On Singular Control for Lévy Processes

We revisit the classical singular control problem of minimizing running and controlling costs. Existing studies have shown the optimality of a barrier strategy when driven by Brownian motion or Lévy processes with one-sided jumps. Under the assumption that the running cost function is convex, we show the optimality of a barrier strategy for a general class of Lévy processes. Funding: This work was supported by the Japan Society for the Promotion of Science [Grants 18J12680, 19H01791, 20K035758, 21K13807, and JPJSBP120209921] and a University of Queensland start-up grant.