Partial regularity of semiconvex viscosity supersolutions to fully nonlinear elliptic HJB equations and applications to stochastic control

TL;DR

This paper proves partial regularity of semiconvex viscosity supersolutions to fully nonlinear elliptic HJB equations, with implications for stochastic control, including feedback control construction and value function differentiability.

Contribution

It establishes differentiability properties of supersolutions along certain directions, extending regularity results and applications in stochastic control problems.

Findings

Supersolutions are differentiable along the range of the second-order coefficient.

Provides a method to construct classical feedback controls in drift-control problems.

Shows value functions are differentiable when the second-order term is nondegenerate.

Abstract

In this note, we demonstrate that a locally semiconvex viscosity supersolution to a possibly degenerate fully nonlinear elliptic Hamilton-Jacobi-Bellman (HJB) equation is differentiable along the directions spanned by the range of the coefficient associated with the second-order term. The proof leverages techniques from convex analysis combined with a contradiction argument. This result has significant implications for various stationary stochastic control problems. In the context of drift-control problems, it provides a pathway to construct a candidate optimal feedback control in the classical sense and establish a verification theorem. Furthermore, in optimal stopping and impulse control problems, when the second-order term is nondegenerate, the value function of the problem is shown to be differentiable.