考虑时滞效应与均值-方差效用的非零和投资与再保险博弈

doi:10.15960/j.cnki.issn.1007-6093.2021.02.003

摘要/Abstract

摘要：

在考虑时滞效应的影响下研究了非零和随机微分投资与再保险博弈问题。以最大化终端绝对财富和相对财富的均值-方差效用为目标，构建了两个相互竞争的保险公司之间的非零和投资与再保险博弈模型，分别在经典风险模型和近似扩散风险模型下探讨了博弈的Nash均衡策略。借助随机控制理论以及相应的广义Hamilton-Jacobi-Bellman（HJB）方程，得到了均衡投资与再保险策略和值函数的显式表达。最后，通过数值例子分析了模型中相关参数变动对均衡策略的影响。

关键词: 投资与再保险, 非零和博弈, 时滞效应, 均值-方差效用, 广义Hamilton-Jacobi-Bellman方程

Abstract:

This paper investigates a non-zero-sum stochastic differential investment and reinsurance game with delay effect between two competitive insurers, who aim to maximize the mean-variance utility of his terminal wealth relative to that of his competitor. By applying stochastic control theory, corresponding extended Hamilton-Jacobi-Bellman (HJB) system of equations are established. Furthermore, closed-form expressions for the Nash equilibrium investment and reinsurance strategies and the corresponding value functions are derived both in the classical risk model and its diffusion approximation. Finally, some numerical examples are conducted to illustrate the influence of model parameters on the equilibrium investment and reinsurance strategies and draw some economic interpretations from these results.

Key words: investment and reinsurance, non-zero-sum game, delay effect, meanvariance utility, extended Hamilton-Jacobi-Bellman equation

中图分类号:

F830.59
F224

朱怀念, 钟慧, 宾宁. 考虑时滞效应与均值-方差效用的非零和投资与再保险博弈[J]. 运筹学学报, 2021, 25(2): 35-54.

Huainian ZHU, Hui ZHONG, Ning BIN. Non-zero-sum investment and reinsurance game with delay effect and mean-variance utility[J]. Operations Research Transactions, 2021, 25(2): 35-54.

图/表 5

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图4

参考文献 45

1	Meng H , Zhang X . Optimal risk control for the excess of loss reinsurance policies[J]. ASTIN Bulletin, 2010, 40 (1): 179- 197. doi: 10.2143/AST.40.1.2049224
2	Chen S , Li Z , Li K . Optimal investment-reinsurance policy for an insurance company with VaR constraint[J]. Insurance: Mathematics and Economics, 2010, 47 (2): 144- 153. doi: 10.1016/j.insmatheco.2010.06.002
3	Luo S . On proportional reinsurance with a linear transaction rate[J]. Risk and Decision Analysis, 2012, 3 (1-2): 115- 137. doi: 10.3233/RDA-2011-0034
4	Bai L , 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. doi: 10.1016/j.insmatheco.2013.09.008
5	Zhou M , Cai J . Optimal dynamic risk control for insurers with state-dependent income[J]. Journal of Applied Probability, 2014, 51 (2): 417- 435. doi: 10.1239/jap/1402578634
6	Zhang N , Jin Z , Li S , et al. Optimal reinsurance under dynamic VaR constraint[J]. Insurance: Mathematics and Economics, 2016, 71, 232- 243. doi: 10.1016/j.insmatheco.2016.09.011
7	Liang Z , Yuen K C , Guo J . Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process[J]. Insurance: Mathematics and Economics, 2011, 49 (2): 207- 215. doi: 10.1016/j.insmatheco.2011.04.005
8	Gu A , Guo X , Li Z , et al. Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model[J]. Insurance: Mathematics and Economics, 2012, 51 (3): 674- 684. doi: 10.1016/j.insmatheco.2012.09.003
9	Zhao H , Rong X , Zhao Y . Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model[J]. Insurance: Mathematics and Economics, 2013, 53 (3): 504- 514. doi: 10.1016/j.insmatheco.2013.08.004
10	Guan G , Liang Z . Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks[J]. Insurance: Mathematics and Economics, 2014, 55, 105- 115. doi: 10.1016/j.insmatheco.2014.01.007
11	Yuen K C , Liang Z , Zhou M . Optimal proportional reinsurance with common shock dependence[J]. Insurance: Mathematics and Economics, 2015, 64, 1- 13. doi: 10.1016/j.insmatheco.2015.04.009
12	Liang Z , Yuen K C . Optimal dynamic reinsurance with dependent risks: variance premium principle[J]. Scandinavian Actuarial Journal, 2016, 2016 (1): 18- 36. doi: 10.1080/03461238.2014.892899
13	李启才, 顾孟迪. 指数均值回复金融市场下的最优投资和最优再保险策略[J]. 管理工程学报, 2016, 30 (4): 79- 84.
14	Chen L , Qian L , Shen Y , et al. Constrained investment-reinsurance optimization with regime switching under variance premium principle[J]. Insurance: Mathematics and Economics, 2016, 71, 253- 267. doi: 10.1016/j.insmatheco.2016.09.009
15	Xu L , Zhang L , Yao D . Optimal investment and reinsurance for an insurer under Markovmodulated financial market[J]. Insurance: Mathematics and Economics, 2017, 74, 7- 19. doi: 10.1016/j.insmatheco.2017.02.005
16	Zeng Y , Li Z , Lai Y . Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps[J]. Insurance: Mathematics and Economics, 2013, 52 (3): 498- 507. doi: 10.1016/j.insmatheco.2013.02.007
17	Chen P , Yam S C P . Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers[J]. Insurance: Mathematics and Economics, 2013, 53 (3): 871- 883. doi: 10.1016/j.insmatheco.2013.10.004
18	Bi J , Meng Q , Zhang Y . Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer[J]. Annals of Operations Research, 2014, 212 (1): 43- 59. doi: 10.1007/s10479-013-1338-z
19	Lin X , Qian Y . Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model[J]. Scandinavian Actuarial Journal, 2016, 2016 (7): 646- 671. doi: 10.1080/03461238.2015.1048710
20	Li D , Zeng Y , Yang H . Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps[J]. Scandinavian Actuarial Journal, 2018, 2018 (2): 145- 171. doi: 10.1080/03461238.2017.1309679
21	Chang M H , Pang T , Yang Y . A stochastic portfolio optimization model with bounded memory[J]. Mathematics of Operations Research, 2011, 36 (4): 604- 619. doi: 10.1287/moor.1110.0508
22	Federico S . A stochastic control problem with delay arising in a pension fund model[J]. Finance and Stochastics, 2011, 15 (3): 421- 459. doi: 10.1007/s00780-010-0146-4
23	Elsanosi I , Øksendal B , Sulem A . Some solvable stochastic control problems with delay[J]. Stochastics: An International Journal of Probability and Stochastic Processes, 2000, 71 (1/2): 69- 89.
24	Larssen B . Dynamic programming in stochastic control of systems with delay[J]. Stochastics: An International Journal of Probability and Stochastic Processes, 2002, 74 (3/4): 651- 673.
25	Shen Y , Zeng Y . Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach[J]. Insurance: Mathematics and Economics, 2014, 57, 1- 12. doi: 10.1016/j.insmatheco.2014.04.004
26	A C , Li Z . Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model[J]. Insurance: Mathematics and Economics, 2015, 61, 181- 196. doi: 10.1016/j.insmatheco.2015.01.005
27	杨潇潇, 梁志彬, 张彩斌. 基于时滞和多维相依风险模型的最优期望-方差比例再保险[J]. 中国科学: 数学, 2017, 47 (6): 723- 756.
28	A C , Lai Y , Shao Y . Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model[J]. Journal of Computational and Applied Mathematics, 2018, 342, 317- 336. doi: 10.1016/j.cam.2018.03.035
29	Deng C , Bian W , Wu B . Optimal reinsurance and investment problem with default risk and bounded memory[J]. International Journal of Control, doi: 10.1080/00207179.2019.1573320
30	Wang S , Rong X , Zhao H . Optimal time-consistent reinsurance-investment strategy with delay for an insurer under a defaultable market[J]. Journal of Mathematical Analysis and Applications, 2019, 474 (2): 1267- 1288. doi: 10.1016/j.jmaa.2019.02.016
31	Browne S . Stochastic differential portfolio games[J]. Journal of Applied Probability, 2000, 37 (1): 126- 147. doi: 10.1239/jap/1014842273
32	Bensoussan A , Siu C C , Yam S C P , et al. A class of non-zero-sum stochastic differential investment and reinsurance games[J]. Automatica, 2014, 50 (8): 2025- 2037. doi: 10.1016/j.automatica.2014.05.033
33	Meng H , Li S , Jin Z . A reinsurance game between two insurance companies with nonlinear risk processes[J]. Insurance: Mathematics and Economics, 2015, 62, 91- 97. doi: 10.1016/j.insmatheco.2015.03.008
34	Yan M , Peng F , Zhang S . A reinsurance and investment game between two insurance companies with the different opinions about some extra information[J]. Insurance: Mathematics and Economics, 2017, 75, 58- 70. doi: 10.1016/j.insmatheco.2017.04.002
35	Siu C C , Yam S C P , Yang H , et al. A class of nonzero-sum investment and reinsurance games subject to systematic risks[J]. Scandinavian Actuarial Journal, 2017, 2017 (8): 670- 707. doi: 10.1080/03461238.2016.1228542
36	Chen S , Yang H , Zeng Y . Stochastic differential games between two insurers with generalized mean-variance premium principle[J]. ASTIN Bulletin, 2018, 48 (1): 413- 434. doi: 10.1017/asb.2017.35
37	Deng C , Zeng X , Zhu H . Non-zero-sum stochastic differential reinsurance and investment games with default risk[J]. European Journal of Operational Research, 2018, 264 (3): 1144- 1158. doi: 10.1016/j.ejor.2017.06.065
39	Wang N , Zhang N , Jin Z , et al. Robust non-zero-sum investment and reinsurance game with default risk[J]. Insurance: Mathematics and Economics, 2019, 84, 115- 132. doi: 10.1016/j.insmatheco.2018.09.009
	Chen L , Shen Y . Stochastic Stackelberg differential reinsurance games under time-inconsistent mean-variance framework[J]. Insurance: Mathematics and Economics, 2019, 88, 120- 137. doi: 10.1016/j.insmatheco.2019.06.006
40	Zhu H , Cao M , Zhang C . Time-consistent investment and reinsurance strategies for meanvariance insurers with relative performance concerns under the Heston model[J]. Finance Research Letters, 2019, 30, 280- 291. doi: 10.1016/j.frl.2018.10.009
41	Espinosa G E , Touzi N . Optimal investment under relative performance concerns[J]. Mathematical Finance, 2015, 25 (2): 221- 257. doi: 10.1111/mafi.12034
42	DeMarzo P M , Kaniel R , Kremer I . Relative wealth concerns and financial bubbles[J]. Review of Financial Studies, 2008, 21 (1): 19- 50. doi: 10.1093/rfs/hhm032
43	Basak S , Makarov D . Strategic asset allocation in money management[J]. Journal of Finance, 2014, 69 (1): 179- 217. doi: 10.1111/jofi.12106
44	Björk T, Murgoci A. A general theory of Markovian time inconsistent stochastic control problems[EB/OL]. (2019-12-02)[2010-09-17]. http://dx.doi.org/10.2139/ssrn.1694759.
45	Grandell J . Aspects of Risk Theory[M]. New York: Springer, 1990.

保险公司1的参数		保险公司2的参数
λ₁	3	λ₂	4
μ₁	2	μ₂	2.500
σ₁	2	σ₂	3
η₁	0.400	η₂	0.800
δ₁	0.050	δ₂	0.030
h₁	2	h₂	3
ω₁	0.200	ω₂	0.244
κ₁	0.300	κ₂	0.700
γ₁	0.200	γ₂	0.900

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