  We revisit an absolutely-continuous version of the stochastic control problem
driven by a L\'evy process. A strategy must be absolutely continuous with
respect to the Lebesgue measure and the running cost function is assumed to be
convex. We show the optimality of a refraction strategy, which adjusts the
drift of the state process at a constant rate whenever it surpasses a certain
threshold. The optimality holds for a general L\'evy process, generalizing the
spectrally negative case presented in Hern\'andez-Hern\'andez et al.(2016).
