Stochastic Feedback Control With One-Dimensional Degenerate Diffusions and Nonsmooth Value Functions

We study the stochastic control problems with degenerate semimartingales and nonsmooth value functions. We first derive the optimality conditions for finite horizon problems with nonsmooth value functions and singular control of infinite horizon discounted problems. Then, we show that the local time at the degenerate point is zero, so if the value function has nonsmooth first-order derivatives at the degenerate points of all policies, the nonsmoothness can be ignored and the optimality equation for stochastic control, including optimal stopping, is simply the classical Hamilton–Jacobi–Bellman equation at the smooth points. Furthermore, we model the singular control by the reflecting force in the Skorokhod problem and show that there are two classes of degenerate points; in the first class, singular control at a degenerate point can be treated in the same way as regular control; and in the second class, the density of the singular control force may be of the order <inline-formula><tex-math notation="LaTeX">$(\sqrt{dt})^{1+\gamma }$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$0<\gamma <1$</tex-math></inline-formula>, with <inline-formula><tex-math notation="LaTeX">$\gamma =0$</tex-math></inline-formula> for nondegenerate points. We apply the direct-comparison-based approach to derive all the results. In all the analysis, viscosity solution is not needed.