多损失WCVaR模型的等价性定理
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摘要
研究了多概率分布簇下的多损失下的WCVaR (Multi Worst Conditional Valueat-Risk)模型等价性定理,根据概率分布簇的VaR测度值,定义了多损失下的WCVaR风险测度值和对应的多目标优化模型(MWCVaR),证明了多目标优化模型(MWCVaR)等价另一个多目标优化模型求解.对于有限分布簇情形,在一定条件下,证明了用有限个分布簇就可以近似计算多损失(MWCVaR)优化模型.
Under multi-probability distribution cluster, equivalence of multi-loss WCVaR(Worst Conditional Value-at-Risk, MCVaR) problem is studied. According to the VaR measure under multi-probability distribution cluster, the multi-objective optimization problem(MWCVaR) is defined by the WCVaR risk measurement. It is proved that the optimization problem(MWCVaR) is equivalent to solve another multi-objective optimization problem. In the case of a finite distributions cluster, under certain conditions, we prove that the Pareto effective solution of the problem(MWCVaR) can be approximated by a finite distributions cluster.
Under multi-probability distribution cluster, equivalence of multi-loss WCVaR(Worst Conditional Value-at-Risk, MCVaR) problem is studied. According to the VaR measure under multi-probability distribution cluster, the multi-objective optimization problem(MWCVaR) is defined by the WCVaR risk measurement. It is proved that the optimization problem(MWCVaR) is equivalent to solve another multi-objective optimization problem. In the case of a finite distributions cluster, under certain conditions, we prove that the Pareto effective solution of the problem(MWCVaR) can be approximated by a finite distributions cluster.
引文
[1] Rockafellar R T, Uryasev S. Optimization of conditional value-at-risk[J]. The Journal of Risk,2000, 2:21-41.
[2] Rockafellar R T, Uryasev S. Conditional value-at-risk for general loss distributions[J]. Journal of Banking&Finance, 2002, 26:1443-1471.
[3] Alexander Gordon J, Baptista A M. A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model[J]. Management Science, 2004, 50:1261-1273.
[4] Huang Dashan, Zhu Shushang, Frank J F, et al. Portfolio selection with uncertain exit time:A robust CVaR approach[J]. Journal of Economic Dynamics&Control, 2008, 32:594-623.
[5] Zhu Shushang, Masao Fukushima. Worst-case conditional value-at-risk with application to robust portfolio management[J]. Operations Research, 2009, 57:1155-1168.
[6] Zhu Shushang, Li Duan, Wang Shouyang. Robust portfolio selection under downside risk measures[J]. Quantitative Finance, 2009, 9:869-885.
[7] Huang Dashan, Zhu Shushang, Frank J F, et al. Portfolio selection under distributional uncertainty:A relative robust CVaR approach[J]. European Journal of Operational Research, 2010,203:185-194.
[8] Akiko Takeda, Takafumi Kanamori. A robust approach based on conditional value-at-risk measure to statistical learning problems[J]. European Journal of Operational Research, 2009,198:287-296.
[9] Gotoh Junya, Takano Yuichi. Newsvendor solutions via conditional value-at-risk minimization[J]. European Journal of Operational Research, 2007, 179:80-96.
[10] Zhou Yanju, Chen Xiaohong, Wang Zongrun. Optimal ordering quantities for multi-products with stochastic demand:Return-CVaR model[J]. International Journal of Production Economics,2008, 112:782-795.
[11] Qiu Ruozhen, Shang Jennifer, Huang Xiaoyuan. Robust inventory decision under distribution uncertainty:A CVaR-based optimization approach[J]. International Journal of Production Economics, 2014, 153:13-23.
[12]蒋敏,孟志青.多目标条件风险值理论[M].北京:科学出版社,2014.
[13] Sion M. On general minimax theorems[J]. Pacific Journal Mathematics, 1958, 8(1):171-176.
[2] Rockafellar R T, Uryasev S. Conditional value-at-risk for general loss distributions[J]. Journal of Banking&Finance, 2002, 26:1443-1471.
[3] Alexander Gordon J, Baptista A M. A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model[J]. Management Science, 2004, 50:1261-1273.
[4] Huang Dashan, Zhu Shushang, Frank J F, et al. Portfolio selection with uncertain exit time:A robust CVaR approach[J]. Journal of Economic Dynamics&Control, 2008, 32:594-623.
[5] Zhu Shushang, Masao Fukushima. Worst-case conditional value-at-risk with application to robust portfolio management[J]. Operations Research, 2009, 57:1155-1168.
[6] Zhu Shushang, Li Duan, Wang Shouyang. Robust portfolio selection under downside risk measures[J]. Quantitative Finance, 2009, 9:869-885.
[7] Huang Dashan, Zhu Shushang, Frank J F, et al. Portfolio selection under distributional uncertainty:A relative robust CVaR approach[J]. European Journal of Operational Research, 2010,203:185-194.
[8] Akiko Takeda, Takafumi Kanamori. A robust approach based on conditional value-at-risk measure to statistical learning problems[J]. European Journal of Operational Research, 2009,198:287-296.
[9] Gotoh Junya, Takano Yuichi. Newsvendor solutions via conditional value-at-risk minimization[J]. European Journal of Operational Research, 2007, 179:80-96.
[10] Zhou Yanju, Chen Xiaohong, Wang Zongrun. Optimal ordering quantities for multi-products with stochastic demand:Return-CVaR model[J]. International Journal of Production Economics,2008, 112:782-795.
[11] Qiu Ruozhen, Shang Jennifer, Huang Xiaoyuan. Robust inventory decision under distribution uncertainty:A CVaR-based optimization approach[J]. International Journal of Production Economics, 2014, 153:13-23.
[12]蒋敏,孟志青.多目标条件风险值理论[M].北京:科学出版社,2014.
[13] Sion M. On general minimax theorems[J]. Pacific Journal Mathematics, 1958, 8(1):171-176.