Stochastic Order Of Convex Risk Measure Comonotonic Intended

With the market deregulation and the economic globalization, investors and regulators are taking more and more attention to measure market risk. And after decades of innovation and development, the financial risk measure theories and methods:VaR, CVaR, coherence risk measure, convex risk measure, quasi-convex risk measure and so on, have achieved fruitful results, Especially the convex and quasi-convex risk measures have touted by the academia, because they play an important role in characterizing risk diversification and risk liquidity. We all know that the convex and quasi-convex risk measures must meet the convexity and quasi-convexity constraints. But when the risky assets are not independent, the risk measure will not satisfied the convexity and quasi-convexity constraints, in other words, the convex and quasi-convex risk measure can not measure the risk of all risk assets. Therefore comonotonic convex risk measures have come into being. The comonotonic convex risk measures expend the ordinary monotonic to the stochastic and stop-loss order monotonic, to make what convex risk measure studies more widely and deeply.Song Y., Yan Jia’an and other professors studied deeply the comonotonic convex risk measure in2009, and gave the representation theorem of comonotonic convex risk measures by using Choquet integral.Based on the ideas of Song Y.,Yan Jia’an and other professors, this paper studies comonotonic quasi-convex risk measures, and arrives at the corresponding representation theorem and so on.Firstly, based on cash additivity, this paper proposes a new risk measure which is not only comonotonic quasi-convex, but also respects the (first) stochastic dominance or stop-loss (second) order, We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, prove it and discuss some properties of them. In addition, we study and give equivalence relations of convex, quasi-convex, comonotonic convex and comonotonic quasi-convex in certain circumstances.Secondly, based on cash subadditivity, we study the risk measure which is not only comonotonic quasi-convex, but also respects the (first) stochastic dominance or stop-loss (second) order. We also give their representations in terms of Choquet integrals w.r.t. distorted probabilities, prove it and discuss some properties of them. However some equivalence relation is no longer valid.Some examples are given in the end. The results obtained in this paper can be regarded as a natural generalization of Song Y., Yan Jia’an and other professors’works.