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Convergent Theorems for the Generalized G-expectation

Author: YuanJinJian
Tutor: ShiYuFeng
School: Shandong University
Course: Financial Mathematics and Financial Engineering
Keywords: Backward Stochastic Differential Equations g-expectation General g-expectation Comparison theorem Convergence theorem
CLC: O211.63
Type: Master's thesis
Year: 2008
Downloads: 82
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Abstract


This paper discusses the infinite horizon backward stochastic differential equations, and g-expectation of some nature, given the definition of a general g-expectation, and prove a general g-expectation meet Levi theorems, Fatou theorem and Lebesgue dominated convergence theorem. Backward stochastic differential equations (abbreviated as BSDE) was first in the literature [9] introduced by Pardoux and Peng. Forward stochastic differential equation solution to determine the state (initial conditions) today becomes tomorrow's general state of uncertainty, to study the statistical laws; Backward Stochastic Differential Equations tomorrow (generally uncertain) goals into today to determine the status, in order to develop today's decision-making. Backward stochastic differential equations theory has been widely accepted and applied, interesting because of its theoretical specific system properties, and because it was found important applications: it provides a solution to the mathematical financial problems theoretical price of an effective framework, such as found in [2] El Karoui Quenez, many important financial markets, derivative securities (such as options, futures, etc.) can be solved using backward stochastic differential equations; [10] Duffie and Epstein can use it to describe the uncertain economic environment, consumer preferences (ie utility function theory that is the foundation of econometrics): Peng in [11] by backward stochastic differential equations to obtain the nonlinear Feynman-Kac formula, which can be used to deal with such as reaction-diffusion equation and the Navier-Stokes equations and other well-known nonlinear partial differential equations. Peng in the literature [1] first proposed the concept of g-expectation and conditional g-expectation, Peng, Briand, Coquet then study some properties of g-expectations and conditions of g-expectations, and successfully promote these nature to nonlinear mathematical expectation, has made remarkable achievements. As backward promotion of the theory of stochastic differential equations, of Chen Zengjing and Wang Bo in the literature [7] proved the existence and uniqueness of solution of the infinite horizon backward stochastic differential equations, and at this time given by equation g - the definition of expectations. In [6] for general integrable random variable Chen Zengjing with another way to define the g-expectation: not by solving the corresponding backward stochastic differential equations, but to use the operator extension, Peng given square integrable random variable g-expectation Extension to integrable random variable space. The combination of these two results, the money is quietly waiting in the literature [8] using a similar method to promote the results of [6] to the infinite time interval, promotion g-expectation defined. Comparison theorem is a classic result of backward stochastic differential equation theory, it is peng [12] on the basis of Pardoux-Peng [13] and El Karoui,-Peng-Quenez [2] summarized out. The role of the theorem is: When the backward stochastic differential equation can compare two different final value conditions and generate a factor of the theorem, we can compare their solutions. Application similar to Peng's method of proof in this article prove the infinite time interval comparison theorems of backward stochastic differential equations. With the top of the content as the basis, this paper presents a general g-expectation defined and simple to prove some properties, the method of proof of these properties and the classical convergence theorem, given the general g-expectation 3 convergence The proof of the theorem.

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CLC: > Mathematical sciences and chemical > Mathematics > Probability Theory and Mathematical Statistics > Theory of probability ( probability theory, probability theory ) > Random process > Stochastic differential equation
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