An Optimal Multibarrier Strategy for a Singular Stochastic Control Problem with a State-Dependent Reward

We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish sufficient conditions for determining the optimality of the one-barrier strategy when the uncontrolled process X follows a spectrally negative Lévy process with a Lévy measure defined by a completely monotone density. Secondly, to verify the optimality of the \((2n+1)\) $(2n+1)$ -barrier strategy when X is a Brownian motion with a drift. Additionally, we provide an algorithm to compute the barrier values in the latter case.