Representation Theorem for Generators of BSDEs Driven by 𝐺-Brownian Motion and Its Applications

Abstract

We obtain a representation theorem for the generators of BSDEs driven by 𝐺-Brownian motions and then we use the representation theorem to get a converse comparison theorem for 𝐺-BSDEs and some equivalent results for nonlinear expectations generated by 𝐺-BSDEs.

1. Introduction

Let (Ω,ℱ,𝑃) be a probability space, and, for fixed 𝑇∈[0,+∞), let (𝐵_𝑡)_{0≤𝑡≤𝑇} be a standard Brownian motion and let ℱ_𝑡 be the augmentation of 𝜎{𝐵_𝑠,0≤𝑠≤𝑡}. Then Pardoux and Peng [1] introduced the backward stochastic differential equations (BSDEs) and proved the existence and uniqueness result of the BSDEs. In 1997, Peng [2] promoted 𝑔-expectations based on BSDEs. One of the important properties of 𝑔-expectations is comparison theorem or monotonicity. Chen [3] first considers a converse result of BSDEs under equal case. After that, Briand et al. [4] obtained a converse comparison theorem for BSDEs under general case. They also derived a representation theorem for the generator 𝑔. Following this paper, Jiang [5] discussed a more general representation theorem then, in his another paper [6], showed a more general converse comparison theorem. Here the representation theorem is an important method in solving the converse comparison problem and other problems (see Jiang [7]).

Peng [8–13] defined the 𝐺-expectations and 𝐺-Brownian motions (𝐺-BMs) and proved the representation theorem of 𝐺-expectation by a set of singular probabilities, which differs from nonlinear 𝑔-expectations because 𝑔-expectations are equivalent with a group of absolutely continuous probabilities with respect to the probability measure 𝑃. Soner et al. [14] obtained an existence and uniqueness result of 2 BSDEs. Recently, Hu et al. [15] proved another existence and uniqueness result on BSDEs driven by 𝐺-Brownian motions (𝐺-BSDEs).

An important advantage of 𝐺-BSDEs is the easiness to define the nonlinear expectations. Hu et al. in [16] gave a comparison theorem for 𝐺-BSDEs and talked about the properties of corresponding nonlinear expectations. In this paper, we consider the representation theorem for generators of 𝐺-BSDEs and then consider the converse comparison theorem of 𝐺-BSDEs and some equivalent results for nonlinear expectations generated by 𝐺-BSDEs. In the following, in Section 2, we review some basic concepts and results about 𝐺-expectations. We give the representation theorem of 𝐺-BSDEs in Section 3. In Section 4, we consider the applications of representation theorem of 𝐺-BSDEs, which contain the converse comparison theorem and some equivalent results for nonlinear expectations generated by 𝐺-BSDEs.

2. Preliminaries

We review some basic notions and results of 𝐺-expectation, the related spaces of random variables, and the backward stochastic differential equations driven by a 𝐺-Brownian motion. The readers may refer to [10, 13, 15, 17–19] for more details.

Definition 1. Let Ω be a given set and let ℋ be a vector lattice of real valued functions defined on Ω, namely, 𝑐∈ℋ for each constant 𝑐 and |𝑋|∈ℋ if 𝑋∈ℋ. ℋ is considered as the space of random variables. A sublinear expectation 𝔼 on ℋ is a functional 𝔼∶ℋ→ℝ satisfying the following properties: for all 𝑋,𝑌∈ℋ, one has (a)monotonicity: if 𝑋≥𝑌, then 𝔼[𝑋]≥𝔼[𝑌];(b)constant preservation: 𝔼[𝑐]=𝑐;(c)subadditivity: 𝔼[𝑋+𝑌]≤𝔼[𝑋]+𝔼[𝑌];(d)positive homogeneity: 𝔼[𝜆𝑋]=𝜆𝔼[𝑋] for each 𝜆≥0. (Ω,ℋ,𝔼) is called a sublinear expectation space.

Definition 2. Let 𝑋₁ and 𝑋₂ be two 𝑛-dimensional random vectors defined, respectively, in sublinear expectation spaces (Ω₁,ℋ₁,𝔼₁) and (Ω₂,ℋ₂,𝔼₂). They are called identically distributed, denoted by 𝑋=₁^𝑑𝑋₂, if 𝔼₁[𝜑(𝑋₁)]=𝔼₂[𝜑(𝑋₂)], for all𝜑∈𝐶_𝑏⋅Lip(ℝ^𝑛), where 𝐶_𝑏⋅Lip(ℝ^𝑛) denotes the space of bounded and Lipschitz functions on ℝ^𝑛.

Definition 3. In a sublinear expectation space (Ω,ℋ,𝔼), a random vector 𝑌=(𝑌₁,…,𝑌_𝑛), 𝑌_𝑖∈ℋ, is said to be independent of another random vector 𝑋=(𝑋₁,…,𝑋_𝑚), 𝑋_𝑖∈ℋ under 𝔼[⋅], denoted by 𝑌⟂𝑋, if for every test function 𝜑∈𝐶_𝑏⋅Lip(ℝ^𝑚×ℝ^𝑛) one has 𝔼[𝜑(𝑋,𝑌)]=𝔼[𝔼[𝜑(𝑥,𝑌)]_𝑥=𝑋].

Definition 4 (𝐺-normal distribution). A 𝑑-dimensional random vector 𝑋=(𝑋₁,…,𝑋_𝑑) in a sublinear expectation space (Ω,ℋ,𝔼) is called 𝐺-normally distributed if for each 𝑎,𝑏≥0 one has 𝑎𝑋+𝑏𝑋=^𝑑𝑎²+𝑏²𝑋, (1) where 𝑋 is an independent copy of 𝑋; that is, 𝑋=^𝑑𝑋 and 𝑋⟂𝑋. Here, the letter 𝐺 denotes the function 𝐺(𝐴)1∶=2𝔼[⟨𝐴𝑋,𝑋⟩]∶𝕊_𝑑⟶ℝ, (2) where 𝕊_𝑑 denotes the collection of 𝑑×𝑑 symmetric matrices.

Peng [13] showed that 𝑋=(𝑋₁,…,𝑋_𝑑) is 𝐺-normally distributed if and only if for each 𝜑∈𝐶_𝑏⋅Lip(ℝ^𝑑), 𝑢(𝑡,𝑥)∶=𝔼[𝜑(𝑥+√𝑡𝑋)], (𝑡,𝑥)∈[0,∞)×ℝ^𝑑, is the solution of the following 𝐺-heat equation: 𝜕_𝑡𝑢−𝐺𝐷²_𝑥𝑢=0,𝑢(0,𝑥)=𝜑(𝑥). (3)

The function 𝐺(⋅)∶𝕊_𝑑→ℝ is a monotonic, sublinear mapping on 𝕊_𝑑 and 𝐺(𝐴)=(1/2)𝔼[⟨𝐴𝑋,𝑋⟩]≤(1/2)|𝐴|𝔼[|𝑋|²] implies that there exists a bounded, convex, and closed subset Γ⊂𝕊⁺_𝑑 such that 𝐺(𝐴)=12sup_𝛾∈Γtr𝛾𝐴, (4) where 𝕊⁺_𝑑 denotes the collection of nonnegative elements in 𝕊_𝑑.

In this paper, we only consider nondegenerate 𝐺-normal distribution; that is, there exists some 𝜎²>0 such that 𝐺(𝐴)−𝐺(𝐵)≥𝜎²tr[𝐴−𝐵] for any 𝐴≥𝐵.

Definition 5. (i) Let Ω=𝐶^𝑑₀(ℝ⁺) denote the space of ℝ^𝑑-valued continuous functions on [0,∞) with 𝜔₀=0 and let 𝐵_𝑡(𝜔)=𝜔_𝑡 be the canonical process. Set 𝐿_𝑖𝑝(Ω)∶=𝜑𝐵_𝑡1,…,𝐵_𝑡𝑛∶𝑛≥1,𝑡₁,…,𝑡_𝑛∈[0,∞),𝜑∈𝐶_𝑏⋅Lipℝ^𝑑×𝑛. (5) Let 𝐺∶𝕊_𝑑→ℝ be a given monotonic and sublinear function. 𝐺-expectation is a sublinear expectation defined by 𝔼[𝑋]=𝔼𝜑𝑡₁−𝑡₀𝜉₁,…,𝑡_𝑚−𝑡_𝑚−1𝜉_𝑚, (6) for all 𝑋=𝜑(𝐵_𝑡1−𝐵_𝑡0,𝐵_𝑡2−𝐵_𝑡1,…,𝐵_𝑡𝑚−𝐵_𝑡𝑚−1), where 𝜉₁,…,𝜉_𝑛 are identically distributed 𝑑-dimensional 𝐺-normally distributed random vectors in a sublinear expectation space (Ω,ℋ,𝔼) such that 𝜉_𝑖+1 is independent of (𝜉₁,…,𝜉_𝑖) for every 𝑖=1,…,𝑚−1. The corresponding canonical process 𝐵_𝑡=(𝐵^𝑖_𝑡)^𝑑_𝑖=1 is called a 𝐺-Brownian motion.
(ii) For each fixed 𝑡∈[0,∞), the conditional 𝐺-expectation 𝔼_𝑡 for 𝜉=𝜑(𝐵_𝑡1−𝐵_𝑡0,𝐵_𝑡2−𝐵_𝑡1,…,𝐵_𝑡𝑚−𝐵_𝑡𝑚−1)∈𝐿_𝑖𝑝(Ω), where without loss of generality we suppose 𝑡_𝑖=𝑡, is defined by 𝔼_𝑡𝜑𝐵_𝑡1−𝐵_𝑡0,𝐵_𝑡2−𝐵_𝑡1,…,𝐵_𝑡𝑚−𝐵_𝑡𝑚−1=𝜓𝐵_𝑡1−𝐵_𝑡0,𝐵_𝑡2−𝐵_𝑡1,…,𝐵_𝑡𝑖−𝐵_𝑡𝑖−1, (7) where 𝜓𝑥₁,…,𝑥_𝑖=𝔼𝜑𝑥₁,…,𝑥_𝑖,𝐵_𝑡𝑖+1−𝐵_𝑡𝑖,…,𝐵_𝑡𝑚−𝐵_𝑡𝑚−1. (8)

For each fixed 𝑇>0, we set 𝐿_𝑖𝑝Ω_𝑇∶=𝜑𝐵_𝑡1,…,𝐵_𝑡𝑛∶𝑛≥1,𝑡₁,…,𝑡_𝑛∈[0,𝑇],𝜑∈𝐶_𝑏⋅Lipℝ^𝑑×𝑛. (9) For each 𝑝≥1, we denote by 𝐿^𝑝_𝐺(Ω) (resp., 𝐿^𝑝_𝐺(Ω_𝑇)) the completion of 𝐿_𝑖𝑝(Ω) (resp., 𝐿_𝑖𝑝(Ω_𝑇)) under the norm ‖𝜉‖_𝑝,𝐺=(𝔼[|𝜉|^𝑝])^1/𝑝. It is easy to check that 𝐿^𝑞_𝐺(Ω)⊂𝐿^𝑝_𝐺(Ω) for 1≤𝑝≤𝑞 and 𝔼_𝑡[⋅] can be extended continuously to 𝐿¹_𝐺(Ω).

For each fixed 𝐚∈ℝ^𝑑, 𝐵^𝐚_𝑡=⟨𝐚,𝐵_𝑡⟩ is a 1-dimensional 𝐺_𝐚-Brownian motion, where 𝐺_𝐚(𝛼)=(1/2)(𝜎²_𝐚𝐚𝑇𝛼⁺−𝜎²_{−𝐚𝐚𝑇}𝛼⁻), 𝜎²_𝐚𝐚𝑇=2𝐺(𝐚𝐚^𝑇) and 𝜎²_{−𝐚𝐚𝑇}=−2𝐺(−𝐚𝐚^𝑇). Let 𝜋^𝑁_𝑡={𝑡^𝑁₀,…,𝑡^𝑁_𝑁}, 𝑁=1,2,…, be a sequence of partitions of [0,𝑡] such that 𝜇(𝜋^𝑁_𝑡)=max{|𝑡^𝑁_𝑖+1−𝑡^𝑁_𝑖|∶𝑖=0,…,𝑁−1}→0; the quadratic variation process of 𝐵^𝐚 is defined by 𝐵^𝐚_𝑡=lim_{𝜇𝜋𝑁𝑡→0}^𝑁−1_𝑗=0𝐵^𝐚_{𝑡𝑁𝑗+1}−𝐵^𝐚_𝑡𝑁𝑗². (10) For each fixed 𝐚,𝐚∈ℝ^𝑑, the mutual variation process of 𝐵^𝐚 and 𝐵^𝐚 is defined by 𝐵^𝐚,𝐵^𝐚_𝑡=14𝐵^𝐚+𝐚_𝑡−𝐵^𝐚−𝐚_𝑡. (11)

Definition 6. For fixed 𝑇>0, let 𝑀⁰_𝐺(0,𝑇) be the collection of processes in the following form: for a given partition {𝑡₀,…,𝑡_𝑁}=𝜋_𝑇 of [0,𝑇], 𝜂_𝑡(𝜔)=^𝑁−1_𝑗=0𝜉_𝑗𝐼_{[𝑡𝑗,𝑡𝑗+1)}(𝑡), (12) where 𝜉_𝑗∈𝐿_𝑖𝑝(Ω_𝑡𝑗), 𝑗=0,1,2,…,𝑁−1. For 𝑝≥1, one denotes by 𝐻^𝑝_𝐺(0,𝑇), 𝑀^𝑝_𝐺(0,𝑇) the completion of 𝑀⁰_𝐺(0,𝑇) under the norms ‖𝜂‖_𝐻^𝑝_𝐺={𝔼[(∫^𝑇₀|𝜂_𝑠|²𝑑𝑠)^𝑝/2]}^1/𝑝, ‖𝜂‖_𝑀^𝑝_𝐺={𝔼[∫^𝑇₀|𝜂_𝑠|^𝑝𝑑𝑠]}^1/𝑝, respectively.

For each 𝜂∈𝑀¹_𝐺(0,𝑇), we can define the integrals ∫^𝑇₀𝜂_𝑡𝑑𝑡 and ∫^𝑇₀𝜂_𝑡𝑑⟨𝐵^𝐚,𝐵^𝐚⟩_𝑡 for each 𝐚, 𝐚∈ℝ^𝑑. For each 𝜂∈𝐻^𝑝_𝐺(0,𝑇;ℝ^𝑑) with 𝑝≥1, we can define Itô's integral ∫^𝑇₀𝜂_𝑡𝑑𝐵_𝑡.

Let 𝑆⁰_𝐺(0,𝑇)={ℎ(𝑡,𝐵_𝑡1∧𝑡,…,𝐵_{𝑡𝑛∧𝑡})∶𝑡₁,…,𝑡_𝑛∈[0,𝑇],ℎ∈𝐶_𝑏,Lip(ℝ^𝑛+1)}. For 𝑝≥1 and 𝜂∈𝑆⁰_𝐺(0,𝑇), set ‖𝜂‖_𝑆^𝑝_𝐺={𝔼[sup_{𝑡∈[0,𝑇]}|𝜂_𝑡|^𝑝]}^1/𝑝. Denote by 𝑆^𝑝_𝐺(0,𝑇) the completion of 𝑆⁰_𝐺(0,𝑇) under the norm ‖⋅‖_𝑆^𝑝_𝐺.

We consider the following type of 𝐺-BSDEs (in this paper, we always use Einstein convention): 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝑠+^𝑇_𝑡𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠−𝐾_𝑇−𝐾_𝑡, (13) where 𝑓𝑡,𝜔,𝑦,𝑧,𝑔_𝑖𝑗𝑡,𝜔,𝑦,𝑧∶[0,𝑇]×Ω_𝑇×ℝ×ℝ^𝑑⟶ℝ, (14) satisfy the following properties.(H1)There exists some 𝛽>1 such that for any 𝑦,𝑧,𝑓(⋅,⋅,𝑦,𝑧),𝑔_𝑖𝑗(⋅,⋅,𝑦,𝑧)∈𝑀^𝛽_𝐺(0,𝑇).(H2)There exists some 𝐿>0 such that 𝑓𝑡,𝑦,𝑧−𝑓𝑡,𝑦^′,𝑧^′+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑡,𝑦,𝑧−𝑔_𝑖𝑗𝑡,𝑦^′,𝑧^′≤𝐿𝑦−𝑦^′+𝑧−𝑧^′. (15)

For simplicity, we denote by 𝔖^𝛼_𝐺(0,𝑇) the collection of processes (𝑌,𝑍,𝐾) such that 𝑌∈𝑆^𝛼_𝐺(0,𝑇), 𝑍∈𝐻^𝛼_𝐺(0,𝑇;ℝ^𝑑), 𝐾 is a decreasing 𝐺-martingale with 𝐾₀=0 and 𝐾_𝑇∈𝐿^𝛼_𝐺(Ω_𝑇).

Definition 7. Let 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇) and 𝑓 and 𝑔_𝑖𝑗 satisfy (H1) and (H2) for some 𝛽>1. A triplet of processes (𝑌,𝑍,𝐾) is called a solution of (13) if for some 1<𝛼≤𝛽 the following properties hold:(a)(𝑌,𝑍,𝐾)∈𝔖^𝛼_𝐺(0,𝑇);(b)𝑌_𝑡=𝜉+∫^𝑇_𝑡𝑓(𝑠,𝑌_𝑠,𝑍_𝑠)𝑑𝑠+∫^𝑇_𝑡𝑔_𝑖𝑗(𝑠,𝑌_𝑠,𝑍_𝑠)𝑑⟨𝐵^𝑖,𝐵^𝑗⟩_𝑠−∫^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠−(𝐾_𝑇−𝐾_𝑡).

Theorem 8 (see [15]). Assume that 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇) and 𝑓 and 𝑔_𝑖𝑗 satisfy (H1) and (H2) for some 𝛽>1. Then, (13) has a unique solution (𝑌,𝑍,𝐾). Moreover, for any 1<𝛼<𝛽, one has 𝑌∈𝑆^𝛼_𝐺(0,𝑇), 𝑍∈𝐻^𝛼_𝐺(0,𝑇;ℝ^𝑑), and 𝐾_𝑇∈𝐿^𝛼_𝐺(Ω_𝑇).

We have the following estimates.

Proposition 9 (see [15]). Let 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇) and 𝑓, 𝑔_𝑖𝑗 satisfy (H1) and (H2) for some 𝛽>1. Assume that (𝑌,𝑍,𝐾)∈𝔖^𝛼_𝐺(0,𝑇) for some 1<𝛼<𝛽 is a solution of (13). Then, there exists a constant 𝐶_𝛼>0 depending on 𝛼, 𝑇, 𝐺, 𝐿 such that 𝑌_𝑡^𝛼≤𝐶_𝛼𝔼_𝑡𝜉^𝛼+^𝑇_𝑡ℎ⁰_𝑠𝑑𝑠^𝛼,𝔼^𝑇₀𝑍_𝑠²𝑑𝑠^𝛼/2≤𝐶_𝛼𝔼sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼+𝔼sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼1/2×𝔼^𝑇₀ℎ⁰_𝑠𝑑𝑠^𝛼1/2,𝔼𝐾_𝑡^𝛼≤𝐶_𝛼𝔼sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼+𝔼^𝑇₀ℎ⁰_𝑠𝑑𝑠^𝛼, (16) where ℎ⁰_𝑠=|𝑓(𝑠,0,0)|+∑^𝑑_𝑖,𝑗=1|𝑔_𝑖𝑗(𝑠,0,0)|.

Proposition 10 (see [15, 20]). Let 𝛼≥1 and 𝛿>0 be fixed. Then, there exists a constant 𝐶 depending on 𝛼 and 𝛿 such that 𝔼sup_{[]𝑡∈0,𝑇}𝔼_𝑡𝜉^𝛼≤𝐶𝔼𝜉^{𝛼+𝛿𝛼/(𝛼+𝛿)}+𝔼𝜉^𝛼+𝛿,∀𝜉∈𝐿_𝐺^𝛼+𝛿Ω_𝑇. (17)

Theorem 11 (see [16]). Let (𝑌^𝑙,𝑍^𝑙,𝐾^𝑙), 𝑙=1,2, be the solutions of the following 𝐺-BSDEs: 𝑌^𝑙_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌^𝑙_𝑠,𝑍^𝑙_𝑠𝑑𝑠+^𝑇_𝑡𝑔_𝑖𝑗𝑠,𝑌^𝑙_𝑠,𝑍^𝑙_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠+𝑉^𝑙_𝑇−𝑉^𝑙_𝑡−^𝑇_𝑡𝑍^𝑙_𝑠𝑑𝐵_𝑠−𝐾^𝑙_𝑇−𝐾^𝑙_𝑡, (18) where 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇), 𝑓 and 𝑔_𝑖𝑗 satisfy (H1) and (H2) for some 𝛽>1 and (𝑉^𝑙_𝑡)_𝑡≤𝑇 are RCLL processes in 𝑀^𝛽_𝐺(0,𝑇) such that 𝔼[sup_{𝑡∈[0,𝑇]}|𝑉^𝑙_𝑡|^𝛽]<∞. If   𝑉¹_𝑡−𝑉²_𝑡 is an increasing process, then 𝑌¹_𝑡≥𝑌²_𝑡 for 𝑡∈[0,𝑇].

In this paper, we also need the following assumptions for 𝐺-BSDE (13).(H3)For each fixed (𝜔,𝑦,𝑧)∈Ω_𝑇×ℝ×ℝ^𝑑, 𝑡→𝑓(𝑡,𝜔,𝑦,𝑧) and 𝑡→𝑔_𝑖𝑗(𝑡,𝜔,𝑦,𝑧) are continuous.(H4)For each fixed (𝑡,𝑦,𝑧)∈[0,𝑇)×ℝ×ℝ^𝑑, 𝑓(𝑡,𝑦,𝑧), 𝑔_𝑖𝑗(𝑡,𝑦,𝑧)∈𝐿^𝛽_𝐺(Ω_𝑡), and lim_𝜀→0+1𝜀𝔼_𝑡^𝑡+𝜀𝑓𝑢,𝑦,𝑧−𝑓𝑡,𝑦,𝑧^𝛽+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑢,𝑦,𝑧−𝑔_𝑖𝑗𝑡,𝑦,𝑧^𝛽𝑑𝑢=0. (19)(H5)For each (𝑡,𝜔,𝑦)∈[0,𝑇]×Ω_𝑇×ℝ, 𝑓(𝑡,𝜔,𝑦,0)=𝑔_𝑖𝑗(𝑡,𝜔,𝑦,0)=0.

Assume that 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇); 𝑓 and 𝑔_𝑖𝑗 satisfy (H1), (H2), and (H5) for some 𝛽>1. Let (𝑌^𝑇,𝜉,𝑍^𝑇,𝜉,𝐾^𝑇,𝜉) be the solution of 𝐺-BSDE (13) corresponding to 𝜉, 𝑓, and 𝑔_𝑖𝑗 on [0,𝑇]. It is easy to check that 𝑌^𝑇,𝜉=𝑌^𝑇′,𝜉 on [0,𝑇] for 𝑇^′>𝑇. Following [16], we can define consistent nonlinear expectation 𝔼_𝑡𝜉=𝑌_𝑡^𝑇,𝜉for𝑡∈[0,𝑇] (20) and set 𝔼[𝜉]=𝔼₀[𝜉]=𝑌₀^𝑇,𝜉.

3. Representation Theorem of Generators of 𝐺-BSDEs

We consider the following type of 𝐺-FBSDEs: 𝑋_𝑠^𝑡,𝑥=𝑥+^𝑠_𝑡𝑏𝑋_𝑢^𝑡,𝑥+𝑑𝑢_𝑠𝑡ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥𝑑𝐵^𝑖,𝐵^𝑗_𝑢+^𝑠_𝑡𝜎𝑋_𝑢^𝑡,𝑥𝑑𝐵_𝑢,_𝜀𝑌_𝑠^{𝑡,𝑥,𝑦,𝑝}=𝑦+𝑝,𝑋^𝑡,𝑥_𝑡+𝜀−𝑥+_𝑠^𝑡+𝜀𝑓𝑢,^𝜀𝑌_𝑢^{𝑡,𝑥,𝑦,𝑝},^𝜀𝑍_𝑢^{𝑡,𝑥,𝑦,𝑝}+𝑑𝑢_{𝑠𝑡+𝜀}𝑔_𝑖𝑗𝑢,^𝜀𝑌_𝑢^{𝑡,𝑥,𝑦,𝑝},^𝜀𝑍_𝑢^{𝑡,𝑥,𝑦,𝑝}𝑑𝐵^𝑖,𝐵^𝑗_𝑢−_𝑠^{𝜀𝑡+𝜀}𝑍_𝑢^{𝑡,𝑥,𝑦,𝑝}𝑑𝐵_𝑢−^𝜀𝐾^{𝑡,𝑥,𝑦,𝑝}_𝑡+𝜀−^𝜀𝐾_𝑠^{𝑡,𝑥,𝑦,𝑝}, (21)(22) where ℎ_𝑖𝑗=ℎ_𝑗𝑖 and 𝑔_𝑖𝑗=𝑔_𝑗𝑖, 1≤𝑖,𝑗≤𝑑.

We now give the main result in this section.

Theorem 12. Let 𝑏∶ℝ^𝑛→ℝ^𝑛, ℎ_𝑖𝑗∶ℝ^𝑛→ℝ^𝑛, and 𝜎∶ℝ^𝑛→ℝ^𝑛×𝑑 be Lipschitz functions and let 𝑓 and 𝑔_𝑖𝑗 satisfy (H1), (H2), (H3), and (H4) for some 𝛽>1. Then, for each (𝑡,𝑥,𝑦,𝑝)∈[0,𝑇)×ℝ^𝑛×ℝ×ℝ^𝑛 and 𝛼∈(1,𝛽), one has 𝐿^𝛼_𝐺−lim_𝜀→0+1𝜀^𝜀𝑌_𝑡^{𝑡,𝑥,𝑦,𝑝}−𝑦=𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+(𝑥𝑔𝑝,𝑏)+2𝐺_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+𝑝,ℎ_𝑖𝑗(𝑥)^𝑑_𝑖,𝑗=1. (23)

Proof. For each fixed (𝑡,𝑥,𝑦,𝑝)∈[0,𝑇)×ℝ^𝑛×ℝ×ℝ^𝑛, we write (𝑌^𝜀,𝑍^𝜀,𝐾^𝜀) instead of (^𝜀𝑌^{𝑡,𝑥,𝑦,𝑝},^𝜀𝑍^{𝑡,𝑥,𝑦,𝑝},^𝜀𝐾^{𝑡,𝑥,𝑦,𝑝}) for simplicity. We have 𝔼[|𝑋^𝑡,𝑥_𝑡+𝜀|^𝛾]<∞ for each 𝛾≥1 (see [16, 19]). Thus, by Theorem 8, 𝐺-BSDE (22) has a unique solution (𝑌^𝜀,𝑍^𝜀,𝐾^𝜀) and 𝑌^𝜀_𝑡∈𝐿^𝛼_𝐺(Ω_𝑡). We set, for 𝑠∈[𝑡,𝑡+𝜀], 𝑌^𝜀_𝑠=𝑌^𝜀_𝑠−𝑦+𝑝,𝑋_𝑠^𝑡,𝑥,𝑍−𝑥_𝜀𝑠=𝑍^𝜀_𝑠−𝜎^𝑇𝑋_𝑠^𝑡,𝑥𝑝,𝐾^𝜀_𝑠=𝐾^𝜀_𝑠. (24) Applying Itô’s formula to 𝑌^𝜀_𝑠 on [𝑡,𝑡+𝜀], it is easy to verify that (𝑌^𝜀,𝑍^𝜀,𝐾^𝜀) solves the following 𝐺-BSDE: 𝑌^𝜀_𝑠=_𝑠^𝑡+𝜀𝑓𝑢,𝑌^𝜀_𝑢+𝑦+𝑝,𝑋_𝑢^𝑡,𝑥,𝑍−𝑥_𝜀𝑢+𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑑𝑢_{𝑠𝑡+𝜀}𝑝,𝑏𝑋_𝑢^𝑡,𝑥+𝑑𝑢_{𝑠𝑡+𝜀}𝑔_𝑖𝑗𝑢,𝑌^𝜀_𝑢+𝑦+𝑝,𝑋_𝑢^𝑡,𝑥,𝑍−𝑥_𝜀𝑢+𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑠^𝑡+𝜀𝑝,ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥𝑑𝐵^𝑖,𝐵^𝑗_𝑢−_𝑠^𝑡+𝜀𝑍^𝜀_𝑢𝑑𝐵_𝑢−𝐾^𝜀_𝑡+𝜀−𝐾^𝜀_𝑠. (25) From Proposition 9, 𝑌^𝜀_𝑠^𝛼≤𝐶_𝛼𝔼_𝑠𝑠^𝑡+𝜀𝑓𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑝,𝑏𝑋_𝑢^𝑡,𝑥+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑝,ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥𝑑𝑢^𝛼,𝔼_𝑡^𝑡+𝜀𝑍^𝜀_𝑢²𝑑𝑢^𝛼/2≤𝐶_𝛼𝔼_𝑡^𝑡+𝜀𝑓𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑝,𝑏𝑋_𝑢^𝑡,𝑥+𝑝,ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥,𝜎−𝑥_𝑇𝑋_𝑢^𝑡,𝑥𝑝𝑑𝑢^𝛼+𝔼sup_{[]𝑠∈𝑡,𝑡+𝜀}𝑌^𝜀_𝑠^𝛼, (26) hold for some constant 𝐶_𝛼>0, only depending on 𝛼, 𝑇, 𝐺, and 𝐿. By Proposition 10 and the Lipschitz assumption, we obtain 𝔼sup_{[]𝑠∈𝑡,𝑡+𝜀}𝑌^𝜀_𝑠^𝛼+_𝑡^𝑡+𝜀𝑍^𝜀_𝑢²𝑑𝑢^𝛼/2≤𝐶₁𝜀^𝛼𝔼11+𝜀_𝑡^𝑡+𝜀𝑓(𝑢,0,0)^𝛽+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗(𝑢,0,0)^𝛽𝑑𝑢^𝛼/𝛽+sup_{[]𝑠∈𝑡,𝑡+𝜀}𝑋_𝑠^{𝑡,𝑥𝛽}, (27) where 𝐶₁ is a constant depending on 𝑥, 𝑦, 𝑝, 𝛼, 𝛽, 𝑇, 𝐺, and 𝐿. Noting that 𝔼[sup_{𝑠∈[𝑡,𝑡+𝜀]}|𝑋_𝑠^𝑡,𝑥|^𝛽]≤𝐶₂(1+|𝑥|^𝛽) (see [16, 19]), where 𝐶₂ depends on 𝑇 and 𝐿, and the following inequality holds: _𝑡^𝑡+𝜀𝑓(𝑢,0,0)^𝛽+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗(𝑢,0,0)^𝛽𝑑𝑢≤2^𝛽−1𝜀𝑓(𝑡,0,0)^𝛽+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗(𝑡,0,0)^𝛽+_𝑡^𝑡+𝜀𝑓(𝑢,0,0)−𝑓(𝑡,0,0)^𝛽+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗(𝑢,0,0)−𝑔_𝑖𝑗(𝑡,0,0)^𝛽𝑑𝑢. (28) Together with assumption (H4), we get 𝔼sup_{[]𝑠∈𝑡,𝑡+𝜀}𝑌^𝜀_𝑠^𝛼+_𝑡^𝑡+𝜀𝑍^𝜀_𝑢²𝑑𝑢^𝛼/2≤𝐶₃𝜀^𝛼, (29) where 𝐶₃ depends on 𝑥, 𝑦, 𝑝, 𝛼, 𝛽, 𝑇, 𝐺, and 𝐿. Now, we prove (23). Let us consider 1𝜀𝑌^𝜀_𝑡=1−𝑦𝜀𝑌^𝜀_𝑡=1𝜀𝔼_𝑡𝑌^𝜀_𝑡+𝐾^𝜀_𝑡+𝜀−𝐾^𝜀_𝑡=1𝜀𝔼_𝑡𝑡^𝑡+𝜀𝑓𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑑𝑢_{𝑡𝑡+𝜀}𝑝,𝑏𝑋_𝑢^𝑡,𝑥𝑑𝑢+_𝑡^𝑡+𝜀𝑔_𝑖𝑗𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝×𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥𝑑𝐵^𝑖,𝐵^𝑗_𝑢+𝐿_𝜀, (30) where 𝐿_𝜀=1𝜀𝔼_𝑡∫_𝑡^𝑡+𝜀𝑓𝑢,𝑌^𝜀_𝑢+𝑦+𝑝,𝑋_𝑢^𝑡,𝑥,𝑍−𝑥_𝜀𝑢+𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑑𝑢_{𝑡𝑡+𝜀}𝑝,𝑏𝑋_𝑢^𝑡,𝑥+𝑑𝑢_{𝑡𝑡+𝜀}𝑔_𝑖𝑗𝑢,𝑌^𝜀_𝑢+𝑦+𝑝,𝑋_𝑢^𝑡,𝑥,𝑍−𝑥_𝜀𝑢+𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥𝑑𝐵^𝑖,𝐵^𝑗_𝑢−𝔼_𝑡𝑡^𝑡+𝜀𝑓𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑑𝑢_{𝑡𝑡+𝜀}𝑝,𝑏𝑋_𝑢^𝑡,𝑥+𝑑𝑢_{𝑡𝑡+𝜀}𝑔_𝑖𝑗𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝×𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥𝑑⟨𝐵^𝑖,𝐵^𝑗⟩_𝑢. (31) It is easy to check that |𝐿_𝜀|≤(𝐶₄/𝜀)𝔼_𝑡[∫_𝑡^𝑡+𝜀(|𝑌^𝜀_𝑢|+|𝑍^𝜀_𝑢|)𝑑𝑢], where 𝐶₄ depends on 𝐺, 𝐿, and 𝑇. Thus, by (29), we get 𝔼𝐿_𝜀^𝛼≤𝐶^𝛼₄𝜀^𝛼𝔼_𝑡^𝑡+𝜀𝑌^𝜀_𝑢+𝑍^𝜀_𝑢𝑑𝑢^𝛼≤2^𝛼−1𝐶^𝛼₄𝜀^𝛼𝔼_𝑡^𝑡+𝜀𝑌^𝜀_𝑢𝑑𝑢^𝛼+_𝑡^𝑡+𝜀𝑍^𝜀_𝑢𝑑𝑢^𝛼≤2^𝛼−1𝐶^𝛼₄𝔼sup_{[]𝑠∈𝑡,𝑡+𝜀}𝑌^𝜀_𝑠^𝛼+𝜀^−𝛼/2𝔼_𝑡^𝑡+𝜀𝑍^𝜀_𝑢²𝑑𝑢^𝛼/2≤2^𝛼−1𝐶^𝛼₄𝐶₃𝜀^𝛼+𝜀^𝛼/2, (32) which implies 𝐿^𝛼_𝐺−lim_𝜀→0+𝐿_𝜀=0. We set 𝑀_𝜀=1𝜀𝔼_𝑡𝑡^𝑡+𝜀𝑓𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝+𝑑𝑢_{𝑡𝑡+𝜀}𝑝,𝑏𝑋_𝑢^𝑡,𝑥+𝑑𝑢_{𝑡𝑡+𝜀}𝑔_𝑖𝑗𝑢,𝑦+𝑝,𝑋_𝑢^𝑡,𝑥−𝑥,𝜎^𝑇𝑋_𝑢^𝑡,𝑥𝑝×𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗𝑋_𝑢^𝑡,𝑥𝑑𝐵^𝑖,𝐵^𝑗_𝑢−𝔼_𝑡𝑡^𝑡+𝜀𝑓𝑢,𝑦,𝜎^𝑇(𝑥)𝑝(𝑥𝜀+𝑑𝑢+𝑝,𝑏)_{𝑡𝑡+𝜀}𝑔_𝑖𝑗𝑢,𝑦,𝜎^𝑇(𝑥)𝑝𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗(𝑥)𝑑𝐵^𝑖,𝐵^𝑗_𝑢. (33) By the Lipschitz condition, we can get |𝑀_𝜀|≤(𝐶₅/𝜀)𝔼_𝑡[∫_𝑡^𝑡+𝜀|𝑋_𝑢^𝑡,𝑥−𝑥|𝑑𝑢], where 𝐶₅ depends on 𝑝, 𝐺, 𝐿, and 𝑇. Noting that 𝔼[sup_{𝑠∈[𝑡,𝑡+𝜀]}|𝑋_𝑠^𝑡,𝑥−𝑥|^𝛼]≤𝐶₆(1+|𝑥|^𝛼)𝜀^𝛼/2 (see [16, 19]), where 𝐶₆ depends on 𝐿, 𝐺, and 𝛼, we obtain 𝔼𝑀_𝜀^𝛼≤𝐶^𝛼₅𝔼sup_{[]𝑠∈𝑡,𝑡+𝜀}𝑋_𝑠^𝑡,𝑥−𝑥^𝛼≤𝐶^𝛼₅𝐶₆1+|𝑥|^𝛼𝜀^𝛼/2, (34) which implies 𝐿^𝛼_𝐺−lim_𝜀→0+𝑀_𝜀=0. Now, we set 𝑁_𝜀=1𝜀𝔼_𝑡𝑡^𝑡+𝜀𝑓𝑢,𝑦,𝜎^𝑇(𝑥)𝑝(𝑥𝜀+𝑑𝑢+𝑝,𝑏)_{𝑡𝑡+𝜀}𝑔_𝑖𝑗𝑢,𝑦,𝜎^𝑇(𝑥)𝑝𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗(𝑥)𝑑𝐵^𝑖,𝐵^𝑗_𝑢−𝔼_𝑡𝑡^𝑡+𝜀𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝(𝑥𝜀+𝑑𝑢+𝑝,𝑏)_{𝑡𝑡+𝜀}𝑔_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗(𝑥)𝑑𝐵^𝑖,𝐵^𝑗_𝑢. (35) It is easy to deduce that |𝑁_𝜀|≤(𝐶₇/𝜀)𝔼_𝑡[∫_𝑡^𝑡+𝜀(|𝑓(𝑢,𝑦,𝜎^𝑇(𝑥)𝑝)−𝑓(𝑡,𝑦,𝜎^𝑇(𝑥)𝑝)|+∑^𝑑_𝑖,𝑗=1|𝑔_𝑖𝑗(𝑢,𝑦,𝜎^𝑇(𝑥)𝑝)−𝑔_𝑖𝑗(𝑡,𝑦,𝜎^𝑇(𝑥)𝑝)|)𝑑𝑢], where 𝐶₇ depends on 𝐺. Then, 𝔼𝑁_𝜀^𝛼≤𝐶^𝛼₇1𝜀𝔼_𝑡^𝑡+𝜀𝑓𝑢,𝑦,𝜎^𝑇(𝑥)𝑝−𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑢,𝑦,𝜎^𝑇(𝑥)𝑝−𝑔_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝^𝛼𝑑𝑢≤𝐶^𝛼₇1𝜀𝔼_𝑡^𝑡+𝜀𝑓𝑢,𝑦,𝜎^𝑇(𝑥)𝑝−𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑢,𝑦,𝜎^𝑇(𝑥)𝑝−𝑔_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝^𝛽𝑑𝑢^𝛼/𝛽. (36) Take limit on both sides of the above inequality and use assumption (H4); then, we have 𝐿^𝛼_𝐺−lim_𝜀→0+𝑁_𝜀=0. (37) On the other hand, 𝔼_𝑡𝑡^𝑡+𝜀𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝(𝑥𝜀+𝑑𝑢+𝑝,𝑏)_{𝑡𝑡+𝜀}𝑔_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝𝑑𝐵^𝑖,𝐵^𝑗_𝑢+_𝑡^𝑡+𝜀𝑝,ℎ_𝑖𝑗(𝑥)𝑑𝐵^𝑖,𝐵^𝑗_𝑢=𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝(𝑥𝜀+𝔼𝜀+𝑝,𝑏)_𝑡𝑔_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+𝑝,ℎ_𝑖𝑗(𝑥×𝐵_𝑖),𝐵^𝑗_𝑡+𝜀−𝐵^𝑖,𝐵^𝑗_𝑡=𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+(𝑥𝑔𝑝,𝑏)+2𝐺_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+𝑝,ℎ_𝑖𝑗(𝑥)^𝑑_𝑖,𝑗=1𝜀. (38) Then, we have 𝐿^𝛼_𝐺−lim_𝜀→0+1𝜀𝑌^𝜀_𝑡−𝑦=𝑓𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+(𝑥𝑔𝑝,𝑏)+2𝐺_𝑖𝑗𝑡,𝑦,𝜎^𝑇(𝑥)𝑝+𝑝,ℎ_𝑖𝑗(𝑥)^𝑑_𝑖,𝑗=1. (39) The proof is complete.

4. Some Applications

4.1. Converse Comparison Theorem for 𝐺-BSDEs

We consider the following 𝐺-BSDEs: 𝑌_𝑡^𝑙,𝜉=𝜉+^𝑇_𝑡𝑓^𝑙𝑠,𝑌_𝑠^𝑙,𝜉,𝑍_𝑠^𝑙,𝜉+𝑑𝑠_𝑇𝑡𝑔^𝑙_𝑖𝑗𝑠,𝑌_𝑠^𝑙,𝜉,𝑍_𝑠^𝑙,𝜉𝑑𝐵^𝑖,𝐵^𝑗_𝑠−^𝑇_𝑡𝑍_𝑠^𝑙,𝜉𝑑𝐵_𝑠−𝐾_𝑇^𝑙,𝜉−𝐾_𝑡^𝑙,𝜉,𝑙=1,2, (40) where 𝑔^𝑙_𝑖𝑗=𝑔^𝑙_𝑗𝑖.

We first generalized the comparison theorem in [16].

Proposition 13. Let 𝑓^𝑙 and 𝑔^𝑙_𝑖𝑗 satisfy (H1) and (H2) for some 𝛽>1, 𝑙=1,2. If 𝑓²−𝑓¹+2𝐺((𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗)^𝑑_𝑖,𝑗=1)≤0, then, for each 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇), one has 𝑌_𝑡^1,𝜉≥𝑌_𝑡^2,𝜉 for 𝑡∈[0,𝑇].

Proof. From the above 𝐺-BSDEs, we have 𝑌_𝑡^2,𝜉=𝜉+^𝑇_𝑡𝑓²𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉+𝑑𝑠_𝑇𝑡𝑔²_𝑖𝑗𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉𝑑𝐵^𝑖,𝐵^𝑗_𝑠−^𝑇_𝑡𝑍_𝑠^2,𝜉𝑑𝐵_𝑠−𝐾_𝑇^2,𝜉−𝐾_𝑡^2,𝜉=𝜉+^𝑇_𝑡𝑓¹𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉+𝑑𝑠_𝑇𝑡𝑔¹_𝑖𝑗𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉𝑑𝐵^𝑖,𝐵^𝑗_𝑠+𝑉_𝑇−𝑉_𝑡−^𝑇_𝑡𝑍_𝑠^2,𝜉𝑑𝐵_𝑠−𝐾_𝑇^2,𝜉−𝐾_𝑡^2,𝜉, (41) where 𝑉_𝑡=^𝑡₀𝑓²−𝑓¹𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉+𝑑𝑠_𝑡0𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉𝑑𝐵^𝑖,𝐵^𝑗_𝑠=^𝑡₀𝑓²−𝑓¹+2𝐺𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗^𝑑_𝑖,𝑗=1𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉𝑑𝑠+^𝑡₀𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉𝑑𝐵^𝑖,𝐵^𝑗_𝑠−^𝑡₀2𝐺𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗^𝑑_𝑖,𝑗=1𝑠,𝑌_𝑠^2,𝜉,𝑍_𝑠^2,𝜉𝑑𝑠. (42) By the assumption, it is easy to check that (𝑉_𝑡)_𝑡≤𝑇 is a decreasing process. Thus, using Theorem 11, we obtain 𝑌_𝑡^1,𝜉≥𝑌_𝑡^2,𝜉 for 𝑡∈[0,𝑇].

Remark 14. Suppose 𝑑=1 and let 𝑓¹=10|𝑧|, 𝑓²=|𝑧|, 𝑔¹=|𝑧|, and 𝑔²=2|𝑧|. It is easy to check that 𝑓²−𝑓¹+2𝐺(𝑔²−𝑔¹)≤0. Thus, 𝑓²−𝑓¹+2𝐺((𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗)^𝑑_𝑖,𝑗=1)≤0 does not imply 𝑓²≤𝑓¹ and (𝑔²_𝑖𝑗)^𝑑_𝑖,𝑗=1≤(𝑔¹_𝑖𝑗)^𝑑_𝑖,𝑗=1.

Now, we give the converse comparison theorem.

Theorem 15. Let 𝑓^𝑙 and 𝑔^𝑙_𝑖𝑗 satisfy (H1), (H2), (H3), (H4), and (H5) for some 𝛽>1, 𝑙=1,2. If 𝑌_𝑡^1,𝜉≥𝑌_𝑡^2,𝜉 for each 𝑡∈[0,𝑇] and 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇), then 𝑓²−𝑓¹+2𝐺((𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗)^𝑑_𝑖,𝑗=1)≤0 q.s..

Proof. For simplicity, we take the notation 𝔼^𝑙_𝑡[𝜉]=𝑌_𝑡^𝑙,𝜉, 𝑙=1,2. For each fixed (𝑡,𝑦,𝑧)∈[0,𝑇)×ℝ×ℝ^𝑑, let us consider 𝜂_𝜀=𝑦+𝑧,ℎ_𝑖𝑗𝐵^𝑖,𝐵^𝑗_𝑡+𝜀−𝐵^𝑖,𝐵^𝑗_𝑡+𝑧,𝐵_𝑡+𝜀−𝐵_𝑡, (43) where ℎ_𝑖𝑗=ℎ_𝑗𝑖∈ℝ^𝑑. By Theorem 12, we have, for each 𝛼∈(1,𝛽), 𝐿^𝛼_𝐺−lim_𝜀→0+1𝜀𝔼^𝑙_𝑡𝜂_𝜀−𝑦=𝑓^𝑙𝑡,𝑦,𝑧+2𝐺𝑔^𝑙_𝑖𝑗𝑡,𝑦,𝑧+𝑧,ℎ_𝑖𝑗^𝑑_𝑖,𝑗=1. (44) Since 𝔼¹_𝑡[𝜂_𝜀]≥𝔼²_𝑡[𝜂_𝜀], 𝑓¹𝑡,𝑦,𝑧+2𝐺𝑔¹_𝑖𝑗𝑡,𝑦,𝑧+𝑧,ℎ_𝑖𝑗^𝑑_𝑖,𝑗=1≥𝑓²𝑡,𝑦,𝑧+2𝐺𝑔²_𝑖𝑗𝑡,𝑦,𝑧+𝑧,ℎ_𝑖𝑗^𝑑_𝑖,𝑗=1q.s. (45) Take a ℎ_𝑖𝑗 such that ⟨𝑧,ℎ_𝑖𝑗⟩=−𝑔¹_𝑖𝑗(𝑡,𝑦,𝑧). Therefore, {𝑓²−𝑓¹+2𝐺((𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗)^𝑑_𝑖,𝑗=1)}(𝑡,𝑦,𝑧)≤0 q.s. By the assumptions (H2) and (H3), it is easy to deduce that 𝑓²−𝑓¹+2𝐺((𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗)^𝑑_𝑖,𝑗=1)≤0 q.s.

In the following, we use the notation 𝔼^𝑙_𝑡[𝜉]=𝑌_𝑡^𝑙,𝜉, 𝑙=1,2.

Corollary 16. Let 𝑓^𝑙 and 𝑔^𝑙_𝑖𝑗 be deterministic functions and satisfy (H1), (H2), (H3), and (H5) for some 𝛽>1, 𝑙=1,2. If 𝔼¹[𝜉]≥𝔼²[𝜉] for each 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇), then 𝑓²−𝑓¹+2𝐺((𝑔²_𝑖𝑗−𝑔¹_𝑖𝑗)^𝑑_𝑖,𝑗=1)≤0.

Proof. Taking 𝜂_𝜀 as in Theorem 15, since 𝑓^𝑙 and 𝑔^𝑙_𝑖𝑗 are deterministic, we could get 𝔼^𝑙_𝑡[𝜂_𝜀]=𝔼^𝑙[𝜂_𝜀], for 𝑙=1, 2. And the proof in Theorem 15 still holds true.

4.2. Some Equivalent Relations

We consider the following 𝐺-BSDE: 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝑠+^𝑇_𝑡𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠−𝐾_𝑇−𝐾_𝑡, (46) where 𝑔_𝑖𝑗=𝑔_𝑗𝑖. We use the notation 𝔼_𝑡[𝜉]=𝑌_𝑡.

Proposition 17. Let 𝑓 and 𝑔_𝑖𝑗 satisfy (H1), (H2), (H3), (H4), and (H5) for some 𝛽>1 and fix 𝛼∈(1,𝛽). Then, one has(1)𝔼_𝑡[𝜉+𝜂]=𝔼_𝑡[𝜉]+𝜂 for 𝑡∈[0,𝑇], 𝜉∈𝐿^𝛼_𝐺(Ω_𝑇), and 𝜂∈𝐿^𝛼_𝐺(Ω_𝑡) if and only if for each 𝑡∈[0,𝑇], 𝑦, 𝑦^′∈ℝ, 𝑧∈ℝ^𝑑,𝑓𝑡,𝑦,𝑧−𝑓𝑡,𝑦^′𝑔,𝑧+2𝐺_𝑖𝑗𝑡,𝑦,𝑧−𝑔_𝑖𝑗𝑡,𝑦^′,𝑧^𝑑_𝑖,𝑗=1=0; (47)(2)𝔼_𝑡[𝜉+𝜂]≤𝔼_𝑡[𝜉]+𝔼_𝑡[𝜂] for 𝑡∈[0,𝑇], 𝜉∈𝐿^𝛼_𝐺(Ω_𝑇), and 𝜂∈𝐿^𝛼_𝐺(Ω_𝑇) if and only if for each 𝑡∈[0,𝑇], 𝑦, 𝑦^′∈ℝ, 𝑧, 𝑧^′∈ℝ^𝑑,0≥𝑓𝑡,𝑦+𝑦^′,𝑧+𝑧^′−𝑓𝑡,𝑦,𝑧−𝑓𝑡,𝑦^′,𝑧^′+2𝐺𝑔_𝑖𝑗𝑡,𝑦+𝑦^′,𝑧+𝑧^′−𝑔_𝑖𝑗𝑡,𝑦,𝑧−𝑔_𝑖𝑗𝑡,𝑦^′,𝑧^′𝑑_𝑖,𝑗=1; (48)(3)𝔼_𝑡[𝜆𝜉+(1−𝜆)𝜂]≤𝜆𝔼_𝑡[𝜉]+(1−𝜆)𝔼_𝑡[𝜂] for 𝑡∈[0,𝑇], 𝜆∈[0,1], 𝜉∈𝐿^𝛼_𝐺(Ω_𝑇), and 𝜂∈𝐿^𝛼_𝐺(Ω_𝑇) if and only if for each 𝑡∈[0,𝑇], 𝑦, 𝑦^′∈ℝ, 𝑧, 𝑧^′∈ℝ^𝑑, 𝜆∈[0,1],0≥𝑓𝑡,𝜆𝑦+(1−𝜆)𝑦^′,𝜆𝑧+(1−𝜆)𝑧^′−𝜆𝑓𝑡,𝑦,𝑧−(1−𝜆)𝑓𝑡,𝑦^′,𝑧^′+2𝐺𝑔_𝑖𝑗𝑡,𝜆𝑦+(1−𝜆)𝑦^′,𝜆𝑧+(1−𝜆)𝑧^′−𝜆𝑔_𝑖𝑗𝑡,𝑦,𝑧−(1−𝜆)𝑔_𝑖𝑗𝑡,𝑦^′,𝑧^′𝑑_𝑖,𝑗=1; (49)(4)𝔼_𝑡[𝜆𝜉]=𝜆𝔼_𝑡[𝜉] for 𝑡∈[0,𝑇], 𝜆≥0, and 𝜉∈𝐿^𝛼_𝐺(Ω_𝑇) if and only if for each 𝑡∈[0,𝑇], 𝑦∈ℝ, 𝑧∈ℝ^𝑑, 𝜆≥0,𝑓𝑡,𝜆𝑦,𝜆𝑧−𝜆𝑓𝑡,𝑦,𝑧=2𝐺𝜆𝑔_𝑖𝑗𝑡,𝑦,𝑧−𝑔_𝑖𝑗𝑡,𝜆𝑦,𝜆𝑧^𝑑_𝑖,𝑗=1=−2𝐺𝑔_𝑖𝑗𝑡,𝜆𝑦,𝜆𝑧−𝜆𝑔_𝑖𝑗𝑡,𝑦,𝑧^𝑑_𝑖,𝑗=1. (50)

Proof. (1) “⇒” part. For each fixed 𝑡∈[0,𝑇), 𝑦, 𝑦^′∈ℝ, 𝑧∈ℝ^𝑑, we take 𝜉_𝜀=𝑦+𝑧,ℎ_𝑖𝑗𝐵^𝑖,𝐵^𝑗_𝑡+𝜀−𝐵^𝑖,𝐵^𝑗_𝑡+𝑧,𝐵_𝑡+𝜀−𝐵_𝑡,𝜂=𝑦^′−𝑦, (51) where ℎ_𝑖𝑗=ℎ_𝑗𝑖∈ℝ^𝑑. Then, by Theorem 12 and 𝔼_𝑡[𝜉_𝜀+𝜂]=𝔼_𝑡[𝜉_𝜀]+𝜂, we can obtain 𝑓𝑡,𝑦^′,𝑧+2𝐺𝑔_𝑖𝑗𝑡,𝑦^′,𝑧+𝑧,ℎ_𝑖𝑗^𝑑_𝑖,𝑗=1=𝑓𝑡,𝑦,𝑧+2𝐺𝑔_𝑖𝑗𝑡,𝑦,𝑧+𝑧,ℎ_𝑖𝑗^𝑑_𝑖,𝑗=1. (52) We choose ℎ_𝑖𝑗 such that 𝑔_𝑖𝑗(𝑡,𝑦^′,𝑧)+⟨𝑧,ℎ_𝑖𝑗⟩=0, which implies (47).
“⇐” part. Let (𝑌,𝑍,𝐾) be the solution of 𝐺-BSDE (46) corresponding to terminal condition 𝜉. We claim that (𝑌_𝑠+𝜂,𝑍_𝑠,𝐾_𝑠)_{𝑠∈[𝑡,𝑇]} is the solution of 𝐺-BSDE (46) corresponding to terminal condition 𝜉+𝜂 on [𝑡,𝑇]. For this, we only need to check that, for 𝑠∈[𝑡,𝑇], ^𝑇_𝑠𝑓𝑢,𝑌_𝑢,𝑍_𝑢𝑑𝑢+^𝑇_𝑠𝑔_𝑖𝑗𝑢,𝑌_𝑢,𝑍_𝑢𝑑𝐵^𝑖,𝐵^𝑗_𝑢=^𝑇_𝑠𝑓𝑢,𝑌_𝑢+𝜂,𝑍_𝑢+𝑑𝑢_𝑇𝑠𝑔_𝑖𝑗𝑢,𝑌_𝑢+𝜂,𝑍_𝑢𝑑𝐵^𝑖,𝐵^𝑗_𝑢. (53) By (47) we can get ^𝑇_𝑠𝑔_𝑖𝑗𝑢,𝑌_𝑢,𝑍_𝑢−𝑔_𝑖𝑗𝑢,𝑌_𝑢+𝜂,𝑍_𝑢𝑑𝐵^𝑖,𝐵^𝑗_𝑢−2^𝑇_𝑠𝐺𝑔_𝑖𝑗𝑢,𝑌_𝑢,𝑍_𝑢−𝑔_𝑖𝑗𝑢,𝑌_𝑢+𝜂,𝑍_𝑢^𝑑_𝑖,𝑗=1=𝑑𝑢_𝑇𝑠𝑔_𝑖𝑗𝑢,𝑌_𝑢,𝑍_𝑢−𝑔_𝑖𝑗𝑢,𝑌_𝑢+𝜂,𝑍_𝑢𝑑𝐵^𝑖,𝐵^𝑗_𝑢+^𝑇_𝑠𝑓𝑢,𝑌_𝑢,𝑍_𝑢−𝑓𝑢,𝑌_𝑢+𝜂,𝑍_𝑢𝑑𝑢≤0,_𝑇𝑠𝑔_𝑖𝑗𝑢,𝑌_𝑢+𝜂,𝑍_𝑢−𝑔_𝑖𝑗𝑢,𝑌_𝑢,𝑍_𝑢𝑑𝐵^𝑖,𝐵^𝑗_𝑢−2^𝑇_𝑠𝐺𝑔_𝑖𝑗𝑢,𝑌_𝑢+𝜂,𝑍_𝑢−𝑔_𝑖𝑗𝑢,𝑌_𝑢,𝑍_𝑢^𝑑_𝑖,𝑗=1=𝑑𝑢_𝑇𝑠𝑔_𝑖𝑗𝑢,𝑌_𝑢+𝜂,𝑍_𝑢−𝑔_𝑖𝑗𝑢,𝑌_𝑢,𝑍_𝑢𝑑𝐵^𝑖,𝐵^𝑗_𝑢+^𝑇_𝑠𝑓𝑢,𝑌_𝑢+𝜂,𝑍_𝑢−𝑓𝑢,𝑌_𝑢,𝑍_𝑢𝑑𝑢≤0, (54) which implies (53). The proof of (1) is complete.
(2) “⇒” part. For each fixed 𝑡∈[0,𝑇), 𝑦, 𝑦^′∈ℝ, 𝑧, 𝑧^′∈ℝ^𝑑, we consider 𝜉_𝜀=𝑦+⟨𝑧,ℎ_𝑖𝑗⟩(⟨𝐵^𝑖,𝐵^𝑗⟩_𝑡+𝜀−⟨𝐵^𝑖,𝐵^𝑗⟩_𝑡)+⟨𝑧,𝐵_𝑡+𝜀−𝐵_𝑡⟩ and 𝜂_𝜀=𝑦^′+⟨𝑧^′,ℎ^′_𝑖𝑗⟩(⟨𝐵^𝑖,𝐵^𝑗⟩_𝑡+𝜀−⟨𝐵^𝑖,𝐵^𝑗⟩_𝑡)+⟨𝑧^′,𝐵_𝑡+𝜀−𝐵_𝑡⟩, where ℎ_𝑖𝑗=ℎ_𝑗𝑖∈ℝ^𝑑 and ℎ^′_𝑖𝑗=ℎ^′_𝑗𝑖∈ℝ^𝑑. Then, by Theorem 12 and 𝔼_𝑡[𝜉_𝜀+𝜂_𝜀]=𝔼_𝑡[𝜉_𝜀]+𝔼_𝑡[𝜂_𝜀], we obtain 𝑓𝑡,𝑦+𝑦^′,𝑧+𝑧^′+2𝐺𝑔_𝑖𝑗𝑡,𝑦+𝑦^′,𝑧+𝑧^′+𝑧,ℎ_𝑖𝑗+𝑧^′,ℎ^′_𝑖𝑗^𝑑_𝑖,𝑗=1≤𝑓𝑡,𝑦,𝑧+𝑓𝑡,𝑦^′,𝑧^′+2𝐺𝑔_𝑖𝑗𝑡,𝑦,𝑧+𝑧,ℎ_𝑖𝑗^𝑑_𝑖,𝑗=1+2𝐺𝑔_𝑖𝑗𝑡,𝑦^′,𝑧^′+𝑧^′,ℎ^′_𝑖𝑗^𝑑_𝑖,𝑗=1. (55) We choose ℎ_𝑖𝑗, ℎ^′_𝑖𝑗 such that 𝑔_𝑖𝑗(𝑡,𝑦,𝑧)+⟨𝑧,ℎ_𝑖𝑗⟩=0 and 𝑔_𝑖𝑗(𝑡,𝑦^′,𝑧^′)+⟨𝑧^′,ℎ^′_𝑖𝑗⟩=0, which implies (48).
“⇐” part. Let (𝑌,𝑍,𝐾) and (𝑌^′,𝑍^′,𝐾^′) be the solutions of 𝐺-BSDE (46) corresponding to terminal condition 𝜉 and 𝜂, respectively. Then, (𝑌+𝑌^′,𝑍+𝑍^′,𝐾) solves the following 𝐺-BSDE: 𝑌_𝑡+𝑌^′_𝑡=𝜉+𝜂+^𝑇_𝑡𝑓𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠+𝑑𝑠_𝑇𝑡𝑔_𝑖𝑗𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠+𝑉_𝑇−𝑉_𝑡−^𝑇_𝑡𝑍_𝑠+𝑍^′_𝑠𝑑𝐵_𝑠−𝐾_𝑇−𝐾_𝑡, (56) where 𝑉_𝑡=−𝐾^′_𝑡−^𝑡₀𝑓𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠−𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^′_𝑠,𝑍^′_𝑠−𝑑𝑠_𝑡0𝑔_𝑖𝑗𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠−𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑖𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠=−𝐾^′_𝑡−^𝑡₀𝑔_𝑖𝑗𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠−𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑖𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠−2^𝑡₀𝐺𝑔_𝑖𝑗𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠−𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑖𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠^𝑑_𝑖,𝑗=1−𝑑𝑠_𝑡0𝑓𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠−𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^′_𝑠,𝑍^′_𝑠+2𝐺𝑔_𝑖𝑗𝑠,𝑌_𝑠+𝑌^′_𝑠,𝑍_𝑠+𝑍^′_𝑠−𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑖𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠^𝑑_𝑖,𝑗=1𝑑𝑠. (57) By (48), it is easy to check that 𝑉_𝑡 is an increasing process. Then, by Theorem 11, we can get 𝔼_𝑡[𝜉+𝜂]≤𝔼_𝑡[𝜉]+𝔼_𝑡[𝜂]. The proof of (2) is complete.
Finally, we could prove (3) as in (2) and (4) as in (1).

Proposition 18. One has the following.(1)If 𝐺(𝐴)+𝐺(−𝐴)>0 for any 𝐴∈𝕊_𝑑 and 𝐴≠0, then (47) holds if and only if 𝑓 and 𝑔_𝑖𝑗 are independent of 𝑦.(2)If there exists an 𝐴∈𝕊_𝑑 with 𝐴≠0 such that 𝐺(𝐴)+𝐺(−𝐴)=0 and 𝐺(𝐴)≠0, then, for any fixed 𝑔(𝑡,𝑦,𝑧) satisfying (H1)–(H5), one has 𝑓(𝑡,𝑦,𝑧)=−2𝐺(𝐴)𝑔(𝑡,𝑦,𝑧) and (𝑔_𝑖𝑗(𝑡,𝑦,𝑧))^𝑑_𝑖,𝑗=1=𝑔(𝑡,𝑦,𝑧)𝐴 satisfying (47).

Proof. It is easy to verify (2), and we only need to prove (1). If (47) holds, it is easy to check that 𝐺((𝑔_𝑖𝑗(𝑡,𝑦,𝑧)−𝑔_𝑖𝑗(𝑡,0,𝑧))^𝑑_𝑖,𝑗=1)+𝐺((𝑔_𝑖𝑗(𝑡,0,𝑧)−𝑔_𝑖𝑗(𝑡,𝑦,𝑧))^𝑑_𝑖,𝑗=1)=0 holds. Then, from the assumption, we get 𝑔_𝑖𝑗(𝑡,𝑦,𝑧)=𝑔_𝑖𝑗(𝑡,0,𝑧). Therefore, by (47), we have 𝑓(𝑡,𝑦,𝑧)=𝑓(𝑡,0,𝑧), which implies that 𝑓 and 𝑔_𝑖𝑗 are independent of 𝑦. The converse part is obvious.

Acknowledgments

K. He acknowledges the financial support from the National Natural Science Foundation of China (Grant nos. 11301068 and 11171062) and the Innovation Program of Shanghai Municipal Education Commission (Grant no. 12ZZ063). M. Hu acknowledges the financial support from the National Natural Science Foundation of China (Grant nos. 11201262 and 11101242) and the Scientific Research Foundation for the Excellent Middle-Aged and Young Scientists of Shandong Province of China (Grant no. BS2013SF020).