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

Optimizing dividends and capital injections limited by bankruptcy, and practical approximations for the Cram\'er-Lundberg process