Computation Of A Class Of Choquet Integrals And Their Application To Risk Measurement

In recent years,with the continuous development of economy and society,the financial market has become increasingly complex and changeable,people then have gradually found that there are a lot of non-additive and uncertain phenomena in the financial market.In this case,the traditional additive measure and linear expecta-tion theory cannot accurately describe the uncertainty of financial risks.Therefore,in order to measure and analyze highly dynamic and complex financial risks,scholars at home and abroad began to actively explore and study non-additive measures and nonlinear integrals,and applied them to financial risk measurement.Among these non-additive measures and nonlinear integrals,fuzzy measure and Choquet integral are more studied.As a nonlinear integral,Choquet integral with respect to fuzzy measure is actually a generalization of the classical Lebesgue integral.Based on its good properties,Choquet integral has been widely used in financial asset pricing,multi-criteria decision making,game theory and insurance contract.In this thesis,we generalize the Choquet integral based on the distorted prob-ability measure and study the calculation of the Choquet integral with respect to distorted probability measure in continuous cases.What's more,we also use the Choquet integral with respect to distorted probability measure to represent risk measure and introduce the distortion risk measure.So far,many studies on Cho-quet integrals have been focused on discrete cases.It was not until the concept of the derivative of a function with respect to fuzzy measure was introduced that some new progress was made in the study of continuous Choquet integral.In fact,results of this thesis are the further supplement and improvement of Choquet inte-gral theory in the continuous case.In order to facilitate the research,we introduce the basic knowledge of fuzzy measure and Choquet integral theory first.Then,we give the representation results about the calculation of the Choquet integral with respect to distorted probability measure of monotone functions and non-monotone functions on the nonnegative real line respectively,and this calculation method is demonstrated by examples.In the case of non-monotone functions,we use the as-cending or descending arrangement of a non-monotone function to convert it into a monotone function.Finally,we introduce the application of Choquet integral with respect to distorted probability measure in risk measure and give several examples of the distortion risk measure.