Lipschitz continuity and semiconcavity properties of the value function of a stochastic control problem

We investigate the Cauchy problem for a nonlinear parabolic partial differential equation of Hamilton–Jacobi–Bellman type and prove some regularity results, such as Lipschitz continuity and semiconcavity, for its unique viscosity solution. Our method is based on the possibility of representing such a solution as the value function of the associated stochastic optimal control problem. The main feature of our result is the fact that the solution is shown to be jointly regular in space and time without any strong ellipticity assumption on the Hamilton–Jacobi–Bellman equation.