American Institute of Mathematical Sciences

Advanced
September  2013, 18(7): 1889-1907. doi: 10.3934/dcdsb.2013.18.1889

Optimal stochastic differential games with VaR constraints

Jingzhen Liu 1, and  Ka-Fai Cedric Yiu 2,

1. 

School of Insurance, Central University Of Finance and Economics, Beijing 100081, China
2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  May 2011 Revised  January 2013 Published  May 2013

The nonlinear dynamic games between competing insurance companies are interesting and important problems because of the general practice of using re-insurance to reduce risks in the insurance industry. This problem becomes more complicated if a proper risk control is imposed on all the involving companies. In order to understand the dynamical properties, we consider the stochastic differential game between two insurance companies with risk constraints. The companies are allowed to purchase proportional reinsurance and invest their money into both risk free asset and risky (stock) asset. The competition between the two companies is formulated as a two player (zero-sum) stochastic differential game. One company chooses the optimal reinsurance and investment strategy in order to maximize the expected payoff, and the other one tries to minimize this value. For the purpose of risk management, the risk arising from the whole portfolio is constrained to some level. By the principle of dynamic programming, the problem is reduced to solving the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations for Nash equilibria. We derive the Nash equilibria explicitly and obtain closed form solutions to HJBI under different scenarios.
Keywords: dynamic programming., risk constraint, stochastic differential game, HJBI equations, Nash equilibria, proportional reinsurance, Optimal investment.
Mathematics Subject Classification: Primary: 93E20, 91A25; Secondary: 49l2.
Citation: Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
References:
[1]

N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems," North Holland, 1981.  Google Scholar

[2]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975. doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

N. Bauerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165. doi: 10.1007/s00186-005-0446-1.  Google Scholar

[4]

R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk, in "Proceedings of the Casualty Actuarial Society Casualty Actuarial Society," (Arlington, Virginia, 2000), LXXXVII, 31-78. Google Scholar

[5]

S. Bouma, "Risk Management in the Insurance Industry and Solvency II," European Survey, Capgemini, November 7, 2006. Google Scholar

[6]

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, 20 (1995), 937-958. doi: 10.1287/moor.20.4.937.  Google Scholar

[7]

S. Browne, Stochastic differential portfolio games, J. Appl. Prob., 37 (2000), 126-147. doi: 10.1239/jap/1014842273.  Google Scholar

[8]

R. J. Elliott and T. K. Siu, On risk minimizing portfolio choice under Markovian regime-switching model, Ann. Oper. Res., 176 (2010), 271-291. doi: 10.1007/s10479-008-0448-5.  Google Scholar

[9]

R. J. Elliott and T. K. Siu, A stochastic differential game for optimal investment of an insurer with regime switching, Quant. Finance, 11 (2011), 365-380. doi: 10.1080/14697681003591704.  Google Scholar

[10]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scand. Actuar. J., 22 (1998), 166-180.  Google Scholar

[11]

J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547. doi: 10.3934/jimo.2012.8.531.  Google Scholar

[12]

J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal Portfolios with stress analysis and the effect of a CVaR constraint, Pac. J. Optim., 7 (2011), 83-95.  Google Scholar

[13]

S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance Math. Econom., 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002.  Google Scholar

[14]

B. Øksendal and A. Sulem, "A Game Theoretic Approach to Amrtingale Measures in Incomplete Markets," University of Oslo and INRIA, 2006. Google Scholar

[15]

D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  Google Scholar

[16]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 2001 (2001), 55-68. doi: 10.1080/034612301750077338.  Google Scholar

[17]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907. doi: 10.1214/aoap/1031863173.  Google Scholar

[18]

J. Suijs, A. De Waegenaere and P. Borm, Stochastic cooperative games in insurance, Insurance Math. Econom., 22 (1998), 209-228. doi: 10.1016/S0167-6687(97)00038-3.  Google Scholar

[19]

K. L. Teo, D. W. Reid and I. E. Boyd, Stochastic optimal control theory and its computational method, Internat. J. Systems Sci., 11 (1980), 77-95. doi: 10.1080/00207728008966998.  Google Scholar

[20]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[21]

K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint, J. Econom. Dynam. Control, 28 (2004), 1317-1334. doi: 10.1016/S0165-1889(03)00116-7.  Google Scholar

[22]

K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 979-989. doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

[23]

X. D. Zeng, A stochastic differential reinsurance game, J. Appl. Prob., 47 (2010), 335-349. doi: 10.1239/jap/1276784895.  Google Scholar

[24]

X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45 (2009), 81-88. doi: 10.1016/j.insmatheco.2009.04.001.  Google Scholar

show all references

References:
[1]

N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems," North Holland, 1981.  Google Scholar

[2]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975. doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

N. Bauerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165. doi: 10.1007/s00186-005-0446-1.  Google Scholar

[4]

R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk, in "Proceedings of the Casualty Actuarial Society Casualty Actuarial Society," (Arlington, Virginia, 2000), LXXXVII, 31-78. Google Scholar

[5]

S. Bouma, "Risk Management in the Insurance Industry and Solvency II," European Survey, Capgemini, November 7, 2006. Google Scholar

[6]

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, 20 (1995), 937-958. doi: 10.1287/moor.20.4.937.  Google Scholar

[7]

S. Browne, Stochastic differential portfolio games, J. Appl. Prob., 37 (2000), 126-147. doi: 10.1239/jap/1014842273.  Google Scholar

[8]

R. J. Elliott and T. K. Siu, On risk minimizing portfolio choice under Markovian regime-switching model, Ann. Oper. Res., 176 (2010), 271-291. doi: 10.1007/s10479-008-0448-5.  Google Scholar

[9]

R. J. Elliott and T. K. Siu, A stochastic differential game for optimal investment of an insurer with regime switching, Quant. Finance, 11 (2011), 365-380. doi: 10.1080/14697681003591704.  Google Scholar

[10]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scand. Actuar. J., 22 (1998), 166-180.  Google Scholar

[11]

J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547. doi: 10.3934/jimo.2012.8.531.  Google Scholar

[12]

J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal Portfolios with stress analysis and the effect of a CVaR constraint, Pac. J. Optim., 7 (2011), 83-95.  Google Scholar

[13]

S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance Math. Econom., 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002.  Google Scholar

[14]

B. Øksendal and A. Sulem, "A Game Theoretic Approach to Amrtingale Measures in Incomplete Markets," University of Oslo and INRIA, 2006. Google Scholar

[15]

D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  Google Scholar

[16]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 2001 (2001), 55-68. doi: 10.1080/034612301750077338.  Google Scholar

[17]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907. doi: 10.1214/aoap/1031863173.  Google Scholar

[18]

J. Suijs, A. De Waegenaere and P. Borm, Stochastic cooperative games in insurance, Insurance Math. Econom., 22 (1998), 209-228. doi: 10.1016/S0167-6687(97)00038-3.  Google Scholar

[19]

K. L. Teo, D. W. Reid and I. E. Boyd, Stochastic optimal control theory and its computational method, Internat. J. Systems Sci., 11 (1980), 77-95. doi: 10.1080/00207728008966998.  Google Scholar

[20]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[21]

K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint, J. Econom. Dynam. Control, 28 (2004), 1317-1334. doi: 10.1016/S0165-1889(03)00116-7.  Google Scholar

[22]

K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 979-989. doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

[23]

X. D. Zeng, A stochastic differential reinsurance game, J. Appl. Prob., 47 (2010), 335-349. doi: 10.1239/jap/1276784895.  Google Scholar

[24]

X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45 (2009), 81-88. doi: 10.1016/j.insmatheco.2009.04.001.  Google Scholar

[1]

Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2195-2211. doi: 10.3934/jimo.2019050
[2]

Jingzhen Liu, Lihua Bai, Ka-Fai Cedric Yiu. Optimal investment with a value-at-risk constraint. Journal of Industrial & Management Optimization, 2012, 8 (3) : 531-547. doi: 10.3934/jimo.2012.8.531
[3]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537
[4]

Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models†. Journal of Dynamics & Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012
[5]

Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020143
[6]

Hiroaki Hata, Li-Hsien Sun. Optimal investment and reinsurance of insurers with lognormal stochastic factor model. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021033
[7]

Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080
[8]

Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025
[9]

Tao Chen, Wei Liu, Tao Tan, Lijun Wu, Yijun Hu. Optimal reinsurance with default risk: A reinsurer's perspective. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2971-2987. doi: 10.3934/jimo.2020103
[10]

Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial & Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743
[11]

Zuo Quan Xu, Fahuai Yi. An optimal consumption-investment model with constraint on consumption. Mathematical Control & Related Fields, 2016, 6 (3) : 517-534. doi: 10.3934/mcrf.2016014
[12]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013
[13]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008
[14]

Yan Zhang, Peibiao Zhao. Optimal reinsurance-investment problem with dependent risks based on Legendre transform. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1457-1479. doi: 10.3934/jimo.2019011
[15]

Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial & Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044
[16]

Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021003
[17]

Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009
[18]

Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control & Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003
[19]

Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021120
[20]

Xia Han, Zhibin Liang, Yu Yuan, Caibin Zhang. Optimal per-loss reinsurance and investment to minimize the probability of drawdown. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021145

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy