Singular control problems appear in a large class of applications, like in queueing, mathematical finance and inventory systems. We revisit a stochastic control problem of optimally modifying the underlying spectrally negative Lévy process. A strategy must be absolutely continuous with respect to the Lebesgue measure, and the objective is to minimize the total costs of the running and controlling costs. Under the assumption that the running cost function is convex, we show the optimality of a refraction strategy. Convergence to the reflection strategy and the relation with singular control problems will be described. 
Joint work with J.L. Pérez and K. Yamazaki.