The Classical Stochastic Impulse Control Problem

TL;DR

This paper provides elementary proofs of regularity estimates for solutions to stochastic impulse control problems, including the classical obstacle problem, and extends these results to fully nonlinear operators and free boundary analysis.

Contribution

It offers a new, elementary proof of the sharp $C_{loc}^{1,1}$ regularity estimate and generalizes it to fully nonlinear operators, also deriving new free boundary regularity results.

Findings

Established sharp $C_{loc}^{1,1}$ regularity for solutions.

Extended regularity results to fully nonlinear operators.

Derived new estimates for the free boundary in classical cases.

Abstract

In this paper we study regularity estimates for the solution to an obstacle problem arising in stochastic impulse control theory. We prove using elementary methods the known sharp $C_{loc}^{1,1}$ estimate for the solution. The new proof is also easily generalizable to stochastic impulse control problems with fully noninear operators. Moreover we obtain new regularity estimates for the free boundary in the classical case.