Abstract
Numerous researchers have applied the martingale approach for models driven by Lévy processes to study optimal investment problems. This paper considers an insurer who wants to maximize the expected utility of terminal wealth by selecting optimal investment and proportional reinsurance strategies. The insurer’s risk process is modeled by a Lévy process and the capital can be invested in a security market described by the standard Black-Scholes model. By the martingale approach, the closed-form solutions to the problems of expected utility maximization are derived. Numerical examples are presented to show the impact of model parameters on the optimal strategies.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S Browne. Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 1995, 20(5): 937–958.
C Hipp, M Plum. Optimal investment for insurers, Insurance: Mathematics and Economics, 2000, 27(2): 215–228.
H Schmidli. On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 2002, 12(3): 890–907.
C S Liu, H L Yang. Optimal investment for an insurer to minimize its probability of ruin, North American Acturial Journal, 2004, 8(2): 11–31.
S E Shreve. Stochastic Calculus for Finance: Continuous-Time Model, Berlin: Spring-Verlag, 2004.
H L Yang, L H Zhang. Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 2005, 37(3): 617–634.
Z W Wang, J M Xia, L H Zhang. Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, 2007, 40(2): 322–334.
I Karatzas, J P Lehoczky, S E Shreve, G L Xu. Martingale and duality methods for utility maximization in incomplete markets, SIAM Journal on Control and Optimization, 1991, 29(3): 702–730.
M D Gu, Y P Yang, J Y Zhang. Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance: Mathematics and Economics, 2010, 46(3): 580–587.
B Zou, A Cadenillas. Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance: Mathematics and Economics, 2014, 58: 57–67.
W J Guo. Optimal portfolio choice for an insurer with loss aversion, Insurance: Mathematics and Economics, 2014, 58: 217–222.
Z B Liang, K C Yuen, K C Cheung. Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Applied Stochastic Models in Business and Industry, 2011, 28(6): 585–597.
Y Zeng, Z F Li. Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance: Mathematics and Economics, 2011, 49(1): 145–154.
N Bäuerle. Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 2005, 62: 159–165.
P Chen, S C P Yam. Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers, Insurance: Mathematics and Economics, 2013, 53(3): 871–883.
L H Bai, J Y Guo. Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 2008, 42(3): 968–975.
B Øksendal, A Sulem. Applied Stochastic Control of Jump Diffusions, New York: In: Universitext, Springer-Verlag, Heidelberg, 2005.
R Cont, P Tankov. Financial Modelling With Jump Processes, In: Chapman and Hall/CRC Financial Mathematics Series, 2003.
Acknowledgement
The authors thank the reviewers for their comments and suggestions, which have greatly helped in the better presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (71471081), Teaching Reform Project of Nanjing University of Finance and Economics (JGY034) and Degree and Graduate Education Project of Nanjing University of Finance and Economics (Y18005).
Rights and permissions
About this article
Cite this article
Liu, Ss., Guo, Wj. & Tong, Xl. Martingale method for optimal investment and proportional reinsurance. Appl. Math. J. Chin. Univ. 36, 16–30 (2021). https://doi.org/10.1007/s11766-021-3463-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-021-3463-8