Analytical Solutions of Insurer and Reinsurer Strategy with Environmental Noise under Logarithm Utility
Abstract
In this paper, the insurer and reinsurer’s strategy and the reinsurer’s surplus were studied in the presence of some random environmental noise on the risky asset under logarithm utility function. A portfolio with one risky and risk free asset was considered such that the risky asset follows the geometric Brownian motion (GBM). It was also assume that the claim process of the insurer is a stochastic differential equation, and the reinsurer can buy proportional reinsurance policy as a backup for their investment. The maximum principle theory and Ito’s lemma were used to derive our optimization problem. The Legendre transformation and dual theory with variable separation technique were used to solve the optimization problem under logarithm utility to obtain the optimal reinsurer strategy (ORS), optimal reinsurer policy (ORP) and the reinsurer’s surplus. More so, some numerical analyses were presented to discuss the effectof somesensitive parameters on the ORS and ORP. The ORS was observed to be a decreasing function of the instantaneous volatility, risk free interest rate but an increasing function of the appreciation rate of the of the risky asset. Furthermore, the relationship between the surplus process and time, risky asset and environmental noise was also given.
Keywords
Surplus processInsurer and reinsurance strategyLegendre transformsLogarithm
References
Adeosun,M.E.,Edeki,S.O.&Ugbebor,O.O.(2015).Stochastic Analysis of Stock Market Price Models: A case Study of the Nigerian Stock Exchange(NSE). WSEAS transactions on Mathematics, 14, pp. 353-363. Akpanibah, E. E. & Ini U. O. (2020). Portfolio strategy for an investor with logarithm utility and stochastic interest rate under constant elasticity of variance model. Journal of the Nigerian Society of Physical Sciences, 2(3), 186-196. Akpanibah E. E, &Ogunmodimu D. C. (2023). Mathematical analysis of stock prices prediction in a financial market using geometric Brownian Motion., 7, 7, pp. 78-84 Akpanibah E. E. & Osu B. O. (2017). Portfolio strategy for an investor with stochastic premium under exponential utility via legendre transform and dual theory. International Journal of Advances in Mathematics, 6, pp. 27-35. Akpanibah, E. E., Osu, B. O., Oruh, B. I. & Obi, C. N. (2019). Strategic optimal portfolio management for a DC pension scheme with return of premium clauses. Transaction of the Nigerian association of mathematical physics, 8, 1, pp. 121-130. Akpanibah, E. E. & Samaila, S. K. (2017). Stochastic strategies for optimal investment in a defined contribution (DC) pension fund. International Journal of Applied Science and Mathematical Theory, 3, 3, pp. 48-55. Amadi U. I., Ogbogbo, C. P. & Osu, B. O. (2022). Stochastic analysis of stock price changes as markov chain in finite states. Global journal of pure and applied sciences, 28, 1, pp. 91-98. Cao Y. & Wan N. (2009). Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman Chunxiang, A., Lai, Y & Shao, Y. (2018). Optimal excess-of-loss reinsurance and investment problem with delay and jump diffusion risk process under the CEV model, Journal of Computational and Applied Mathematics, 342, pp. 317-336. Davies, I., Amadi, I. U. &Ndu, R. I. (2019). Stability Analysis of Stochastic Model for Stock Market Prices. International Journal of Mathematics and Computational Methods, 4,7, pp. 9-86. Deng, C., Bian, W. & Wu, B. (2019). Optimal reinsurance and investment problem with default risk and bounded memory. International Journal of Control, 2019, pp. 1-113. Egbe, G. A, Awogbemi, C. A. & Osu, B. O. (2013). Portfolio optimization of pension fund contribution in Nigeria. Mathematical theory and Modelling, 3, 8, pp. 42-52. Gu, A., Guo, X., Li, Z. & Zeng, Y. (2012). Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model. Insurance: Mathematics and Economics, 51, 3, pp. 674-684. Ihedioha, S. A. (2015). Optimal Portfolios of an Insurer and a Reinsurer under Proportional Reinsurance and Power Utility Preference, Open Access Library Journal,, 2, 12, pp. 1- Ini U. O., Udoh N. A., Njoku, K. N. C. &Akpanibah, E. E. (2021). Mathematical modelling of an insurer's portfolio and reinsurance strategy under the CEV model and CRRA utility. Nigerian Journal of Mathematics and Applications, 31, pp. 38-56. Jonsson, M. & Sircar, R. (2002) Optimal Investment Problems and Volatility Homogenization Approximations. Modern Methods in Scientific Computing and Applications , 75, 255- Li, Q. and Gu, M. (2013). Optimization Problems of Excess-of-Loss Reinsurance and Investment under the CEV Model. Mathematical Analysis, 2013, pp. 1-10. Li, D., Rong, X. & Zhao, H. (2013). Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance model. Transaction on Mathematics, 12(3), 243-255. Li D., Rong X. & Zhao H. (2014). Optimal reinsurance-investment problem for maximizing the product of the insurer's and the reinsurer's utilities under a CEV model, Journal of Computational and Applied Mathematics, 255, pp. 671-683. Li D., Rong X. & Zhao H. (2015). Optimal investment problem for an insurer and a reinsurer, Journal of Systems Science and Complexity, 28, 6, pp. 1326-1343. Lin X. and Li Y. (2011). Optimal reinsurance and investment for a jump diffusion risk process under the CEV model, North American Actuarial Journal, 15, 3, pp. 417-431. Malik M., Abas S., Sukono M. & Prabowo A. (2020). Optimal reinsurance and investment strategy under CEV model with fractional power utility function, Engineering Letters, 28, 4, pp. 1-6. Mwanakatwe P. K., Wang X. & Su Y. (2019). Optimal investment and risk control strategies for an insurance fund in stochastic framework, Journal of Mathematical Finance, 9, 3, pp. 254-265. Nozadi A. E. (2014). Optimal constrained investment and reinsurance in lunberg insurance models, doctoral dissertation. Verlagnichtermittelbar. Osu, B. O., Akpanibah, E. E. & Njoku, K.N.C. (2017). On the Effect of Stochastic Extra Contribution on Optimal Investment Strategies for Stochastic Salary under the Affine Interest Rate Model in a DC Pension Fund, General Letters in Mathematics, 2, 3, pp. 138-149. Osu, B. O., Amadi I.U. & Azor, P.A. (2022). Analytical solution of time-varying investment returns with random parameters. Sebha University Journal of Pure & Applied Sciences, 21, 2, pp. 5-11. Osu B. O &Ihedioha S. A. (2013). Optimal investment for an insurer to minimize its probability of ruin. International journal of statistika and mathematika, 8, 1, pp. 42-50. Osu, B.O, Okoroafor, A.C. &Olunkwa, C. (2009). Stability Analysis of Stochastic model of Stock market price. African Journal of Mathematics and Computer Science, 2, 6, pp. 98-103. Sheng D. (2016). Explicit Solution of the Optimal Reinsurance-Investment Problem with Promotion Budget, Journal of Systems Science and Information, 4, 2, pp. 131-148. Sultan H. & Aqsa P. (2017). Wealth investment strategy for insurance company and probability of ruin, Iranian Journal of Science and Technology, Ttransaction A: Science 42, pp. 1555-1561. Wang, Y., Rong, X. & Zhao, H. (2018). Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, Journal of Computational and Applied Mathematics, 328, pp. 414-431. Xiao, H. Ren, T. Bai, Y. & Zhou, Z. (2019). Time-consistent investment reinsurance strategies for the insurer and the reinsurer under the generalized mean-variance criteria. Mathematics, 7, pp. 1-25.
More Articles from INTERNATIONAL JOURNAL OF APPLIED SCIENCES AND MATHEMATICAL THEORY
5-Point Block Backward Differentiation Formulae for Solving Stiff Chemical Reaction Problems of Ordinary Differential Equations
Author: Samson Yunusa, Imande Terdon Terlumun,, Etuk Emmanuel Dan, and Micah Michael
Analytical–Numerical Solution of a Temperature-Dependent Volterra Integro-Differential Model for Avian Influenza Transmission with Saturated Incidence and Environmental Feedback Via an Enhanced Adomian Decomposition Method
Author: Ejes, Valentine, Liberty Ebiwareme
Modelling Dengue Transmission Dynamics using a Two-Time Delay Differential Equation
Author: Uwakwe, J. I., Anyanwu, E., Inalegwu, N.
Intelligent Diagnosis and Dynamic Differentiated Instruction for High-School Mathematics Concepts Based on APOS Theory - Model Development, Implementation Pathway, And Validation Plan
Author: Xi Yang, Wenzhuo Zhang
Distribution-Free Error Control for Fraud Alert Triage: Conformal Anomaly P-Values Under Class Imbalance, Concept Drift, and Calibration Contamination
Author: Olamiji Onafowokan1, Olawale Fadugba2, Deborah Okunola1, Alex Mendy1