一致风险度量和锥优化分析

一致风险理论的公理系统为风险分析建立了坚实的基础，然而它背后的数学却和凸优化理论思想密切相关，特别是对偶理论. 本文在有限维空间中，利用锥优化的对偶定理给出了一致风险度量的一般表达式的简单证明. 分析了可接受集的概念在一致风险度量中的中心作用，根据锥优化的对偶关系，探索了常用风险度量的性质. 尽管可接受集的大小能够表达风险控制的强弱，但是我们不知道如何定量地表示. 本文提出用相对熵控制风险度量松紧度的方法和意义. 另外，根据一致风险度量的灵活的结构，给出了无套利条件的一种放松，这一结果可用于不完全市场中的期权定价问题.

The Cone Optimization Analysis of Coherent Risk Measures

The axiomatic system of coherent risk measures has set up by Artzner et.al. owever, the mathematics underlying it is closely related to the ideas of convex otimization, specifically the duality theory. This paper simply proved the general expression of coherent risk measures by the cone duality theorem in finite dimension space.We have analyzed the core role of acceptable set in coherent risk measure and the propertiesof conventional risk measures. In spite of the size of acceptable set may be used for controllingthe risk management strictly or loosely, but we do not know how to express it quantitatively. The paper suggest In addition we suggest relaxing no arbitrage condition in view of the flexibility of coherent risk measures and the achieved results may be used to pricing option in incomplete market.