Optimal investment and reinsurance strategies for an insurer with stochastic economic factor

Abstract

This work considers optimal investment and reinsurance strategies for an insurer with stochastic economic factor. In our mathematical model, a risk-free asset and a risky asset are assumed to rely on a stochastic economic factor which is described by a diffusion process. We generalize the claim process to a compound Poisson process with the stochastic economic factor. Using expected utility maximization, we characterize the optimal strategy of investment-reinsurance under the power utility function. We use dynamic programming principle to derive the Hamilton–Jacobi–Bellman (HJB) equation. Then, by analysing the solution of the HJB equation, the optimal investment-reinsurance strategy is obtained and given in the verification theorem. Finally, sensitivity analysis is given to show the economic behavior of the optimal investment and reinsurance strategies.

Keywords

• Stochastic control
• investment-reinsurance strategy
• stochastic economic factor
• Lévy processes
• HJB equation

Supporting Institution

National Natural Science Foundation of China； Guangzhou University

Project Number

National Natural Science Foundation of China (Grant No. 61973096)； Guangzhou University (2021GDJC-D03)

Thanks

This work was supported in part by the National Natural Science Foundation of China (Grant No. 61973096) and the postgraduate innovative ability training program of Guangzhou University (2021GDJC-D03).

References

1. [1] D. Becherer and M. Schweizer, Classical solutions to reaction-diffusion systems for hedging problems with interacting Itô and point processes, Ann. Appl. Probab. 15 (2), 1111-1144, 2005.
2. [2] L. Bo and S.Wang, Optimal investment and risk control for an insurer with stochastic factor, Oper. Res. Lett. 45 (3), 259-265, 2017.
3. [3] M. Brachetta and C. Ceci, Optimal proportional reinsurance and investment for stochastic factor models, Insurance Math. Econom. 87, 15-33, 2019.
4. [4] M. Brachetta and C. Ceci, Optimal excess-of-loss reinsurance for stochastic factor risk models, Risks 7 (2), 48, 2019.
5. [5] C. Ceci, K. Colaneri and A. Cretarola, Optimal reinsurance and investment under common shock dependence between financial and actuarial markets, Insurance Math. Econom. 105, 252-278, 2022.
6. [6] J. Cao, D. Landriault and B. Li, Optimal reinsurance-investment strategy for a dynamic contagion claim model, Insurance Math. Econom. 93, 206-215, 2020.
7. [7] J.C. Cox and C.F. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory 49, 33-83, 1989.
8. [8] A.Y. Golubin, Optimal insurance and reinsurance policies in the risk process, Astin Bull. 38 (2), 383-397, 2008.

9. [9] G. Guan and Z. Liang, Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks, Insurance Math. Econom. 55, 105-115, 2014.
10. [10] J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Springer, Berlin, 1987.
11. [11] I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer, New York, 1998.
12. [12] R. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory 3, 373-413, 1971.
13. [13] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 2003.
14. [14] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, Berlin, 2005.
15. [15] H. Pham, Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints, Appl. Math. Optim. 46, 55-78, 2002.
16. [16] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J. 2001 (1), 55-68, 2001.
17. [17] Y. Shen and Y. Zeng, Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance Math. Econom. 62, 118-137, 2015.
18. [18] Z. Sun and J. Guo, Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility, Math. Methods Oper. Res. 88, 59-79, 2018.
19. [19] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math. 328, 414-431, 2018.
20. [20] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom. 37 (3), 615-634, 2005.
21. [21] T. Zariphopoulou, A solution approach to valuation with unhedgeable risks, Finance Stoch. 5, 61-82, 2001.