Before revision:  We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as uniqueness of the solution to the Hamilton--Jacobi--Bellman equation, and study its qualitative properties both analytically and numerically .
After revision:  We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as uniqueness of the solution to the Hamilton--Jacobi--Bellman equation, and we provide an efficient and convergent numerical scheme for finding the solution .