Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments
In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black⁻Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate an...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-02-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/24/1/21 |
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| author | Christian Kasumo |
| author_facet | Christian Kasumo |
| author_sort | Christian Kasumo |
| collection | DOAJ |
| description | In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black⁻Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton⁻Jacobi⁻Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed individual claim amount distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price. |
| first_indexed | 2024-04-11T22:55:59Z |
| format | Article |
| id | doaj.art-ff8b8391804d448ca2b3b0201f948a24 |
| institution | Directory Open Access Journal |
| issn | 2297-8747 |
| language | English |
| last_indexed | 2024-04-11T22:55:59Z |
| publishDate | 2019-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-ff8b8391804d448ca2b3b0201f948a242022-12-22T03:58:25ZengMDPI AGMathematical and Computational Applications2297-87472019-02-012412110.3390/mca24010021mca24010021Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and InvestmentsChristian Kasumo0Department of Science and Mathematics, School of Science, Engineering and Technology, Mulungushi University, P.O. Box 80415 Kabwe, ZambiaIn this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black⁻Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton⁻Jacobi⁻Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed individual claim amount distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.https://www.mdpi.com/2297-8747/24/1/21ruin probabilityjump-diffusionHJB equationVolterra equationblock-by-block methodproportional reinsuranceinvestments |
| spellingShingle | Christian Kasumo Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments Mathematical and Computational Applications ruin probability jump-diffusion HJB equation Volterra equation block-by-block method proportional reinsurance investments |
| title | Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments |
| title_full | Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments |
| title_fullStr | Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments |
| title_full_unstemmed | Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments |
| title_short | Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments |
| title_sort | minimizing an insurer s ultimate ruin probability by reinsurance and investments |
| topic | ruin probability jump-diffusion HJB equation Volterra equation block-by-block method proportional reinsurance investments |
| url | https://www.mdpi.com/2297-8747/24/1/21 |
| work_keys_str_mv | AT christiankasumo minimizinganinsurersultimateruinprobabilitybyreinsuranceandinvestments |