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Optimal stochastic differential games with VaR constraints
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 
References:
[1] 
N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems," North Holland, 1981. 
[2] 
L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and noshorting constraint, Insurance Math. Econom., 42 (2008), 968975. doi: 10.1016/j.insmatheco.2007.11.002. 
[3] 
N. Bauerle, Benchmark and meanvariance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159165. doi: 10.1007/s0018600504461. 
[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, 3178. 
[5] 
S. Bouma, "Risk Management in the Insurance Industry and Solvency II," European Survey, Capgemini, November 7, 2006. 
[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), 937958. doi: 10.1287/moor.20.4.937. 
[7] 
S. Browne, Stochastic differential portfolio games, J. Appl. Prob., 37 (2000), 126147. doi: 10.1239/jap/1014842273. 
[8] 
R. J. Elliott and T. K. Siu, On risk minimizing portfolio choice under Markovian regimeswitching model, Ann. Oper. Res., 176 (2010), 271291. doi: 10.1007/s1047900804485. 
[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), 365380. doi: 10.1080/14697681003591704. 
[10] 
B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scand. Actuar. J., 22 (1998), 166180. 
[11] 
J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a valueatrisk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531547. doi: 10.3934/jimo.2012.8.531. 
[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), 8395. 
[13] 
S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance Math. Econom., 42 (2008), 434444. doi: 10.1016/j.insmatheco.2007.04.002. 
[14] 
B. Øksendal and A. Sulem, "A Game Theoretic Approach to Amrtingale Measures in Incomplete Markets," University of Oslo and INRIA, 2006. 
[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), 109128. 
[16] 
H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 2001 (2001), 5568. doi: 10.1080/034612301750077338. 
[17] 
H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890907. doi: 10.1214/aoap/1031863173. 
[18] 
J. Suijs, A. De Waegenaere and P. Borm, Stochastic cooperative games in insurance, Insurance Math. Econom., 22 (1998), 209228. doi: 10.1016/S01676687(97)000383. 
[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), 7795. doi: 10.1080/00207728008966998. 
[20] 
H. Yang and L. Zhang, Optimal investment for insurer with jumpdiffusion risk process, Insurance Math. Econom., 37 (2005), 615634. doi: 10.1016/j.insmatheco.2005.06.009. 
[21] 
K. F. C. Yiu, Optimal portfolio under a valueatrisk constraint, J. Econom. Dynam. Control, 28 (2004), 13171334. doi: 10.1016/S01651889(03)001167. 
[22] 
K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regimeswitching and valueatrisk constraint, Automatica, 46 (2010), 979989. doi: 10.1016/j.automatica.2010.02.027. 
[23] 
X. D. Zeng, A stochastic differential reinsurance game, J. Appl. Prob., 47 (2010), 335349. doi: 10.1239/jap/1276784895. 
[24] 
X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45 (2009), 8188. doi: 10.1016/j.insmatheco.2009.04.001. 
show all references
References:
[1] 
N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems," North Holland, 1981. 
[2] 
L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and noshorting constraint, Insurance Math. Econom., 42 (2008), 968975. doi: 10.1016/j.insmatheco.2007.11.002. 
[3] 
N. Bauerle, Benchmark and meanvariance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159165. doi: 10.1007/s0018600504461. 
[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, 3178. 
[5] 
S. Bouma, "Risk Management in the Insurance Industry and Solvency II," European Survey, Capgemini, November 7, 2006. 
[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), 937958. doi: 10.1287/moor.20.4.937. 
[7] 
S. Browne, Stochastic differential portfolio games, J. Appl. Prob., 37 (2000), 126147. doi: 10.1239/jap/1014842273. 
[8] 
R. J. Elliott and T. K. Siu, On risk minimizing portfolio choice under Markovian regimeswitching model, Ann. Oper. Res., 176 (2010), 271291. doi: 10.1007/s1047900804485. 
[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), 365380. doi: 10.1080/14697681003591704. 
[10] 
B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scand. Actuar. J., 22 (1998), 166180. 
[11] 
J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a valueatrisk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531547. doi: 10.3934/jimo.2012.8.531. 
[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), 8395. 
[13] 
S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance Math. Econom., 42 (2008), 434444. doi: 10.1016/j.insmatheco.2007.04.002. 
[14] 
B. Øksendal and A. Sulem, "A Game Theoretic Approach to Amrtingale Measures in Incomplete Markets," University of Oslo and INRIA, 2006. 
[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), 109128. 
[16] 
H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 2001 (2001), 5568. doi: 10.1080/034612301750077338. 
[17] 
H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890907. doi: 10.1214/aoap/1031863173. 
[18] 
J. Suijs, A. De Waegenaere and P. Borm, Stochastic cooperative games in insurance, Insurance Math. Econom., 22 (1998), 209228. doi: 10.1016/S01676687(97)000383. 
[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), 7795. doi: 10.1080/00207728008966998. 
[20] 
H. Yang and L. Zhang, Optimal investment for insurer with jumpdiffusion risk process, Insurance Math. Econom., 37 (2005), 615634. doi: 10.1016/j.insmatheco.2005.06.009. 
[21] 
K. F. C. Yiu, Optimal portfolio under a valueatrisk constraint, J. Econom. Dynam. Control, 28 (2004), 13171334. doi: 10.1016/S01651889(03)001167. 
[22] 
K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regimeswitching and valueatrisk constraint, Automatica, 46 (2010), 979989. doi: 10.1016/j.automatica.2010.02.027. 
[23] 
X. D. Zeng, A stochastic differential reinsurance game, J. Appl. Prob., 47 (2010), 335349. doi: 10.1239/jap/1276784895. 
[24] 
X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45 (2009), 8188. doi: 10.1016/j.insmatheco.2009.04.001. 
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