We consider a boundary-noise and boundary-control problem for an equation of parabolic type in a bounded domain in R-d with a Neumann boundary condition. We study the Hamilton-Jacobi-Bellman equation associated to this problem and introduce an appropriate notion of viscosity solution. We prove the comparison principle for viscosity subsolutions and viscosity supersolutions of the Hamilton-Jacobi-Bellman equation. We also prove various continuity properties of the value function, show that it satisfies the dynamic programming principle, and prove that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.