基于凸概度分布簇下WCVaR模型的有限逼近性
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摘要
如何求解实际问题中Worst条件风险值模型是一个非常困难的问题,研究了凸概率分布簇下的WCVaR(Worst Conditional Value-at-Risk)模型等价性及其在序列分布簇下的有限逼近性,根据概率分布簇的VaR测度值,定义了WCVaR风险测度值和对应的WCVaR模型,证明了WCVaR模型等价一个另一个数学规划问题求解.在一定条件下,证明了在损失有界情形用有限个分布簇就可以足够近似计算WCVaR模型的最优解,因此,对于解决稳健型条件风险值模型具重要的实际价值.
How to solve the worst conditional value at risk model in practical problems is a very difficult problem under the convex probability distribution cluster. In this paper, the equivalence of WCVaR(Worst Conditional Value-at-Risk) model under the convex probability distribution cluster and its properties under the sequence distribution cluster are studied.According to the VaR measure under the convex probability distribution cluster, the WCVaR optimization problem is defined by the WCVaR risk measurement. It is proved that the WCVaR optimization problem is equivalent to solve a nonlinear optimization problem.Under certain conditions, we prove that the optimal solution of the WCVaR problem can be approximated by a finite distributions cluster. Therefore, it has important practical value for solving the conditional conditional value at risk model.
How to solve the worst conditional value at risk model in practical problems is a very difficult problem under the convex probability distribution cluster. In this paper, the equivalence of WCVaR(Worst Conditional Value-at-Risk) model under the convex probability distribution cluster and its properties under the sequence distribution cluster are studied.According to the VaR measure under the convex probability distribution cluster, the WCVaR optimization problem is defined by the WCVaR risk measurement. It is proved that the WCVaR optimization problem is equivalent to solve a nonlinear optimization problem.Under certain conditions, we prove that the optimal solution of the WCVaR problem can be approximated by a finite distributions cluster. Therefore, it has important practical value for solving the conditional conditional value at risk model.
引文
[1] Rockafellar R T, and Uryasev S. Optimization of conditional value-at-risk[J]. The Journal of Risk,2000, 2:21-41.
[2] Rockafellar R T,and Uryasev S. Conditional value-at-risk for general loss distributions[J]. Journal of Banking Finance, 2002, 26:1443-1471.
[3] Alexander G 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 D S, Zhu S S, Fabozzi F J, Fukushima M. Portfolio selection with uncertain exit time:A robust CVaR approach[J]. Journal of Economic Dynamics Control, 2008, 32:594-623.
[5] Zhu S S, Fukushima M. Worst-case conditional value-at-risk with application to robust portfolio management[J]. Operations Research, 2009, 57:1155-1168.
[6] Zhu S S, Li D, Wang S Y. Robust portfolio selection under downside risk measures[J]. Quantitative Finance, 2009, 9:869-885.
[7] Huang D S, Zhu S S, Fabozzi F J, Fukushima M. Portfolio selection under distributional uncertainty:A relative robust CVaR approach[J]. European Journal of Operational Research, 2010, 203:185-194.
[8] Takeda A,Kanamori T. 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 J Y, and Takano Y. Newsvendor solutions via conditional value-at-risk minimization[J].European Journal of Operational Research, 2007, 179:80-96.
[10] Zhou Y J, Chen X H, and Wang Z R. Optimal ordering quantities for multi-products with stochastic demand:Return-CVaR model[J]. International Journal of Production Economics, 2008, 112:782-795.
[11] Qiu R Z, Shang J, Huang X Y. Robust inventory decision under distribution uncertainty:A CVaRbased optimization approach[J]. International Journal of Production Economics, 2014, 153:13-23.
[12] Delage E, Ye Y Y. Distributionally Robust optimization under moment uncertainty with application to data-driven problems[J]. Operations Research, 2010, 58(3):595-612.
[13] Sion M. On general minimax theorems[J]. Pacific Journal of Mathematics, 1958, 8(1):171-176.
[2] Rockafellar R T,and Uryasev S. Conditional value-at-risk for general loss distributions[J]. Journal of Banking Finance, 2002, 26:1443-1471.
[3] Alexander G 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 D S, Zhu S S, Fabozzi F J, Fukushima M. Portfolio selection with uncertain exit time:A robust CVaR approach[J]. Journal of Economic Dynamics Control, 2008, 32:594-623.
[5] Zhu S S, Fukushima M. Worst-case conditional value-at-risk with application to robust portfolio management[J]. Operations Research, 2009, 57:1155-1168.
[6] Zhu S S, Li D, Wang S Y. Robust portfolio selection under downside risk measures[J]. Quantitative Finance, 2009, 9:869-885.
[7] Huang D S, Zhu S S, Fabozzi F J, Fukushima M. Portfolio selection under distributional uncertainty:A relative robust CVaR approach[J]. European Journal of Operational Research, 2010, 203:185-194.
[8] Takeda A,Kanamori T. 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 J Y, and Takano Y. Newsvendor solutions via conditional value-at-risk minimization[J].European Journal of Operational Research, 2007, 179:80-96.
[10] Zhou Y J, Chen X H, and Wang Z R. Optimal ordering quantities for multi-products with stochastic demand:Return-CVaR model[J]. International Journal of Production Economics, 2008, 112:782-795.
[11] Qiu R Z, Shang J, Huang X Y. Robust inventory decision under distribution uncertainty:A CVaRbased optimization approach[J]. International Journal of Production Economics, 2014, 153:13-23.
[12] Delage E, Ye Y Y. Distributionally Robust optimization under moment uncertainty with application to data-driven problems[J]. Operations Research, 2010, 58(3):595-612.
[13] Sion M. On general minimax theorems[J]. Pacific Journal of Mathematics, 1958, 8(1):171-176.