Central Moment And Original Moment Based On G-expectation

In 1990.Pardoux and Peng introduced the following backward stochastic differential equation (BSDE):and proved there exists unique adopted solution to this equation.From then on,lots of scholars have interested in this field and now this result have been applied to finance,cconomic and the other branches of mathematics. Furthcrmore,in1997.Peng found that the solutions of BSDE equipped very good properties when generators satisfies special condition:g(y,0. t) — 0 (t,y) (?) [0, T] ×R.Thus he introduced a nonlinear mathematics expectation g-expectation .which can define conditional expectation, according to the solution of BSDE:In this paper .1 defined some new operators:g-variance,conditional g-variance ,higher central moment and higher original moment based on g-expectation.Moreover.I pay more attention to the following questions:l.When does the operator g-variance retain the classical variance's properties?If g does not equal to 0 and satisfies some special conditions.does it show the special properties belonging to itself? Can we find the relationship between g-probability, g-expectation and g-variance?2.Is there the reverse theorem of conditional g-variance?Can the space L~4Ω,F, P be instead by a large one?3.Obviously,the related coefficient based on g-expectation does not reflect to the linear relationship between two random variables.Which kinds of relationships between the two random variables does it reflect .including linear relationship or not?...