Optimizing Dividends and Capital Injections Limited by Bankruptcy, and Practical Approximations for the Cramér-Lundberg Process

Florin Avram (), Dan Goreac, Rim Adenane and Ulyses Solon
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Florin Avram: Université de Pau
Dan Goreac: Shandong University
Rim Adenane: Université Ibn-Tofail
Ulyses Solon: Université de Pau

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 4, 2339-2371

Abstract: Abstract The recent papers Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017) and Avram et al. (Mathematics 9(9):931, 2021) cost induced dichotomy for optimal dividends in the cramr-lundberg model. Avram et al. (Mathematics 9(9):931, 2021) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-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 et al. cost induced dichotomy for optimal dividends in the Cramér-Lundberg model (Avram et al. in Mathematics 9(9):931, 2021). Secondly, it extends the results to the case when a final penalty P is taken into consideration as well besides a proportional cost $$k>1$$ k > 1 for capital injections. This requires amending the “scale and Gerber-Shiu functions” already introduced in Gajek and Kucinsky (Insur Math Econ 73:1–19, 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.

Keywords: Dividend problem; Capital injections; Penalty at default; Scale functions; Lambert-W function; De Vylder-type approximations; Rational Laplace transform (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11009-021-09916-z

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