Additivity Of G-Expectation For Affine-ralated Random Variables

In 1990, Pardoux and Peng introduced the following backward stochastic differential equation (BSDE):and proved there exists unique adopted solution to this equation. Thus he introduced a nonlinear mathematics expectation — g-expectation, which can define conditional expectation, according to the solution of BSDE:From then on, lots of scholars have interested in this field and now this result have been applied to finance, economic and the other branches of mathematics.As a nonlinear mathematics expectation, g-expectation is not additive for general random variables. We know some nonlinear mathematics expectations is additive for special random variables, for example, Minimum expectation is additive for all affine-ralated random variables. But the additivity of g-expectation is very rigorous, it is not additive even for affine-ralated random variables. A natural problem is can we make g-expectation additive for affine-ralated random variables by restrict the form of g? It is just the main question we research in this paper.Consider the case of 1-dim, when g does not contains random item, g(y, z, t): R~2×[0, T]→ R~1. We study additivity of g-expectation for affine-ralated variables . The main idea is : if g-expectation satisfies positive homogeneity and translability, g-expectation is additive for affine-ralated variables; but under condition that (?)(y, t) ∈ R~m ×[0, T] have g(y, 0, t) = 0 (i.e. condition (A3) in this paper), when g does not depend on y, g-expectation satisfies translability, when g is positive homogeneity with z, g-expectation satisfies positive homogeneity, so we guess whether when g does not depend on y and positive homogeneity with z, g-expectation is additive for affine-ralated variables under (A3). This paper affirms the supposition and point out it is just the necessary and sufficient condition for g-expectation is additive for affine-ralated random variables(Chapter 3 Theorem 3.1.1).Theorem 0.1 Soppose g satisfies (A1)-(A3). If ε_g[·] is additive for all affine-ralated random variables in L~2(Ω,F_T, P) (i.e. (?)ξ, η∈L~2(Ω,F_T,P) and ξ,η are affine-ralated, we have ε_g[ξ + η] = ε_g[ξ| + ε_g[η]), thenghave formg = μ_t|z_t| +v(t)z_t, (0.1)where μ_t, v(t)are continuous functions on [0,T]. Contrarily,if g has the form (0.1), ε_g[·]is additive for all affine-ralated random variables in L~2(Ω, F_T, P).We konw,(A3) is a very strong condition, for example, when we use BSDE into option pricing, the generator g may not satisfies g(y, 0, t) ≡ 0, (?)t ∈ [0, T], so we study the necessary and sufficient condition for g-expectation is additive for affine-ralated random variables without (A3)(Chapter 3 Theorem 3.1.2).Theorem 0.2 Soppose g satisfies (A1)-(A3). If ε_g[·] is additive for all affine-ralated random variables in L~2(Ω, F_T, P), thenghave formg = μ_t|z_t|+v(t)z_t + v'_ty_t, (0.2)where μ_t, v(t) are continuous functions on [0,T]. Contrarily,if g has the form (0.2), ε_g[·] is additive for all affine-ralated random variables in L~2(Ω , F_T, P).Then we compare g-expectation which have the form of g above and Minimum expectation, and point out g-expectation can not cover Minimum expectation. (These productions above have been incepted by Journal of Shandong University (Natural Science).) At last we generalize these productions into conditional g-expectation, and get conclusions similar with Theorem 0.1 and Theorem 0.2, and study the relationship between additivity of g-expectation for atfine-ralated random variables and additivity of conditional g-expectation for affine-ralated random variables.