Choquet Integrals And The Applications In Insurance And Asset Pricing

Choquet integral theory is an integration with respect to a non-additive measure. It is a nonlinear integral and it can be used to solve lots of economic, financial and insurance problems which can not be explained very clearly by classical probability. For example, Choquet expected utility theory under the framework of Choquet integral can explain the Allais paradox and the Ellsberg paradox very well. Therefore, further studies of Choquet integral have its theoretical value. And the study of further applications of Choquet integral in finance, insurance and so on is necessary.There is one kind of very useful Choquet capacity with respect to the special non-additive measure, that is the distorted probability. It includes different forms, such as PH transform and Wang transform. In view of this, firstly the paper summarizes the advantages and disadvantages of the two transforms, and gives the applications of the two transforms in insurance pricing and asset pricing. Secondly, we propose a new more common distorted probability which is defined by F,and prove its properties. In the last part we also study the properties and the convergences of the symmetric Choquet integral. The main contents are as follows:In the first part, we shall introduce the basic theory of the Choquet integral which focuses on capacity, the definitions and the basic properities of the Choquet integral. Also some financial and economic fundamentals shall be described.In part two, firstly we shall discuss the Choquet integral asset pricing theory, then in-troduce the forms, properties and the applications of the PH-transform and Wang-transform. Mainly we shall give a new transform that is defined by F-transform, and prove its some properties:Wang-transform is a special type of F transform. Part three will study the symmetry Choquet integral. Firstly we shall introduce a com-mon integral that is asymmetric Choquet integral, then give the definition and study the basic properties of the symmetric Choquet integral. Finally we shall discuss the convergences of the symmetry Choquet integral.