Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment
Abstract
1. Introduction
2. The Model
3. Integral Differential Equations for
4. Sinc Asymptotic Analysis
4.1. Approximate Solution of
4.2. Error Analysis
5. Numerical Analysis
5.1. Error Analysis of a Special Case
5.2. Examples
5.2.1. The Exponential Distribution Case
5.2.2. The Lognormal Distribution Case
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Paulsen, J.; Gjessing, H.K. Ruin theory with stochastic return on investments. Adv. Appl. Probab. 1997, 29, 965–985. [Google Scholar] [CrossRef]
- Zhang, Z.; Yang, H.; Li, S. The perturbed compound Poisson risk model with two-sided jumps. J. Comput. Appl. Math. 2010, 233, 1773–1784. [Google Scholar] [CrossRef]
- Zhang, Z.; Han, X. The compound Poisson risk model under a mixed dividend strategy. Appl. Math. Comput. 2017, 315, 1–12. [Google Scholar] [CrossRef]
- Liu, Z.; Chen, P.; Hu, Y. On the dual risk model with diffusion under a mixed dividend strategy. Appl. Math. Comput. 2020, 376, 125115. [Google Scholar] [CrossRef]
- Gerber, H.U.; Shiu, E.S. On the time value of ruin. N. Am. Actuar. J. 1998, 2, 48–72. [Google Scholar] [CrossRef]
- Lin, X.S.; Pavlova, K.P. The compound Poisson risk model with a threshold dividend strategy. Insur. Math. Econ. 2006, 38, 57–80. [Google Scholar] [CrossRef]
- Martín-González, E.M.; Murillo-Salas, A.; Pantí, H. Gerber-Shiu function for a class of Markov-modulated Lévy risk processes with two-sided jumps. Methodol. Comput. Appl. Probab. 2022, 24, 2779–2800. [Google Scholar] [CrossRef]
- Yu, W.; Guo, P.; Wang, Q.; Guan, G.; Yang, Q.; Huang, Y.; Yu, X.; Jin, B.; Cui, C. On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model. Mathematics 2020, 8, 511. [Google Scholar] [CrossRef]
- Wang, C.; Xu, J.; Deng, N.; Wang, S. Two-sided jumps risk model with proportional investment and random observation periods. AIMS Math. 2023, 8, 22301–22318. [Google Scholar] [CrossRef]
- Cai, J.; Feng, R.; Willmot, G.E. Analysis of the compound Poisson surplus model with liquid reserves, interest and dividends. ASTIN Bull. 2009, 39, 225–247. [Google Scholar] [CrossRef]
- Peng, D.; Liu, D.; Hou, Z. Absolute ruin problems in a compound Poisson risk model with constant dividend barrier and liquid reserves. Adv. Differ. Equ. 2016, 2016, 1–15. [Google Scholar] [CrossRef]
- Zhang, Y.; Mao, L.; Kou, B. A Perturbed Risk Model with Liquid Reserves, Credit and Debit Interests and Dividends Under Absolute Ruin. In Advances in Computational Science and Computing; Springer: Berlin, Germany, 2019; Volume 877, pp. 340–350. [Google Scholar]
- Chen, X.; Ou, H. A compound Poisson risk model with proportional investment. J. Comput. Appl. Math. 2013, 242, 248–260. [Google Scholar] [CrossRef]
- Wang, C.; Wang, S.; Xu, J.; Li, S. Numerical method for a compound Poisson risk model with liquid reserves and proportional investment. AIMS Math. 2024, 9, 10893–10910. [Google Scholar] [CrossRef]
- Wan, N. Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion. Insur. Math. Econ. 2007, 40, 509–523. [Google Scholar] [CrossRef]
- Zhi, H.; Pu, J. On a dual risk model perturbed by diffusion with dividend threshold. Chin. Ann. Math. Ser. B 2016, 37, 777–792. [Google Scholar] [CrossRef]
- Yang, L.; He, C. Absolute ruin in the compound Poisson model with credit and debit interests and liquid reserves. Appl. Stoch. Model. Bus. Ind. 2014, 30, 157–171. [Google Scholar] [CrossRef]
- Lu, Y.; Li, Y. Dividend payments in a perturbed compound Poisson model with stochastic investment and debit interest. Ukr. Math. J. 2019, 71, 718–734. [Google Scholar] [CrossRef]
- Gao, S.; Liu, Z. The perturbed compound Poisson risk model with constant interest and a threshold dividend strategy. J. Comput. Appl. Math. 2010, 233, 2181–2188. [Google Scholar] [CrossRef]
- Stenger, F. Summary of sinc numerical methods. J. Comput. Appl. Math. 2000, 121, 379–420. [Google Scholar] [CrossRef]
- Wang, C.; Deng, N.; Shen, S. Numerical method for a perturbed risk model with proportional investment. Mathematics 2023, 11, 43. [Google Scholar] [CrossRef]
- Stenger, F. Handbook of Sinc Numerical Methods; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Stenger, F. Numerical Methods Based on Sinc and Analytic Functions; Springer: New York, NY, USA, 1993. [Google Scholar]
Existing Literature | Model | Penalty Function | Sinc | Error Analysis | ||
---|---|---|---|---|---|---|
Liquid Reserve | Investment | Threshold Strategy | ||||
Wang et al. [9] | ✓ | ✓ | ✓ | ✓ | ✓ | |
Wan [15] | ✓ | ✓ | ||||
Zhi and Pu [16] | ✓ | |||||
Peng et al. [11] | ✓ | ✓ | ✓ | ✓ | ||
Cai et al. [10] | ✓ | ✓ | ✓ | ✓ | ||
Yang and He [17] | ✓ | ✓ | ✓ | |||
Lu and Li [18] | ✓ | ✓ | ||||
Chen and Ou [13] | ✓ | ✓ | ✓ | ✓ | ||
Zhang et al. [12] | ✓ | ✓ | ✓ | ✓ | ||
Gao and Liu [19] | ✓ | ✓ | ✓ | |||
This paper | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
u | 2.00 | 2.05 | 2.10 | 2.15 | 2.20 | 2.25 | 2.30 | 2.35 | 2.40 |
ES | 0.4199 | 0.4109 | 0.4021 | 0.3935 | 0.3850 | 0.3767 | 0.3686 | 0.3607 | 0.3530 |
SA | 0.4055 | 0.3999 | 0.3945 | 0.3891 | 0.3838 | 0.3785 | 0.3734 | 0.3684 | 0.3634 |
RE(%) | −3.44 | −2.67 | −1.90 | −1.12 | −0.32 | 0.48 | 1.29 | 2.12 | 2.96 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Wang, C.; Wang, S.; Xu, J.; Li, S. Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment. Symmetry 2024, 16, 612. https://doi.org/10.3390/sym16050612
Wang C, Wang S, Xu J, Li S. Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment. Symmetry. 2024; 16(5):612. https://doi.org/10.3390/sym16050612
Chicago/Turabian StyleWang, Chunwei, Shujing Wang, Jiaen Xu, and Shaohua Li. 2024. "Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment" Symmetry 16, no. 5: 612. https://doi.org/10.3390/sym16050612
APA StyleWang, C., Wang, S., Xu, J., & Li, S. (2024). Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment. Symmetry, 16(5), 612. https://doi.org/10.3390/sym16050612