Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences
Abstract
:1. Introduction
2. Problem Formulation
2.1. Model Formulation
2.2. The Utility Maximization Problem
3. The Pure Reinsurance Problem
4. Reduction to an Optimal Stopping Problem
5. The Optimal Stopping Problem
- If the stopping region is not empty, that is , , we know that , hence , which implies and is optimal for problem (24).
- If the stopping region is not empty, for , we have that , otherwise, by continuity of both the functions, if (or ), the same inequality holds in a neighborhood of , which contradicts that , . Then, and is optimal for problem (24).
- If the continuation region is not empty, that is , , repeating the localization argument with the stopping time , we get
- Finally, for , by assumption, , , is optimal for problem (24) and this concludes the proof.
- 1.
- Ifthen and , so that , implying that .
- 2.
- Ifthen and ; in this case .
- 3.
- Ifthen and , so that .
- If , then .
- If , then there exists such that and .
- (i)
- When , we have that by Remark 3 and it easy to verify that H is increasing in , while it is decreasing in . Hence, it takes the maximum value at . As a consequence, if we have that , being .Otherwise, if there exists such that , that is , and , that is .
- (ii)
- When , by Lemma 2 we get that H is increasing in and we can repeat the same arguments as in the previous case to distinguish the two casese and , obtaining the same results.
- (iii)
- When , by Remark 3, we know that H is decreasing in , so that , that is , . Moreover, in this case, .
- If , then .
- If , then , where is the unique solution to equation
- (1)
- If , then the continuation region is , the value function is
- (2)
- If , then , where is the unique solution to , the value function is
6. Solution to the Original Problem
- (1)
- If , then , that is no reinsurance is purchased.
- (2)
- If , then , that is the optimal choice for the insurer consists in stipulating the contract at the initial time, selecting the optimal retention level (as in the pure reinsurance problem).
- If , then
7. Numerical Simulations
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Brachetta, M.; Ceci, C. Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences. Mathematics 2021, 9, 295. https://doi.org/10.3390/math9040295
Brachetta M, Ceci C. Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences. Mathematics. 2021; 9(4):295. https://doi.org/10.3390/math9040295
Chicago/Turabian StyleBrachetta, Matteo, and Claudia Ceci. 2021. "Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences" Mathematics 9, no. 4: 295. https://doi.org/10.3390/math9040295
APA StyleBrachetta, M., & Ceci, C. (2021). Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences. Mathematics, 9(4), 295. https://doi.org/10.3390/math9040295