Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

Abstract

We consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

1. Introduction

Nonlinear BSDEs in the framework of linear expectation were introduced by Pardoux and Peng [1] in 1990. Then, a lot of researches were studied by many authors, and they provided various applications of BSDEs in stochastic control, finance, stochastic differential games, and second-order partial differential equations theory; see [2–9].

The notion of sublinear expectation space was introduced by Peng [10–12], which is a generalization of classical probability space. The G-expectation, a type of sublinear expectation, has played an important role in the researches of sublinear expectation space recently. It can be regarded as a counterpart of the Wiener probability space in the linear case. Within this G-expectation framework, the G-Brownian motion is the canonical process. Besides, the notions of the G-martingales and the Itô integral with respect to G-Brownian motion were also derived. There are some new structures in these notions and some new applications in the financial models with volatility uncertainty; see Peng [12, 13].

In the G-expectation framework, thanks to a series of studies [14–17], the complete representation theorem for G-martingales has been obtained by Peng et al. [18]. Due to this contribution, a natural formulation of BSDEs driven by G-Brownian motion was found by Hu et al. [19]. In addition, the existence and uniqueness of the solution to the BSDEs driven by G-Brownian motion have been proved. They also have given the comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion in [20]. So the complete theory of BSDEs driven by G-Brownian motion has been established.

An important application of BSDEs is that we can define the recursive utility functions from BSDEs, which can index scaling risks in the study of economics and finance [21–24]. Based on these results, a type of significant stochastic optimal control problems under linear expectation with a BSDE as cost function was studied [2, 4, 7–9]. Under G-expectation, the similar problems will be useful in the future studies of finance models with volatility uncertainty. So we arise a natural question: can we construct the similar results in G-expectation framework? In [25, 26], Zhang have given the study about the stochastic control problems under G-expectation based on the preliminary theory of BSDEs driven by G-Brownian motion. When the complete results about BSDEs driven by G-Brownian motion were established in [19, 20], we tried to prove the complete results of stochastic optimization theory of BSDEs driven by G-Brownian motion in this paper.

In this paper, we investigate the stochastic optimal control problems with a BSDE driven by G-Brownian motion constructed in [19, 20] as cost function. Based on the results in [19, 20], we obtain the dynamic programming principle under G-expectation. Besides, the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

The rest of the paper is organized as follows. In Section 2, we recall the G-expectation framework and adapt it according to our objective. Besides, we give the related properties of forward and backward stochastic differential equations driven by G-Brownian motion, which will be needed in the sequel sections. In Section 3, the stochastic optimal control problems with a BSDE driven by G-Brownian motion as cost function are investigated and a dynamic programming principle under G-expectation is obtained. In Section 4, The value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

2. Preliminaries

In this section, we recall the G-expectation framework established by Peng [10–12, 27]. Besides, we give some results about forward and backward stochastic differential equations driven by G-Brownian motion, which we need in the following sections. Some details can be found in [19, 20].

2.1. G-Expectation and G-Martingales

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

Definition 2 (G-normal distribution). A 𝑑-dimensional random vector 𝑋=(𝑋₁,…,𝑋_𝑑) on a sublinear expectation space (Ω,ℋ,𝔼) is called G-normally distributed if for each 𝑎,𝑏≥0, we have 𝑋+𝑏𝑋=^𝑑𝑎²+𝑏²𝑋, (1) where 𝑋 is an independent copy of 𝑋; that is, 𝑋 and 𝑋 are identically distributed, and 𝑋 is independent of 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 all 𝑑×𝑑 symmetric matrices.

Proposition 3. Let 𝑋 be G-normal distributed. The distribution of X is characterized by 𝑢(𝑡,𝑥)=𝔼𝜑𝑥+√𝑡𝑋,𝜑∈𝐶_{𝑏,𝐿𝑖𝑝}ℝ^𝑑. (3) In particular, 𝔼[𝜑(𝑋)]=𝑢(1,0), where 𝑢 is the unique viscosity solution of the following parabolic PDE defined on [0,∞)×ℝ^𝑑: 𝜕_𝑡𝑢−𝐺𝐷²𝑢=0,𝑢|_𝑡=0=𝜑, (4) where 𝐺 is defined by (2).

Remark 4. It is easy to check that 𝐺 is a monotonic sublinear function defined on 𝕊(𝑑) and 𝐺(𝐴)∶=(1/2)𝔼[(𝐴𝑋,𝑋)]≤(1/2)|𝐴|𝔼[|𝑋|²]=(1/2)|𝐴|𝜎² implies that there exists a bounded, convex, and closed subset Γ⊂𝕊⁺_𝑑 such that 𝐺(𝐴)=12sup_𝛾∈ΓTr𝛾𝐴, (5) where 𝕊⁺_𝑑 denotes the collection of nonnegative elements in 𝕊_𝑑. If there exist some 𝛽>0 such that 𝐺(𝐴)−𝐺(𝐵)≥𝛽tr[𝐴−𝐵] for any 𝐴≥𝐵, we call the G-normal distribution nondegenerate, which is the case we consider throughout this paper.

Definition 5. Let Ω=𝐶^𝑑₀([0,𝑇]), that is, the space of all ℝ^𝑑-valued continuous paths (𝜔_𝑡)_{𝑡∈[0,𝑇]} with 𝜔₀=0. The corresponding canonical process is 𝐵_𝑡(𝜔)=𝜔_𝑡, 𝑡∈[0,𝑇]. 𝑃₀ is wiener measure. 𝔽={ℱ^𝐵_𝑡}_𝑡≥0 is the filtration generated by 𝐵. We let ℋ∶=𝐿_𝑖𝑝(Ω_𝑇) be a linear space of random variables for each fixed 𝑇≥0, where 𝐿_𝑖𝑝(Ω_𝑇)∶={𝜑(𝐵_𝑡1,…,𝐵_𝑡𝑛)∶𝑛≥1,𝑡₁,…,𝑡_𝑛∈[0,𝑇],𝜑∈𝐶_{𝑏,𝐿𝑖𝑝}(ℝ^𝑑×𝑛)}.
(i) The G-expectation 𝐸 is a sublinear expectation defined by 𝐸[𝑋]∶=𝐸𝜑𝑡₁−𝑡₀𝜉₁,…,𝑡_𝑛−𝑡_𝑛−1𝜉_𝑛, (6) for each 𝑋=𝜑(𝐵_𝑡1−𝐵_𝑡0,𝐵_𝑡2−𝐵_𝑡1,…,𝐵_𝑡𝑛−𝐵_𝑡𝑛−1), where (𝜉_𝑖)^𝑛_𝑖=1 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 each 𝑖=1,2,…,𝑛−1. (Ω,ℋ,𝐸) is called G-expectation space and the canonical process {𝐵_𝑡}_{𝑡∈[0,𝑇]} in the sublinear space (Ω,ℋ,𝐸) is called a G-Brownian motion.
(ii) The conditional G-expectation 𝐸_𝑡 of 𝑋∈𝐿_𝑖𝑝(Ω_𝑇) is defined by 𝐸_𝑡𝑗𝜑𝐵_𝑡1−𝐵_𝑡0,𝐵_𝑡2−𝐵_𝑡1,…,𝐵_𝑡𝑛−𝐵_𝑡𝑛−1∶=𝜓𝐵_𝑡1−𝐵_𝑡0,…,𝐵_𝑡𝑗−𝐵_𝑡𝑗−1, (7) where 𝜓(𝑥₁,…,𝑥_𝑗) = 𝐸[𝜑(𝑥₁,…,𝑥_𝑗, 𝑡_𝑗+1−𝑡_𝑗𝜉_𝑗+1,…, 𝑡_𝑛−𝑡_𝑛−1𝜉_𝑛)].

We denote by 𝐿^𝑝_𝐺(Ω_𝑇), 𝑝≥1, the completion of G-expectation space 𝐿_𝑖𝑝(Ω_𝑇) under the norm ‖𝑋‖_𝑝,𝐺∶=(𝐸[|𝑋|^𝑝])^1/𝑝. For all 𝑡∈[0,𝑇], 𝐸[⋅] and 𝐸_𝑡[⋅] are continuous mappings on 𝐿_𝑖𝑝(Ω_𝑇) endowed with the norm ‖⋅‖_1,𝐺. Therefore, it can be extended continuously to 𝐿^𝑝_𝐺(Ω_𝑇).

Definition 6. A process {𝑀_𝑡}_𝑡≥0 is called a G-martingale if for each 𝑡∈[0,𝑇], 𝑀_𝑡∈𝐿¹_𝐺(Ω_𝑡), and for each 𝑠∈[0,𝑡], we have 𝐸_𝑠[𝑀_𝑡]=𝑀_𝑠.

Now we introduce the Itô integral and quadratic variation process with respect to G-Brownian motion in G-expectation space.

Definition 7. Let 𝑝≥1 be fixed. For a given partition 𝜋_𝑇={𝑡₀,…,𝑡_𝑁} of [0,𝑇], we denote 𝑀_𝐺^𝑝,0(0,𝑇) as the collection of the following type of simple processes: 𝜂_𝑡(𝜔)=^𝑁−1_𝑘=0𝜉_𝑘(𝜔)1_{[𝑡𝑘,𝑡𝑘+1)}(𝑡), (8) where 𝜉_𝑘∈𝐿_𝑖𝑝(Ω_𝑡𝑘), 𝑘=0,1,2,…,𝑁−1. We denote by 𝑀^𝑝_𝐺(0,𝑇) the completion of 𝑀_𝐺^𝑝,0(0,𝑇) under the norm ‖⋅‖_𝑀^𝑝_𝐺(0,𝑇)∶={𝐸[∫^𝑇₀|⋅|^𝑝𝑑𝑡]}^1/𝑝.

Definition 8. For each 𝜂∈𝑀_𝐺^2,0(0,𝑇), we define 𝐼𝜂=^𝑇₀𝜂_𝑡𝑑𝐵_𝑡∶=^𝑁−1_𝑗=0𝜉_𝑗𝐵_𝑡𝑗+1−𝐵_𝑡𝑗. (9) The mapping 𝐼∶𝑀_𝐺^2,0(0,𝑇)→𝐿²_𝐺(Ω_𝑇) is continuous and thus can be continuously extended to 𝑀²_𝐺(0,𝑇).

Definition 9. The quadratic variation process of G-Brownian motion is defined by ⟨𝐵⟩_𝑡∶=𝐵²_𝑡−2^𝑡₀𝐵_𝑠𝑑𝐵_𝑠, (10) which is a continuous, nondecreasing process.

Definition 10. We now define the integral of a process 𝜂∈𝑀¹_𝐺(0,𝑇) with respect to ⟨𝐵⟩ as follows: 𝑄_0,𝑇𝜂=^𝑇₀𝜂_𝑡𝑑⟨𝐵⟩_𝑡∶=^𝑁−1_𝑗=0𝜉_𝑗⟨𝐵⟩_𝑡𝑗+1−⟨𝐵⟩_𝑡𝑗∶𝑀_𝐺^1,0(0,𝑇)⟶𝐿¹_𝐺Ω_𝑇. (11) The mapping is continuous and can be extended to 𝑀¹_𝐺(0,𝑇) uniquely.

Then, we detail some results about the quasianalysis theory constructed in [27].

Theorem 11. There exists a weakly compact probability measures family 𝒫 on (Ω,ℬ(Ω)) such that 𝐸[𝑋]=sup_𝑃∈𝒫𝐸_𝑝[𝑋],∀𝑋∈𝐿_𝑖𝑝Ω_𝑇, (12)𝒫 is called a set of probability measures that represents 𝐸.

Definition 12. We define the capacity associated to 𝒫, which is a weakly compact family of probability measure that represents 𝐸, as follows: 𝑐̂(𝐴)∶=sup_𝑃∈𝒫𝑃(𝐴),𝐴∈ℬΩ_𝑇; (13)𝑐̂ is also called the capacity induced by 𝐸.

Let (Ω⁰,ℱ⁰={ℱ⁰_𝑡},ℱ,𝑃⁰) be a filtered probability space, and let {𝑊_𝑡} be a 𝑑-dimensional Brownian motion under 𝑃⁰. [27] proved that 𝒫_𝑀∶={𝑃₀∘𝑋⁻¹∣𝑋_𝑡=∫^𝑡₀ℎ_𝑠𝑑𝑊_𝑠,ℎ∈𝐿²_ℱ0([0,𝑇];Γ^1/2)} represents G-expectation 𝐸, where Γ^1/2∶={𝛾^1/2∣𝛾∈Γ} and Γ is the set in the representation of 𝐺(⋅) of the formula (5).

Definition 13. (i) Let 𝑐̂  be the capacity induced by 𝐸. A set 𝐴⊂Ω is polar if 𝑐(̂𝐴)=0. A property holds “quasi-surely” (q.s. for short) if it holds outside a polar set.
(ii) Let 𝑋 and 𝑌 be two random variables; we say that 𝑋 is a version of 𝑌 if 𝑋=𝑌 q.s.

Let ‖𝜓‖_𝑝,𝐺=[𝐸(|𝜓|^𝑝)]^1/𝑝 for 𝜓∈𝐶_𝑏(Ω_𝑇). The completion of 𝐶_𝑏(Ω_𝑇) and 𝐿_𝑖𝑝(Ω_𝑇) under ‖⋅‖_𝑝,𝐺 is the same, and we denote them by 𝐿^𝑝_𝐺(Ω_𝑇).

2.2. Forward and Backward Stochastic Differential Equations Driven by G-Brownian Motion

We consider the following stochastic differential equations driven by 𝑑 dimensional G-Brownian motion (G-SDE): 𝑋_𝑡=𝑋₀+^𝑡₀𝑏𝑠,𝑋_𝑠𝑑𝑠+^𝑑_𝑖,𝑗=1^𝑡₀ℎ_𝑖𝑗𝑠,𝑋_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠+^𝑑_𝑗=1^𝑡₀𝜎_𝑗𝑠,𝑋_𝑠𝑑𝐵^𝑗_𝑠, (14) where 𝑡∈[0,𝑇], the initial condition 𝑋₀∈ℝ^𝑛 is a given constant, 𝑏,ℎ_𝑖𝑗,𝜎_𝑗 are given functions satisfying 𝑏(⋅,𝑥), and ℎ_𝑖𝑗(⋅,𝑥), 𝜎_𝑗(⋅,𝑥)∈𝑀²_𝐺(0,𝑇;ℝ^𝑛) for each 𝑥∈ℝ^𝑛 and the Lipschitz condition, that is, |𝜙(𝑡,𝑥)−𝜙(𝑡,𝑥^′)|≤𝐾|𝑥−𝑥^′|, for each 𝑡∈[0,𝑇], 𝑥, 𝑥^′∈ℝ^𝑛, 𝜙=𝑏, ℎ_𝑖𝑗, and 𝜎_𝑗, respectively. The solution is a process 𝑋∈𝑀²_𝐺(0,𝑇;ℝ^𝑛) satisfying the G-SDE (14).

Theorem 14 (see [12]). There exists a unique solution 𝑋∈𝑀²_𝐺(0,𝑇;ℝ^𝑛) of the stochastic differential equation (14).

Now, we give the results about BSDEs driven by G-Brownian motion in the G-expectation space (Ω_𝑇,𝐿¹_𝐺(Ω_𝑇),𝐸) with Ω_𝑇=𝐶₀([0,𝑇],ℝ^𝑑) and 𝜎²=𝐸[𝐵²₁]≥−𝐸[−𝐵²₁]=𝜎²>0. We consider the following type of G-BSDEs (we always use Einstein convention): 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝑠+^𝑑_𝑖,𝑗=1^𝑇_𝑡𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠−𝐾_𝑇−𝐾_𝑡, (15) where 𝑓(𝑡,𝜔,𝑦,𝑧), 𝑔_𝑖𝑗(𝑡,𝜔,𝑦,𝑧)∶[0,𝑇]×Ω_𝑇×ℝ×ℝ^𝑑→ℝ satisfy the following properties: there exist some 𝛽>1 such that(H1)for any 𝑦, 𝑧, 𝑓(⋅,⋅,𝑦,𝑧), 𝑔_𝑖𝑗(⋅,⋅,𝑦,𝑧)∈𝑀^𝛽_𝐺(0,𝑇);(H2)for some 𝐿>0, 𝑓𝑡,𝜔,𝑦,𝑧−𝑓𝑡,𝜔,𝑦^′,𝑧^′+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑡,𝜔,𝑦,𝑧−𝑔_𝑖𝑗𝑡,𝜔,𝑦^′,𝑧^′≤𝐿𝑦−𝑦^′+𝑧−𝑧^′. (16)

For simplicity, we denote by 𝔊(0,𝑇) the collection of processes (𝑌,𝑍,𝐾) such that 𝑌∈𝑆^𝛼_𝐺(0,𝑇), 𝑍∈𝐻^𝛼_𝐺(0,𝑇); 𝐾 is a decreasing G-martingale with 𝐾₀=0 and 𝐾_𝑇∈𝐿^𝛼_𝐺(Ω_𝑇). Here, 𝑆^𝛼_𝐺(0,𝑇) is the completion of 𝑆⁰_𝐺(0,𝑇)={ℎ(𝑡,𝐵_{𝑡1∧𝑡,…,𝐵𝑡𝑛∧𝑡})∶𝑡₁,…,𝑡_𝑛∈[0,𝑇],ℎ∈𝐶_{𝑏,𝐿𝑖𝑝(ℝ}^𝑛+1₎} under ‖⋅‖_𝑠^𝑝_𝐺={𝐸[sup_{𝑡∈[0,𝑇]}|𝜂_𝑡|^𝑝]}^1/𝑝, and 𝐻^𝑝_𝐺(0,𝑇) is the completion of 𝑀⁰_𝐺(0,𝑇) under ‖⋅‖_𝐻^𝑃_𝐺={𝐸[(∫^𝑇₀|𝜂_𝑠|²𝑑𝑠)^𝑝/2]}^1/𝑝.

Definition 15. Let 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇) with 𝛽>1; 𝑓 and 𝑔_𝑖𝑗 satisfy (H1) and (H2). A triplet of processes (𝑌,𝑍,𝐾) is called a solution of (15) if for some 1<𝛼≤𝛽 the following properties hold:
(a)  (𝑌,𝑍,𝐾)∈𝔊^𝛼_𝐺(0,𝑇);
(b)𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝑠+^𝑑_𝑖,𝑗=1^𝑇_𝑡𝑔_𝑖𝑗𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠−𝐾_𝑇−𝐾_𝑡. (17)

Theorem 16 (see [19]). Assume that 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇) and 𝑓,𝑔 satisfy (𝐻1) and (𝐻2) for some 𝛽>1. Then, (15) has a unique solution (𝑌,𝑍,𝐾). Moreover, for any 1<𝛼<𝛽, we have 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 17 (see [19]). Let 𝜉∈𝐿^𝛽_𝐺(Ω_𝑇), and 𝑓, 𝑔_𝑖𝑗 satisfy (𝐻1) and (𝐻2) for some 𝛽>1. For some 1<𝛼<𝛽, (𝑌,𝑍,𝐾)∈𝔊^𝛼_𝐺(0,𝑇) is solution of equation (15). Then(i)There exists a constant 𝐶_𝛼∶=𝐶(𝛼,𝑇,𝐺,𝐿)>0 such that𝑌_𝑡^𝛼≤𝐶_𝛼𝐸_𝑡𝜉^𝛼+^𝑇_𝑡ℎ⁰_𝑠^𝛼,𝐸𝑑𝑠_𝑇0𝑍_𝑠²𝑑𝑠^𝛼/2≤𝐶_𝛼𝐸sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼+𝐸sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼1/2𝐸^𝑇₀ℎ⁰_𝑠𝑑𝑠^𝛼1/2,𝐸𝐾_𝑇^𝛼≤𝐶_𝛼𝐸sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼+𝐸^𝑇₀ℎ⁰_𝑠𝑑𝑠^𝛼, (18)where ℎ⁰_𝑠=|𝑓(𝑠,0,0)|+∑^𝑑_𝑖,𝑗=1|𝑔_𝑖𝑗(𝑠,0,0)|.(ii)For any given 𝛼<𝛼^′<𝛽, there exists a constant 𝐶_𝛼,𝛼′ depending on 𝛼,𝛼^′,𝑇,𝐺,𝐿 such that𝐸sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼≤𝐶_𝛼,𝛼′𝐸sup_{[]𝑡∈0,𝑇}𝐸_𝑡𝜉^𝛼+𝐸sup_{[]𝑡∈0,𝑇}𝐸_𝑡^𝑇₀ℎ⁰_𝑠𝑑𝑠^{𝛼′𝛼/𝛼′}+𝐸sup_{[]𝑡∈0,𝑇}𝐸_𝑡^𝑇₀ℎ⁰_𝑠𝑑𝑠^𝛼′. (19)

Proposition 18 (see [20]). Let 𝜉^𝑖∈𝐿^𝛽_𝐺(Ω_𝑇), 𝑖=1,2, and 𝑓^𝑖, 𝑔^𝑖_𝑖𝑗 satisfy (H1) and (H2) for some 𝛽>1. For some 1<𝛼<𝛽, (𝑌^𝑖,𝑍^𝑖,𝐾^𝑖)∈𝔊^𝛼_𝐺(0,𝑇) are solutions of (15) corresponding to 𝜉^𝑖, 𝑓^𝑖, and 𝑔^𝑖_𝑖𝑗. Set 𝑌_𝑡=𝑌¹_𝑡−𝑌²_𝑡, 𝑍_𝑡=𝑍¹_𝑡−𝑍²_𝑡, and 𝐾_𝑡=𝐾¹_𝑡−𝐾²_𝑡. (i)There exists a constant 𝐶_𝛼∶=𝐶(𝛼,𝑇,𝐺,𝐿)>0 such that𝑌_𝑡^𝛼≤𝐶_𝛼𝐸_𝑡𝜉̂^𝛼+^𝑇_𝑡ℎ_𝑠^𝛼𝑑𝑠, (20)where 𝜉̂=𝜉¹−𝜉², ℎ_𝑠=|𝑓¹(𝑠,𝑌²_𝑠,𝑍²_𝑠)−𝑓²(𝑠,𝑌²_𝑠,𝑍²_𝑠)|+∑^𝑑_𝑖,𝑗=1|𝑔¹_𝑖𝑗(𝑠,𝑌²_𝑠,𝑍²_𝑠)−𝑔²_𝑖𝑗(𝑠,𝑌²_𝑠,𝑍²_𝑠)|.(ii)For any given 𝛼^′ with 𝛼<𝛼^′<𝛽, there exists a constant 𝐶_𝛼,𝛼′ depending on 𝛼,𝛼^′,𝑇,𝐺,𝐿 such that𝐸sup_{[]𝑡∈0,𝑇}𝑌_𝑡^𝛼≤𝐶_𝛼,𝛼′𝐸sup_{[]𝑡∈0,𝑇}𝐸_𝑡𝜉̂^𝛼+𝐸sup_{[]𝑡∈0,𝑇}𝐸_𝑡^𝑇₀ℎ_𝑠𝑑𝑠^{𝛼′𝛼/𝛼′}+𝐸sup_{[]𝑡∈0,𝑇}𝐸_𝑡^𝑇₀ℎ_𝑠𝑑𝑠^𝛼′. (21)

Theorem 19 (see [20]). Let (𝑌^𝑖_𝑡,𝑍^𝑖_𝑡,𝐾^𝑖_𝑡)_𝑡≤𝑇, 𝑖=1,2, be the solutions of the following G-BSDEs: 𝑌^𝑖_𝑡=𝜉^𝑖+^𝑇_𝑡𝑓_𝑖𝑠,𝑌^𝑖_𝑠,𝑍^𝑖_𝑠𝑑𝑠+^𝑇_𝑡𝑔_𝑖𝑠,𝑌^𝑖_𝑠,𝑍^𝑖_𝑠𝑑⟨𝐵⟩_𝑠−^𝑇_𝑡𝑍^𝑖_𝑠𝑑𝐵_𝑠−𝐾^𝑖_𝑇−𝐾^𝑖_𝑡, (22) where 𝜉^𝑖∈𝐿^𝛽_𝐺(Ω_𝑇), 𝑓_𝑖, 𝑔_𝑖 satisfy (H1) and (H2) with 𝛽>1. If 𝜉¹≥𝜉², 𝑓₁≥𝑓₂, 𝑔₁≥𝑔₂, then 𝑌¹_𝑡≥𝑌²_𝑡.

Theorem 20 (see [20]). Let (𝑌^𝑖_𝑡,𝑍^𝑖_𝑡,𝐾^𝑖_𝑡)_𝑡≤𝑇, 𝑖=1,2, be the solutions of the following G-BSDEs: 𝑌^𝑖_𝑡=𝜉^𝑖+^𝑇_𝑡𝑓_𝑖𝑠,𝑌^𝑖_𝑠,𝑍^𝑖_𝑠𝑑𝑠+^𝑇_𝑡𝑔_𝑖𝑠,𝑌^𝑖_𝑠,𝑍^𝑖_𝑠𝑑⟨𝐵⟩_𝑠−^𝑇_𝑡𝑍^𝑖_𝑠𝑑𝐵_𝑠−𝐾^𝑖_𝑇−𝐾^𝑖_𝑡+𝑉^𝑖_𝑇−𝑉^𝑖_𝑡, (23) where 𝜉^𝑖∈𝐿^𝛽_𝐺(Ω_𝑇), 𝑓_𝑖, 𝑔_𝑖 satisfy (𝐻1) and (𝐻2), (𝑉^𝑖_𝑡)_𝑡≤𝑇 are RCLL processes such that 𝐸[sup_{𝑡∈[0,𝑇]}|𝑉^𝑖_𝑡|^𝛽]<∞ with 𝛽>1. If 𝜉¹≥𝜉², 𝑓₁≥𝑓₂, 𝑔₁≥𝑔₂, and 𝑉¹_𝑡−𝑉²_𝑡 is an increasing process, then 𝑌¹_𝑡≥𝑌²_𝑡.

3. A DPP for Stochastic Optimal Control Problems under G-Expectation

Now we introduce the setting for stochastic optimal control problems under 𝐺-expectation. We suppose that the control state space 𝑉 is a compact metric space. Let the set of admissible control processes 𝒰 for the player be a set of 𝑉-valued stochastic processes in 𝑀²_𝐺([𝑡,𝑇];ℝ^𝑛), 𝑡∈[0,𝑇]. For a given admissible control 𝜐(⋅)∈𝒰, the corresponding orbit, which regards 𝑡 as the initial time and 𝜉∈𝐿²_𝐺(Ω_𝑡;ℝ^𝑛) as the initial state, is defined by the solution of the following type of G-SDE: 𝑑𝑋_𝑠^{𝑡,𝜉;𝜐}=𝑏𝑠,𝑋_𝑠^{𝑡,𝜉;𝜐},𝜐_𝑠𝑑𝑠+^𝑑_𝑖,𝑗=1ℎ_𝑖𝑗𝑠,𝑋_𝑠^{𝑡,𝜉;𝜐},𝜐_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠+^𝑑_𝑗=1𝜎_𝑗𝑠,𝑋_𝑠^{𝑡,𝜉;𝜐},𝜐_𝑠𝑑𝐵^𝑗_𝑠,𝑠∈[𝑡,𝑇],𝑋_𝑡^{𝑡,𝜉;𝜐}=𝜉, (24) where 𝑏, ℎ_𝑖𝑗, 𝜎_𝑗∶[0,𝑇]×ℝ^𝑛×𝒰→ℝ^𝑛 are deterministic functions and satisfy the following conditions (H3): (A1)ℎ_𝑖𝑗=ℎ_𝑗𝑖 for 1≤𝑖,𝑗≤𝑑;(A2)For every fixed (𝑥,𝜐)∈ℝ^𝑛×𝒰, 𝑏(⋅,𝑥,𝜐), ℎ_𝑖𝑗(⋅,𝑥,𝜐), 𝜎_𝑗(⋅,𝑥,𝜐) are continuous in 𝑡;(A3)There exists a constant 𝐿>0, for any 𝑡∈[0,𝑇], 𝑥, 𝑥^′∈ℝ^𝑛, 𝜐, 𝜐^′∈𝒰 such that 𝑏(𝑡,𝑥,𝜐)−𝑏𝑡,𝑥^′,𝜐^′+^𝑑_𝑖,𝑗=1ℎ_𝑖𝑗(𝑡,𝑥,𝜐)−ℎ_𝑖𝑗𝑡,𝑥^′,𝜐^′+^𝑑_𝑗=1𝜎_𝑗(𝑡,𝑥,𝜐)−𝜎_𝑗𝑡,𝑥^′,𝜐^′≤𝐿𝑥−𝑥^′+𝜐−𝜐^′. (25)

From the assumption (H3), we can get global linear growth conditions for 𝑏, ℎ_𝑖𝑗, 𝜎_𝑗; that is, there exists 𝐶>0 such that, for 𝑡∈[0,𝑇], 𝑥∈ℝ^𝑛, 𝜐∈𝒰, |𝑏(𝑡,𝑥,𝜐)|+∑^𝑑_𝑖,𝑗=1|ℎ_𝑖𝑗(𝑡,𝑥,𝜐)|+∑^𝑑_𝑗=1|𝜎_𝑗(𝑡,𝑥,𝜐)|≤𝐶(1+|𝑥|+|𝜐|). Obviously, under the above assumptions, for any 𝜐(⋅)∈𝒰, G-SDE (24) has a unique solution. Moreover, we have the following estimates.

Proposition 21. Let 𝜉, 𝜉^′∈𝐿^𝑝_𝐺(Ω_𝑡;ℝ^𝑛) with 𝑝≥2, 𝜐(⋅), 𝜐^′(⋅)∈𝒰, 𝑡∈[0,𝑇], and 𝛿∈[0,𝑇−𝑡]; then we have 𝐸_𝑡𝑋^{𝑡,𝜉;𝜐}_𝑡+𝛿−𝑋^{𝑡,𝜉′;𝜐′}_𝑡+𝛿^𝑝≤𝐶𝜉−𝜉^′𝑝+_𝑡^𝑡+𝛿𝐸_𝑡𝜐_𝑟−𝜐^′_𝑟^𝑝,𝐸𝑑𝑟_𝑡𝑋^{𝑡,𝜉;𝜐}_𝑡+𝛿^𝑝≤𝐶1+𝜉^𝑝,𝐸_𝑡sup_{[]𝑠∈𝑡,𝑡+𝛿}𝑋_𝑠^{𝑡,𝜉;𝜐}−𝜉^𝑝≤𝐶1+𝜉^𝑝𝛿^𝑝/2, (26) where 𝐶 depends on 𝐿,𝐺,𝑝,𝑛,𝑇.

Proof. The proof is similar to the proof of Proposition 4.1 in [20].

Now we give bounded functions Φ∶ℝ^𝑛→ℝ, 𝑓∶[0,𝑇]×ℝ^𝑛×ℝ×ℝ^𝑑×𝒰→ℝ, 𝑔_𝑖𝑗∶[0,𝑇]×ℝ^𝑛×ℝ×ℝ^𝑑×𝒰→ℝ that satisfy the following conditions: (H4) (i)𝑔_𝑖𝑗=𝑔_𝑗𝑖 for 1≤𝑖,𝑗≤𝑑.(ii)For every fixed (𝑥,𝑦,𝑧,𝜐)∈ℝ^𝑛×ℝ×ℝ^𝑛×𝒰, 𝑓(⋅,𝑥,𝑦,𝑧,𝜐) and 𝑔_𝑖𝑗(⋅,𝑥,𝑦,𝑧,𝜐) are continuous in 𝑡, 1≤𝑖,𝑗≤𝑑.(iii)There exist a constant 𝐿>0, for 𝑡∈[0,𝑇], 𝑥, 𝑥^′∈ℝ^𝑛, 𝑦, 𝑦^′∈ℝ, 𝑧, 𝑧^′∈ℝ^𝑑, 𝜐, 𝜐^′∈𝒰, such that Φ(𝑥)−Φ𝑥^′≤𝐿𝑥−𝑥^′,𝑓𝑡,𝑥,𝑦,𝑧,𝜐−𝑓𝑡,𝑥^′,𝑦^′,𝑧^′,𝜐^′+^𝑑_𝑖,𝑗=1𝑔_𝑖𝑗𝑡,𝑥,𝑦,𝑧,𝜐−𝑔_𝑖𝑗𝑡,𝑥^′,𝑦^′,𝑧^′,𝜐^′≤𝐿𝑥−𝑥^′+𝑦−𝑦^′+𝑧−𝑧^′+𝜐−𝜐^′. (27)

From (H4), we have that Φ, 𝑓, and 𝑔_𝑖𝑗 also satisfy global linear growth condition in 𝑥; that is, there exists 𝐶>0, such that for all 0≤𝑡≤𝑇, 𝜐∈𝒰, 𝑥∈ℝ^𝑛, |Φ(𝑥)|+𝑓(𝑡,𝑥,0,0,𝜐)+𝑔_𝑖𝑗((|𝑥|+|𝜐.𝑡,𝑥,0,0,𝜐)≤𝐶1+|) (28) For any 𝜐∈𝒰 and 𝜉∈𝐿²_𝐺(Ω_𝑡,ℝ^𝑛), the mappings 𝑓(𝑠,𝑥,𝑦,𝑧,𝜐)∶=𝑓(𝑠,𝑋_𝑠^{𝑡,𝜉;𝜐},𝑦,𝑧,𝜐_𝑠) and 𝑔_𝑖𝑗(𝑠,𝑥,𝑦,𝑧,𝜐)=𝑔_𝑖𝑗(𝑠,𝑋_𝑠^{𝑡,𝜉;𝜐},𝑦,𝑧,𝜐_𝑠), where (𝑠,𝑦,𝑧)∈[0,𝑇]×ℝ×ℝ^𝑑 satisfy the conditions of Theorem 16 on the interval [𝑡,𝑇]. Therefore, there exists a unique solution for the following G-BSDE: 𝑌_𝑠^{𝑡,𝜉;𝜐}=Φ𝑋_𝑇^{𝑡,𝜉;𝜐}+^𝑇_𝑠𝑓𝑟,𝑋_𝑟^{𝑡,𝜉;𝜐},𝑌_𝑟^{𝑡,𝜉;𝜐},𝑍_𝑟^{𝑡,𝜉;𝜐},𝜐_𝑟−𝑑𝑟_𝑇𝑠𝑍_𝑟^{𝑡,𝜉;𝜐}𝑑𝐵_𝑟−𝐾_𝑇^{𝑡,𝜉;𝜐}−𝐾_𝑠^{𝑡,𝜉;𝜐}+^𝑑_𝑖,𝑗=1^𝑇_𝑠𝑔_𝑖𝑗𝑟,𝑋_𝑟^{𝑡,𝜉;𝜐},𝑌_𝑟^{𝑡,𝜉;𝜐},𝑍_𝑟^{𝑡,𝜉;𝜐},𝜐_𝑟𝑑𝐵^𝑖,𝐵^𝑗_𝑟, (29) where 𝑋^{𝑡,𝜉;𝜐} is introduced by (24).

Proposition 22. For each 𝜉, 𝜉^′∈𝐿^𝑝_𝐺(Ω_𝑡;ℝ^𝑛) with 𝑝≥2 and 𝜐(⋅), 𝜐^′(⋅)∈𝒰, we have 𝑌_𝑡^{𝑡,𝜉;𝜐}−𝑌^𝑡,𝜉′_𝑡^;𝜐≤𝐶𝜉−𝜉^′,𝑌_𝑡^{𝑡,𝜉;𝜐}𝜉,𝑌≤𝐶1+_{𝑡𝑡,𝜉;𝜐}−𝑌^{𝑡,𝜉;𝜐′}_𝑡≤𝐶^𝑇_𝑡𝐸_𝑡𝜐(𝑟)−𝜐^′(𝑟)²𝑑𝑟^1/2, (30) where 𝐶 depends on 𝐿, 𝐺, 𝑛, and 𝑇.

Proof. The proof is similar to the Proposition 4.2 in [20].

Given a control process 𝜐(⋅)∈𝒰, we introduce an associated cost functional 𝐽(𝑡,𝑥;𝜐)=𝑌_𝑡^{𝑡,𝑥;𝜐},(𝑡,𝑥)∈[0,𝑇]×ℝ^𝑛, (31) where the process 𝑌_𝑡^{𝑡,𝜉;𝜐} is defined by G-BSDE (29). Similar to the proof of Theorem 4.4 in [20], we have that for 𝑡∈[0,𝑇], 𝜉∈𝐿²_𝐺(Ω_𝑡,ℝ^𝑛), 𝐽(𝑡,𝜉;𝜐)∶=𝑌_𝑡^{𝑡,𝜉;𝜐}. (32) But we are more interested in the case when 𝜉=𝑥.

Now we define the value function as follows: 𝑢(𝑡,𝑥)∶=sup_{𝜐(⋅)∈𝒰}𝐽(𝑡,𝑥;𝜐). (33)

Proposition 23. 𝑢(𝑡,𝑥) is a deterministic function of (𝑡,𝑥).

Proof. For a partition of [𝑡,𝑠]: 𝑡=𝑡₀<𝑡₁<⋯<𝑡_𝑁=𝑠, 𝑝≥2, 𝑡≤𝑠≤𝑇, we denote 𝐿_𝑖𝑝(Ω^𝑡_s)∶={𝜑(𝐵_𝑡1−𝐵_𝑡,…,𝐵_𝑡𝑛−𝐵_𝑡)∶𝑛≥1,𝑡₁,…,𝑡_𝑛∈[𝑡,𝑠],𝜑∈𝐶_{𝑏,𝐿𝑖𝑝}(ℝ^𝑑×𝑛)}, 𝑀_𝐺^𝑝,0,𝑡(𝑡,𝑠;ℝ^𝑛) by the collection of simple processes 𝜂(𝑟)=∑^𝑁−1_𝑘=0𝜉_𝑘1_{[𝑡𝑘,𝑡𝑘+1)}(𝑟), where 𝜉_𝑘∈𝐿_𝑖𝑝(Ω^𝑡_𝑡𝑘;ℝ^𝑛), 𝑘=0,1,2,…,𝑁−1, and 𝑀_𝐺^𝑝,𝑡(𝑡,𝑠;ℝ^𝑛) by the completion of 𝑀_𝐺^𝑝,0,𝑡(𝑡,𝑠;ℝ^𝑛) under the norm ‖𝜂‖_𝑀^𝑝_{𝐺(𝑡,𝑠;ℝ𝑛)}∶={𝐸[∫^𝑠_𝑡|𝜂(𝑟)|^𝑝𝑑𝑟]}^1/𝑝. Use the similar method in Lemma 43 of [27]; we can prove that 𝜐∈𝑀²_𝐺(𝑡,𝑠;ℝ^𝑛) is a 𝑉-valued process; there exists {𝑢=∑^𝑁_𝑖=11_𝐴𝑖𝑢^𝑖}_𝑁∈ℕ; 𝑢^𝑖∈𝑀_𝐺^2,𝑡(𝑡,𝑠;ℝ^𝑛) is a 𝑉-valued process, 𝐴_𝑖 is a partition of ℬ(Ω_𝑡) such that 𝑢→𝑣 under probability measure 𝑃∈𝒫_𝑀. When 𝜐(𝑠)∈𝑀_𝐺^2,𝑡(𝑡,𝑠;ℝ^𝑛), we note that 𝐽(𝑡,𝑥;𝜐) is a deterministic function of (𝑡,𝑥) because 𝑏, ℎ_𝑖𝑗, 𝜎_𝑗, Φ, 𝑓, and 𝑔_𝑖𝑗 are deterministic functions, and 𝐵_𝑠∶=𝐵_𝑡+𝑠−𝐵_𝑡 is a G-Brownian motion. So we need to construct a sequence of admissible controls {𝜐^𝑖(⋅)} of the form 𝜐^𝑖_𝑠=^𝑁𝑖_𝑗=1𝜐_𝑠^𝑖𝑗1_𝐴𝑖𝑗 (34) satisfying lim_𝑖→∞𝐽(𝑡,𝑥;𝜐^𝑖(⋅))=𝑢(𝑡,𝑥), where 𝜐^𝑖𝑗(⋅)∈𝑀_𝐺^2,𝑡(𝑡,𝑠;ℝ^𝑛) is a 𝑉-valued processes and {𝐴_𝑖𝑗}^𝑁𝑖_𝑗=1 is a partition of ℬ(Ω_𝑡). Firstly, there exists {𝜐^𝑘}_𝑘≥1⊂𝒰, such that 𝑢(𝑡,𝑥)=sup_𝑘≥1𝐽(𝑡,𝑥;𝜐^𝑘). Then we define 𝜐,𝜐^′∈𝒰, 𝜐∨𝜐^′_𝑠=[];𝜐0,𝑠∈0,𝑡_𝑠,𝑠∈(𝑡,𝑇],on𝐽(𝑡,𝑥;𝜐)≥𝐽𝑡,𝑥;𝜐^′;𝜐^′_𝑠,𝑠∈(𝑡,𝑇],on𝐽(𝑡,𝑥;𝜐)<𝐽𝑡,𝑥;𝜐^′. (35) Therefore, 𝐽𝑡,𝑥;𝜐∨𝜐^′≥𝐽(𝑡,𝑥;𝜐)∨𝐽𝑡,𝑥;𝜐^′. (36) Set 𝜐¹∶=𝜐¹∨𝜐¹, 𝜐^𝑘∶=𝜐^𝑘−1∨𝜐^𝑖, 𝑖≥2. So 𝑢(𝑡,𝑥)=lim_𝑘→∞𝐽(𝑡,𝑥;𝜐^𝑘). Without loss of generality, suppose 𝐸[(𝑢(𝑡,𝑥)−𝐽(𝑡,𝑥;𝜐^𝑘))²]≤1/𝑘,𝑘≥1. We denote 𝜐^𝑘_𝑠=^𝑁𝑖−1_𝑗,𝑘=0𝜐_𝑗,𝑘(𝑠)1_𝐴^𝑘_𝑗, (37) where 𝜐_𝑗,𝑘∈𝑀_𝐺^2,𝑡(𝑡,𝑠;ℝ^𝑛) is a 𝑉-valued process and {𝐴^𝑘_𝑗}_{0≤𝑗≤𝑁𝑘−1} is a partition of ℬ(Ω_𝑡). Then we can suppose for 𝑘≥1, 𝐸[∫^𝑇_𝑡|𝜐^𝑘_𝑠−𝜐^𝑘_𝑠|²𝑑𝑠]≤1/𝐶𝑘. From Proposition 22, we have 𝐸𝐽𝑡,𝑥;𝜐^𝑘−𝐽𝑡,𝑥;𝜐^𝑘2≤𝐶𝐸^𝑇_𝑡𝜐^𝑘_𝑠−𝜐^𝑘_𝑠²≤1𝑑𝑠𝑘. (38) Therefore, 𝐸[|𝑢(𝑡,𝑥)−𝐽(𝑡,𝑥;𝜐^𝑘)|²]≤4/𝑘. Then, we have 𝐽𝑡,𝑥;𝜐^𝑘=^𝑁𝑘−1_𝑗=01_𝐴^𝑘_𝑗𝐽𝑡,𝑥;𝜐_𝑗,𝑘≤𝑢(𝑡,𝑥). (39) Now, we suppose that 𝐽(𝑡,𝑥,𝜐)≤max_{0≤𝑗≤𝑁𝑘−1}𝐽𝑡,𝑥;𝜐_𝑗,𝑘=𝐽𝑡,𝑥;𝜐_𝑗′,𝑘. (40) Because 𝐸[|𝐽(𝑡,𝑥;𝜐^𝑘)−𝑢(𝑡,𝑥)|²]→0, we have 𝑢(𝑡,𝑥)=lim_𝑘→∞𝐽𝑡,𝑥;𝜐_𝑗′,𝑘,q.s.. (41) Hence, 𝐸[𝑢(𝑡,𝑥)]=𝑢(𝑡,𝑥). We have finished the proof.

Lemma 24. For any 𝑡∈[0,𝑇], 𝑥, 𝑥^′∈ℝ^𝑛, we have 𝑢(𝑡,𝑥)−𝑢𝑡,𝑥^′≤𝐶𝑥−𝑥^′|𝑢((|𝑥.,𝑡,𝑥)|≤𝐶1+|) (42)(43)

Proof. By Proposition 22, we have for 𝜐(⋅)∈𝒰, |𝐽((⋅(|𝑥,𝐽((⋅𝑡,𝑥;𝜐))|≤𝐶1+|)𝑡,𝑥;𝜐))−𝐽𝑡,𝑥^′;𝜐(⋅)≤𝐶𝑥−𝑥^′. (44) Then, ∀𝜀>0, there exist 𝜐(⋅), 𝜐^′(⋅)∈𝒰 such that 𝐽((⋅()((⋅𝐽𝑡,𝑥;𝜐))≤𝑢𝑡,𝑥≤𝐽𝑡,𝑥;𝜐))+𝜀,𝑡,𝑥^′;𝜐^′(⋅)≤𝑢𝑡,𝑥^′≤𝐽𝑡,𝑥^′;𝜐^′(⋅)+𝜀. (45) Now, we have (|𝑥((⋅()((⋅(|𝑥−𝐶1+|)≤𝐽𝑡,𝑥;𝜐))≤𝑢𝑡,𝑥≤𝐽𝑡,𝑥;𝜐))+𝜀≤𝐶1+|)+𝜀. (46) So, we get (43). Similarly, we obtain 𝐽𝑡,𝑥;𝜐^′(⋅)−𝐽𝑡,𝑥^′;𝜐^′(⋅)()−𝜀≤𝑢𝑡,𝑥−𝑢𝑡,𝑥^′≤𝐽(𝑡,𝑥;𝜐(⋅))−𝐽𝑡,𝑥^′;𝜐(⋅)+𝜀. (47) Then, −𝐶𝑥−𝑥^′−𝜀≤𝑢(𝑡,𝑥)−𝑢𝑡,𝑥^′≤𝐶𝑥−𝑥^′+𝜀. (48) Thus, we have proved (42).

Lemma 25. For any 𝑡∈[0,𝑇], 𝜁∈𝐿²_𝐺(Ω_𝑡;ℝ^𝑛), and 𝜁 is ℱ^𝐵_𝑡 measurable; we have ∀𝜐(⋅)∈𝒰, 𝑢(𝑡,𝜁)≥𝑌_𝑡^{𝑡,𝜁;𝜐}. (49) Conversely, ∀𝜀>0, there exists a 𝜐(⋅)∈𝒰, such that 𝑢(𝑡,𝜁)≤𝑌_𝑡^{𝑡,𝜁;𝜐}+𝜀. (50)

Proof. We already know that 𝑢(𝑡,𝑥) is continuous with respect to 𝑥 and 𝑌_𝑡^{𝑡,𝜁;𝜐} is continuous with respect to (𝜁,𝜐(⋅)). We want to prove (49) and only need to discuss the simple random variables 𝜁 of the form 𝜁=^𝑁_𝑖=11_𝐴𝑖𝑥_𝑖, (51) and 𝜐(⋅) of the form 𝜐(⋅)=^𝑁_𝑖=11_𝐴𝑖𝜐^𝑖(⋅). (52) Here, 𝑖=1,2,…,𝑁, 𝑥_𝑖∈ℝ^𝑛, 𝜐^𝑖∈𝑀_𝐺^2,𝑡(𝑡,𝑠;ℝ^𝑛), and {𝐴_𝑖}^𝑁_𝑖=1 is a ℬ(Ω_𝑡)-partition. Then, from the same technique used in the proof of Theorem 4.4 in [20], we have 𝑌_𝑡^{𝑡,𝜁;𝜐}=^𝑁_𝑖=11_𝐴𝑖𝑌^{𝑡,𝑥𝑖;𝜐𝑖}_𝑡≤^𝑁_𝑖=11_𝐴𝑖𝑢𝑡,𝑥_𝑖=𝑢𝑡,^𝑁_𝑖=11_𝐴𝑖𝑥_𝑖=𝑢(𝑡,𝜁). (53) So we have proved (49). Now we prove (50) in a similar way. We first construct a random variable 𝜂∈𝐿²_𝐺(Ω_𝑡;ℝ^𝑛), 𝜂=^𝑁_𝑖=1𝑥^𝑖_𝑠1_𝐴𝑖, (54) where (𝐴_𝑖)^𝑁_𝑖=1 is a ℬ(Ω_𝑡)-partition and 𝑥_𝑖∈ℝ^𝑛, such that |𝜂−𝜁|≤𝜀/3𝐶. Then, we have 𝑌_𝑡^{𝑡,𝜂;𝜐}−𝑌_𝑡^{𝑡,𝜁;𝜐}≤𝜀3,𝑢()≤𝜀𝑡,𝜁−𝑢𝑡,𝜂3, (55) for 𝜐(⋅)∈𝒰. Now, we choose a control 𝜐^𝑖(⋅)∈𝑀_𝐺^2,𝑡(𝑡,𝑠;ℝ^𝑛), such that 𝑢(𝑡,𝑥_𝑖)≤𝑌^{𝑡,𝑥𝑖;𝜐𝑖}_𝑡+𝜀/3. Set 𝜐(⋅)∶=∑^𝑁_𝑖=1𝜐^𝑖(⋅)1_𝐴𝑖. Finally, we get 𝑌_𝑡^{𝑡𝜁;𝜐}≥−𝑌_𝑡^{𝑡,𝜂;𝜐}−𝑌_𝑡^{𝑡,𝜁;𝜐}+𝑌_𝑡^{𝑡,𝜂;𝜐}𝜀≥−3+^𝑁_𝑖=1𝑌^{𝑡,𝑥𝑖;𝜐𝑖}_𝑡1_𝐴𝑖𝜀≥−3+^𝑁_𝑖=1𝑢𝑡,𝑥_𝑖−𝜀31_𝐴𝑖=−2𝜀3+^𝑁_𝑖=1𝑢𝑡,𝑥_𝑖1_𝐴𝑖=−2𝜀3+𝑢𝑡,𝜂≥−𝜀+𝑢(𝑡,𝜁). (56) So, we have (50).

Now, we give a type of DPP for our stochastic optimal control problems. Firstly, we define a family of backward semigroups associated with the G-BSDE (29). Given the initial data (𝑡,𝑥), a positive number 𝛿≤𝑇−𝑡, and a random variable 𝜂∈𝐿^𝑝_𝐺(Ω;ℝ) with 𝑝>1, we set 𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝜂∶=𝑌_𝑠^{𝑡,𝑥;𝜐}, (57) where (𝑌_𝑠^{𝑡,𝑥;𝜐})_{𝑡≤𝑠≤𝑡+𝛿} is the solution of the following G-BSDE with the time horizon 𝑡+𝛿: 𝑌_𝑠^{𝑡,𝑥;𝜐}=𝜂+_𝑠^𝑡+𝛿𝑓𝑟,𝑋_𝑟^{𝑡,𝑥;𝜐},𝑌_𝑟^{𝑡,𝑥;𝜐},𝑍_𝑟^{𝑡,𝑥;𝜐},𝜐_𝑟−𝑑𝑟_{𝑠𝑡+𝛿}𝑍_𝑟^{𝑡,𝑥;𝜐}𝑑𝐵_𝑟−𝐾_𝑇^{𝑡,𝑥;𝜐}−𝐾_𝑡^{𝑡,𝑥;𝜐}+^𝑑_{𝑖,𝑗=1𝑠}^𝑡+𝛿𝑔_𝑖𝑗𝑟,𝑋_𝑟^{𝑡,𝑥;𝜐},𝑌_𝑟^{𝑡,𝑥;𝜐},𝑍_𝑟^{𝑡,𝑥;𝜐},𝜐_𝑟𝑑𝐵^𝑖,𝐵^𝑗_𝑟. (58) Obviously, for the solution 𝑌^{𝑡,𝑥;𝜐} of G-BSDE (29), we have 𝐺^{𝑡,𝑥;𝜐}_𝑡,𝑇Φ𝑋_𝑇^{𝑡,𝑥;𝜐}=𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑌^{𝑡,𝑥;𝜐}_𝑡+𝛿. (59) Then, we can obtain the DPP for our stochastic optimal control problems as follows.

Theorem 26. The value function 𝑢(𝑡,𝑥) has the following proposition: for every 0≤𝛿≤𝑇−𝑡, we have 𝑢(𝑡,𝑥)=sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿. (60)

Proof. We have 𝑢(𝑡,𝑥)=sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_𝑡,𝑇Φ𝑋_𝑇^{𝑡,𝑥;𝜐}=sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑌^{𝑡+𝛿,𝑋𝑡,𝑥;𝜐𝑡+𝛿;𝜐}_𝑡+𝛿. (61) Obviously, 𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿 is ℱ^𝐵_𝑡+𝛿 measurable. So, by Lemma 25 and Theorem 19, we have 𝑢(𝑡,𝑥)≤sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿. (62) Besides, for 𝜀>0, there exists an admissible control 𝜐(⋅)∈𝒰 such that 𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿≤𝑌^{𝑡+𝛿,𝑋𝑡,𝑥;𝜐𝑡+𝛿;𝜐}_𝑡+𝛿+𝜀. (63) Then, 𝑢(𝑡,𝑥)≥sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿−𝜀≥sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿−𝐶𝜀. (64) Because 𝜀 can be arbitrarily small, we get (60).

Proposition 27. 𝑢(𝑡,𝑥) is 1/2-Hölder continuous in 𝑡.

Proof. For any given (𝑡,𝑥)∈[0,𝑇]×ℝ^𝑛 and 𝛿>0(𝑡+𝛿≤𝑇), from Theorem 26, we know that for 𝜀>0, there exists a 𝜐(⋅)∈𝒰 such that 𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿+𝜀≥𝑢(𝑡,𝑥)≥𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿. (65) Then, we need to prove 𝑢(𝑡,𝑥)−𝑢(𝑡+𝛿,𝑥)≤𝐶𝛿^1/2resp.,≥−𝐶𝛿^1/2. (66) We only check the first inequality in (66). The second can be proved similarly. We have ∀𝜀>0, 𝑢(𝑡,𝑥)−𝑢(𝑡+𝛿,𝑥)≤𝐼¹_𝛿+𝐼²_𝛿+𝜀, (67) where 𝐼¹_𝛿=𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿−𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}[𝑢(,𝐼𝑡+𝛿,𝑥)]_2𝛿=𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}[𝑢(𝑡+𝛿,𝑥)]−𝑢(𝑡+𝛿,𝑥). (68) From Proposition 21, we have 𝐸_𝑡𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿−𝑥²≤𝐶1+|𝑥|²𝛿. (69) By Proposition 22 and Lemma 24, we deduce that 𝐼¹_𝛿≤𝐶𝐸_𝑡𝑢𝑡+𝛿,𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿−𝑢(𝑡+𝛿,𝑥)^21/2≤𝐶𝐸_𝑡𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿−𝑥^21/2≤𝐶^′𝛿^1/2. (70) Based on the definition of 𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}, we get 𝐼²_𝛿=𝐸_𝑡𝑢(𝑡+𝛿,𝑥)+_𝑡^𝑡+𝛿𝑓𝑠,𝑋_𝑠^{𝑡,𝑥;𝜐},𝑌_𝑠^{𝑡,𝑥;𝜐},𝑍_𝑠^{𝑡,𝑥;𝜐},𝜐_𝑠+𝑑𝑠_{𝑑𝑖,𝑗=1𝑡𝑡+𝛿}𝑔_𝑖𝑗𝑠,𝑋_𝑠^{𝑡,𝑥;𝜐},𝑌_𝑠^{𝑡,𝑥;𝜐},𝑍_𝑠^{𝑡,𝑥;𝜐},𝜐_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠−_𝑡^𝑡+𝛿𝑍_𝑠^{𝑡,𝑥;𝜐}𝑑𝐵_𝑠−𝐾_𝑇^{𝑡,𝑥;𝜐}−𝐾_𝑡^{𝑡,𝑥;𝜐}()=𝐸−𝑢𝑡+𝛿,𝑥_{𝑡𝑡𝑡+𝛿}𝑓𝑠,𝑋_𝑠^{𝑡,𝑥;𝜐},𝑌_𝑠^{𝑡,𝑥;𝜐},𝑍_𝑠^{𝑡,𝑥;𝜐},𝜐_𝑠+𝑑𝑠_{𝑑𝑖,𝑗=1𝑡𝑡+𝛿}𝑔_𝑖𝑗𝑠,𝑋_𝑠^{𝑡,𝑥;𝜐},𝑌_𝑠^{𝑡,𝑥;𝜐},𝑍_𝑠^{𝑡,𝑥;𝜐},𝜐_𝑠𝑑𝐵^𝑖,𝐵^𝑗_𝑠≤𝐶^′𝛿^1/2×1+𝐸_𝑡𝑡^𝑡+𝛿𝑋_𝑠^{𝑡,𝑥;𝜐2}+𝑌_𝑠^{𝑡,𝑥;𝜐2}+𝑍_𝑠^{𝑡,𝑥;𝜐2}𝑑𝑠^1/2. (71) By Proposition 22, we can prove the following inequality easily by the similar method in Proposition 3.5 of [11] 𝐸_𝑡𝑡^𝑡+𝛿𝑍_𝑠^{𝑡,𝑥;𝜐2}𝑑𝑠^1/2≤𝐶(1+|𝑥|). (72) So we have 𝐼²_𝛿≤𝐶^′𝛿^1/2. Hence, by (67) we have 𝑢(𝑡,𝑥)−𝑢(𝑡+𝛿,𝑥)≤𝐶^′𝛿^1/2+𝜀. (73) Let 𝜀→0; we obtain the first inequality of (66). The proof is completed.

4. Value Function and Viscosity Solution of Fully Nonlinear Second-Order Partial Differential Equation

In this section, we consider the following fully nonlinear second-order partial differential equation 𝜕_𝑡𝑢+𝐹𝐷²_𝑥𝑢,𝐷_𝑥𝑢,𝑢,𝑥,𝑡=0,(𝑡,𝑥)∈[0,𝑇]×ℝ^𝑛,𝑢(𝑇,𝑥)=Φ(𝑥), (74) where 𝐹𝐷²_𝑥𝑢,𝐷_𝑥𝑢,𝑢,𝑥,𝑡=sup_𝜐∈𝑉𝐺𝐻𝐷²_𝑥𝑢,𝐷_𝑥𝑢,𝑢,𝑥,𝑡,𝜐+𝑏(𝑡,𝑥,𝜐),𝐷_𝑥𝑢+𝑓𝑡,𝑥,𝑢,𝜎₁(𝑡,𝑥),𝐷_𝑥𝑢,…,𝜎_𝑑(𝑡,𝑥),𝐷_𝑥𝑢,𝐻,𝜐_𝑖𝑗𝐷²_𝑥𝑢,𝐷_𝑥=𝐷𝑢,𝑢,𝑥,𝑡,𝜐_2𝑥𝑢⋅𝜎_𝑖(𝑡,𝑥,𝜐),𝜎_𝑗(𝐷𝑡,𝑥,𝜐)+2_𝑥𝑢,ℎ_𝑖𝑗(𝑡,𝑥,𝜐)+2𝑔_𝑖𝑗𝑡,𝑥,𝑢,𝜎₁(𝑡,𝑥,𝜐),𝐷_𝑥𝑢,…,𝜎_𝑑(𝑡,𝑥,𝜐),𝐷_𝑥𝑢,𝜐. (75)

Remark 28. The definition and uniqueness of the viscosity solution of above second-order partial differential equation can be found in Appendix C in Peng [12]. So, we only need to prove that 𝑢(𝑡,𝑥) is a viscosity solution of (74). Besides, from the result of Section 3, we can have that 𝑢(𝑡,𝑥) is continuous in [0,𝑇]×ℝ^𝑛.

Definition 29. A real-valued continuous function 𝑢(𝑡,𝑥)∈𝐶([0,𝑇]×ℝ^𝑛), 𝑢(𝑇,𝑥)≤Φ(𝑥), for any 𝑥∈ℝ^𝑛, is called a viscosity subsolution (super-solution) of (74); if for all functions 𝜑∈𝐶^2,3([0,𝑇]×ℝ^𝑛) satisfy 𝜑≥𝑢 and 𝜑(𝑡,𝑥)=𝑢(𝑡,𝑥) at fixed (𝑡,𝑥)∈[0,𝑇)×ℝ^𝑛, we have 𝜕_𝑡𝜑(𝑡,𝑥)+𝐹𝐷²_𝑥𝜑(𝑡,𝑥),𝐷_𝑥𝜑()()().𝑡,𝑥,𝜑𝑡,𝑥,𝑥,𝑡≥0≤0 (76)

Theorem 30. Under the assumptions (H3) and (H4), the value function 𝑢(𝑡,𝑥) defined by (33) is a viscosity solution of (74).

In order to prove the Theorem, we need three lemmas. Firstly, we set 𝐹₁()𝑟,𝑥,𝑦,𝑧,𝜐=⟨𝑏𝑟,𝑥,𝜐,𝐷_𝑥𝜑(𝑟,𝑥)⟩+𝜕_𝑡𝜑()(),𝜎𝑡,𝑥+𝑓𝑟,𝑥,𝑦+𝜑𝑟,𝑥𝑧+₁(𝑡,𝑥,𝜐),𝐷_𝑥𝜑(,𝜎𝑟,𝑥)…,_𝑑(𝑡,𝑥,𝜐),𝐷_𝑥𝜑(,𝐹𝑟,𝑥),𝜐_2𝑖𝑗=𝐷𝑟,𝑥,𝑦,𝑧,𝜐_𝑥𝜑(𝑟,𝑥),ℎ_𝑖𝑗(+1𝑟,𝑥,𝜐)2𝐷²_𝑥𝜑(𝑟,𝑥)𝜎_𝑖(𝑟,𝑥,𝜐),𝜎_𝑗(𝑟,𝑥,𝜐)+𝑔_𝑖𝑗(),𝜎𝑟,𝑥,𝑦+𝜑𝑟,𝑥𝑧+₁(𝑡,𝑥,𝜐),𝐷_𝑥𝜑(,𝜎𝑟,𝑥)…,_𝑑(𝑟,𝑥,𝜐),𝐷_𝑥𝜑(𝑟,𝑥),𝜐. (77) Then, we consider a G-BSDE defined on the interval [𝑡,𝑡+𝛿](0<𝛿≤𝑇−𝑡): 𝑌_𝑠^1,𝜐=_𝑠^𝑡+𝛿𝐹₁𝑟,𝑋_𝑟^{𝑡,𝑥;𝜐},𝑌_𝑟^1,𝜐,𝑍_𝑟^1,𝜐,𝜐_𝑟+𝑑𝑟_{𝑠𝑡+𝛿}𝑍_𝑟^1,𝜐𝑑𝐵_𝑟−𝐾¹_𝑡+𝛿−𝐾¹_𝑠−^𝑑_{𝑖,𝑗=1𝑠}^𝑡+𝛿𝐹₂^𝑖𝑗𝑟,𝑋_𝑟^{𝑡,𝑥;𝜐},𝑌_𝑟^1,𝜐,𝑍_𝑟^1,𝜐,𝜐_𝑟𝑑𝐵^𝑖,𝐵^𝑗_𝑟, (78) where 𝜐(⋅)∈𝒰 and 𝑋_𝑠^{𝑡,𝑥;𝜐} is defined by (24).

Lemma 31. For 𝑠∈[𝑡,𝑡+𝛿], we have 𝐺^{𝑡,𝑥;𝜐}_{𝑠,𝑡+𝛿}𝜑𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿,𝑡+𝛿−𝜑𝑋_𝑠^{𝑡,𝑥;𝜐},𝑠 (79) which is the solution of (78).

Proof. From the definition of 𝐺^{𝑡,𝑥;𝜐}_{𝑠,𝑡+𝛿}, we know that 𝐺^{𝑡,𝑥;𝜐}_{𝑠,𝑡+𝛿}[𝜑(𝑋^{𝑡,𝑥;𝜐}_𝑡+𝛿,𝑡+𝛿)] is the solution of G-BSDE (29) on [𝑡,𝑡+𝛿] with terminal condition 𝜑(X^{𝑡,𝑥;𝜐}_𝑡+𝛿,𝑡+𝛿). Applying Itô’s formula to 𝜑(𝑋_𝑠^{𝑡,𝑥;𝜐},𝑠), we can obtain the result.

Now we construct a simple G-BSDE by replacing the driving process 𝑋_𝑠^{𝑡,𝑥;𝜐} by its deterministic initial value 𝑥 as follows: 𝑌_𝑠^2,𝜐=_𝑠^𝑡+𝛿𝐹₁𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟+𝑑𝑟_{𝑑𝑖,𝑗=1𝑠𝑡+𝛿}𝐹₂^𝑖𝑗𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟𝑑𝐵^𝑖,𝐵^𝑗_𝑟−_𝑠^𝑡+𝛿𝑍_𝑟^2,𝜐𝑑𝐵_𝑟−𝐾²_𝑡+𝛿−𝐾²_𝑠. (80)

Lemma 32. We have the following estimate, for 𝜐(⋅)∈𝒰, 𝑌_𝑡^1,𝜐−𝑌_𝑡^2,𝜐≤𝐶𝛿^3/2, (81) where 𝐶 is independent of the control processes 𝜐(⋅).

Proof. By Proposition 21, we have the estimate for 𝑝≥2𝐸_𝑡sup_{[]𝑠∈𝑡,𝑡+𝛿}𝑋_𝑠^{𝑡,𝑥;𝜐}−𝑥^𝑝≤𝐶1+|𝑥|^𝑝𝛿^𝑝/2. (82) By Proposition 18, we get for fixed 𝑝>2 and 2<𝑝<𝛽, 𝑌_𝑡^1,𝜐−𝑌_𝑡^2,𝜐2≤𝐸sup_{[]𝑠∈𝑡,𝑡+𝛿}𝑌_𝑡^1,𝜐−𝑌_𝑡^2,𝜐2≤𝐶𝐸sup_{[]𝑠∈𝑡,𝑡+𝛿}𝐸_𝑠𝑡^𝑡+𝛿𝐹_𝑟𝑑𝑟^𝑝2/𝑝+𝐸sup_{[]𝑠∈𝑡,𝑡+𝛿}𝐸_𝑠𝑡^𝑡+𝛿𝐹_𝑟𝑑𝑟^𝑝, (83) where 𝐹_𝑟=𝐹₁𝑟,𝑋_𝑟^{𝑡,𝑥;𝜐},𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟−𝐹₁𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟+^𝑑_𝑖,𝑗=1𝐹₁^𝑖,𝑗𝑟,𝑋_𝑟^{𝑡,𝑥;𝜐},𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟−𝐹₂^𝑖,𝑗𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟. (84) It is easy to prove that 𝐹_𝑟≤𝐶𝑋_𝑟^{𝑡,𝑥;𝜐}−𝑥. (85) Then, we can deduce that |𝑌_𝑡^1,𝜐−𝑌_𝑡^2,𝜐|≤𝐶𝛿^3/2.

Lemma 33. We have sup_{𝜐(⋅)∈𝒰}𝑌_𝑡^2,𝜐=𝑌⁰(𝑡), (86) where 𝑌₀(⋅) is the solution of the following ODE: −𝑑𝑌⁰_𝑠=𝐹⁰𝑠,𝑥,𝑌⁰_𝑟[],𝑌,0𝑑𝑠,𝑠∈𝑡,𝑡+𝛿_0𝑡+𝛿=0, (87) where 𝐹⁰(𝑟,𝑥,𝑦,𝑧)=sup_𝜐∈𝑉{𝐹₁(𝑟,𝑥,𝑦,𝑧,𝜐)+2𝐺[(𝐹₂^𝑖𝑗(𝑟,𝑥,𝑦,𝑧,𝜐))^𝑑_𝑖,𝑗=1]}.

Proof. By Theorem 16, we know that the G-BSDE (80) have a unique solution (𝑌,𝑍,𝐾). Hence, there exists a process 𝑉_𝑠^2,𝜐=^𝑑_𝑖,𝑗=1^𝑠_𝑡𝐹₂^𝑖𝑗𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐𝑑𝐵^𝑖,𝐵^𝑗_𝑟−^𝑠_𝑡2𝐺𝐹₂^𝑖𝑗𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐^𝑑_𝑖,𝑗=1𝑑𝑟. (88) Here, 𝑉_𝑠^2,𝜐, 𝑠∈[𝑡,𝑡+𝛿] is a decreasing and continuous process by [28]. Besides, it satisfies 𝐸[sup_{𝑠∈[𝑡,𝑡+𝛿]}|𝑉_𝑠^2,𝜐|^𝛽]<∞ obviously. So, 𝑌_𝑠^2,𝜐 is the solution of the following G-BSDE: 𝑌_𝑠^2,𝜐=_𝑠^𝑡+𝛿𝐹₁𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟+2𝐺𝐹₂^𝑖𝑗𝑟,𝑥,𝑌_𝑟^2,𝜐,𝑍_𝑟^2,𝜐,𝜐_𝑟^𝑑_𝑖,𝑗=1−𝑑𝑟_{𝑠𝑡+𝛿}𝑍_𝑟^2,𝜐𝑑𝐵_𝑟−𝐾²_𝑡+𝛿−𝐾²_𝑠+𝑉^2,𝜐_𝑡+𝛿−𝑉_𝑠^2,𝜐, (89) where 𝜐(⋅)∈𝒰. In addition, we have 𝑌⁰_𝑡=_𝑠^𝑡+𝛿𝐹⁰𝑟,𝑥,𝑌⁰_𝑟,𝑍⁰_𝑟−𝑑𝑟_{𝑠𝑡+𝛿}𝑍⁰_𝑟𝑑𝐵_𝑟−𝐾⁰_𝑡+𝛿−𝐾⁰_𝑠+𝑉⁰_𝑡+𝛿−𝑉⁰_𝑠, (90) where (𝑍,𝐾,𝑉)=0. By the comparison of Theorem 20 and the definition of 𝐹⁰, we have for 𝜐(⋅)∈𝒰, 𝑌_𝑠^2,𝜐≤𝑌⁰_𝑠,𝑠∈[𝑡,𝑡+𝛿]. (91) On the other hand, there exists a measurable function 𝜐^′(𝑟,𝑥,𝑦,𝑧)∶[𝑡,𝑇]×ℝ^𝑛×ℝ×ℝ^𝑑×ℝ→𝑉 such that 𝐹⁰𝑟,𝑥,𝑦,𝑧=𝐹₁𝑟,𝑥,𝑦,𝑧,𝜐^′+2𝐺𝐹₂^𝑖𝑗𝑟,𝑥,𝑦,0,𝜐^′𝑑_𝑖,𝑗=1. (92) Then, we have 𝜐^′(𝑟,𝑥,𝑌⁰_𝑟,𝑍⁰_𝑟)∈𝒰, and 𝑌⁰_𝑡 is the solution of the following G-BSDE: 𝑌⁰_𝑠=_𝑠^𝑡+𝛿𝐹₁𝑟,𝑥,𝑌⁰_𝑟,𝑍⁰_𝑟,𝜐^′_𝑟+𝑑𝑟_{𝑑𝑖,𝑗=1𝑠𝑡+𝛿}𝐹₂^𝑖𝑗𝑟,𝑥,𝑌⁰_𝑟,𝑍⁰_𝑟,𝜐^′_𝑟𝑑𝐵^𝑖,𝐵^𝑗_𝑟−_𝑠^𝑡+𝛿𝑍⁰_𝑟𝑑𝐵_𝑟−𝐾⁰_𝑡+𝛿−𝐾⁰_𝑠, (93) where 𝑍⁰_𝑟,𝜐=0, 𝐾⁰_𝑠=^𝑑_𝑖,𝑗=1^𝑠_𝑡𝐹₂^𝑖𝑗𝑟,𝑥,𝑌⁰_𝑟,0,𝜐^′𝑑𝐵^𝑖,𝐵^𝑗_𝑟−^𝑠_𝑡2𝐺𝐹₂^𝑖𝑗𝑟,𝑥,𝑌⁰_𝑟,0,𝜐^′𝑑_𝑖,𝑗=1𝑑𝑟. (94) So, 𝑌⁰_𝑡≤sup_{𝜐(⋅)∈𝒰}𝑌_𝑡^2,𝜐. Now, we have proved the lemma.

Then, we give the proof of Theorem 30:

Proof. We set 𝜑∈𝐶^2,3([0,𝑇]×ℝ^𝑛) and 𝜑(𝑡,𝑥)=𝑢(𝑡,𝑥) for fixed (𝑡,𝑥)∈[0,𝑇]×ℝ^𝑛. From Theorem 26, we know 𝜑(𝑡,𝑥)=𝑢(𝑡,𝑥)=sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑋^𝑡,𝑥_𝑡+𝛿,𝑡+𝛿. (95) By 𝜑≥𝑢(𝜑≤𝑢) and the definition of 𝐺, sup_{𝜐(⋅)∈𝒰}𝐺^{𝑡,𝑥;𝜐}_{𝑡,𝑡+𝛿}𝑢𝑋^𝑡,𝑥_𝑡+𝛿,𝑡+𝛿−𝜑(𝑡,𝑥)≥0(≤0). (96) Then, form Lemma 31, sup_{𝜐(⋅)∈𝒰}𝑌_𝑡^1,𝜐≥0(≤0). (97) Besides, from Lemma 32, sup_{𝜐(⋅)∈𝒰}𝑌_𝑡^2,𝜐≥𝐶𝛿^3/2≤𝐶𝛿^3/2. (98) Finally, Lemma 33 implies 𝑌⁰(𝑡)≥𝐶𝛿^3/2≤𝐶𝛿^3/2. (99) So, 𝐹⁰(𝑟,𝑥,0,0)≥0(≤0) and from the definition of viscosity solution of (74), we know 𝑢(𝑡,𝑥) is a viscosity solution of (74).

Acknowledgment

This work was supported by the National Basic Research Program of China (Project no. 973-2007CB814901).