  We revisit a stochastic control problem of optimally modifying the underlying
spectrally negative Levy 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. We also obtain the convergence of the optimal refraction strategies
and the value functions, as the control set is enlarged, to those in the
relaxed case without the absolutely continuous assumption. Numerical results
are further given to confirm these analytical results.
