Partial Regularity of Semiconvex Viscosity Supersolutions to Fully Nonlinear Elliptic HJB Equations and Applications to Stochastic Control

In this note, we show that a locally semiconvex viscosity supersolution of a possibly degenerate fully nonlinear elliptic Hamilton–Jacobi–Bellman equation is differentiable along the directions spanned by the range of the coefficient matrix associated with the second-order term. The proof combines tools from convex analysis with a contradiction argument. This result has significant implications for stationary stochastic control problems. In drift-control problems, it enables the construction of a candidate optimal feedback control in the classical sense and the establishment of a verification theorem. Moreover, in optimal stopping and impulse control problems, when the second-order term is nondegenerate, the value function is shown to be continuously differentiable.