The Classical Stochastic Impulse Control Problem

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.