On stochastic optimal control problems with state constraint

We consider optimal control problems with state constraint, where states X-t given as solutions of controlled stochastic differential equations are required to satisfy the constraint described either by the condition that Xt is an element of G ($) over bar for all t > 0 or by the condition that X-t is an element of G for all t > 0, with G being a given open subset of RN. Under suitable assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-Jacobi-Bellman equation, which is fully nonlinear degenerate second order elliptic equation, Lipschitz and Holder regularity results for the viscosity solution of the state constraint problem, and that the value functions V associated with the constraint G, V, of the relaxed problem associated with the constraint G, and Vg associated with the constraint G, satisfy in the viscosity sense the state constraint problem and hence are identified with its unique viscosity solution.