Some Properties and Convergence Theorems of Conditional g-expectation

The research is sponsored by the national natural science fund (NO. 70971079). Abstract Peng Shige introduced the notions of g-expectation and conditional g-expectation, which is the nonlinear extension of classic mathematical expectation and conditional mathematical expectation. Just as g-expectation preserves most properties of classic mathematical expectation except the linearity, conditional g-expectation preserves most properties of classic conditional mathematical expectation except the linearity. Based on the properties and theorems of classic conditional mathematical expectation, this paper will discuss and sum up some properties and convergence theorems of conditional g-expectation.

Let (Ω, F, P) be a probability space, in which {B t } 0≤t≤T is a d-dimensional standard Brownian motion.Let {F t } 0≤t≤T be the filtration generated by this Brownian motion denoted by F t = σ{B s , s ≤ t}.For ∀T ∈ [0, ∞), we define the following two spaces first: L 2 (0, F t , P; R) := {X : X t is a F t -adapted process and X 2 L := E T 0 |X s | 2 ds < ∞}; L 2 (Ω, F t , P; R) := {ξ : ξ is a F T -measurable random variable and E|ξ| 2 ds < ∞}.
Definition 1.1(g-expectation and conditional g-expectation) For any given T ≥ 0, ξ ∈ L 2 (Ω, F T , P; R), we assume the generator g satisfis (H 1 ) and (H 2 ), (y ξ,g,T t , z ξ,g,T t ) is the adapted solution to BSDE(1), then we can call ε g [ξ] = y ξ,g,T 0 and ε g [ξ\F t ] = y ξ,g,T t are the mathematical expectation and conditional mathematical expectation generated by function g, denoted by g-expectation and conditional g-expectation shortly.
Since g-expectation and conditional g-expectation can be considered as the extension of classic mathematical expectation and conditional mathematical expectation, they preserve most properties of classic mathematical expectation and conditional mathematical expectation except the linearity.Many scholars have studied all kinds of properties and convergence theorems and Jensen inequality of g-expectation, such as Peng Shige (Peng, 1997, p. 141-159), Cheng Zengjing (Chen, 1999, p. 175-180), Jang Long (Jiang, 2003, p. 13-17)and (Jiang, 2004, p. 401-412), Zhang Hui (Zhang, 2005, p. 29-30), Lin Qun (Lin, 2008, p. 28-30)and so on.This paper will discuss and sum up some properties and convergence theorems of conditional g-expectation basing on the properties and theorems of the classic conditional mathematical expectation given by the previous studies.
Under the above assumptions(g satisfies the assumptions (H 1 ), (H 2 ), (H 3 )) and the definition of g-expectation, let ξ n , n ≥ 1 be a random variable sequence and ξ, η ∈ L 2 (Ω, F T , P), we can have the following theorems: Theorem 3.2 (Monotone convergence theorem of conditional g-expectation) Proof: Here we just proof the situation of(i), the situation of(ii)can be proved similarly.
From the monotonicity of conditional g-expectation we know that ε g [ξ n \F t ] progressively increase with n.
For ∀B ∈ F t , from properties(v)and(ii)we have: , so the theorem is proved.
Theorem 3.3 (Fatou lemma of conditional g-expectation) Proof: Here we also just proof the situation of(i), the situation of(ii)can be proved similarly.
Let η n = inf k≥n ξ k , then we have η ≤ η n ↑ lim n ξ n a.s.. Besides, from lemma 3.1 we have Then the inequality is proved.
Theorem 3.4 (Lebesgue dominated convergence theorem of conditional g-expectation) If |ξ n | ≤ η and ξ n → ξ a.s.when n → ∞, then we have that lim Then the theorem is proved.
For classic conditional mathematical expectation we have the following Jensen inequality (Wang, J. G, 2005): If ξ and ϕ(ξ) are integrable random variables and ϕ(ξ) is a convex function in R, then we have To proof it, we need an important property of classic conditional expectation: However, for general g, conditional g-expectation doesn't have the property: for ∀η ∈ F t