On singular control for L\'evy processes

We revisit the classical singular control problem of minimizing running and controlling costs. The problem arises in inventory control, as well as in healthcare management and mathematical finance. Existing studies have shown the optimality of a barrier strategy when driven by the Brownian motion or L\'evy processes with one-side jumps. Under the assumption that the running cost function is convex, we show the optimality of a barrier strategy for a general class of L\'evy processes.