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

Salvatore Federico; Giorgio Ferrari; Mauro Rosestolato

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arXiv · 2501.15285

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

Salvatore Federico·Giorgio Ferrari·Mauro Rosestolato

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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.

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Salvatore Federico, Giorgio Ferrari, Mauro Rosestolato. 2025-01-25. Partial regularity of semiconvex viscosity supersolutions to fully nonlinear elliptic HJB equations and applications to stochastic control. https://arxiv.org/abs/2501.15285

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