Existence of Solutions to the Hamilton–Jacobi–Bellman Equation under Quadratic Growth Conditions

In this paper we show the existence of solutions with quadratic growth to Hamilton–Jacobi–Bellman equations. We assume that the Hamiltonian has the quadratic growth both in x and p. We obtain the explicit Lipschitz bound of solutions in the weighted sup-norm. Also, we discuss the vanishing viscosity limit of solutions and show the existence of viscosity solutions to the resulting Hamilton–Jacobi equation.