Pseudoconvex Risk Measure Representation Theorem And Its Nature

Financial risk management is one of the most important theories of modern finances and economics, while measuring financial risk with risk measures is the fundamental work. In this study, we have done some deeper research in risk measures, which is the study of quasi-convex risk measure.Firstly, we conclude the acceptance sets, dual representation theorem and penalty functions of coherent risk measure and convex risk measure.Secondly, we have done some specific works:introduce a generalized CVaR (conditional value at risk), denoted by GCVaR, which is satisfied with the definition of convex risk measure, and get the formulation of its penalty function. We introduce another convex risk measure based of entropy function, and discuss the formulation of this kind of convex risk measures and its penalty functions, when the weight in the entropy function has several different forms. And then study the portfolio problem based on convex risk measures, give some optimal propositions with dual programming and an example of portfolio problem based on CVaR.Thirdly, we have the most important findings of this paper. We introduce the definition of quasi-convex risk measure, and give representation theorem by dual theory. We study the relationship between quasi-convex risk measure and convex risk measure. We also study the characters of the optimal problem based on quasi-convex risk measure using multiplier theory and dual theory.