  The purpose of this article is to prove existence, uniqueness and uniform
gradient estimates for unbounded classical solutions of a
Hamilton-Jacobi-Bellman equation. Such an equation naturally arises in
stochastic control problems. Contrary to the classical literature which handles
the case of bounded regular coefficients, we only impose Lipschitz regularity
conditions, allowing for a linear growth of coefficients. This latter
regularity assumption is natural in a probabilistic setting. In principle, this
assumption is compatible with global Lipschitz regularity for the solution.
However, to the best of our knowledge, this useful result had not been
established before. The proof that we provide relies on classical methods from
the viscosity solution theory, combining the Ishii-Lions method [IL90] for
uniformly elliptic equations with ideas from the weak Bernstein method [Bar91].
