solve, in a mild sense, Hamilton-Jacobi-Bellman equations in Banach spaces. We apply the results to controlled stochastic heat and wave equations with a cost functional that is well defined on continuous functions, and to delay equations in spaces of p-integrable functions.