A class of stochastic optimal control problems with state constraint

We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state X_t is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that X_t e Ḡ for all t > 0 or by the condition that X_t ∈ G for all t > 0, where G be a given open subset of R^N. Under the assumption that for each z ∈ ∂G there exists a_z ∈ A, where A denotes the control set, such that the diffusion matrix σ(x, a) vanishes for a = a_z and for x ∈ ∂G in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = a_z and x = z as well as some other mild assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-Jacobi-Bellman equation, prove that the value functions V associated with the constraint Ḡ, V_r of the relaxed problem associated with the constraint Ḡ, and V₀ associated with the constraint G, satisfy in the viscosity sense the state constraint problem, and establish Holder regularity results for the viscosity solution of the state constraint problem.