Risk Measurement Problem Based On The Signed Choquet Integral

In recent years,with the rapid development of the economy and financial mar-ket,the volatility of financial market has become more and more serious.The financial risk has become more and more serious.Risk is not only existing in the financial market,but also everywhere in our daily life.It is so small that we have to bump into each other in the daily life,such as sudden COVID-19.The objective existence and destructiveness of risk are all revealed.How to accurately measure risk and avoid risk in time has become a concern of people.Risk in essence can be understood as the uncertainty of the change of things.Because of the uncertainty of risk,for a long time,people hope to have a kind of expression to express the risk clearly and intuitively,and all kinds of methods of risk measure come into being at the historic moment.With the continuous exploration and innovation of researchers,there are more and more kinds of risk measure methods,VAR has been widely used and promoted.Artzner et al.proposed that a good risk measure should follow the principle of monotonicity.However,in real life,many risk measurement methods does not have monotonicity.And for that,in this paper,I introduce signed Choquet integral,study the basic properties of signed Choquet integral,and study the upper and lower bounds of risk aggregation of signed Choquet integral with dependent uncertainty.Firstly,on the basis of the known equivalent conditions of the convexity of signed Choquet integral,with the help of the related derivative properties of convex and concave order and hardy-littlewood-polya inequality,the convexity order con-sistency of signed Choquet integral is given,and the equivalent conditions of the concavity of signed Choquet integral are obtained.Secondly,by using the equivalent condition of the concavity and convexity of the signed Choquet integral,we obtain the upper and lower bounds of the risk aggrega-tion of variables with comonotonicity and counter-monotonic respectively when the twist function has concavity and convexity.By using the known extreme-aggregation measure,we obtain that Ih*is the smallest law-invariant convex functional diminat-ing and is the largest law-invariant concave functional diminating Ih.With the help of the above conclusions,the best and worst-case values of risk aggrega-tion under signed Choquet integral dependent uncertainty are obtained by using the mathematical tools such as joint mixability and conditional joint mixability.Finally,the paper analyzes and summarizes the main achievements and future development direction.