【作者】 张静；

【导师】 陈增敬；

【作者基本信息】 山东大学 ， 概率统计， 2007， 硕士

【摘要】 Choquet(1953)把概率测度扩展到一类非线性测度，称之为容度。并通过下面的式子定义了一类非线性数学期望-Choquet期望：C(ξ)：=integral from n=-∞to 0[V(ξ≥t)-1]dt+integral from n=0 to∞V(ξ≥t)dt。Choquet期望处理不确定性问题，在统计，经济，金融和物理中都有非常广泛的应用。许多文章研究了它的性质和应用，具体可见Anger(1977)，Dellacherie(1970)，Graf(1980)，Sarin and Wakker(1992)，Schmeidler(1989)，Wakker(2001)，Wasserman and Kadane(1990)等等。Choquet容度在稳健性分析，决策论和对策论中也有很广泛的应用。其中一类重要的Choquet容度是对称相关容度，在稳健性分析中许多容度都是对称相关容度，或者可以通过一对一光滑映射到对称相关容度。[Buja(1986)，Huber and Strassen(1973)，Wasserman and Kadane(1990)and Fortini and Ruggeri(1994)]。许多其他文章也研究了对称容度，具体可见Armstrong(1990)，Dempster(1967，1968)，Anger and Lembcke(1985)，Walley(1991)，Talagrand(1978)，Wasserman and Kadane(1992)等等。Peng and Pardoux(1990)提出倒向随机微分方程(以下简写BSDE).Peng(1997)通过BSDE的解定义了g-期望和条件g-期望，并证明了在生成元g和终端值ξ满足一定条件时，g-期望和条件g-期望保持了经典数学期望除了线性性以外的一切性质。g-期望在金融中得到了广泛的应用，可参见Chen and Epstein(2002)。Chen(2005)研究了g-期望和Choquet期望之间的关系，并给出了一个充分必要条件。在这篇文章里，我们给出了g-概率的对称性的概念。在g-期望中，有一类g-期望和Choquet期望等价，我们就研究用这类g-期望定义的g-概率的对称性问题。更多还原

【Abstract】 Choquet (1953) extended the probability measure P in (0,1) to a nonlinear probability measure V (also called the capacity) and obtain the following definition C(ξ) of nonlinear mathematical expectations (called the Choquet expectation):Choquet expectations, which are used to deal with uncertain phenomena, have many applications in statistics, economics, finance and physics. Many papers study the Choquet expectation and its applications, see, for example, Anger (1977),Dellacherie (1970), Graf (1980), Sarin and Wakker (1992), Schmeidler (1989), Wakker (2001), Wasserman and Kadane (1990) and the references therein.Choquet capacities are widely used in robustness, decision theory and game theory. A kind of Choquet capacities, which is called symmetric,coherent is really important. Many capacities that arise in robustness are symmetric, coherent capacities or can be transformed into the same by a smooth,one-to-one mapping.[Buja (1986),Huber and Strassen (1973), Wasserman and Kadane (1990) and Fortini and Ruggeri(1994)]. Many other papers study symmetric capacities, see, for examples, Armstrong (1990), Dempster (1967,1968), Anger and Lembcke (1985), Walley (1991), Talagrand (1978), Wasserman and Kadane (1992) and so on.Peng and Pardoux (1990) introduced a kind of equation called backward stochastic differential equation (BSDE).Peng (1997) introduced the notion of g-expectation via backward stochastic differential equation. He showed that under suitable square integrability assumptions on the coefficient g and the terminal value ξ, the g-expectation of random variable ξ preserve many of the basic properties (except linearity) of the convenient mathematical expectation. g-expectation has been applied in finance, see, for example, Chen and Epstein (2002).Chen (2005) studied the relation between g-expectation and Choquet expectation and provided a necessary and sufficient condition.In this paper, we define symmetric property of g-probability. Chen (2005) proved that a kind of g-expectation is equal to Choquet expectation. We study the symmetric property of g-probability, which is defined by this kind of g-expectation.更多还原