Reasearch About Non-linear Expectation And Related Questions

Derivative securities have been mentioned as one of major causes of the continuing global financial crisis. Major banks, financial institutions, and insurance companies, such as American International Group (AIG), Lehman Brothers, and Merrill Lynch, collapsed or experienced financial problems partly because; of their massive exposures to the risks from trading derivative securities. There is a huge profit in trading derivative securities, but, trading derivative securities is risky. It is, therefore, important to develop appro-priate methods to assess, manage and control the risks inherent from trading derivative securities. One of the challenging issues for evaluating and managing risk of derivative securities is that the risk behavior of these securities is nonlinear. However, it seems that rigorous theories for describing, evaluating and managing nonlinear risk be-havior are lacking or still at their infancy. This motivates the quest for rigorous theories for describing and understanding nonlinear risk.The future uncertain risk is regarded as a random variable X, and then given a functional E, the functional E[X] is known as a risk measure or mathematical expecta-tion. Generally speaking, the study of financial derivatives risk of nonlinear behavior is the study of nonlinear mathematical expectation.Pardoux and Peng (1990) first studied the following nonlinear backward stochastic differential equation (BSDE for short): and showed that the above BSDE has a unique solution. Based on further investigating the properties of BSDEs, Peng (1997) introduced the notions of g-expectations and con-ditional g-expectations via a class of BSDEs. It is worth noting that Peng g-expectation is the first suitable one to describe the nonlinear behavior of the risk, because it has all the properties of linear expectation except linearity. And conditions of g-expectation is the first dynamically consistent nonlinear expectations.To study stock market under volatility uncertainty, Peng proposed the general non-linear expectations without a probability space. Peng introduced the notions of G-normal ditribution, G-Brownian motion, constructed the Ito’s integral with respect to G-Brownian motion, obtained G-Ito’s formula and a series of results, established a set of complete theoretical framework.It is that nonlinear probability theory is closely related to nonlinear expectations theory, sometimes, we also known as capacity theory. For the capacity theory, in 1953, there are two important paper:one is the Choquet (1953) and the other one is Shaply (1953). From the potential theory, Choquet introduced and understand the capacity theory, and puts forward the definition of Choquet expectations. From the cooperative game, Shaply introduced and understand the capacity theory, puts forward Shaply value which has been widely applied in the field of cooperative game.In the nonlinear expectation theory frame, there are many theoretical issues need to study, and an important research issue is limits theory. Peng has proved that a weak law of large Numbers and central limit theorem in the nonlinear expectation theory frame, see Peng (2010). In Chen (2010), Professor Chen has proved strong law of large numbers in the nonlinear expectation theory frame, known as " the strong law of large Numbers for capacities." Professor Chen asserted that the sample mean, don’t like in the classical theory of probability convergence in expectations, but falls in the interval between upper and lower expectations of a random variable. Since then, many scholars begin to pay close attention to this field about limits theory in the nonlinear expectation theory frame. Based on those paper above, this paper is go on to study limits theory in the nonlinear expectation theory frame and G-Brownian motion.In this doctoral thesis, the main research contents include:(A) We introduce the concept of comonotone random set, and based on this, we obtain some properties of Choquet integral of set-valued mappings:(1) the Choquet integral of set-valued mappings has comonotonous additive properties, positive homogeneity, translation invariant, sup-norm continuity, etc. (2) the Fubini theorem of the Choquet integral of set-valued mappings.(B) We study the weak law of large numbers for sublinear expectations. The assump-tions in our theorem is consistent with the classical weak law of large numbers. We also gives some specific nonlinear expectations, such as the mean deviation function-al and one-sided moment coherent risk measure, and studied the weak law of large number in these specific sublinear expectations.(C) We introduce the negatively dependent notion in the capacity frame and give an example to show the possibility of this concept. Based on the negatively dependent notion, we study the strong law of large numbers for capacities, and give some applications of the strong law of large numbers:renewal theorem, sub-entropy and sup-entropy, invariance principle. We also obtain the second Borel-Cantelli lemma without assumption of independence and sub-linearity. Also, in no independence, no subadditivity of nonlinear expectation of assumptions, we got(D) Since Peng introduced the G Brownian motion, the research on G-Brownian mo-tion has been very active. We will study the Hausdorff dimension of G-Brownian motion and extend G-Ito formula:(1) in the capacity theory frame, we get that the Hausdorff dimension of one-dimensional G-Brownian motion; (2) we obtain an extension of G-Ito’s formula for the G-Brownian motion and study the G-Ito’s for-mula for a function φ which be an absolutely continuous function with locally square integrable derivative.