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

Salvatore Federico, Giorgio Ferrari and Mauro Rosestolato
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Salvatore Federico: Center for Mathematical Economics, Bielefeld University
Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University
Mauro Rosestolato: Center for Mathematical Economics, Bielefeld University

No 744, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

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 dif- ferentiable 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 con- tradiction 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.

Keywords: viscosity solution; semiconvexity; Hamilton-Jacobi-Bellman equation; stochastic control; optimal stopping; impulse stochastic control; feedback control; smooth-fit principle. (search for similar items in EconPapers)
Pages: 14
Date: 2025-08-18
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