On the Fragility of the Basis on the Hamilton-Jacobi-Bellman Equation in Economic Dynamics

The paper proves (i) a counterexample where the HJB has many classical solutions but the value function is not even a viscosity solution, and (ii) an equivalence/uniqueness theory under added assumptions (Theorems 1–3). The candidate solution reconstructs Theorems 2–3: from a classical HJB solution it builds a closed-loop optimal policy and shows uniqueness among concave nondecreasing classical solutions. The logical structure (Peano existence, linear growth comparison via an affine bound on f, verification with a transversality limit, and the bounding argument for uniqueness) closely matches the paper’s proofs. Minor issues in the candidate’s write-up include an over-strong claim that sup u(c)=+∞ (not implied by u′(R++)=R++) and an unproven boundary step asserting V′(0+)=∞ implies b(k)≥0 near 0. These are easily repaired by adopting the paper’s positivity and comparison arguments. Overall, both are correct; the proof strategies are substantially the same, with the paper being more careful on technical points.