Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model
We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absol...
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MDPI AG
2021-04-01
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| author | Florin Avram Dan Goreac Juan Li Xiaochi Wu |
| author_facet | Florin Avram Dan Goreac Juan Li Xiaochi Wu |
| author_sort | Florin Avram |
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| description | We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>−</mo><mi>a</mi><mo>,</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula> policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit <i>a</i>, and only pay dividends when the reserve reaches an upper barrier <i>b</i>. An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) or Shreve-Lehoczy-Gaver (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition Løkka-Zervos (2008). In the first case, companies start paying dividends at the barrier <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>b</mi><mo>*</mo></msup><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, while in the second they must wait for reserves to build up to some (fully determined) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>b</mi><mo>*</mo></msup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> before paying dividends. |
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| spelling | doaj.art-413ea6e8e00a47a2bcd01454cc9b728c2023-11-21T16:38:44ZengMDPI AGMathematics2227-73902021-04-019993110.3390/math9090931Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg ModelFlorin Avram0Dan Goreac1Juan Li2Xiaochi Wu3Laboratoire de Mathématiques Appliquées, Université de Pau, F-64012 Pau, FranceSchool of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, ChinaSchool of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, ChinaSchool of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, ChinaWe investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>−</mo><mi>a</mi><mo>,</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula> policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit <i>a</i>, and only pay dividends when the reserve reaches an upper barrier <i>b</i>. An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) or Shreve-Lehoczy-Gaver (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition Løkka-Zervos (2008). In the first case, companies start paying dividends at the barrier <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>b</mi><mo>*</mo></msup><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, while in the second they must wait for reserves to build up to some (fully determined) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>b</mi><mo>*</mo></msup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> before paying dividends.https://www.mdpi.com/2227-7390/9/9/931optimal dividendscapital injectionsCramér-Lundberg modelbuffered reflectionscale functionsabsolutely continuous supersolutions |
| spellingShingle | Florin Avram Dan Goreac Juan Li Xiaochi Wu Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model Mathematics optimal dividends capital injections Cramér-Lundberg model buffered reflection scale functions absolutely continuous supersolutions |
| title | Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model |
| title_full | Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model |
| title_fullStr | Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model |
| title_full_unstemmed | Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model |
| title_short | Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model |
| title_sort | equity cost induced dichotomy for optimal dividends with capital injections in the cramer lundberg model |
| topic | optimal dividends capital injections Cramér-Lundberg model buffered reflection scale functions absolutely continuous supersolutions |
| url | https://www.mdpi.com/2227-7390/9/9/931 |
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