Uniformly Convex (quasi-convex) The Basic Characteristics Of Risk Measure

In modern financial economics, the financial risk is one of important research fields. As a basic means of risk management, the financial risk measurement measures all kinds of financial risks effectively by different measure methods.In this study, we have done some deeper research in coherent measures of risk, convex risk measures and quasi-convex risk measures on the basis of the existing research of them. And then we also build the definition of uniformly convex risk measure and uniformly quasi-convex risk measure and give the basic characteristics of them.Firstly, we give some necessary notations and terminologies, such as the acceptable set, dual representation theorem, and list some related properties of the currency risk measure, consistent risk measure, convex risk measure and quasi-convex risk measure.Secondly, this paper gives the definition of the uniformly convex risk measure and then studies some basic properties of uniformly convex risk measure by using the definition and properties of generalized convex functions.Finally, on the basis of uniformly convex risk measure, this paper gives the definition of the uniformly quasi-convex risk measure and also studies some basic properties. At the same time, we also study the equivalence relationship of the uniformly quasi-convex risk measure and convex risk measure.The uniformly convex (quasi-convex) risk measures obtained in this paper can be regarded as a natural generalization of convex (quasi-convex) risk measures.