  We obtain weighted uniform estimates for the gradient of the solutions to a
class of linear parabolic Cauchy problems with unbounded coefficients. Such
estimates are then used to prove existence and uniqueness of the mild solution
to a semi-linear backward parabolic Cauchy problem, where the differential
equation is the Hamilton-Jacobi-Bellman equation of a suitable optimal control
problem. Via backward stochastic differential equations, we show that the mild
solution is indeed the Value Function of the controlled equation and that the
feedback law is verified.
