JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (5): 49-57.doi: 10.6040/j.issn.1671-9352.0.2016.244

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Diffusion approximation and optimal investment for modern risk model

ZHANG Jie-song   

  1. School of Management, Huaibei Normal University, Huaibei 235000, Anhui, China
  • Received:2016-06-07 Online:2017-05-20 Published:2017-05-15

Abstract: Based on the modern risk model based on the policy entry process, this paper describes the surplus process of the insurance company, and proves that the risk process can be approximated to the diffusion process with time-varying drift rate by using the martingale central limit theorem. On this basis, it is assumed that the insurers buy a fixed proportion of reinsurance and invest in the Black-Scholes financial market and study the investment decision-making problem that minimizes the probability of ruin. The explicit expression of the optimal strategy and the value function is obtained by using the dynamic programming principle.

Key words: policy entrance process, martingale central limit theorem, diffusion approximation, optimal investment

CLC Number: 

  • F840.32
[1] GRANDEII J. Aspects of risk theory[M]. Berlin: Springer, 1991.
[2] KASS R, GOOVAERTS M, DHAENE J, et al. Modern actuarial risk theory[M]. Dordrecht: Kluwer Academic Publishers, 2001.
[3] 白建明, 尹晓玲. 小额索赔情形下现代风险模型的破产概率上界[J]. 系统工程学报, 2015, 30(001): 86-93. BAI Jianming, YIN Xiaoling. Upper bound of ruin probability for modern risk model with small claim condition[J]. Journal of Systems Engineering, 2015, 30(001): 86-93.
[4] LI Z H, ZHU J X, CHEN F. Study of a risk model based on the entrance process[J]. Statistics & probability letters, 2005, 72(1): 1-10.
[5] XIAO H M, LI Z H, LIU W W. The limit behavior of a risk model based on entrance processes[J]. Computers & Mathematics With Applications, 2008, 56(5): 1434-1440.
[6] XIAO H M, LI Z H, LIU W W. The finite time ruin probability of a new risk model based on entrance process[J]. Communications in Statistics-Theory and Methods, 2013, 42(2): 336-345.
[7] CHEN F, ZHU J X, LI Z H. Upper bounds for the ruin probabilities of the entrance-based risk model[J]. Communications in Statistics-Theory and Methods, 2008, 37(16): 2634-2652.
[8] 肖鸿民, 白建明. 重尾索赔条件下基于进入过程的保险风险模型的破产概率[J]. 山东大学学报: 理学版, 2010, 45(10): 122-126. XIAO Hongmin, BAI Jianming. Properties of ruin probability for a risk model based on the policy entrance process under heavily-tailed claims[J]. Journal of Shandong University(Natural Science), 2010, 45(10): 122-126.
[9] 唐风琴, 李泽慧, 陈进源. 一类基于进入过程的风险模型的精细大偏差[J]. 数学物理学报, 2011, 31(3): 737-751. TANG Fengqin, LI Zehui, CHEN Jinyuan. The precise large deviations for a risk model based on the policy entrance process[J]. Acta Mathematica Scientia, 2011, 31(3): 737-751.
[10] LI Z H, KONG X B. A new risk model based on policy entrance process and its weak convergence properties [J]. Applied Stochastic Models in Business and Industry, 2007, 23(3): 235-246.
[11] 王丽珍, 李静. 政策约束下基于风险调整报酬率的保险投资策略研究[J]. 中国管理科学, 2012, 20(1): 16-22. WANG Lizhen, LI Jing. Research on insurance portfolio selection under based on risk-adjusted return under the constraint of policy[J]. Chinese Journal of Management Science, 2012, 20(1): 16-22.
[12] 周明, 陈建成, 董洪斌. 风险调整资本收益率下的最优再保险策略[J]. 系统工程理论与实践, 2010, 30(11): 1931-1937. ZHOU Ming, CHEN Jiancheng, DONG Hongbin. Optimal reinsurance strategies under return on risk-adjusted capital rate[J]. Systems Engineering—Theory & Practice, 2010, 30(11): 1931-1937.
[13] 罗琰, 杨招军. 最小化破产概率的最优投资[J]. 管理科学学报, 2011, 14(5): 77-85. LUO Yan, YANG Zhaojun. Optimal investment for minimizing the probability of bankruptcy[J]. Journal of Management Science in China, 2011, 14(5): 77-85.
[14] LIU C S, YANG H L. Optimal investment for an insurer to minimize its probability of ruin[J]. North American Actuarial Journal, 2004, 8(2): 11-31.
[15] IGLEHART D L. Diffusion approximations in collective risk theory[J]. Journal of Applied Probability, 1969, 6(2): 285-292.
[16] 陈树敏, 李仲飞. 保险公司实业项目投资策略研究[J]. 系统科学与数学, 2010, 30(10): 1293-1303. CHEN Shumin, LI Zhongfei. The optimal policy for insurance company with real investment[J]. Journal of Systems Science and Mathematical Sciences, 2010, 30(10): 1293-1303.
[17] SCHMIDLI H. Optimal proportional reinsurance policies in a dynamic setting[J]. Scandinavian Actuarial Journal, 2001, 2001(1): 55-68.
[18] PROMISLOW D, YOUNG V R. Minimizing the probability of ruin when claims follow Brownian motion with drift[J]. North American Actuarial Journal, 2005, 9(3): 110-128.
[19] BI X C, ZHANG S G. Minimizing the risk of absolute ruin under a diffusion approximation model with reinsurance and investment[J]. Journal of Systems Science and Complexity, 2015, 28(1): 144-155.
[20] BAI L H, CAI J, ZHOU M. Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting[J]. Insurance: Mathematics and Economics, 2013, 53(3): 664-670.
[21] ASMUSSEN S, ALBRECHER H. Ruin probabilities[M]. London: World Scientific, 2010.
[22] PHILIP P. Stochastic integration and differential equations [M]. 2nd ed. Berlin: Springer-Verlag, 2005.
[23] FLEMING W H, SONER H M. Controlled Markov processes and viscosity solutions[M]. Netherlands:Springer Science & Business Media, 2006.
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