Existence and global Lipschitz estimates for unbounded classical solutions of a Hamilton–Jacobi equation

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. These Lipschitz assumptions are natural in a probabilistic setting. In principle, these assumptions are compatible with global Lipschitz regularity for the solution. However, to the best of our knowledge, this useful result had not been established before. Our proofs rely on the Ishii–Lions method [36]. We combine several elements from the viscosity solution theory to obtain estimates at the edges of what seems possible.