Reflected Quadratic BSDEs driven by $G$-Brownian Motions

TL;DR

This paper studies reflected quadratic G-Brownian motion driven backward stochastic differential equations, establishing existence, uniqueness, and linking solutions to nonlinear PDEs via a nonlinear Feynman-Kac formula.

Contribution

It introduces a method for solving reflected quadratic G-BSDEs and connects these solutions to fully nonlinear PDEs in a Markovian setting.

Findings

Existence of solutions via the penalty method

Uniqueness of solutions through a priori estimates

Nonlinear Feynman-Kac formula linking G-BSDEs and PDEs

Abstract

In this paper, we consider a reflected backward stochastic differential equation driven by a $G$-Brownian motion ($G$-BSDE), with the generator growing quadratically in the second unknown. We obtain the existence by the penalty method, and a priori estimates which implies the uniqueness, for solutions of the $G$-BSDE. Moreover, focusing our discussion at the Markovian setting, we give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.

Equations

$$Y_t=\xi +{\int }_t^{T}f(s,Y_s,Z_s)ds-{\int }_t^T{Z}_{s}d{W}_s,t\in [0,T].$$

$$Y_t=\xi +{\int }_t^{T}g(s,Y_s,Z_s)ds+{\int }_t^{T}f(s,Y_s,Z_s)d{⟨B⟩}_s-{\int }_t^T{Z}_{s}d{B}_s-{\int }_t^{T}d{K}_s,t\in [0,T],$$

$$\{\begin{array}{l}{Y}_t=\xi +{\int }_t^{T}g(s,Y_s,Z_s)ds+{\int }_t^{T}f(s,Y_s,Z_s)d{⟨B⟩}_s\\ -{\int }_t^T{Z}_{s}d{B}_s+{\int }_t^{T}d{A}_s,t\in [0,T];\\ Y_t\ge S_t,0\le t\le T;{\int }_0^{\cdot }(S_s-Y_s)d{A}_s\text{ is a non-increasing }G\text{-martingale.}\end{array}$$

$$|\varphi (x)-\varphi (y)|\le C(1+|x{|}^k+|y{|}^k)|x-y|,\forall x,y\in {\mathbb{R}}^n,$$

$$\partial_{t}u-G\big{(}D_{x}^{2}u\big{)}=0;~{}u(0,x)=\phi(x).$$

$$G(A):=\frac{1}{2}\hat{\mathbb{E}}[⟨AX,X⟩]:{\mathbb{S}}_d\to \mathbb{R},$$

$$G(A)=\frac{1}{2}\hat{\mathbb{E}}[⟨AX,X⟩]\le \frac{1}{2}|A|\hat{\mathbb{E}}[|X{|}^2]:=\frac{1}{2}|A|{\bar{\sigma }}^2$$

$$G(A)=\frac{1}{2}\underset{\gamma \in \Gamma }{\sup }\text{tr}[\gamma A],$$

$$Lip(\Omega_{T}):=\big{\{}\phi(B_{t_{1}},B_{t_{2}}-B_{t_{1}},\cdots,B_{t_{n}}-B_{t_{n-1}}):0\leq t_{1}\leq t_{2}\leq\cdots\leq t_{n}\leq T,~{}\phi\in\mathcal{C}_{l,lip}(\mathbb{R}^{d\times n})\big{\}}$$

$$\varphi (B_{t_1},B_{t_2}-B_{t_1},\cdots ,B_{t_n}-B_{t_{n-1}}),\varphi \in {\mathcal{C}}_{l,lip}({\mathbb{R}}^{d\times n}),$$

$${\hat{\mathbb{E}}}_{t_i}=[\varphi (B_{t_1},B_{t_2}-B_{t_1},\cdots ,B_{t_n}-B_{t_{n-1}})]=\tilde{\varphi }(B_{t_1},B_{t_2}-B_{t_1},\cdots ,B_{t_i}-B_{t_{i-1}}),$$

$$\tilde{\varphi }(x_1,x_2,\cdots ,x_i)=\hat{\mathbb{E}}[\varphi (x_1,x_2,\cdots ,x_i,B_{t_{i+1}}-B_{t_i},\cdots ,B_{t_n}-B_{t_{n-1}})].$$

$$\xi =\hat{\varphi }(B_{t_1},B_{t_2}-B_{t_1},\cdots ,B_t-B_{t_i},B_{t_{i+1}}-B_t,\cdots ,B_{t_n}-B_{t_{n-1}}),\hat{\varphi }\in {\mathcal{C}}_{l,lip}({\mathbb{R}}^{d\times (n+1)}).$$

$$\hat{\mathbb{E}}[\xi ]=\underset{\mathbb{P}\in \mathcal{P}}{\sup }{E}^{\mathbb{P}}[\xi ],\forall \xi \in L_G^1({\Omega }_T),$$

$${\hat{\mathbb{E}}}_t[\xi ]=\underset{{\mathbb{P}}^{\prime }\in \mathcal{P}(t,\mathbb{P})}{\mathrm{esssup}}{E}_t^{{\mathbb{P}}^{\prime }}[\xi ],\mathbb{P}\text{-a.s.},$$

$$\mathcal{P}(t,\mathbb{P}):=\{{\mathbb{P}}^{\prime }\in \mathcal{P}:{\mathbb{P}}^{\prime }=\mathbb{P}\text{on}{\mathcal{F}}_t\}.$$

$${\hat{\mathbb{E}}}_t[{\eta }_t+M_T-M_t]={\eta }_t.$$

$${\Vert \xi \Vert }_{L_G^{\infty }}:=\inf \{M\ge 0:|\xi |\le M,\text{q.s.}\}.$$

$$\left\|\xi\right\|_{\alpha,\mathcal{E}}^{\alpha}\leq\gamma^{*}\big{\{}\left\|\xi\right\|^{\alpha}_{L_{G}^{\alpha+\delta}}+14^{1/\gamma}C_{\beta/\gamma}\left\|\xi\right\|^{(\alpha+\beta)/\gamma}_{L_{G}^{\alpha+\delta}}\big{\}},\quad\forall\xi\in Lip(\Omega_{T}),$$

$$\hat{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}\hat{\mathbb{E}}_{t}[|\xi|^{\alpha}]\Big{]}\leq C_{1}\big{\{}\hat{\mathbb{E}}[|\xi|^{\alpha+\delta}]^{\alpha\over\alpha+\delta}+\hat{\mathbb{E}}[|\xi|^{\alpha+\delta}]\big{\}}.$$

$${⟨B^a⟩}_t=\underset{\mu ({\pi }_t^N)\to 0}{\lim }\sum _{j=0}^{N-1}(B_{t_{j+1}^N}^a-B_{t_j^N}^a{)}^2,$$

$$\left\langle B^{a},B^{\bar{a}}\right\rangle_{t}={1\over 4}\big{[}\left\langle B^{a+\bar{a}}\right\rangle_{t}-\left\langle B^{a-\bar{a}}\right\rangle_{t}\big{]}.$$

$${\eta }_t(\omega )=\sum _{j=0}^{N-1}{\xi }_j(\omega ){\text{1}}_{[t_j,t_{j+1})}(t),$$

$${\Vert \eta \Vert }_{H_G^p}:=\hat{\mathbb{E}}{\left[{\left({\int }_0^T|{\eta }_s{|}^{2}ds\right)}^{\frac{p}{2}}\right]}^{\frac{1}{p}}\mathrm{and}{\Vert \eta \Vert }_{M_G^p}:=\hat{\mathbb{E}}{\left[\left({\int }_0^T|{\eta }_s{|}^{p}ds\right)\right]}^{\frac{1}{p}}.$$

$$\underline{\sigma}^{p}c_{p}\hat{\mathbb{E}}_{t}\Big{[}\Big{(}\int_{t}^{T}|\eta_{s}|^{2}ds\Big{)}^{p\over 2}\Big{]}\leq\hat{\mathbb{E}}_{t}\Big{[}\sup_{u\in[t,T]}\Big{|}\int_{t}^{u}\eta_{s}dB_{s}\Big{|}^{p}\Big{]}\leq\bar{\sigma}^{p}C_{p}\hat{\mathbb{E}}_{t}\Big{[}\Big{(}\int_{t}^{T}|\eta_{s}|^{2}ds\Big{)}^{p\over 2}\Big{]},~{}~{}\text{q.s.}$$

$$S_G^0(0,T)$$

$${\Vert \eta \Vert }_{S_G^p}:=\hat{\mathbb{E}}{\left[\underset{t\in [0,T]}{\sup }|{\eta }_t{|}^p\right]}^{\frac{1}{p}}.$$

$$F(Y):=\limsup_{\varepsilon\to 0}\hat{\mathbb{E}}\big{[}\sup_{t\in[0,T]}\sup_{s\in[t,t+\varepsilon]}|Y_{t}-Y_{s}|^{p}\big{]}^{1\over p}=0.$$

$$Y_t=\xi +{\int }_t^{T}g(s,{\omega }_{\cdot \wedge s},Y_s,Z_s)ds+{\int }_t^{T}f(s,{\omega }_{\cdot \wedge s},Y_s,Z_s)d{⟨B⟩}_s-{\int }_t^T{Z}_{s}d{B}_s-{\int }_t^{T}d{K}_s,\text{q.s.},$$

$$\underset{y,z\in \mathbb{R}}{\sup }|f(t,\omega ,y,z)-f(t^{\prime },{\omega }^{\prime },y,z)|\le w(|t-t^{\prime }|+{\Vert \omega -{\omega }^{\prime }\Vert }_{\infty }),$$

Full text

1 Introduction

A general backward stochastic differential equation (BSDE) takes the following form:

[TABLE]

The function $f$ is conventionally called the generator and the random variable $\xi$ is called the terminal value. Bismut [2, 3] initially gave a complete linear theory, where the generator is linear in both unknown variables, and derived the stochastic Riccati equation as a particular nonlinear BSDE where the generator is quadratic in the second unkown variable. Pardoux and Peng [29] established the existence and uniqueness result when the generator $f$ is uniformly Lipschitz continuous in both unknown variables and the terminal value $\xi$ is square integrable. Subsequently, an intensive attention has been given to relax the assumption of the uniformly Lipschitz continuity on the generator. In particular, the one-dimensional BSDE with a quadratic generator (i.e., the so-called quadratic BSDE) was studied by Kobylanski [18] for a bounded terminal value $\xi$, and by Briand and Hu [5, 6] for an unbounded terminal value $\xi$ of some suitable exponential moments. The multi-dimensional quadratic BSDE was discussed by Tang [41] and Hu and Tang [16].

As a constrained BSDE, a reflected backward stochastic differential equation (RBSDE) was formulated and studied by El Karoui et al. [11], where the first unknown $Y$ is required to stay up a given continuous process $S$ and an additional increasing process which satisfies the Skorohod condition, is thus introduced into the equation. Subsequently, much efforts have been made to relax the Lipschitz assumption on the generator. For the quadratic case, see Kobylanski et al. [19] with bounded terminal values, and Lepeltier and Xu [21] with unbounded terminal values.

To incorporate the Knightian uncertainty, Peng [32, 33, 34, 35] introduced the notion of $G$-expectation as a time-consistent sub-linear expectation, and constructed (via a fully nonlinear PDE) the so-called $G$-Brownian motion $\{B_{t},~{}t\in[0,+\infty)\}$, whose quadratic variation process $\left\langle B\right\rangle$—in contrast to the classical Brownian motion—is not deterministic. The stochastic integral with respect to the $G$-Brownian motion and its quadratic variation were also discussed by Peng [32]. Denis et al. [10] proves that the $G$-expectation is in fact the upper expectation over a collection of mutually singular martingale measures $\mathcal{P}$. Hu et al. [12] showed that there is a unique triple of processes $(Y,Z,K)$ in a proper Banach space satisfying the following scalar-valued BSDE driven by the $G$-Brownian motion $B$:

[TABLE]

where $f$ and $g$ are uniformly Lipshchitz in both unknown variables. Hu et al. [15] proved the existence and uniqueness for adapted solutions to the scalar-valued $z$-quadratic BSDE (1.1) driven by the $G$-Brownian motion $B$ for a bounded terminal value $\xi$. Very recently, Li, Peng, and Soumana Hima [22] discuss a reflected BSDE driven by the $G$-Brownian motion subject to a lower obstacle under the uniformly Lipschitz condition, where a $G$-martingale condition rather than the conventional Skorohod condition, is used to characterize the unknown bounded variational process which is introduced into the equation to keep the first unknown process stay up the lower obstacle under the $G$-expectation. More precisely, they showed that there is a unique triple $(Y,Z,A)$ of processes satisfying the following equation:

[TABLE]

A subsequent study of Li and Peng [23] reported the following unexpected observation on the upper obstacle problem for the reflected BSDE driven by a $G$-Brownian motion: the proof of the uniqueness of solutions in the lower obstacle problem turns out to be difficult to be adapted to the upper obstacle problem. Since the preceding two equations hold $\mathbb{P}$-$a.s.$ for each $\mathbb{P}\in\mathcal{P}$, they are also associated to second order BSDEs, which have been discussed by Cheridito et al. [8], Soner et al. [38], and Possamaï and Zhou [36]. Moreover, Matoussi, Piozin and Possamaï [26] and Matoussi, Possamaï and Zhou [27, 28] discuss the reflected second order BSDEs. In the context of a $G$-BSDE, the solution is universally discussed in a “better” space of processes, and its existence naturally requires more regularity of the coefficients.

As a generalized counterpart of the classical reflected quadratic BSDEs, the existence and uniqueness result for reflected quadratic BSDEs driven by $G$-Brownian motions still remains to be studied. The main objective of this paper is to provide the well-posedness of the reflected $G$-BSDE (1.2) when the generator has a quadratic growth and the terminal value $\xi$ is bounded. As noted in Li, Peng and Soumana Hima [22] and Possamaï and Zhou [36], the dominated convergence theorem does not hold under the $G$-framework, and a bounded sequence in $M_{G}^{p}(0,T)$ does not necessarily have the weak compactness. These striking differences prevent us from adapting the method of Kobylanski et al. [19] to approximate the quadratic generator with Lipshcitz ones and then to prove the solutions of the approximating reflected BSDEs to converge to that of the original reflected quadratic BSDE. Instead in this paper, we use a penalty method in the spirit of El Karoui et al. [11] (for a BSDE in a Wiener space) and Li, Peng and Soumana Hima [22] (for a $G$-BSDE). Since our generator is allowed to grow quadratically in the second unknown variable, the terminal value $\xi$ is assumed to be bounded for simplicity of exposition, and then the symmetric martingale part of the underlying BSDE is discussed in the BMO space.

As in Hu et al. [13] and Li, Peng and Soumana Hima [22], the solution of a forward backward differential equations driven by $G$-Brownian motion ($G$-FBSDEs in short) can be interpreted as a viscosity solution of a PDE. We first prove the existence of the quadratic $G$-BSDEs in a Markovian setting. We then give the nonlinear Feynman-Kac formula for a fully nonlinear parabolic variational in equality via the quadratic $G$-BSDEs and the reflected quadratic $G$-BSDEs.

The paper is organized as follows. Section 2 is dedicated to preliminaries on the $G$-framework, the formulation of reflected $G$-BSDEs, $G$-BMO martingales and $G$-Girsanov Theorem. In Section 3, we introduce some priori estimates for quadratic reflected $G$-BSDEs through the $G$-Girsanov transformation, which yields the uniqueness in a straightforward way. In Section 4, we establish the approximation method via penalization. We state some convergence properties of the solutions to the penalized $G$-BSDEs. In Section 5, we prove our main result and a comparison theorem. Finally, in Section 6, we give a nonlinear Feynmann-Kac formula and address the relation between quadratic $G$-BSDEs and nonlinear parabolic PDEs.

2 Preliminaries

2.1 Notations and results on $G$-expectation and quadratic $G$-BSDEs

In this section, we first recall notations and basic results concerning $G$-expectation, $G$-Brownian motion and related $G$-stochastic calculus, and quadratic $G$-BSDEs. More details can be found in [12], [13], [25], [32], [33], and [34].

Let $\Omega$ be a complete separable metric space, and let $\mathcal{H}$ be a linear space of real-valued functions defined on $\Omega$ satisfying $c\in\mathcal{H}$ for each constant $c$ and $|X|\in\mathcal{H}$ if $X\in\mathcal{H}.$ $\mathcal{H}$ is considered as the space of random variables.

Definition 2.1**.**

(Sublinear expectation space). A sublinear expectation $\hat{\mathbb{E}}[\cdot]$ is a functional $\hat{\mathbb{E}}:\mathcal{H}\to\mathbb{R}$ satisfying the following properties: for all $X,Y\in\mathcal{H},$ we have

Monotonicity: if $X\geq Y$, then $\hat{\mathbb{E}}[X]\geq\hat{\mathbb{E}}[Y]$; 2. 2.

Constant preservation: $\hat{\mathbb{E}}[c]=c,~{}c\in\mathbb{R}$; 3. 3.

Sub-additivity: $\hat{\mathbb{E}}[X+Y]\leq\hat{\mathbb{E}}[X]+\hat{\mathbb{E}}[Y]$; 4. 4.

Positive homogeneity: $\hat{\mathbb{E}}[\lambda X]=\lambda\hat{\mathbb{E}}[X]$, for all $\lambda\geq 0$.

We call the triple $(\Omega,\mathcal{H},\hat{\mathbb{E}})$ a sublinear expectation space.

Definition 2.2**.**

(Independence). In a sublinear expectation space $(\Omega,\mathcal{H},\hat{\mathbb{E}})$, a random vector $Y=(Y_{1},Y_{2},\cdots,Y_{n}),~{}Y_{i}\in\mathcal{H}$ is said to be independent of another random vector $X=(X_{1},X_{2},\cdots,X_{m}),~{}X_{i}\in\mathcal{H}$, if $\hat{\mathbb{E}}[\phi(X,Y)]=\hat{\mathbb{E}}[\hat{\mathbb{E}}[\phi(x,Y)]|_{x=X}],$ for all $\phi\in\mathcal{C}_{l,lip}(\mathbb{R}^{m+n})$, where $\mathcal{C}_{l,lip}(\mathbb{R}^{n})$ is the space of real continuous functions defined on $\mathbb{R}^{n}$ such that

[TABLE]

where $k$ and $C$ depend only on $\phi.$

Definition 2.3**.**

($G$-normal distribution). We say the random vector $X=(X_{1},X_{2},\cdots,X_{d})$ is $G$-normally distributed, if for any function $\phi\in\mathcal{C}_{l,lip}(\mathbb{R}^{d})$, the function $u$ defined by u(t,x):=\hat{\mathbb{E}}\big{[}\phi\big{(}x+\sqrt{t}X\big{)}\big{]},~{}(t,x)\in[0,+\infty)\times\mathbb{R}^{d}, is a viscosity of $G$-heat equation:

[TABLE]

Here $G$ denotes the function

[TABLE]

where $\mathbb{S}_{d}$ denotes the collection of $d\times d$ symmetric matrices.

The function $G(\cdot)$ is a monotonic, sublinear mapping on $\mathbb{S}_{d}$  and

[TABLE]

implies that there exists a bounded, convex and closed subset $~{}\Gamma\subseteq\mathbb{S}_{d}^{+}~{}$ such that

[TABLE]

where $~{}\mathbb{S}_{d}^{+}~{}$ denotes the collection of nonnegative elements in $~{}\mathbb{S}_{d}$.

In this paper, we only consider a non-degenerate $G$-normal distribution, i.e, there exists some $\underline{\sigma}>0$ such that $G(A)-G(B)\geq\underline{\sigma}^{2}\text{tr}[A-B]$ for any $A\geq B.$

We now fix $\Omega:=C_{0}([0,\infty);\mathbb{R}^{d})$, the space of all $\mathbb{R}^{d}$-valued continuous functions $\{\omega_{t},~{}t\in[0,+\infty)\}$ with $\omega_{0}=0.$ Let $\mathcal{F}=\{\mathcal{F}_{t},~{}t\in[0,+\infty)\}$ be the nature filtration generated by the canonical process $\{B_{t},~{}t\in[0,+\infty)\}$, i.e., $B_{t}(\omega)=\omega_{t}$ for $(t,\omega)\in[0,\infty)\times\Omega.$ Set $\Omega_{T}:=C_{0}([0,T];\mathbb{R}^{d})$. Let us consider the function spaces defined by

[TABLE]

for $T>0$, and $Lip(\Omega)=\bigcup\limits_{n=1}^{\infty}Lip(\Omega_{n})$.

Definition 2.4**.**

($G$-Brownian motion and $G$-expectation). On the sublinear expectation space $(\Omega,Lip(\Omega),\hat{\mathbb{E}})$, the canonical process $\{B_{t},~{}t\in[0,+\infty)\}$ is called $G$-Brownian motion if the following properties are satisfied:

$B_{0}=0$; 2. 2.

For each $t,s>0$, the increment $B_{t+s}-B_{t}$ is independent of $(B_{t_{1}},B_{t_{2}},\cdots,B_{t_{n}})$, for each $n\in\mathbb{N}$ and $0\leq t_{1}\leq t_{2}\leq\cdots\leq t_{n}\leq t;$ 3. 3.

$B_{t+s}-B_{t}$* is $G$-normally distributed.*

Moreover, the sublinear expectation $\hat{\mathbb{E}}[\cdot]$ is called $G$-expectation.

Definition 2.5**.**

(Conditional $G$-expectation). For the random variable $\xi\in Lip(\Omega_{T})$ of the following form:

[TABLE]

the conditional $G$-expectation $\hat{\mathbb{E}}_{t_{i}}[\cdot]$, $i=1,2,\cdots,n$, is defined as follows:

[TABLE]

where

[TABLE]

If $t\in(t_{i},t_{i+1})$, the conditional $G$-expectation $\hat{\mathbb{E}}_{t}[\xi]$ could be defined by reformulating $\xi$ as

[TABLE]

For $\xi\in Lip(\Omega_{T})$ and $p\geq 1$, we consider the norm \left\|\xi\right\|_{L_{G}^{p}}=\big{(}\hat{\mathbb{E}}[|\xi|^{p}]\big{)}^{1\over p}. Denote by $L_{G}^{p}(\Omega_{T})$ the Banach completion of $Lip(\Omega_{T})$ under $\left\|\cdot\right\|_{L_{G}^{p}}$. It is easy to check that the conditional $G$-expectation $\hat{\mathbb{E}}_{t}[\cdot]:~{}Lip(\Omega_{T})\to Lip(\Omega_{t})$ is a continuous mapping and thus can be extended to $\hat{\mathbb{E}}_{t}:~{}L_{G}^{p}(\Omega_{T})\to L_{G}^{p}(\Omega_{t})$.

Definition 2.6**.**

($G$-martingale). A process $\{M_{t},~{}t\in[0,T]\}$ is called a $G$-martingale if

1. (i)

$M_{t}\in L_{G}^{1}(\Omega_{t})$, for any $t\in[0,T]$; 2. (ii)

$\hat{\mathbb{E}}_{s}[M_{t}]=M_{s}$, for all $0\leq s\leq t\leq T$.

The process $\{M_{t},~{}t\in[0,T]\}$ is called a symmetric $G$-martingale if $-M$ is also a $G$-martingale.

The following representation result of $G$-expectation on $L_{G}^{1}(\Omega_{T})$, can be found in Denis et al. [10, Propositions 49 and 50, page 157-158] and Hu and Peng [14, Theorem 3.5, page 544].

Theorem 2.1**.**

There exists a weakly compact set $\mathcal{P}\subseteq\mathcal{M}_{1}(\Omega_{T})$ (i.e., the set of all probability measures on $(\Omega_{T},\mathcal{B}(\Omega_{T}))$), such that

[TABLE]

where $E^{\mathbb{P}}[\cdot]$ is the expectation operator with respect to probability $\mathbb{P}$. Such $\mathcal{P}$ is called a representative set of $\hat{\mathbb{E}}$.

Let $\mathcal{P}$ be a weakly compact set that represents $\hat{\mathbb{E}}$. For this $\mathcal{P}$, we define capacity $c(A):=\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}(A),~{}A\in\mathcal{B}(\Omega_{T})$.

Definition 2.7**.**

(Quasi-sure). A set $A\in\mathcal{B}(\Omega_{T})$ is a polar set if $c(A)=0$. A property holds “quasi-surely” (q.s.) if it holds outside a polar set.

In what follows, two random variables $X$ and $Y$ will not be distinguished if $X=Y$, q.s.

Soner et al. [37, Proposition 3.4, page 272] give the following characterization of the conditional $G$-expectation.

Theorem 2.2**.**

For any $\xi\in L_{G}^{1}(\Omega_{T})$, $t\in[0,T]$ and $\mathbb{P}\in\mathcal{P}$,

[TABLE]

where

[TABLE]

In view of Theorem 2.2, it is easy to check the following property for $G$-martingales.

Proposition 2.3**.**

Assume that $\{M_{s},~{}s\in[0,T]\}$ is a $G$-Martingale and $\{\eta_{s},~{}s\in[0,T]\}$ is a process satisfying $\eta_{s}\in L_{G}^{1}(\Omega_{s})$, for any $s\in[0,T]$. Then we have for any $t\in[0,T]$,

[TABLE]

For the terminal value of quadratic $G$-BSDE, we define the space $L_{G}^{\infty}(\Omega_{T})$ as the completion of $Lip(\Omega_{T})$ under the norm

[TABLE]

For $\xi\in Lip(\Omega_{T})$ and $p\geq 1$, define $\left\|\xi\right\|_{p,\mathcal{E}}=\hat{\mathbb{E}}[\sup_{t\in[0,T]}\hat{\mathbb{E}}_{t}[|\xi|^{p}]]^{1\over p}$ and denote by $L^{p}_{\mathcal{E}}(\Omega_{T})$ the completion of $Lip(\Omega_{T})$ under $\left\|\cdot\right\|_{p,\mathcal{E}}$. Song [40, Theorem 3.4, page 293]) gives the following estimate.

Theorem 2.4**.**

For any $\alpha\geq 1$ and $\delta>0$, $L^{\alpha+\delta}_{G}(\Omega_{T})\subseteq L^{\alpha}_{\mathcal{E}}(\Omega_{T})$. More precisely, for any $1<\gamma<\beta:={(\alpha+\delta)/\alpha}$, $\gamma\leq 2$, we have

[TABLE]

where $C_{\beta/\gamma}=\sum_{i=1}^{\infty}i^{-\beta/\gamma}$, $\gamma^{*}=\gamma/(\gamma-1)$.

Remark 2.5**.**

In view of [12, Remark 2.9], there exists $C_{1}$ depending only on $\alpha$ and $\delta$ such that

[TABLE]

Let $B$ be a $d$-dimensional $G$-Brownian motion. For each fixed $a\in\mathbb{R}^{d}$, $B_{t}^{a}=\left\langle a,B_{t}\right\rangle$ is a 1-dimensional $G_{a}$-Brownian motion, where $G_{a}(\alpha)=G(aa^{T})\alpha^{+}+G(-aa^{T})\alpha^{-}.$ The quadratic variation process of $B^{a}$ is defined by

[TABLE]

where $\pi_{t}^{N}$, $N=1,2,\cdots$, are refining partitions of $[0,t]$. By Peng [34], for all $t,s>0$, $\left\langle B^{a}\right\rangle_{t+s}-\left\langle B^{a}\right\rangle_{t}\in[-2G(-aa^{T})s,2G(aa^{T})s]$, q.s.

For each fixed $a,\bar{a}\in\mathbb{R}^{d}$, the mutual variation process of $B^{a}$ and $B^{\bar{a}}$ is defined by

[TABLE]

Next we discuss the stochastic integrals with respect to the $G$-Brownian motion and its quadratic variation.

Definition 2.8**.**

Let $M_{G}^{0}(0,T)$ be the collection of processes $\eta$ of the following form: for a given partition $\{t_{1},r_{2},\cdots,t_{n}\}=\pi_{T}$ of $[0,T]$,

[TABLE]

where $\xi_{i}\in Lip(\Omega_{t_{i}})$ for $i=0,1,2,\cdots,N-1$. For $p\geq 1$ and $\eta\in M_{G}^{0}(0,T)$, define

[TABLE]

Denote by $H_{G}^{p}(0,T)$ and $M_{G}^{p}(0,T)$ the completion of $M_{G}^{0}(0,T)$ under norms $\left\|\cdot\right\|_{H_{G}^{p}}$ and $\left\|\cdot\right\|_{M_{G}^{p}}$, respectively.

For both processes $\eta\in M_{G}^{2}(0,T)$ and $\xi\in M_{G}^{1}(0,T)$, the $G$-Itô integrals $\{\int_{0}^{t}\eta_{s}dB_{s}^{i},~{}0\leq t\leq T\}$ and $\{\int_{0}^{t}\xi_{s}d\langle B^{i},B^{j}\rangle_{s},~{}0\leq t\leq T\}$ are well defined in [25] and [34]. Moreover, the following BDG inequality can be found in [40, Proposition 4.3, Page 295].

Proposition 2.6**.**

For $\eta\in H_{G}^{\alpha}(0,T)$ with $\alpha\geq 1$ and $p\in(0,\alpha]$, we have

[TABLE]

Denote by \mathcal{C}_{b,lip}(\mathbb{R}^{1+d\times n})\big{\}} the collection of all bounded and Lipschitz functions on $\mathbb{R}^{1+d\times n}$. Define

[TABLE]

For $p\geq 1$ and $\eta\in S_{G}^{0}(0,T)$, set

[TABLE]

Denote by $S_{G}^{p}(0,T)$ the completion of $S_{G}^{0}(0,T)$ under the norm $\left\|\cdot\right\|_{S_{G}^{p}}$. The following continuity of $Y\in S_{G}^{p}(0,T)$ for $p>1$ can be found in Li, Peng, and Song [24, Lemma 3.7, page 12].

Lemma 2.7**.**

For $Y\in S_{G}^{p}(0,T)$ with $p>1$, we have, by setting $Y_{s}=Y_{T}$ for $s>T$,

[TABLE]

Similar to $S_{G}^{p}(0,T)$, we can define the space $S_{G}^{\infty}(0,T)$ as the completion of $S_{G}^{0}(0,T)$ under the norm $\left\|\eta\right\|_{S_{G}^{\infty}}:=\left\|\sup_{t\in[0,T]}|\eta_{t}|\right\|_{L_{G}^{\infty}}$.

We now introduce some results on quadratic $G$-BSDEs in [15]. For simplicity, we assume $d=1$ and consider the following type of equation:

[TABLE]

where the generator $(f,g):[0,T]\times\Omega_{T}\times\mathbb{R}^{2}\to\mathbb{R}^{2}$ and the terminal value $\xi$ are supposed to satisfy the following conditions:

• (H1)

$\int_{0}^{T}|f(t,\omega,0,0)|^{2}dt+\int_{0}^{T}|g(t,\omega,0,0)|^{2}dt+|\xi(\omega)|\leq M_{0}$, q.s.;

• (H2)

The generator $(f,g)$ is uniformly continuous in $(t,\omega)$, i.e. there is a non-decreasing continuous function $w:[0,+\infty)\to[0,+\infty)$ such that $w(0)=0$ and

[TABLE]

[TABLE]

• (H3)

There are two positive constants $L_{y}$ and $L_{z}$ such that for each $(t,\omega)\in[0,T]\times\Omega,$

[TABLE]

Remark 2.8**.**

In [15], the triple $(f,g,\xi)$ is supposed to satisfy the following condition:

• (H1’)

For each $t\in[0,T],\quad|f(t,\omega,0,0)|+|g(t,\omega,0,0)|+|\xi(\omega)|\leq M_{0}$, q.s.

The results there still hold if (H1’) is replaced with (H1), by a similar analysis as in [12] and [15].

Remark 2.9**.**

Assumption (H3) implies the following

[TABLE]

with $h=f,g.$ So $(f,g)$ are linear in $y$ and quadratic in $z$.

For simplicity, we denote by $\mathfrak{G}_{G}^{p}(0,T)$ the collection of process $(Y,Z,K)$ such that $(Y,Z)\in S_{G}^{p}(0,T)\times H_{G}^{p}(0,T)$ and $K$ is a decreasing $G$-martingale with $K_{0}=0$ and $K_{T}\in L_{G}^{p}(\Omega_{T})$. Hu et al. [15, Theorem 5.3, page 22; Eq (3.2) and (3.3), page 13] give the following theorem.

Theorem 2.10**.**

Assume that $\xi\in L_{G}^{\infty}(\Omega_{T})$ and the triple $(f,g,\xi)$ satisfies (H1)-(H3). Then equation (2.1) has a unique solution $(Y,Z,K)\in\mathfrak{G}_{G}^{2}(0,T)$ such that

[TABLE]

and

[TABLE]

where the norm $\left\|\cdot\right\|_{BMO_{G}}$ will be defined in Section 2.3.

2.2 Formulation of the problem

For simplicity, we consider the $G$-expectation space $(\Omega,L_{G}^{1}(\Omega_{T}),\hat{\mathbb{E}})$ for the case of $d=1$ and $\bar{\sigma}^{2}=\hat{\mathbb{E}}[B_{1}^{2}]\geq-\hat{\mathbb{E}}[-B_{1}^{2}]=\underline{\sigma}^{2}>0$. Consider the following equation:

[TABLE]

where the generator $(f,g):[0,T]\times\Omega_{T}\times\mathbb{R}^{2}\to\mathbb{R}^{2}$ and the terminal value $\xi$ are assumed to satisfy (H1)-(H3). Moreover, the obstacle process $\{S_{t},~{}t\in[0,T]\}$ is supposed to satisfy the following conditions:

• (H4)

$S_{\cdot}\in\bigcap\limits_{\alpha>1}S_{G}^{\alpha}(0,T)$ with $S_{T}\leq\xi$, q.s. Furthermore, there is a positive constant $N_{0}$ such that $S_{t}\leq N_{0}$, q.s., for any $t\in[0,T]$.

• (H5)

$S$ is uniformly continuous in $(t,\omega)$, i.e. there is a non-decreasing continuous function $w:[0,+\infty)\to[0,+\infty)$ with $w(0)=0$ such that

[TABLE]

Remark 2.11**.**

Like in [15], Assumptions (H2) and (H5) are used to ensure the existence of solutions to our subsequent penalized quadratic $G$-BSDEs.

A solution of reflected $G$-BSDEs is defined as follows.

Definition 2.9**.**

A triple of processes $(Y,Z,A)$ belongs to $\mathcal{S}_{G}^{\alpha}(0,T)$ for $\alpha>1$ if $(Y,Z)\in S_{G}^{\alpha}(0,T)\times H_{G}^{\alpha}(0,T)$ and $A$ is a continuous nondecreasing process such that $A_{0}=0$ and $A_{T}\in L_{G}^{\alpha}(\Omega_{T})$. The triple (Y,Z,A) is said to be a solution to the reflected $G$-BSDE (2.2) if  $(Y,Z,A)\in\mathcal{S}_{G}^{\alpha}(0,T)$, and satisfies (2.2) for $t\in[0,T]$.

Our objective is to establish the existence and uniqueness result for the quadratic $G$-BSDE (2.2). For simplicity of exposition, we assume that $g\equiv 0$ in what follows. Corresponding results still hold for the case of $g\neq 0.$

2.3 $G$-BMO martingales and $G$-Girsanov Theorem

We now introduce some results of $G$-BMO martingale and $G$-Girsanov Theorem in [15] and [36].

Definition 2.10**.**

For $Z\in H_{G}^{2}(0,T)$, a symmetric $G$-martingale $\int_{0}^{\cdot}Z_{s}dB_{s}$ on $[0,T]$ is called a $G$-BMO martingale if

[TABLE]

where $\mathcal{T}_{0}^{T}$ denotes the totality of all $\mathcal{F}$-stopping times taking values in $[0,T]$ and $\left\|Z\right\|_{BMO(\mathbb{P})}$ stands for the BMO norm of $\int_{0}^{\cdot}Z_{s}dB_{s}$ under probability measure $\mathbb{P}$.

Set

[TABLE]

In a straightforward manner, we have the following important norm estimate for a $G$-BMO martingale $\int_{0}^{\cdot}Z_{s}dB_{s}$.

Lemma 2.12**.**

For $Z\in BMO_{G}$, we have for each  $t\in[0,T]$,

[TABLE]

where $C_{\alpha}$ is a positive constant depending on $\alpha$.

Proof.

Fix some $(t,\mathbb{P})\in[0,T]\times\mathcal{P}$. In view of [17, Corollary 2.1, page 28], for each $\mathbb{P^{\prime}}\in\mathcal{P}(t,\mathbb{P})$ we have

[TABLE]

where $C_{\alpha}$ is a positive constant depending only on $\alpha$. In view of the definition of $\mathcal{P}(t,\mathbb{P})$, we have

[TABLE]

In view of Theorem 2.2 and noting that $C_{\alpha}$ is independent of $\mathbb{P}^{\prime}$, we have

[TABLE]

Notice that $C_{\alpha}$ is independent of $\mathbb{P}$ and we get the lemma. ∎

Like in the classical stochastic analysis, a $G$-BMO martingale can be used to define an exponential $G$-martingale. Hu et al. [15, Lemma 3.2, page 11] give the following lemma.

Lemma 2.13**.**

For $Z\in BMO_{G}$, the process

[TABLE]

is a symmetric $G$-martingale.

Similarly to Possamaï and Zhou [36], we have the following lemmas.

Lemma 2.14**.**

(Reverse Hölder Inequality) Let \phi(x)=\big{\{}1+{1\over x^{2}}\log{2x-1\over 2(x-1)}\big{\}}^{1\over 2}-1 and $1<q<+\infty.$ If  $\left\|Z\right\|_{BMO_{G}}<\phi(q),$ we have

[TABLE]

for a constant $C_{q}>0$ depending only on $q$.

Proof.

For each $\mathbb{P}\in\mathcal{P}$,

[TABLE]

Then, from [17, Theorem 3.1, page 54], we have

[TABLE]

for a positive constant $C_{q}$ which does not dependent on $\mathbb{P}$. ∎

Lemma 2.15**.**

Let $1<r<+\infty.$ If  $\left\|Z\right\|_{BMO_{G}}<{\sqrt{2}\over 2}(\sqrt{r}-1),$

[TABLE]

holds with a constant $C_{r}>0$ depending only on $r$.

Proof.

Similarly to the proof of Lemma 2.14, the desired result is an immediate consequence of [17, Theorem 2.4, page 33] for all $\mathbb{P}\in\mathcal{P}$. ∎

Remark 2.16**.**

Assume $\left\|Z\right\|_{BMO_{G}}<\phi(q)$ for some $q\in(1,+\infty)$. Fix some $(t,\mathbb{P})\in[0,T]\times\mathcal{P}$. In view Lemma 2.14, we have for each $\mathbb{P}^{\prime}\in\mathcal{P}(t,\mathbb{P})$,

[TABLE]

In view of the the definition of $\mathcal{P}(t,\mathbb{P})$, we have

[TABLE]

Thus in view of Theorem 2.2, we get

[TABLE]

Noting that $C_{q}$ is independent of  $\mathbb{P}$, we have the following reverse Hölder inequality,

[TABLE]

Similarly, in view of Theorem 2.2 and Lemma 2.15, we have

[TABLE]

if $\left\|Z\right\|_{BMO_{G}}<{\sqrt{2}\over 2}(\sqrt{r}-1)$ for some $r\in(1,+\infty)$.

Remark 2.17**.**

The reverse Hölder inequality in Remark 2.16 is used in the proof of Hu et al. [15, Lemma 3.4]. We give a proof here for convenience of the reader.

Remark 2.18**.**

Suppose there exist $\{Z^{n}\}_{n\in\mathbb{N}}\subseteq H_{G}^{2}(0,T)$ such that $\left\|Z^{n}\right\|_{BMO_{G}}\leq M$ for all $n\in\mathbb{N}$. Taking $t=0$ in Reamrk 2.16, we can know that there exist $q>1$ and $r>1$ which are depending only on $M$ such that:

[TABLE]

With the exponential martingale, we can generalize the Girsanov theorem. In [15], we know that we can define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ with $\mathscr{E}(Z)$ satisfying

[TABLE]

where $p>{q\over q-1}$ and $q$ is the order in the reverse Hölder inequality for $\mathscr{E}(Z)$. Moreover, the conditional expectation $\tilde{\mathbb{E}}_{t}[\cdot]$ is well-defined following the procedure introduced in [15] and [42]. And we have

[TABLE]

The following two lemmas give the Girsanov theorem in the $G$-framework, and can be found in Hu et al. [15].

Lemma 2.19**.**

Suppose that $Z\in BMO_{G}$. We define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ by $\mathscr{E}(Z)$. Then the process $B-\int Zd\left\langle B\right\rangle$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}[\cdot]$.

Lemma 2.20**.**

Suppose that $Z\in BMO_{G}$. We define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ by $\mathscr{E}(Z)$. Suppose that $K$ is a decreasing $G$-martingale such that $K_{0}=0$ and for some $p>{q\over q-1},K_{t}\in L_{G}^{p}(\Omega_{t}),0\leq t\leq T,$ where $q$ is the order in the reverse Hölder inequality for $\mathscr{E}(Z)$. Then $K$ is a decreasing $G$-martingale under $\tilde{\mathbb{E}}[\cdot]$.

3 A priori estimates for solutions of reflected quadratic $G$-BSDEs

With $G$-BMO martingale and $G$-Girsanov Theorem, we have the following comparison theorem for quadratic $G$-BSDEs.

Theorem 3.1**.**

Let the triplet $(\xi^{i},f^{i},g^{i})$ satisfy (H1)-(H3) for $i=1,2$. Let $(Y^{i},Z^{i},K^{i})\in\mathfrak{G}_{G}^{2}(0,T)$ be the solution to the following $G$-BSDE:

[TABLE]

where $V^{i}$ is a continuous finite variation process, for $i=1,2$. Assume that

[TABLE]

and $K^{i}$ is a decreasing $G$-martingale. If $\xi^{1}\geq\xi^{2}$, $g^{1}\geq g^{2}$, $f^{1}\geq f^{2}$,  q.s. and $V^{1}-V^{2}$ is an increasing process, then we have $Y_{t}^{1}\geq Y_{t}^{2}$,  q.s.,  for any $t\in[0,T]$.

Proof.

Without loss of generality, we assume that $g^{1}=g^{2}=0$.

Define $\hat{\xi}:=\xi^{1}-\xi^{2}$ and for $t\in[0,T]$,

[TABLE]

and $\hat{f_{t}}:=f^{1}(t,Y_{t}^{2},Z_{t}^{2})-f^{2}(t,Y_{t}^{2},Z_{t}^{2}).$ Like in the proof of [15, Proposition 3.5], we use the method of linearization to write

[TABLE]

where for $0\leq s\leq T,$

[TABLE]

for a scalar Lipschitz continuous function $l$ such that $\textbf{1}_{[-\varepsilon,\varepsilon]}(x)\leq l(x)\leq\textbf{1}_{[-2\varepsilon,2\varepsilon]}(x)$ with $x\in(-\infty,+\infty)$. We also have

[TABLE]

Define $\tilde{B}_{t}:=B_{t}-\int_{0}^{t}\hat{b}_{s}^{\varepsilon}d\left\langle B\right\rangle_{s}$ for $t\in[0,T]$. In view of [15, Lemma 3.6], we know that $\hat{b}^{\varepsilon}\in BMO_{G}$. Therefore, we can define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ by $\mathscr{E}(\hat{b}^{\varepsilon})$, such that $\tilde{B}$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}[\cdot]$. Then the last $G$-BSDE reads

[TABLE]

Applying Itô’s formula to $e^{\int_{0}^{t}\hat{a}_{s}^{\varepsilon}d\left\langle B\right\rangle_{s}}\hat{Y}_{t}$,  we have

[TABLE]

So we have

[TABLE]

In view of Hu et al. [12, Lemma 3.4] and Lemma 2.20, we know $\int_{0}^{\cdot}e^{\int_{0}^{s}\hat{a}_{u}^{\varepsilon}d\left\langle B\right\rangle_{u}}dK_{s}^{2}$ is a decreasing $G$-martingale under both $\hat{\mathbb{E}}[\cdot]$ and $\tilde{\mathbb{E}}[\cdot]$. Taking conditional $G$-expectation on both sides, we have

[TABLE]

Since $|\hat{a}_{s}^{\varepsilon}|\leq L_{y}$, we have

[TABLE]

Finally, it remains to prove the limit

[TABLE]

Let \phi(x)=\big{\{}1+{1\over x^{2}}\log{2x\over 2(x-1)}\big{\}}^{1\over 2}-1. We know there exist $p>1$ independent of $\varepsilon$ such that

[TABLE]

where $p^{\prime}={p\over p-1}$. Then according to Lemma 2.14, for $X\in L_{G}^{p}(\Omega_{T})$, we have

[TABLE]

In view of Lemma 2.12, we have

[TABLE]

So we get \lim_{\varepsilon\to 0}\tilde{\mathbb{E}}_{t}\Big{[}\int_{t}^{T}|\hat{m}_{s}^{\varepsilon}|d\left\langle B\right\rangle_{s}\Big{]}=0,~{}~{}\text{q.s.} ∎

Consider the following type of BSDE:

[TABLE]

with $A$ being a continuous nondecreasing process and $A_{0}=0$.

Proposition 3.2**.**

Let $f$ satisfy (H1) and (H3). Assume that $(Y,Z,A)$ solves

[TABLE]

where

[TABLE]

and $A$ is a continuous nondecreasing process with $A_{0}=0$.

Then there exist constant $C_{1}:=C_{1}(\left\|Y\right\|_{S_{G}^{\infty}},T,L_{z},L_{y},M_{0},\bar{\sigma})$ such that

[TABLE]

and constant $C_{2}:=C_{2}(\left\|Y\right\|_{S_{G}^{\infty}},T,L_{z},L_{y},M_{0},\bar{\sigma},\alpha)$ for any $\alpha\geq 1$, such that

[TABLE]

Proof.

For each $\mathbb{P}\in\mathcal{P}$, we know

[TABLE]

Then, for some $a>0~{}$, applying Itô’s formula under $\mathbb{P}$ to $e^{-aY_{t}}$,  we have for each $\tau\in\mathcal{T}_{0}^{T}$,

[TABLE]

Since $A~{}$ is a continuous nondecreasing process, noting $a>0$ and Remark 2.9, we have

[TABLE]

Taking $a=4L_{z}$, noting $Y\in S_{G}^{\infty}(0,T)$ and taking conditional expectations under $\mathbb{P}$ on both sides, we have

[TABLE]

Then with the arbitrariness of $\tau$, we obtain for all $\mathbb{P}\in\mathcal{P}$,

[TABLE]

Finally, with the arbitrariness of $\mathbb{P}$, we get

[TABLE]

Now we get the estimate for $Z$. We have

[TABLE]

In view of BDG inequality and Remark 2.9, we have for each $\alpha\geq 1$

[TABLE]

In view of Lemma 2.12, we have

[TABLE]

Substituting the estimate for $Z$, we get the estimate for $A$. ∎

Proposition 3.3**.**

Let $(\xi,f,S)$ satisfy (H1), (H3) and (H4). Assume that the triplet $(Y,Z,A)\in\mathcal{S}_{G}^{2p}(0,T)$ with some $p>1$, is a solution to the reflected $G$-BSDE with data $(\xi,f,S)$. Moreover, we suppose

[TABLE]

*with $q$ satisfying $p>{q\over q-1}$.

Then there exists a constant $~{}C:=C(T,L_{z},L_{y},\bar{\sigma},N_{0})~{}$ such that*

[TABLE]

Proof.

For some $r>0~{}$, applying Itô’s formula to $e^{rt}|Y_{t}-N_{0}|^{2}$,  we have for each $t\in[0,T]$,

[TABLE]

We have

[TABLE]

where

[TABLE]

with $s\in[0,T]$ and the function $l$ being Lipschitz continuous such that $\textbf{1}_{[-\varepsilon,\varepsilon]}(x)\leq l(x)\leq\textbf{1}_{[-2\varepsilon,2\varepsilon]}(x)$ for $x\in(-\infty,+\infty)$. Moreover,

[TABLE]

In view of [15, Lemma 3.6], we know that $b^{\varepsilon}\in BMO_{G}$. Set $\tilde{B}_{t}:=B_{t}-\int_{0}^{t}b_{s}^{\varepsilon}d\left\langle B\right\rangle_{s}$ for $t\in[0,T]$. Thus we can define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ by $\mathscr{E}(b_{s}^{\varepsilon})$, such that $\tilde{B}$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}[\cdot]$. Then we have for each $t\in[0,T]$,

[TABLE]

Setting $r>\bar{\sigma}^{2}(1+2L_{y})$ and taking conditional expectations on both sides, we have

[TABLE]

From (3.1), we know that $\{-\int_{0}^{t}(Y_{s}-S_{s})dA_{s}\}_{t\in[0,T]}$ is a non-increasing G-martingale under $\hat{\mathbb{E}}[\cdot]$. Moreover,

[TABLE]

Note $p>{q\over q-1}$ and

[TABLE]

In view of Lemma 2.20, we know that $\{-\int_{0}^{t}(Y_{s}-S_{s})dA_{s}\}_{t\in[0,T]}$ is a non-increasing G-martingale under $\tilde{\mathbb{E}}[\cdot]$. Then for each $t\in[0,T]$,

[TABLE]

Let $\varepsilon\to 0$, we have

[TABLE]

So we get the estimate for $Y$. ∎

Proposition 3.4**.**

Let $(\xi^{1},f^{1},S^{1})$ and $(\xi^{2},f^{2},S^{2})$ be two sets of data, each one satisfying (H1), (H3) and (H4). Assume that the triplet $(Y^{i},Z^{i},A^{i})\in\mathcal{S}_{G}^{2p}(0,T)$ with some $p>1$, is a solution of the reflected $G$-BSDE with data $(\xi^{i},f^{i},S^{i}),~{}i=1,2$. Moreover, we suppose

[TABLE]

with $q$ satisfying $p>{q\over q-1}$. Then there exists a constant $~{}C_{1}:=C_{1}(q,T,L_{z},L_{y},N_{0})~{}$ such that for each $t\in[0,T]$,

[TABLE]

where

[TABLE]

Moreover, there exists a constant $~{}C_{2}:=C_{2}(T,L_{z},L_{y},\bar{\sigma},N_{0},M_{0})~{}$ such that

[TABLE]

Proof.

First, with Proposition 3.2 and 3.3, we know that there exists a constant $C:=C(T,L_{z},L_{y},\bar{\sigma},N_{0},M_{0})$ such that

[TABLE]

Define

[TABLE]

With the condition of $f^{1}$ and $f^{2}$, we see that $\hat{\lambda}\in H_{G}^{2p}(0,T).$ As in the proof of Proposition 3.3, define for $0\leq s\leq T,$

[TABLE]

where $l$ is a Lipschitz continuous function such that $\textbf{1}_{[-\varepsilon,\varepsilon]}(x)\leq l(x)\leq\textbf{1}_{[-2\varepsilon,2\varepsilon]}(x)$. Also define $\hat{A}:=A^{1}-A^{2}$. We have

[TABLE]

and for each $s\in[0,T]$,

[TABLE]

Then we have

[TABLE]

where $d\tilde{B}_{s}=dB_{s}-\hat{b}_{s}^{\varepsilon}d\left\langle B\right\rangle_{s}.$ Similarly to the proof of Proposition 3.3, we can define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ by $\mathscr{E}(\hat{b}^{\varepsilon})$, such that $\tilde{B}$ is a $G$-Brownian Motion under $\tilde{\mathbb{E}}[\cdot]$.

For some $r>0~{}$, applying Itô’s formula to $e^{rt}|\hat{Y}_{t}|^{2}$, we have for each $t\in[0,T]$,

[TABLE]

In view of a similar argument as in the proof of Proposition 3.3, we see that $\int_{0}^{\cdot}(S_{s}^{i}-Y_{s}^{i})\,dA_{s}^{i}$ is a non-increasing $G$-martingale on $[0,T]$ under $\tilde{\mathbb{E}}[\cdot]$ for $i=1,2$.

Setting $r>\bar{\sigma}^{2}(2+2L_{y})$ and taking conditional expectations on both sides, we have

[TABLE]

Note that $\left\|\hat{b}_{s}^{\varepsilon}\right\|_{BMO_{G}}\leq\left\|L_{z}(1+|Z_{1}|+|Z_{2}|)\right\|_{BMO_{G}}<\phi(q)<\phi(p^{\prime})$ where $p^{\prime}={p\over p-1}$. Then according to Lemma 2.14, $\forall X\in L_{G}^{p}(\Omega_{T})$, we have for each $t\in[0,T]$,

[TABLE]

So by the Hölder inequality, we have for each $t\in[0,T]$,

[TABLE]

Then there exists a constant $~{}C_{1}:=C_{1}(q,T,L_{z},L_{y},N_{0})~{}$ such that for each $t\in[0,T]$,

[TABLE]

Finally, we just need to prove

[TABLE]

In view of Lemma 2.12, we have

[TABLE]

So we get \lim_{\varepsilon\to 0}\tilde{\mathbb{E}}_{t}\Big{[}\int_{t}^{T}|\hat{m}_{s}^{\varepsilon}|^{2}d\left\langle B\right\rangle_{s}\Big{]}=0. And we get the estimate for $\hat{Y}.$

Then we consider the estimate for $\hat{Z}.$ Applying Itô’s formula to $e^{rt}|\hat{Y}_{t}|^{2}$,  we have for each $t\in[0,T]$,

[TABLE]

Taking conditional expectations on both sides, we have

[TABLE]

Note that $\forall i=1,2,$

[TABLE]

With (3.2), we get the estimate for $\hat{Z}.$ ∎

Remark 3.5**.**

The uniqueness for solutions to the reflected quadratic $G$-BSDE is an immediate consequence of Proposition 3.4.

4 Penalized $G$-BSDEs and their limit

Similar to [22] and [23], we use a penalized method. In this section, we first prove some convergence properties of solutions to the penalized $G$-BSDEs. For $(f,\xi,S)$ satisfying (H1)-(H5) and $n\in\mathbb{N}$, we consider the following penalized $G$-BSDE:

[TABLE]

Define $L_{t}^{n}:=n\int_{0}^{t}(Y_{s}^{n}-S_{s})^{-}ds$ for $t\in[0,T]$. The penalized $G$-BSDE reads:

[TABLE]

From Theorem 2.10, the penalized BSDE (4.1) or (4.2) has a unique solution $(Y^{n},Z^{n},K^{n})\in\mathfrak{G}_{G}^{2}(0,T)$ such that

[TABLE]

and

[TABLE]

Both estimates depend on $n$. In fact, $(Y^{n},Z^{n},K^{n},L^{n})$ is uniformly bounded in $n$.

Lemma 4.1**.**

There exists two positive constants $C$ and $C_{p}$ which are independent of $n$, such that

[TABLE]

and

[TABLE]

Proof.

First we consider the estimate for $Y^{n}$. The proof is very similar to that of Proposition 3.3.

For some $r>0~{}$, applying Itô’s formula to $e^{rt}|Y_{t}-N_{0}|^{2}$,  we have for each $t\in[0,T]$,

[TABLE]

Noting that

[TABLE]

we have

[TABLE]

Similar to the proof of Proposition 3.3, we have for each $s\in[0,T]$,

[TABLE]

where

[TABLE]

So we have

[TABLE]

where $d\tilde{B}_{s}^{n,\varepsilon}=dB_{s}-b_{s}^{n,\varepsilon}d\left\langle B\right\rangle_{s}$. In view of [15, Lemma 3.6], we know that $b^{n,\varepsilon}\in BMO_{G}$. Thus we can define a new $G$-expectation $\tilde{\mathbb{E}}^{n,\varepsilon}[\cdot]$ by $\mathscr{E}(b_{s}^{n,\varepsilon})$, such that $\tilde{B}^{n,\varepsilon}$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}^{n,\varepsilon}[\cdot]$.

In view of Hu et al. [12, Lemma 3.4] and Lemma 2.20, we know that the process

[TABLE]

is a decreasing $G$-martingale under both $\hat{\mathbb{E}}[\cdot]$ and $\tilde{\mathbb{E}}^{n,\varepsilon}[\cdot]$. Setting $r>\bar{\sigma}^{2}(1+2L_{y})$ and taking conditional expectations in the last inequality, we have for each $t\in[0,T]$,

[TABLE]

Then

[TABLE]

Setting $\varepsilon\to 0$, we have

[TABLE]

So we know there exists a constant $C^{\prime}$ independent of $n$ such that $\left\|Y^{n}\right\|_{S_{G}^{\infty}}\leq C^{\prime}$.

Then by Proposition 3.2, we know that there exist two constants $C^{\prime\prime}$ and $C_{p}^{\prime}$ which are independent of $n$, such that

[TABLE]

and

[TABLE]

We have

[TABLE]

with $C=2(C^{\prime}+C^{\prime\prime})$, and

[TABLE]

with $C_{p}=2C_{p}^{\prime}$. ∎

The following lemma plays a key role in the proof of the convergence of $\{Y^{n}\}$. It gives the convergence of $(Y^{n}-S)^{-}$ in $S_{G}^{\alpha}(0,T)$.

Lemma 4.2**.**

For each $\alpha>1$, we have

[TABLE]

Proof.

The lemma has been proved by Li, Peng and Soumana Hima [22, Lemma 4.3] when the generator $(f,g)$ is uniformly Lipschitz continuous. Their arguments can be adapted to our general case.

First, we sketch the main ideas. Under our $z$-quadratic generator, we will still use the method of linearization. By the $G$-Girsanov theorem, we can we rewrite the $G$-BSDE (4.1) so that the generator is independent of $z$ under a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$. Similarly as in [22, Lemma 4.3], the following holds true:

[TABLE]

Then from Lemmas 2.15 and 2.14, we see that $\tilde{\mathbb{E}}[\cdot]$ can be replaced with $\hat{\mathbb{E}}[\cdot]$ in the last limit, which completes the proof.

Now we begin our proof. Similar to the proof of Proposition 3.3 and Lemma 4.1, we first rewrite the $G$-BSDE (4.1) by linearization into the form:

[TABLE]

with

[TABLE]

So the $G$-BSDE (4.1) reads

[TABLE]

where $d\tilde{B}^{n,\varepsilon}_{s}=dB_{s}-b_{s}^{n,\varepsilon}d\left\langle B\right\rangle_{s}$. In view of [15, Lemma 3.6], we know that $b^{n,\varepsilon}\in BMO_{G}$. Thus we can define a new $G$-expectation $\tilde{\mathbb{E}}^{n,\varepsilon}[\cdot]$ by $\mathscr{E}(b^{n,\varepsilon})$, such that $\tilde{B}^{n,\varepsilon}$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}^{n,\varepsilon}[\cdot]$.

We now prove

[TABLE]

Set

[TABLE]

Then we have for each $t\in[0,T]$,

[TABLE]

In view of [13, Theorem 3.6], we have for each $t\in[0,T]$,

[TABLE]

where

[TABLE]

It follows that

[TABLE]

We have for any $\alpha>1$,

[TABLE]

In view of Lemma 4.1, we have

[TABLE]

For $\epsilon>0$, it is straightforward to show for each $t\in[0,T]$,

[TABLE]

For $\delta\in(0,T)$, we have

[TABLE]

Define the function

[TABLE]

In view of Lemma 4.1, we can choose $p>1$ independent of $n$ and $\varepsilon$, such that

[TABLE]

Set $q={p\over p-1}$. Then in view of Lemma 2.14, we have for each $\alpha>1$ and $X\in L_{G}^{q}(\Omega_{T})$,

[TABLE]

where $C_{p}$ depends only on $p$.

In view of Assumption (H4) on $S$, we know

[TABLE]

So we have for all $\alpha>1$,

[TABLE]

and

[TABLE]

From (4.7), we know

[TABLE]

Then, in view of (4.4), (4.9), and Remark 2.5, we have

[TABLE]

where $C^{\prime\prime}$ is independent of $n$, $\delta$ and $\epsilon$. Therefore, in view of Lemma 2.7, setting $\epsilon\to 0$, we have

[TABLE]

In view of Theorem 3.1, we get $Y_{t}^{n}\geq Y_{t}^{1}$ and then obtain

[TABLE]

By Lemma 2.7 again and noting that $(Y_{T}^{1}-S_{T})^{-}=0$, we obtain

[TABLE]

Finally, with (4.8), we derive that

[TABLE]

Let $\delta\to 0$ and we know

[TABLE]

Therefore, we have (4.3).

Next we want to change the $G$-expectation in the last equality. Actually, in view of Lemma 2.15 and Remark 2.18, there exists $r>1$ which is independent of $n$ and $\varepsilon$, such that

[TABLE]

for some positive constant $C_{r}$ which depends only on $r$. Thus, for each $\alpha>1$, we have

[TABLE]

So

[TABLE]

∎

Now we show the convergence of the sequence $\{Y^{n}\}_{n=1}^{\infty}$.

Lemma 4.3**.**

The sequence $\{Y^{n}\}_{n=1}^{\infty}$ is a Cauchy sequence in $S_{G}^{\alpha}(0,T)$ for any $\alpha\geq 2$.

Proof.

For $m,n\in\mathbb{N}$ and each $t\in[0,T]$, set

[TABLE]

We use the method of linearization. Similar to the proof of Proposition 3.3 and Lemma 4.1, $\forall\varepsilon>0$, we write for each $s\in[0,T]$,

[TABLE]

with

[TABLE]

So we have for each $t\in[0,T]$,

[TABLE]

where $d\tilde{B}_{s}^{n,m,\varepsilon}=dB_{s}-b_{s}^{n,m,\varepsilon}\hat{Z}_{s}^{n,m}d\left\langle B\right\rangle_{s}.$ In view of [15, Lemma 3.6], we know that $b^{n,m,\varepsilon}\in BMO_{G}$ and we define a new $G$-expectation $\tilde{\mathbb{E}}^{n,m,\varepsilon}[\cdot]$ by $\mathscr{E}(b^{n,m,\varepsilon})$, such that $\tilde{B}^{n,m,\varepsilon}$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}^{n,m,\varepsilon}[\cdot]$.

For all $\alpha\geq 2$, by applying Itô’s formula to $|\hat{Y}_{t}^{n,m}|^{\alpha}e^{rt}$, we get for each $t\in[0,T]$,

[TABLE]

Let $r>L_{y}\alpha\bar{\sigma}^{2}$. Noting that $|a_{s}^{n,m,\varepsilon}|\leq L_{y}$, we get

[TABLE]

It is easy to check that

[TABLE]

Noting that

[TABLE]

we have

[TABLE]

where

[TABLE]

In view of [12, Lemma 3.3] and [15, Lemma 3.4], we conclude that $M$ is a $G$-martingale under $\tilde{\mathbb{E}}^{n,m,\varepsilon}[\cdot]$. Thus we obtain

[TABLE]

Noting the following estimate

[TABLE]

where $C$ is independent of $n$, $m$ and $\varepsilon$, we deduce from (4.10) that

[TABLE]

Recall that

[TABLE]

In view of Lemma 4.1, we can choose $p>1$ independent of $n$, $m$ and $\varepsilon$, such that

[TABLE]

Set $q={p\over p-1}$. Then in view of Lemma 2.14, we have for each $\alpha>1$ and $X\in L_{G}^{q}(\Omega_{T})$,

[TABLE]

where $C_{p}$ depends only on $p$.

Then we have for some $\beta>1$,

[TABLE]

and

[TABLE]

Moreover, in view of the assumption on $S$ and Lemma 4.1, we know $C_{1}$ and $C_{2}$ are independent of $n$, $m$ and $\varepsilon$. From Remark 2.5, there exists a constant $C^{\prime}$ independent of $n$, $m$ and $\varepsilon$, such that

[TABLE]

Then with (4.11), (4.13) and (4.14), we have

[TABLE]

where $\bar{C}$ is independent of $n$, $m$ and $\varepsilon$.

In view of Lemma 2.15 and Remark 2.18, there is $r>1$ which is independent of $n$, $m$ and $\varepsilon$, such that

[TABLE]

where $C_{r}$ depends only on $r$. Thus, for each $\alpha^{\prime}\geq 2$, we have

[TABLE]

Setting $\alpha=\alpha^{\prime}r>2$ in (4.15), we have

[TABLE]

On the other hand,

[TABLE]

Let $\varepsilon\to 0$ in (4.16). Then in view of Lemma 4.2, we conclude that $(Y^{n})_{n=1}^{\infty}$ is a Cauchy sequence in $S_{G}^{\alpha\prime}(0,T).$ ∎

5 Existence and uniqueness result on reflected quadratic $G$-BSDEs

Our main result in the paper is stated as follows.

Theorem 5.1**.**

Let the triple $(\xi,f,S)$ satisfy (H1)-(H5). Then, the reflected $G$-BSDE (3.1) has a unique solution (Y,Z,A) such that $(Y,Z)\in S_{G}^{\infty}(0,T)\times BMO_{G}$ and $A\in\bigcap_{\alpha\geq 2}S_{G}^{\alpha}(0,T)$.

Proof.

The uniqueness of the solution is referred to Remark 3.5. We now prove the existence.

Recalling the penalized $G$-BSDE (4.2), for $m,n\in\mathbb{N}$ and each $t\in[0,T]$, define

[TABLE]

and

[TABLE]

In view of Lemma 4.3, there exists $Y\in S_{G}^{\alpha}(0,T)$ satisfying

[TABLE]

Note that there is $L:=L(L_{y},L_{z})$ such that for each $t\in[0,T]$,

[TABLE]

Applying Itô’s formula to $|\hat{Y}^{n,m}_{t}|^{2}$, we get for each $t\in[0,T]$,

[TABLE]

Setting $t=0$, we have

[TABLE]

With the B-D-G inequality and Hölder’s inequality, we have

[TABLE]

In view of Lemmas 4.1 and 2.12, there exists a constant $C_{1}$ independent of $m$ and $n$, such that

[TABLE]

In view of Lemma 4.3, we know that $\{Z^{n}\}_{n=1}^{\infty}$ is a Cauchy sequence in $H_{G}^{\alpha}(0,T)$ for $\alpha\geq 2$. Thus there exists $Z\in H_{G}^{\alpha}(0,T)$ satisfying

[TABLE]

Now set $A^{n}:=L^{n}-K^{n}$. It is easy to check that $(A_{t}^{n})_{t\in[0,T]}$ is a nondecreasing process and

[TABLE]

So we get

[TABLE]

From the assumption on $f$, we have

[TABLE]

Then in view of Lemmas 4.1 and 2.12 and inequality (5.1), we know that $\{A^{n}\}_{n=1}^{\infty}$ is a Cauchy sequence in $S_{G}^{\alpha}(0,T)$ for each $\alpha\geq 2$. There exists a nondecreasing process $(A_{t})_{t\in[0,T]}$ such that

[TABLE]

Now, we prove $Y\in S_{G}^{\infty}(0,T)$. In view of Lemma 4.1, we know there exist a constant $C>0$ such that $\left\|Y^{n}\right\|_{S_{G}^{\infty}}\leq C$. Recall that

[TABLE]

From

[TABLE]

we see that for each $\mathbb{P}\in\mathcal{P}$, $\{\sup_{t\in[0,T]}|Y_{t}^{n}|,n=1,2,\ldots\}$ converges in probability $\mathbb{P}$ to $\sup_{t\in[0,T]}|Y_{t}|$. Then, there exits a sub-sequence of $(\sup_{t\in[0,T]}|Y_{t}^{n}|)$ such that ${\mathbb{P}}$-a.s.,

[TABLE]

Since $\sup_{t\in[0,T]}|Y_{t}^{n_{k}}|\leq C$ for a positive constant $C$ independent of $\mathbb{P}$, we have $\sup_{t\in[0,T]}|Y_{t}|\leq C$ $\mathbb{P}\text{\,-a.s.}$ for each $\mathbb{P}\in\mathcal{P}$, and then $\sup_{t\in[0,T]}|Y_{t}|\leq C$, q.s., which yields the inequality $\left\|Y\right\|_{S_{G}^{\infty}}\leq C$. In view of Proposition 3.2, we have $Z\in BMO_{G}$.

From Lemma 4.2, we have $Y_{t}\geq S_{t}$ for $t\in[0,T]$. We claim that $\int_{0}^{\cdot}(S_{s}-Y_{s})\,dA_{s}$ is a non-increasing $G$-martingale on $[0,T]$. Set $\tilde{K}_{t}^{n}=\int_{0}^{t}(Y_{s}-S_{s})dK_{s}^{n}$. Since $Y_{t}\geq S_{t}$ for $t\in[0,T]$ and $K^{n}$ is a decreasing $G$-martingale, then $\tilde{K}^{n}$ is a decreasing $G$-martingale.

We have

[TABLE]

with

[TABLE]

and

[TABLE]

In view of Lemma 4.1 and identically as in the proof of [22, Theorem 5.1], we have

[TABLE]

which implies that $\int_{0}^{\cdot}(S_{s}-Y_{s})\,dA_{s}$ is a non-increasing $G$-martingale on $[0,T]$. ∎

In an identical way, we have the following theorem.

Theorem 5.2**.**

Suppose that $\xi$, $f$, $g$, and $S$ satisfy (H1)-(H5). Then the reflected $G$-BSDE (2.2) has a unique solution (Y,Z,A) such that $(Y,Z)\in L_{G}^{\infty}[0,T]\times BMO_{G}$ and $A\in\bigcap_{\alpha\geq 2}S_{G}^{\alpha}[0,T]$.

We have the following comparison theorem for reflected quadratic $G$-BSDEs.

Theorem 5.3**.**

Let the set $(\xi^{i},f^{i},g^{i},S^{i})$ satisfy (H1)-(H5), and $(Y^{i},Z^{i},A^{i})\in\mathcal{S}_{G}^{2}(0,T)$ be the solution to the following reflected $G$-BSDE:

[TABLE]

with $i=1,2$. Assume that $(Y^{i},Z^{i})\in S_{G}^{\infty}(0,T)\times BMO_{G}$ and $A^{i}_{T}\in\bigcap_{p\geq 1}L_{G}^{p}(\Omega_{T})$ for $i=1,2$. If $\xi^{1}\geq\xi^{2}$, $g^{1}\geq g^{2}$, $f^{1}\geq f^{2}$, and $S^{1}\geq S^{2}$,  q.s., then $Y_{t}^{1}\geq Y_{t}^{2},~{}~{}\text{q.s.}$  for any $t\in[0,T]$.

Proof.

The proof is identical to that of [22, Theorem 5.3].

We first consider the following $G$-BSDE

[TABLE]

for $n=1,2,\cdots$. As before, we have

[TABLE]

Noting that $Y_{t}^{1}\geq S_{t}^{1}$, we can rewrite the equation for $(Y^{1},Z^{1},A^{1})$ as

[TABLE]

Using Theorem 3.1, we have $Y_{t}^{1}\geq y_{t}^{n},~{}~{}\text{q.s.}$  for all $n\in\mathbb{N}$. Letting $n\to\infty$, we conclude that $Y_{t}^{1}\geq Y_{t}^{2},~{}~{}\text{q.s.}$ ∎

6 Relation between quadratic $G$-BSDEs and nonlinear parabolic PDEs

Consider the following PDE:

[TABLE]

where

[TABLE]

for each $(A,p,y,x,t)\in\mathbb{S}_{n}\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\times[0,T]$.

We shall give a nonlinear Feynman-Kac formula for the fully nonlinear PDE (6.1) when the functions $f$ and $g$ are quadratic in the last argument. Similar to Li, Peng and Soumana Hima [22, Section 6], we give the relationship between solutions of the obstacle problem for nonlinear parabolic PDEs and the related reflected quadratic $G$-BSDEs.

In what follows, we consider the $G$-expectation space $(\Omega,L_{G}^{1}(\Omega_{T}),\hat{\mathbb{E}})$ for the case of $d=1$ and $\bar{\sigma}^{2}=\hat{\mathbb{E}}[B_{1}^{2}]\geq-\hat{\mathbb{E}}[-B_{1}^{2}]=\underline{\sigma}^{2}>0$.

6.1 Nonlinear Feynman-Kac formula

Our main assumptions of this section are formulated as follows.

For deterministic functions $b,h,\sigma:[0,T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$, $\phi:\mathbb{R}^{n}\to\mathbb{R}$, and $f,g:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, we make the following assumptions.

• (A1)

The functions $b,h,\sigma,f,g$ are uniformly continuous in $t$, i.e. there is a non-decreasing continuous function $w:[0,+\infty)\to[0,+\infty)$ such that $w(0)=0$ and

[TABLE]

[TABLE]

• (A2)

There exist a positive integer $m$ and a constant $L>0$ such that for each $(t,x,x^{\prime},y,y^{\prime},z,z^{\prime})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}$,

[TABLE]

• (A3)

There is a positive constant $M_{0}$ such that

[TABLE]

• (A4)

There are two constants $\varepsilon>0$ and $K>0$ such that for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

Remark 6.1**.**

Assumption (A4) implies that $\sigma$ is bounded on $[0,T]\times\mathbb{R}^{n}$.

For each $(t,\xi)\in[0,T]\times\bigcap\limits_{p\geq 2}L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$, we consider the following of $G$-SDE:

[TABLE]

Denote by $X^{t,\xi}$ the solution to $G$-SDE (6.2). Then, we have

Proposition 6.2**.**

(See [34, Chapter V]) Let $\xi,\xi^{\prime}\in L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$ with $p\geq 2$. Then we have, for each $\delta\in[0,T-t]$,

[TABLE]

where the constant $C$ depends on $L,G,p,n$ and $T$.

Proposition 6.3**.**

Let the triplet $(b^{i},h^{i},\sigma^{i})$ satisfy (A1)-(A2) for $i=1,2$. For each $(t,\xi)\in[0,T]\times L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$, $p\geq 2$, let $X^{t,\xi,i}$ be the solution to the following $G$-SDE:

[TABLE]

Then for each $\delta\in[0,T-t]$, there exist a constant $C$ depends only on $L,G,p$ and $T$, such that

[TABLE]

where for each $u\in[t,T]$,

[TABLE]

Proof.

For simplicity, we assume $h^{i}=0$. Then we have

[TABLE]

In view of BDG inequality, we have

[TABLE]

By the Gronwall’s inequality, we obtain

[TABLE]

∎

We now consider the following $G$-BSDE.

[TABLE]

We should point out that Theorem 5.3 in Hu et al. [15] can not be used to our case directly. Because it is hard to check the assumption (H2) directly in our Markovian case. We now give the the existence of the solution to $G$-BSDE (6.3) in the spirit of the method in Hu et al. [12] and Hu et al. [15].

Without loss of generality, we assume $h=0$, $g=0$ and $t=0$. For each $x_{0}\in\mathbb{R}^{n}$, we consider the following forward and backward differential equations in the $G$-framework ($G$-FBSDE )

[TABLE]

where $b,\sigma,f$ and $\phi$ satisfy (A1)-(A4).

First, we introduce the following fully nonlinear PDE on $[0,T]$:

[TABLE]

We make the following assumptions on the coefficients of the PDE (6.6).

• (A5)

The function $f(t,x,y,z)$ is continuously differentiable in $(x,y,z)$, differentiable in $t$, and twice differentiable in $(x,y,z)$, where the first-order time derivative of $f$ and the second-order derivatives of $f$ in $(x,y,z)$ are bounded on the set $[0,T]\times\mathbb{R}^{n}\times[-M_{y},M_{y}]\times[-M_{z},M_{z}]$, for any $M_{y},M_{z}>0$.

• (A6)

Both functions $b$ and $\sigma$ are differentiable in $t$ and twice differentiable in $x$, where the first-order time derivative of $(b,\sigma)$ and the second-order spatial derivatives of $(b,\sigma)$ are bounded on the set $[0,T]\times\mathbb{R}^{n}$.

• (A7)

The functions $b$ is bounded on the set $[0,T]\times\mathbb{R}^{n}$. The function $f$ is bounded on the set $[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}$.

• (A8)

There exists a constant $L>0$ such that for each $(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}$,

[TABLE]

Note that Peng [34, Appendix C] used Krylov [20, Theorem 6.4.3] to prove that there is a classical solution to PDE (6.6) when $b=0$, $f=0$, and $\sigma=1$. In a similar way, we prove that there is a classical solution to PDE (6.6) and further that $u(t,\cdot)$ is uniformly Lipschitz continuous.

Proposition 6.4**.**

Assume $b,\sigma,f$ and $\phi$ satisfy (A1)-(A8). Then the PDE (6.6) admits a classical solution $u\in C([0,T]\times\mathbb{R}^{n})$ bounded by $M:=M(M_{0},L)$, and there exists a constant $\alpha\in(0,1)$ such that for each $k\in(0,T)$,

[TABLE]

Moreover, there exists a constant $C>0$ such that for all $t\in[0,T]$,

[TABLE]

Proof.

First, we introduce the truncation function. For each integer $N$, let $\rho_{N}:\mathbb{R}\to\mathbb{R}$ be a a smooth modification of the projection on $[-N,N]$ such that $|\rho_{N}|\leq N$, $|\rho_{N}^{\prime}|\leq 1$ and $\rho_{N}(z)=z$ when $|z|\leq N-1$. We consider the following PDE.

[TABLE]

where $f^{N}$ is defined as

[TABLE]

It is easy to check that $f^{N}$ is uniformly Lipschitz in $z$.

Considering the PDE for the quantity $e^{(L\bar{\sigma}^{2}+1)(t-T)}u(t,x)$ as in Peng [34, Appendix C], in view of Krylov [20, Theorem 6.4.3], we can prove that the PDE (6.7) admits a classical solution $u^{N}\in C([0,T]\times\mathbb{R}^{n})$ dominated by a constant $M:=M(M_{0},L)$, such that for some constant $\alpha\in(0,1)$, the related restriction of $u^{N}$ belong to $C^{1+\alpha/2,2+\alpha}([0,k]\times\mathbb{R}^{n})$ with any $k\in(0,T)$.

We now rewrite PDE (6.7) into a HJB equation, and then estimate the gradient $D_{x}u^{N}$. Since

[TABLE]

the PDE (6.7) is the following HJB equation:

[TABLE]

where the Hamiltonian $H^{N}$ is defined as follows: for $(t,x,y,p,A,v)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{S}_{n}\times[\underline{\sigma},\bar{\sigma}]$,

[TABLE]

with

[TABLE]

This shows that $u^{N}$ is in fact a value function of a control problem.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be the classical Wiener space. Let $W$ be a one-dimension standard Brownian motion under Probability $\mathbb{P}$. For each $(t,x)\in[0,T]\times\mathbb{R}^{n}$, we consider the FBSDE:

[TABLE]

Let $\mathcal{F}^{W}$ be the filtration generated by $W$ and augmented by all $\mathbb{P}$-null sets. Let $\mathcal{V}$ be the set of all $\mathcal{F}^{W}_{t}$-progressively measurable processes valued in $[\underline{\sigma},\bar{\sigma}]$. In view of [31, Theorem 4.2] or [7, Theorem 4.2, Theorem 5.3] and noting that $u^{N}$ is a viscosity solution of the PDE (6.8), we have

[TABLE]

Note that for each $z\in\mathbb{R}$,

[TABLE]

We have for each $(t,x,y,v)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times[\underline{\sigma},\bar{\sigma}]$,

[TABLE]

In view of [5, Lemma 1], there exists a constant $C_{1}$ independent of $v$ and $N$ such that for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

In view of [4, Proposition 2.1], there exists a constant $C_{2}$ independent of $v$ and $N$ such that for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

By a similar stability result as in [1, Theorem 5.1], there exist a constant $C_{3}$ and some $p>2$ which are independent of $v$ and $N$ such that for each $(t,x,x^{\prime})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$,

[TABLE]

where

[TABLE]

Thus we get,

[TABLE]

By inequality (3.3) in [7], there exists a constant $C_{4}$ independent of $v$ and $N$ such that for each $(t,x,x^{\prime})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$,

[TABLE]

Thus there exists a constant $C$ independent of $v$ and $N$ such that for each $(t,x,x^{\prime})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$,

[TABLE]

which means $|u^{N}(t,x)-u^{N}(t,x^{\prime})|\leq C|x-x^{\prime}|$. So we get

[TABLE]

In view of Remark 6.1, we know $\sigma$ is bounded. Let

[TABLE]

It is easy to check that for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

Set $u:=u^{N}$ and we know $u$ is the solution of PDE (6.6). ∎

For each $t\in[0,T]$, we set $Y_{t}:=u(t,X_{t})$, $Z_{t}:=\sigma^{\mathrm{T}}(t,X_{t})D_{x}u(t,X_{t})$ and

[TABLE]

For any $0<k<T$, applying Itô’s formula to $u(t,X_{t})$ for $t\in[0,k]$, we get

[TABLE]

Similarly to [12, inequality (4.3)], in view of Proposition 6.4 and (6.9), we could obtain that there exist a constant $C>0$, for $t_{1},t_{2}\in[0,T]$ and $x_{1},x_{2}\in\mathbb{R}^{n}$,

[TABLE]

Then we can deduce that $(Y,Z,K)\in\bigcap\limits_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$ and $(Y,Z,K)$ is a solution to (6.5). So we have the following lemma.

Lemma 6.5**.**

Assume $b,\sigma,f$, and $\phi$ satisfy assumptions (A1)-(A8). Then $G$-BSDE (6.5) has a solution $(Y,Z,K)\in\bigcap\limits_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$.

As an immediate consequence of both proofs of [15, Proposition 3.5] and Proposition 3.4, we have the following stability property for quadratic $G$-BSDEs.

Proposition 6.6**.**

Let the triplet $(\xi^{i},f^{i},g^{i})$ satisfy assumption (H1) and (H3) for $i=1,2$. Let $(Y^{i},Z^{i},K^{i})\in\mathfrak{G}_{G}^{2}(0,T)$ be the solution to the following $G$-BSDE:

[TABLE]

Moreover, we suppose

[TABLE]

Then for each $p>{q\over q-1}$, there exists a constant $~{}C:=C(p,T,L,M_{0})~{}$ such that for any $t\in[0,T]$,

[TABLE]

Remark 6.7**.**

We still have [15, inequality (3.2)], which means that the constant $C_{1}$ and $p$ in Proposition 6.6 depend only on $T,L$ and $M_{0}$.

The main result of this section is stated as follows.

Theorem 6.8**.**

Assume that $b,\sigma,f$, and $\phi$ satisfy assumptions (A1)-(A4). Then $G$-BSDE (6.5) has a unique solution $(Y,Z,K)\in\bigcap\limits_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$.

Proof.

The uniqueness result directly comes from Proposition 6.6. Now we focus on the existence result. We borrow the idea of Hu et al. [15] to mollify the coefficients of the $G$-FBSDE.

Step 1. We assume $f$ satisfy the following condition.

• (A5’)

The first-order time derivative of $f$ in $t$, the spatial derivatives of $f$ up to the second-order are bounded on the set $[0,T]\times\mathbb{R}^{n}\times[-M_{y},M_{y}]\times[-M_{z},M_{z}]$, for any $M_{y},M_{z}>0$.

We replace (A5) with (A5’). Assume $b,\sigma,f$, and $\phi$ satisfy assumptions (A1)-(A4), assumption (A5’) and assumptions (A6)-(A8). Then we can obtain that $G$-BSDE (6.5) has one a solution $(Y,Z,K)\in\bigcap_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$ with exactly the same method of step 1 in Hu et al. [15, Section 5].

Step 2. We assume $b,\sigma,f$, and $\phi$ satisfy assumptions (A1)-(A4) and assumptions (A6)-(A8). For each $(t,x,y,z)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}$, we define

[TABLE]

where $\rho_{k}$ is a positive smooth function such that its support is contained in a ${1\over k}$-ball in $\mathbb{R}^{n+1}$ and $\int_{\mathbb{R}^{n+1}}\rho_{k}=1$. In addition, we define the extension of $f$ on $\mathbb{R}$, i.e., $f(t,\cdot,\cdot,\cdot)=f(t^{+}\land T,\cdot,\cdot,\cdot)$. We can check that $f^{k}$ satisfies (A5’). Therefore, in view of the result in step 1, we obtain that the $G$-BSDE (6.5) with the coefficients $(\phi,f^{k})$ admits a solution $(Y^{k},Z^{k},K^{k})\in\bigcap_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$. Noting that for each $k_{1}\geq k_{2}$ and $t\in[0,T]$,

[TABLE]

we could deduce that the sequence $\{Y^{k}\}_{k=1}^{\infty}$ is a Cauchy sequence in $S_{G}^{p}(0,T)$ for any $p\geq 2$ by Proposition 6.6 and Remark 6.7. Thus we could conclude that $G$-BSDE (6.5) has a solution $(Y,Z,K)\in\bigcap_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$ in a similar way as in step 1 in Hu et al. [15, Section 5].

Step 3. We assume $b,\sigma,f$, and $\phi$ satisfy assumptions (A1)-(A4) and assumptions (A7)-(A8). For each $(t,x)\in[0,T]\times\mathbb{R}^{n}$, we define

[TABLE]

where $\rho_{k}$ is a positive smooth function such that its support is contained in a ${1\over k}$-ball in $\mathbb{R}^{n+1}$ and $\int_{\mathbb{R}^{n+1}}\rho_{k}=1$. We can check that $b^{k}$ and $\sigma^{k}$ satisfies (A6). Moreover, (A4) still hold here for $\sigma^{k}$ when $k$ is large enough. Actually, if we assume $\sigma$ satisfies (A4) with constants $\varepsilon$ and $K$, we can check that for $k$ large enough and for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

Let $X^{k}$ be the solution of the following $G$-SDE:

[TABLE]

From step 2, we can let $(Y^{k},Z^{k},K^{k})$ be the solution to the following $G$-BSDE:

[TABLE]

For each $k_{1},k_{2}\in\mathbb{N}$ and $t\in[0,T]$, set

[TABLE]

It is easy to check that for $l=b,\sigma$ and each $t\in[0,T]$,

[TABLE]

In view of Proposition 6.3, we obtain for each $p\geq 2$ and $t\in[0,T]$,

[TABLE]

On the other hand, in view of Proposition 6.6 and [15, (3.2)], we obtain that for some $p>1$ and each $t\in[0,T]$,

[TABLE]

Therefore,

[TABLE]

In view of (6.10), we obtain that for each $\delta>0$

[TABLE]

Then in view of Remark 2.5, we know $\{Y^{k}\}_{k=1}^{\infty}$ is a Cauchy sequence in $S_{G}^{p}(0,T)$ for any $p\geq 2$. Thus we could conclude that $G$-BSDE (6.5) has a solution $(Y,Z,K)\in\bigcap_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$.

Step 4. We now consider the situation that $\phi(\cdot)$ and $f(t,\cdot,y,z)$ can be locally Lipschitz. We assume $b,\sigma,f$, and $\phi$ satisfy assumptions (A1)-(A4) and assumption (A7). For each $(t,x,y,z)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}$, we define

[TABLE]

and

[TABLE]

where $\rho_{k}$ is a positive smooth function such that its support is contained in a ${1\over k}$-ball in $\mathbb{R}^{n}$ and $\int_{\mathbb{R}^{n}}\rho_{k}=1$. Noting that

[TABLE]

we obtain that $\phi^{k}$ is Lipschitz in $x$. Similarly, it is easy to check $f^{k}(t,\cdot,y,z)$ is uniformly Lipschitz. Therefore, in view of the result in step 3, we obtain that the $G$-BSDE (6.5) with the coefficients $(\phi^{k},f^{k})$ admits a solution $(Y^{k},Z^{k},K^{k})\in\bigcap_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$. It is easy to check that

[TABLE]

Similarly, for each $t\in[0,T]$,

[TABLE]

In view of Proposition 6.2, we obtain for each $p\geq 2$,

[TABLE]

and

[TABLE]

Then again in view of Remark 2.5, Proposition 6.6 and Remark 6.7, we know $\{Y^{k}\}_{k=1}^{\infty}$ is a Cauchy sequence in $S_{G}^{p}(0,T)$ for any $p\geq 2$. Thus we could conclude that $G$-BSDE (6.5) has a solution $(Y,Z,K)\in\bigcap_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$.

Step 5. Finally, we remove the boundedness condition on $b$ and $f$. We assume $b,\sigma,f$, and $\phi$ satisfy assumptions (A1)-(A4). Set $b^{k}:=[b\lor(-k)]\land k$ and $f^{k}:=[f\lor(-k)]\land f$. It is easy to check that $b^{k},\sigma,f^{k}$, and $\phi$ satisfy assumptions (A1)-(A4) and assumption (A7). Let $(X^{k},Y^{k},Z^{k},K^{k})$ be the solution of the following $G$-FBSDE:

[TABLE]

For each $k_{1},k_{2}\in\mathbb{N}$ and $t\in[0,T]$, set

[TABLE]

and

[TABLE]

Assume $k_{1}<k_{2}$, then for each $\delta>0$ and $t\in[0,T]$, we have

[TABLE]

and

[TABLE]

In view of Proposition 6.3, we obtain for each $p\geq 2$ and $t\in[0,T]$,

[TABLE]

where $\lambda_{T}:=\int_{0}^{T}|b(s,0)|^{p(1+\delta)}ds$. In view of Proposition 6.2, we obtain for each $p\geq 2$,

[TABLE]

On the other hand, for each $p\geq 2$ and $0<\delta\leq 1$,

[TABLE]

Note that [15, (3.2)] still hold here, which means there exists a constant $C=C(M_{0},L,G,T)$, such that

[TABLE]

In view of Lemma 2.12 and (A3), we obtain

[TABLE]

In view of Remark 2.5, inequalities (6.12), (6.13) and (6.14), Proposition 6.6 and Remark 6.7, we see that $\{Y^{k}\}_{k=1}^{\infty}$ is a Cauchy sequence in $S_{G}^{p}(0,T)$ for any $p\geq 2$. Thus we could conclude that $G$-BSDE (6.5) has a solution $(Y,Z,K)\in\bigcap_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$. ∎

Moreover, we have the following result with a similar argument before.

Theorem 6.9**.**

Assume that $\xi\in\bigcap\limits_{p\geq 2}L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$ and $b,h,\sigma,g,f$, and $\phi$ satisfy assumptions (A1)-(A4) . Then $G$-BSDE (6.3) has a unique solution $(Y,Z,K)\in\bigcap\limits_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$.

Remark 6.10**.**

Once we have a solution$(Y,Z,K)\in\bigcap\limits_{p\geq 2}\mathfrak{G}_{G}^{p}(0,T)$ to $G$-BSDE (6.3). In view of [15, (3.2),(3.3)], there are two constants $C1=C1(M_{0},L,G,T)$ and $C2=C2(p,M_{0},L,G,T)$ such that for all $p\geq 2$,

[TABLE]

Now we can give the relationship between quadratic $G$-FBSDEs and parabolic PDEs. For $(t,\xi)\in[0,T]\times\bigcap\limits_{p\geq 2}L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$, denote by $(X^{t,\xi},Y^{t,\xi},Z^{t,\xi},K^{t,\xi})$ the solution to the $G$-FBSDE (6.2)-(6.3).

Proposition 6.11**.**

For each $t\in[0,T]$ and $\xi,\xi^{\prime}\in\bigcap\limits_{p\geq 2}L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$, we have

[TABLE]

where the constant $C$ depends on $L,G$ and $T$.

Proof.

For simplicity, we assume $g=0$ and $h=0$. In view of Proposition 6.6, Remark 6.10, and Proposition 6.2, we obtain

[TABLE]

On the other hand, we get $|Y_{t}^{t,\xi}|\leq C$, q.s., directly from Remark 6.10. ∎

Now for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$, we define $u(t,x):=Y_{t}^{t,x}$. Identically as in [13, Reamrk 4.3], we deduce that $u$ is a deterministic function. In view of Proposition 6.11, we immediately have the following estimates:

[TABLE]

where the constant $C$ depends on $L,G$ and $T$. Moreover, with the same proof [13, Theorem 4.4], we have the following Proposition.

Proposition 6.12**.**

For each $\xi\in\bigcap\limits_{p\geq 2}L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$, we have $u(t,\xi)=Y_{t}^{t,\xi}$.

Now we give the main result of this section.

Theorem 6.13**.**

Let $u(t,x):=Y_{t}^{t,x}$ for $(t,x)\in[0,T]\times\mathbb{R}^{n}$. Then, the function $u$ is a viscosity solution to the PDE (6.1).

Proof.

Without loss of genearlity, we still assume that $h=0$ and $g=0$. First, we show $u$ is a continuous function. Fix some $(t,x)\in[0,T]\times\mathbb{R}^{n}$. In view of $Y_{t+\delta}^{t,x}=Y_{t+\delta}^{t+\delta,X_{t+\delta}^{t,x}}$ and Proposition 6.12, we obtain $Y_{t+\delta}^{t,x}=u(t+\delta,X_{t+\delta}^{t,x})$ for $\delta\in[0,T-t]$. Thus we obtain

[TABLE]

The generator can be written as in Proposition 3.3 the following form: for each $s\in[0,T]$,

[TABLE]

with

[TABLE]

for a Lipschitz continuous function $l$ such that $\textbf{1}_{[-\varepsilon,\varepsilon]}(x)\leq l(x)\leq\textbf{1}_{[-2\varepsilon,2\varepsilon]}(x)$ at each $x\in(-\infty,+\infty)$. Moreover,

[TABLE]

In view of [15, Lemma 3.6] and Remark 6.10, we know that $b^{\varepsilon}\in BMO_{G}$. Set $\tilde{B}_{t}:=B_{t}-\int_{0}^{t}b_{s}^{\varepsilon}d\left\langle B\right\rangle_{s}$ for $t\in[0,T]$. Thus we can define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ by $\mathscr{E}(b^{\varepsilon})$, such that $\tilde{B}$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}[\cdot]$. Thus we have

[TABLE]

where $f_{s}:=f(s,X_{s}^{t,x},0,0)$. Taking $G$-expectation $\tilde{\mathbb{E}}[\cdot]$, we get

[TABLE]

In view of Proposition 6.11, we obtain

[TABLE]

Let $\varepsilon<1$. In view of Remark 6.10 and (A3), there exists a constant $C$ depending on $M_{0},L,G$ and $T$, such that

[TABLE]

Thus we know

[TABLE]

Note that $\left\|b_{s}^{\varepsilon}\right\|_{BMO_{G}}\leq\left\|L(1+|Z^{t,x}|)\right\|_{BMO_{G}}$. Then according to Lemma 2.14 and Remark 6.10, there exists $p>1$ depending on $M_{0},L,G$ and $T$, such that, for each $(s,X)\in[0,T]\times L_{G}^{p}(\Omega_{T})$

[TABLE]

where $C_{p}$ depending only on $p$. In view of Proposition 6.2, we get

[TABLE]

and

[TABLE]

Then from (6.15), we have for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

where $C$ depends on $M_{0},L,G$ and $T$. On the other hand, we get from Proposition 6.11 that for each $(t,x,x^{\prime})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$,

[TABLE]

It follows that $u$ is continuous.

For any fixed $(t_{0},x_{0})\in(0,T)\times\mathbb{R}^{n}$, let $\psi\in C^{1,2}([0,T]\times\mathbb{R}^{n})$ such that for each $(t,x)\in\left([0,T]\times\mathbb{R}^{n}\right)\backslash\{(t_{0},x_{0})\}$

[TABLE]

Without loss of generality, we may assume that there exists some $m_{1}>0$ such that for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

We want to prove that

[TABLE]

Let us assume the inequality before does not hold. Let $O_{\delta}(t_{0},x_{0})$ be a open ball centered at $(t_{0},x_{0})$, with radius $\delta$. By continuity, there exists some $\delta\in(0,T-t_{0})$ such that for each $(t,x)\in O_{\delta}(t_{0},x_{0})$,

[TABLE]

Setting $\tilde{Y}_{t}:=\psi(t,X_{t}^{t_{0},x_{0}})$ and $\tilde{Z}_{t}:=\sigma^{\mathrm{T}}(t,X_{t}^{t_{0},x_{0}})D_{x}\psi(t,X_{t}^{t_{0},x_{0}})$, it is easy to check that for each $t\in[t_{0},T]$,

[TABLE]

For each $t\in[t_{0},T]$, set $\tilde{K}_{t}:=\int_{t_{0}}^{t}F_{s}d\left\langle B\right\rangle_{s}-\int_{t_{0}}^{t}G(2F_{s})ds$, where

[TABLE]

We can check that $\tilde{K}$ is a decreasing $G$-martingale. Noting that $\sigma$ is bounded here, by Proposition 6.2 and inequality (6.17), we deduce that $\tilde{K}_{T}\in\bigcap\limits_{p\geq 1}L_{G}^{p}(\Omega_{T})$. Now we have for each $t\in[t_{0},T]$,

[TABLE]

Now we set $\delta Y:=\tilde{Y}-Y^{t_{0},x_{0}}$, $\delta Z:=\tilde{Z}-Z^{t_{0},x_{0}}$ and for each $s\in[t_{0},T]$,

[TABLE]

Then for each $t\in[t_{0},T]$,

[TABLE]

As what we do in Proposition 3.4, we have for each $s\in[t_{0},T]$,

[TABLE]

where

[TABLE]

In view of [15, Lemma 3.6], Remark 6.10, and inequality (6.17), we know that $b^{\varepsilon}\in BMO_{G}$. Set $\tilde{B}_{t}:=B_{t}-\int_{0}^{t}b_{s}^{\varepsilon}d\left\langle B\right\rangle_{s}$ for $t\in[0,T]$. Thus we can define a new $G$-expectation $\tilde{\mathbb{E}}[\cdot]$ by $\mathscr{E}(b^{\varepsilon})$, such that $\tilde{B}$ is a $G$-Brownian motion under $\tilde{\mathbb{E}}[\cdot]$. Thus the equality (6.18) can be written as

[TABLE]

Applying Itô’s formula to $e^{\int_{0}^{t}a_{s}^{\varepsilon}d\left\langle B\right\rangle_{s}}\delta Y_{t}$,  we have

[TABLE]

Let $\mathcal{P}$ be the weakly compact set that represents $\hat{\mathbb{E}}$. For each $\mathbb{P}\in\mathcal{P}$, let $\tau^{\mathbb{P}}$ be the following stopping time under $\mathbb{P}$:

[TABLE]

By the strict minimum property (6.16), we notice that

[TABLE]

It is easy to check that $\tau^{\mathbb{P}}<T$, $\mathbb{P}\text{\,-a.s.}$ and $(\psi-u)(\tau^{\mathbb{P}},X_{\tau^{\mathbb{P}}}^{t_{0},x_{0}})\geq\eta$, $\mathbb{P}\text{\,-a.s.}$. From equality (6.19), we have for each $t\in[t_{0},T]$,

[TABLE]

Note that for each $(s,\omega)\in[t_{0},T]\times\Omega_{T}$ satisfying $t_{0}\leq s\leq\tau^{\mathbb{P}}(\omega)$, $\tilde{F}_{s}\leq 0$. Thus we have

[TABLE]

Since $\tilde{B}$ is a martingale under the new probability $\mathbb{Q}$ with $d\mathbb{Q}:=\mathscr{E}(b^{\varepsilon})_{T}d\mathbb{P}$, we have in particular

[TABLE]

While $\delta Y_{t_{0}}=(\psi-u)(t_{0},x_{0})$ and $|m_{s}^{\varepsilon}|\leq\rho(\varepsilon)$ for a nonnegative continuous function $\rho$ defined on $\mathbb{R}^{+}$ with $\rho(0)=0$, we have

[TABLE]

for each $\mathbb{P}\in\mathcal{P}$. Consequently, we have

[TABLE]

In view of Lemma 2.20 and Remark 6.10, the process $K^{t_{0},x_{0}}$ is a $G$-martingale under $\tilde{\mathbb{E}}[\cdot]$, and

[TABLE]

Letting $\varepsilon\to 0$ in the last inequality, we have

[TABLE]

which is a contradiction. Hence, $u$ is a viscosity subsolution.

In a similar way, $u$ can be shown to be a viscosity supersolution. ∎

Remark 6.14**.**

When the functions $f$ and $g$ do not depend on $y$, one can get the uniqueness of viscosity solution to PDE by the uniqueness result in Da Lio and Ley [9] concerning Bellman-Isaacs equation.

6.2 Relation between reflected quadratic $G$-BSDEs and obstacle problems for nonlinear parabolic PDEs

With the preceding nonlinear Feynman-Kac formula, we can give the relationship between solutions of the obstacle problem for nonlinear parabolic PDEs and the related reflected quadratic $G$-BSDEs. For each $(t,\xi)\in[0,T]\times\bigcap\limits_{p\geq 2}L_{G}^{p}(\Omega_{t};\mathbb{R}^{n})$, we consider the following $G$-SDE:

[TABLE]

and the following type of reflected $G$-BSDE:

[TABLE]

where $b,h,\sigma,l:[0,T]\times\mathbb{R}^{n}\to\mathbb{R}$, $\phi:\mathbb{R}^{n}\to\mathbb{R}$, $f,g:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are deterministic functions and satisfy (A1)-(A4). Moreover, we have the following assumption on $l$:

• (A9)

The function $l(t,\cdot)$ is uniformly Lipschitz and $l(T,x)\leq\phi(x)$ for any $x\in\mathbb{R}^{n}$. Furthermore, there exists a constant $N_{0}$ such that $l(t,x)\leq N_{0},$ for any $t\in[0,T]$.

• (A10)

The function $l(\cdot,x)$ is uniformly continuous, i.e. there is a non-decreasing continuous function $w:[0,+\infty)\to[0,+\infty)$ such that $w(0)=0$ and

[TABLE]

Remark 6.15**.**

In the Markovian case, Assumptions (H2) and (H5) may not hold directly. However, in view of Remark 2.11, one can still get the results under Assumptions (H1), (H3) and (H4) as long as the penalized quadratic $G$-BSDE has a solution. The reflected $G$-BSDE (6.22) has one solution in the sense of Definition 2.9 and all results in Sections 3-5 still hold here under Assumptions (A1)-(A4) and (A9)-(A10).

Consider the following obstacle problem for a parabolic PDE:

[TABLE]

where

[TABLE]

for each $(A,p,y,x,t)\in\mathbb{S}_{n}\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\times[0,T]$.

We need to recall the equivalent definition of the viscosity solution of the obstacle problem (6.23) as in [22] or [34].

Definition 6.1**.**

Let $u\in C([0,T]\times\mathbb{R}^{n})$ and $(t,x)\in[0,T]\times\mathbb{R}^{n}$. We denote by $\mathcal{P}^{2,+}u(t,x)$ the set of triples $(p,q,A)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{S}_{n}$ satisfying

[TABLE]

Similarly, we define $\mathcal{P}^{2,-}u(t,x):=-\mathcal{P}^{2,+}(-u)(t,x)$.

Definition 6.2**.**

It can be said that $u\in C([0,T]\times\mathbb{R}^{n})$ is a viscosity subsolution of (6.23) if $u(T,x)\leq\phi(x),x\in\mathbb{R}^{n}$, and for each $(t,x)\in[0,T)\times\mathbb{R}^{n}$ and $(p,q,A)\in\mathcal{P}^{2,+}u(t,x)$,

[TABLE]

It can be said that $u\in C([0,T]\times\mathbb{R}^{n})$ is a viscosity supersolution of (6.23) if $u(T,x)\geq\phi(x),x\in\mathbb{R}^{n}$, and for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$ and $(p,q,X)\in\mathcal{P}^{2,-}u(t,x)$,

[TABLE]

$u\in C([0,T]\times\mathbb{R}^{n})$* is said to be a viscosity solution of (6.23) if it is both a viscosity subsolution and supersolution.*

We now define $u(t,x):=Y_{t}^{t,x}$. Similarly as before, we can note that $u$ is a deterministic function. We now should prove that $u\in C([0,T]\times\mathbb{R}^{n})$.

Lemma 6.16**.**

Let assumptions (A1)-(A4) and (A9)-(A10) hold. For each $t\in[0,T]$, $x_{1},x_{2}\in\mathbb{R}^{n}$, we have

[TABLE]

Proof.

Without loss of generality, we assume $h=0$ and $g=0$. In view of Proposition 3.2 and Proposition 3.3, we deduce that there exist a constant $C_{1}:=C_{1}(T,L,G,M_{0},N_{0})$ such that

[TABLE]

and a constant $C_{2}:=C_{2}(T,L,G,M_{0},N_{0},\alpha)$, for any $\alpha\geq 1$, such that

[TABLE]

In view of Proposition 3.4 and its proof and noting that $u$ is deterministic, we obtain that there exist a constant $~{}C:=C(T,L,G,M_{0},N_{0})~{}$ and $p\geq 2$ such that for each $t\in[0,T]$,

[TABLE]

∎

Lemma 6.17**.**

Let assumptions (A1)-(A4) and (A9)-(A10) hold. The function $u(t,x):=Y_{t}^{t,x}$ is continuous in $t$.

Proof.

For simplicity, we assume $h=0$ and $g=0$. We define $X_{s}^{t,x}:=x$, $Y_{s}^{t,x}:=Y_{t}^{t,x}$, $Z_{s}^{t,x}:=0$ and $A_{s}^{t,x}:=0$ for $s\in[0,t]$. It is easy to check that $(Y^{t,x},Z^{t,x},A^{t,x})$ is a solution to the following $G$-BSDE on $[0,T]$:

[TABLE]

where

[TABLE]

Fix $x\in\mathbb{R}^{n}$. As before, in view of Propositions 3.2-3.4, we have for $0\leq t_{1}\leq t_{2}\leq T$ and some $p\geq 2$,

[TABLE]

For each $\alpha\geq 2$, we have

[TABLE]

On the other hand, in view of Proposition 3.3, for each $\alpha\geq 2$,

[TABLE]

Then from (6.25), we know $u$ is continuous in $t$. ∎

Now we consider the penalized $G$-BSDEs:

[TABLE]

We define $u_{n}(t,x):=Y_{t}^{t,x,n}$, $(t,x)\in[0,T]\times\mathbb{R}^{n}$. In view of Theorem 6.13, $u_{n}$ is the viscosity solution to the following PDE:

[TABLE]

where

[TABLE]

for each $(A,p,y,x,t)\in\mathbb{S}_{n}\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\times[0,T]$.

Theorem 6.18**.**

Let assumptions (A1)-(A4) and (A9)-(A10) hold. The function $u(t,x):=Y_{t}^{t,x}$ is a viscosity solution of the obstacle problem (6.23).

Proof.

We follow the procedure of [22, Theorem 6.7], and only sketch the main ideas.

From previous results, we have for each $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

[TABLE]

Moreover, functions $u$ and $u_{n}$ are continuous. Then in view of Dini’s Theorem, $u_{n}$ uniformly converges to $u$ on any compact subset.

We now show $u$ is a viscosity subsolution to (6.23). For each fixed $(t,x)\in[0,T]\times\mathbb{R}^{n}$, let $(p,q,A)\in\mathcal{P}^{2,+}u(t,x)$. We may assume $u(t,x)>l(t,x)$. Similar as in the proof of [22, Theorem 6.7], we deduce that there exist sequences

[TABLE]

where $(p_{j},q_{j},X_{j})\in\mathcal{P}^{2,+}u_{n_{j}}(t_{j},x_{j})$. Since $u_{n}$ is the viscosity solution to (6.26), it follows that for any $j$,

[TABLE]

Noting that $u(t,x)>l(t,x)$, by the uniform convergence of $u_{n}$, we deduce that $u_{j}(t_{j},x_{j})>l(t_{j},x_{j})$ for sufficiently large integer $j$. Thus letting $j\to\infty$, we have

[TABLE]

which means $u$ is a viscosity subsolution to (6.23).

In a similar way, $u$ is proved to be a viscosity supersolution to (6.23). ∎