Studies On Multivariate Convex Risk Measures With Trading Preferences

Nowadays,the research of risk measure has become one of the most popular topics in the field of mathematical finance,even the risk measurement has been praised as ”the third financial revolution of Wall Street” by some scholars.So far,the axiomatic approach and the construction approach to risk measures have been being the most popular approaches.In the recent years,the measures of multidimensional risks(i.e.portfolios)and the set-valued risk measures have been becoming the most active area among the risk measures.Burgert and R¨uschendorf(2006)[1] firstly introduced the real-valued multivariate coherent risk measures and multivariate convex risk measures.In this paper,we construct a new kind of scalar multivariate convex risk measures from a set of acceptable portfolios in terms of multivariate shortfall risk with certain loss level.The dual representation for the multivariate convex risk measures is provided with tractable minimal penalty function.Finally,examples including scalar multivariate entropic risk measures are also given.In Chapter 1,we briefly recall the background of the studies on risk measures,and state what will be studied in the thesis.In Chapter 2,we study the multivariate risk measures with trading preferences.This chapter is devoted to the main results of this thesis.