Symmetric Property Of A Kind Of G-Probability

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.