Sensitivity analysis for HJB equations with an application to a coupled backward-forward system

In this paper, we analyse the dependence of the solution of Hamilton-Jacobi-Bellman equations on a functional parameter. This sensitivity analysis not only has the interest on its own, but also is important for the mean field games methodology, namely for solving a coupled backward-forward system. We show that the unique solution of a Hamilton-Jacobi-Bellman equation and its spacial gradient are Lipschitz continuous uniformly with respect to a functional parameter. In particular, we provide verifiable criteria for the so-called feedback regularity condition.