The Calculation Of Choquet Expectation And Minimax Expectation And The Problem Of ±μ-independence

As an extension of the traditional mathematical expectation, Choquet expectation and minimax expectation have many applications in finance and insurance. In general, because of their nonlinearity, it is hard to calculate the Choquet expectation and minimax expectation. In the first part, the paper will calculate some special Choquet expectations and minimax expectations with the method of theorem 1 of [1] on the base of backward stochastic differential equations (BSDEs). In the second part, also on the base of BSDEs, the paper will consider the problem of independence of events and random variables under two special patterns of g: g =μ|z| and g = -μ|z|. And, the paper proves a necessanty of g-independence of two random variables under the two given patterns using examples.In the first chapter, the paper introduces the basic backgrounds and contents of the Choquet expectation, minimax expectation, and Peng's g-expectation. Especially, after the appearence of BSDEs, g-expectation and conditional g-expectation, they make great contributions on the development of nonlinear expectation. Choquet expectation, minimax expectation and Peng's g-expectation improve each other, and have created great value in mathematics, economics and finance. In addition, this chapter introduces the problem of independence of events and random variables after the appearence of BSDEs and proves the necessarities of g-independence of events and random variables.In the second chapter, the first three sections make particular introductions of Choquet expectation, minimax expectation and Peng's g-expectation. Among them, the third section introduces the Choquet expectation and minimax expectation from the view of option pricing and makes out the fact that Choquet expectation and minimax expectation are equal in numerical value for some European options. The last section calculates the minimax expectations and Choquet expectations of some special...