  The recent papers Gajek-Kucinsky(2017) and Avram-Goreac-Li-Wu(2020)
investigated the control problem of optimizing dividends when limiting capital
injections stopped upon bankruptcy. The first paper works under the spectrally
negative L\'evy model; the second works under the Cram\'er-Lundberg model with
exponential jumps, where the results are considerably more explicit. The
current paper has three purposes. First, it illustrates the fact that quite
reasonable approximations of the general problem may be obtained using the
particular exponential case studied in Avram-Goreac-Li-Wu(2020). Secondly, it
extends the results to the case when a final penalty $P$ is taken into
consideration as well besides a proportional cost $k>1$ for capital injections.
This requires amending the "scale and Gerber-Shiu functions" already introduced
in Gajek-Kucinsky(2017). Thirdly, in the exponential case, the results will be
made even more explicit by employing the Lambert-W function. This tool has
particular importance in computational aspects and can be employed in
theoretical aspects such as asymptotics.
