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

A Semi-Linear Backward Parabolic cauchy Problem with Unbounded Coefficients of Hamilton-Jacobi-Bellman Type and Applications to optimal control