Before revision:  We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as uniqueness of the  Hamilton--Jacobi--Bellman equation, and study  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 Hamilton--Jacobi--Bellman equation, and study its qualitative properties both analytically and numerically.