Partial regularity of semiconvex viscosity supersolutions to fully nonlinear elliptic HJB equations and applications to stochastic control
Salvatore Federico, Giorgio Ferrari, Mauro Rosestolato
TL;DR
This work establishes that locally semiconvex viscosity supersolutions to possibly degenerate fully nonlinear elliptic HJB equations are directionally differentiable along the range of the diffusion coefficient, $R(x) = \mathrm{Span}\{\mathrm{R}(\sigma(x,a)) : a \in A\}$. Using convex-analysis techniques and a contradiction argument, the authors derive that $v$ is differentiable along $R(x)$ with $D_{R(x)}v(x) = P_{R(x)}D^-v(x)$, and, under continuous variation of subspaces, that $D_Sv$ is continuous; in the nondegenerate case, $v$ is $C^1$. They apply these results to stochastic drift-control problems, showing that the value function $V$ is a viscosity (super)solution to the HJB equation and is locally semiconvex under standard assumptions, which yields directional differentiability and, when $R(x)=\mathbb{R}^n$, full $C^1$ regularity. The implications extend to optimal feedback synthesis, and to smooth-fit principles in optimal stopping and impulse control, providing a unified partial regularity framework even in degenerate settings and enabling classical verification theorems and policy characterizations. Overall, the paper links semiconvexity and viscosity supersolutions to partial regularity of value functions across drift-control, stopping, and impulse-control problems, with clear pathways to stronger regularity under hypoellipticity or nondegeneracy.
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.