**7. Concluding Remarks**

In this study, we proved that the optimal strategy for the bail-out dividend problem with fixed transaction costs is given by a reflected (*<sup>c</sup>*1, *<sup>c</sup>*2)-policy. We also characterized the optimal thresholds and gave a semi-explicit form for the value function in terms of the scale functions. In addition, we used the previous results to solve the constrained dividend maximization problem with the restriction that the expected present value of the capital injected is bounded by a given constant. The solution of the constrained problem can provide the insurance company with a guideline to maximize the profits of the shareholders taking into account the risk of bail-out losses.

It is a legitimate and interesting question whether the optimal strategy and the associated value function with transaction costs (i.e., *δ* > 0) converge to the corresponding optimal strategy and its value function without transaction cost as *δ* ↓ 0. Although we conjecture that indeed this is the case, further investigation is needed.

Another interesting generalization would involve considering fixed and proportional costs for the capital injection as well. We conjecture that in this case the optimal strategy would consist in a double band strategy, that is, a band strategy similar to the dividend paymen<sup>t</sup> strategy given in this paper, and a band strategy for the capital injection, which consists in pushing the process to a positive level each time the surplus process tries to cross below 0. We leave this problem as an opportunity for future research.

**Author Contributions:** All authors equally contributed to this paper.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank the anonymous reviewers for their comments and suggestions, which helped to improve significantly the quality of this paper. They would also like to thank Florin Avram who encouraged them to contribute to this special issue of *Risks*.

**Conflicts of Interest:** The authors declare no conflict of interest.
