A Kind Of Multivariable Risk Measures In Frictional Markets Based On Transaction Costs

The measures of financial risk have always been a hot topic in the financial field.Rea-sonable and effective constructions of risk measures models have always been one of the difficult tasks in financial researches.As is known to everyone,classical risk measures gen-erally use cash or a single acceptable asset as a mean of adjusting risk positions.Considering the complexity of financial risk and the fact that effective assets allocation can reduce invest-ment costs,which means the multivariable assets to hedge positions,this paper puts forward the concept of multivariable risk measures based on synchronous adjustment of cash and acceptable assets.Then this paper explores and establishes the representation theorem of the multivariable risk measures and the corresponding convex extensions of it,and gives the relationship between the multivariable risk measures before and after the convex extension.Firstly,the subadditivity and convexity of the multivariable convex risk measures when the acceptable sets are convex are discussed,and the dual representation of the multivariable convex risk measures is given.Secondly,the model of multivariable risk measures in frictional financial markets with transaction costs is studied,that is,the problem of multivariable risk measures corresponding to several types of nonconvex acceptable sets.Multivariable risk measures in accordance with nonconvex acceptable sets are established.The convexity and other basic properties of multivariable risk measures based on different norms are discussed.The convex extensions of the nonconvex multivariable risk measures are studied,and it is proved that the convex extension risk measures can be expressed by the multivariable nonconvex risk measures.Thirdly,by further relaxing the restrictive conditions of nonconvex acceptable sets(re-leasing the weak cone restriction so that the acceptable assets do not need to satisfy the continuity requirement),with the basic techniques of real analysis,it can also be proved that the convex extensions of the multivariable risk measures defined on the new nonconvex acceptable sets can still be expressed by the nonconvex risk measures.Finally,several examples of multivariable risk measure are listed.The convex exten-sions of the multivariable nonconvex risk measures are deducted.