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

TL;DR

This paper explores optimizing dividends and capital injections with bankruptcy constraints in risk models, providing practical approximations, extending existing results to include penalties, and employing Lambert-W functions for explicit solutions.

Contribution

It introduces practical approximations for general models based on exponential cases, extends the control problem to include penalties, and utilizes Lambert-W functions for explicit solutions.

Findings

Approximate solutions using exponential case models.

Extension of control problem to include penalties.

Explicit solutions employing Lambert-W functions.

Abstract

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.