G-expectation

In this paper, we mainly discuss this problem: If random variables ζ, η∈ L² (Ω,F_T ,P,R), and P(ζ ≤ x) = P(η≤ x), (?)x ∈ R, then does there always exist ζ_g[ζ] = ε_g[η]? In fact that is true if and only if ε_g[·] degenerate to E[·].In 1990, Peng and Pardoux gave the well known theorem that one kind of nonlinear backward stochastic differential equation (BSDE in short) has a unique pair of solution (y_t, z_t). Then Peng showed that the solution of BSDE at the time 0 (marked as y₀ ) can be considered as a new kind of expectation. Peng named it g-expectation.Then Chen published a series of papars, which showed that if g satisfy some properties, then the corresponding ε_g[·] has the same properties as the linear expectation E[·].In this papaer, we also compare the g-expectation with the classical expectation, and we will show a property of the g-expectation:suppose that g(y, z, t) is a deterministic function, (?)(y, z, t) ∈R × R^d × [0. T], and g not depend on y, then the following conclusions are equivalent, (i) ε_g[ζ]= ε_g[η], for all ζ,η∈ L²(Ω, F_T, P; R); and ζ, 77 have the same distribution respect with P. (ii) g(z,t) ≡ 0, (?)(z,t) ∈R^d×[0,T],...