Semismooth Potentials of Stochastic Systems With Degenerate Diffusions

We prove that for a number of optimal control problems, including finite horizon, long-run average, and optimal stopping, with one-dimensional degenerate diffusion processes, the potential function (solution to Poisson equation) and, hence, the value function (solution to HJB equation) are semismooth at the degenerate points (i.e., the left- and right-hand side derivatives exist but may not be equal). This allows applying the Ito-Tanaka formula in the direct-comparison-based optimization approach, and previous results on semismooth value functions depend heavily on this property. This result will facilitate further research in stochastic optimal control.