Doubly Reflected Backward SDEs Driven by $G$-Brownian Motions and Fully Nonlinear PDEs with Double Obstacles

In 1997, El Karoui et al., 1997a first introduced the reflected backward stochastic differential equation (RBSDE), where the first component of the solution is constrained to remain above a specified continuous process, known as the obstacle. To enforce this constraint, an additional non-decreasing process is introduced to push the solution upwards, while adhering to the Skorohod condition in a minimal manner. This problem is intimately linked to various fields including optimal stopping problems (see, e.g., Cheng and Riedel, (2013)), pricing for American options (see, e.g., El Karoui et al., 1997b ), and the obstacle problem for partial differential equations (PDEs) (see, e.g., Bally et al., (2002)).

Subsequently, Cvitanic and Karatzas, (1996) extended the above results to encompass scenarios involving two obstacles. In this setting, the solution $Y$ is constrained to remain between two specified continuous processes, known as the lower and upper obstacles. Consequently, two non-decreasing processes are introduced in the doubly RBSDE, with the aim of pushing the solution upwards and pulling it downwards, respectively, while ensuring adherence to the Skorohod conditions. Additionally, they demonstrated that the solution $Y$ coincides with the value function of a Dynkin game. Given its significance in both theoretical analysis and practical applications, numerous studies have been conducted. Interested readers may refer to Crépey and Matoussi, (2008); Dumitrescu et al., (2016); Grigorova et al., (2018); Hamadene and Hassani, (2005); Hamadène et al., (1997); Peng and Xu, (2005) and the references therein for further exploration.

The classical theory is limited to solving financial problems under drift uncertainty and the associated semi-linear PDEs. Motivated by the need to address financial problems under volatility uncertainty and the associated fully nonlinear PDEs, Peng, (2007, 2008, 2019) introduced a novel nonlinear expectation theory known as the $G$-expectation theory. This theory involves the construction of a nonlinear Brownian motion, termed $G$-Brownian motion, and the introduction of corresponding $G$-Itô’s calculus. Building upon the $G$-expectation theory, Hu et al., 2014a investigated BSDEs driven by $G$-Brownian motions ($G$-BSDEs). In comparison with classical results, $G$-BSDEs include an additional non-increasing $G$-martingale $K$ in the equation due to nonlinearity. In Hu et al., 2014a , the authors established the well-posedness of $G$-BSDEs, while the comparison theorem, Feynman-Kac formula, and Girsanov transformation can be found in their companion paper Hu et al., 2014b .

In recent years, Li et al., 2018b introduced reflected $G$-BSDEs with a lower obstacle. Given the presence of a non-increasing $G$-martingale in $G$-BSDEs, the definition deviates from the classical case. Specifically, they amalgamated the non-decreasing process, intended to elevate the solution, with the non-increasing $G$-martingale into a general non-decreasing process that satisfies a martingale condition. Existence was established through approximation via penalization, while uniqueness was derived from a prior estimates. For further insights, readers may refer to Li and Peng, (2020). The study of reflected $G$-BSDEs with two obstacles is undertaken by Li and Song, (2021). They introduced a so-called approximate Skorohod condition and established the well-posedness of doubly reflected $G$-BSDEs when the upper obstacle is a generalized $G$-Itô process.

Three natural questions arise concerning the doubly reflected $G$-BSDE of the following form:

$\displaystyle\begin{cases}Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})d\langle B\rangle_{s}-\int_{t}^{T}Z_{s}dB_{s}+(A_{T}-A_{t}),\vspace{0.2cm}\\ L_{t}\leq Y_{t}\leq U_{t},\quad 0\leq t\leq T,\vspace{0.2cm}\\ (Y,A)\textrm{ satisfies the approximate Skorohod condition with order $\alpha$ }(\textmd{ASC}_{\alpha}),\end{cases}$

whose detailed description and the $\textmd{ASC}_{\alpha}$ are provided in Subsection 3.1. Firstly, what types of fully nonlinear PDEs can be represented by reflected $G$-BSDEs with two obstacles? Secondly, in addition to finding that connection, can we enhance the theory of doubly reflected $G$-BSDEs further at the same time? Lastly, in order to achieve both of these goals, what new mathematical strategies suffice and can be developed? The objective of this paper is to address these three questions.

We discovered that to establish the connection between the solution of the doubly reflected $G$-BSDE and the double obstacle fully nonlinear PDEs, it would be desirable if we can construct a monotone sequence converging to that solution. However, achieving this is challenging. Say, we consider the following penalized reflected $G$-BSDEs with a lower obstacle parameterized by $n\in\mathbb{N}$, which has been a workhorse in existing literature:

$\displaystyle\begin{cases}&\bar{Y}^{n}_{t}=\xi+\int_{t}^{T}f(s,\bar{Y}^{n}_{s},\bar{Z}^{n}_{s})ds+\int_{t}^{T}g(s,\bar{Y}^{n}_{s},\bar{Z}^{n}_{s})d\langle B\rangle_{s}-n\int_{t}^{T}(\bar{Y}^{n}_{s}-U_{s})^{+}ds\\ &\hskip 28.45274pt-\int_{t}^{T}\bar{Z}^{n}_{s}dB_{s}+(\bar{A}^{n}_{T}-\bar{A}^{n}_{t}),\vspace{0.2cm}\\ &\bar{Y}^{n}_{t}\geq L_{t},\quad 0\leq t\leq T,\vspace{0.2cm}\\ &\big{\{}-\int_{0}^{t}(\bar{Y}^{n}_{s}-L_{s})d\bar{A}^{n}_{s}\big{\}}_{t\in[0,T]}\textrm{ is a non-increasing $G$-martingale}.\end{cases}$

By the comparison theorem for reflected $G$-BSDEs, $\bar{Y}^{n}$ is non-increasing in $n$. The purpose of the penalization term is to drive the solution $\bar{Y}^{n}$ downwards so that the limiting process $Y$ (if it exists) remains below $U$. Thus, the remaining challenge is to demonstrate that the sequence $\bar{Y}^{n}$ converges to some process $Y$, which is the first component of solution to the desired doubly reflected $G$-BSDE. However, unlike in Li and Peng, (2020) and Li and Song, (2021), the main problem is that $\bar{A}^{n}$ is no longer a $G$-martingale. Consequently, we are unable to show that $(\bar{Y}^{n}-U)^{+}$ converges to $0$ with the explicit rate $\frac{1}{n}$.

Then we found that for $n,m\in\mathbb{N}$, we could consider the following family of $G$-BSDEs instead:

$$\begin{split}Y^{n,m}_{t}=&\xi+\int_{t}^{T}f(s,Y^{n,m}_{s},Z^{n,m}_{s})ds+\int_{t}^{T}g(s,Y^{n,m}_{s},Z^{n,m}_{s})d\langle B\rangle_{s}-\int_{t}^{T}Z_{s}^{n,m}dB_{s}\\ &+\int_{t}^{T}m(Y_{s}^{n,m}-L_{s})^{-}ds-\int_{t}^{T}n(Y_{s}^{n,m}-U_{s})^{+}ds-(K_{T}^{n,m}-K_{t}^{n,m}),\end{split}$$

and set $A^{n,m,+}_{t}=\int_{0}^{t}m(Y_{s}^{n,m}-L_{s})^{-}ds$ and $A^{n,m,-}_{t}=\int_{0}^{t}n(Y_{s}^{n,m}-U_{s})^{+}ds$. By letting $m$ tend to infinity, we can demonstrate that $(Y^{n,m},Z^{n,m},A^{n,m,+}-K^{n,m})$ converges to $(\bar{Y}^{n},\bar{Z}^{n},\bar{A}^{n})$. Then, as $n$ tends to infinity, $(\bar{Y}^{n},\bar{Z}^{n},\bar{A}^{n})$ converges to $(Y,Z,A)$, the solution of the doubly reflected $G$-BSDE. Specifically, the penalized $G$-BSDEs with parameters $n$ and $m$ enable us to determine the convergence rate of $(Y^{n,m}-U)^{+}$, with an explicit rate of $\frac{1}{n}$, uniformly in $m$ (refer to Lemma 3.4). Consequently, the convergence rate remains consistent for the limit process $(\bar{Y}^{n}-U)^{+}$. However, achieving uniform boundedness of $Y^{n,m}$ requires a different approach than the one used in Li and Song, (2021). Therefore, we abandoned the application of $G$-Itô’s formula and instead resorted to employing comparison results. Although this approach is not novel and dates back decades to Peng and Xu, (2005), its application in the context of reflected $G$-BSDEs is innovative. Our first main result, establishing the well-posedness of doubly reflected $G$-BSDEs and their approximating sequences, is presented in Theorem 3.2.

Indeed, the approach that can be used to answer those three natural questions is considerably more intricate than the methods employed in Li and Song, (2021). However, the existence of the non-increasing $G$-martingale introduces a disparity between reflected $G$-BSDEs with upper and lower obstacles. The inclusion of both the non-increasing $G$-martingale and the non-increasing process for pulling down the solution results in a finite variation process, complicating the derivation of a priori estimates. Consequently, we made every effort to recycle results from Li and Song, (2021) and extend certain preliminary results. For example, Proposition 3.7 extends Proposition 3.1 in Li and Song, (2021) in two aspects. Both propositions aim to assess the difference between the first components of solutions to doubly reflected $G$-BSDEs. Notably, in Proposition 3.7, the obstacles of the doubly reflected $G$-BSDEs are permitted to vary, while in Proposition 3.1 in Li and Song, (2021), equality is assumed for the obstacles, i.e., $L^{1}\equiv L^{2}$ and $U^{1}\equiv U^{2}$, making it a special case of our condition. Moreover, our general conditions are even more relaxed. The advantage of this construction is that $\bar{Y}^{n}$ is non-increasing in $n$ and the solution $\bar{Y}^{n}$ provides a probabilistic representation for the PDE with an obstacle in a Markovian setting, which enable us to establish the connection between doubly reflected $G$-BSDEs and PDEs with two obstacles in the last section. Generally speaking, in a Markovian framework, the solution $Y$ of the doubly reflected $G$-BSDE is the unique viscosity solution of the associated double obstacle PDE, which extends the result in Hamadene and Hassani, (2005) to the fully nonlinear case. Our second main result, the function $u$ defined in (4.3) being the solution to the fully nonlinear obstacle problem (4.4), is presented in Theorem 4.6.

The remaining sections of the paper are structured as follows. In Section 2, we provide an overview of fundamental concepts and findings pertaining to $G$-expectation, $G$-BSDEs, and reflected $G$-BSDEs. In Section 3, we delve into the investigation of doubly reflected $G$-BSDEs and establish their well-posedness. In Section 4, we establish the relationship between fully nonlinear PDEs with double obstacles and doubly reflected $G$-BSDEs. Throughout the paper, the letter $C$, with or without subscripts, will represent a positive constant whose value may vary from line to line.

Let $\Omega_{T}=C_{0}([0,T];\mathbb{R})$, the space of real-valued continuous functions starting from the origin, i.e., $\omega_{0}=0$ for any $\omega\in\Omega_{T}$, be endowed with the supremum norm. Let $\mathcal{B}(\Omega_{T})$ be the Borel set and $B$ be the canonical process. Set

$$L_{ip}(\Omega_{T}):=\Big{\{}\varphi(B_{t_{1}},...,B_{t_{n}}):\ n\in\mathbb{N},\ t_{1},\cdots,t_{n}\in[0,T],\ \varphi\in C_{b,Lip}(\mathbb{R}^{n})\Big{\}},$$

where $C_{b,Lip}(\mathbb{R}^{n})$ denotes the set of all bounded Lipschitz functions on $\mathbb{R}^{n}$. We fix a sublinear and monotone function $G:\mathbb{R}\rightarrow\mathbb{R}$ defined by

$\displaystyle G(a):=\frac{1}{2}(\overline{\sigma}^{2}a^{+}-\underline{\sigma}^{2}a^{-}),$

(2.1)

where $0<\underline{\sigma}^{2}<\overline{\sigma}^{2}$. The associated $G$-expectation on $(\Omega_{T},L_{ip}(\Omega_{T}))$ can be constructed in the following way. Given that $\xi\in L_{ip}(\Omega_{T})$ can be represented as $\xi=\varphi(B_{{t_{1}}},B_{t_{2}},\cdots,B_{t_{n}})$, set for $t\in[t_{k-1},t_{k})$ with $k=1,\cdots,n$,

$$\widehat{\mathbb{E}}_{t}[\varphi(B_{{t_{1}}},B_{t_{2}},\cdots,B_{t_{n}})]:=u_{k}(t,B_{t};B_{t_{1}},\cdots,B_{t_{k-1}}),$$

where $u_{k}(t,x;x_{1},\cdots,x_{k-1})$ is a function of $(t,x)$ parameterized by $(x_{1},\cdots,x_{k-1})$ such that it solves the following fully nonlinear PDE defined on $[t_{k-1},t_{k})\times\mathbb{R}$:

$$\partial_{t}u_{k}+G(\partial_{x}^{2}u_{k})=0,$$

whose terminal conditions are given by

$$\begin{cases}u_{k}(t_{k},x;x_{1},\cdots,x_{k-1})=u_{k+1}(t_{k},x;x_{1},\cdots,x_{k-1},x),\quad k<n,\\ u_{n}(t_{n},x;x_{1},\cdots,x_{n-1})=\varphi(x_{1},\cdots,x_{n-1},x).\end{cases}$$

Hence, the $G$-expectation of $\xi$ is $\widehat{\mathbb{E}}_{0}[\xi]$, denoted as $\widehat{\mathbb{E}}[\xi]$ for simplicity. The triple $(\Omega_{T},L_{ip}(\Omega_{T}),\widehat{\mathbb{E}})$ is called the $G$-expectation space and the process $B$ is the $G$-Brownian motion.

For $\xi\in L_{ip}(\Omega_{T})$ and $p\geq 1$, we define

$$\|\xi\|_{L_{G}^{p}}:=(\widehat{\mathbb{E}}|\xi|^{p}])^{1/p}.$$

The completion of $L_{ip}(\Omega_{T})$ under this norm is denote by $L_{G}^{p}(\Omega)$. For all $t\in[0,T]$, $\widehat{\mathbb{E}}_{t}[\cdot]$ is a continuous mapping on $L_{ip}(\Omega_{T})$ w.r.t the norm $\|\cdot\|_{L_{G}^{1}}$. Hence, the conditional $G$-expectation $\mathbb{\widehat{E}}_{t}[\cdot]$ can be extended continuously to the completion $L_{G}^{1}(\Omega_{T})$. Furthermore, Denis et al., (2011) proved that the $G$-expectation has the following representation.

For $\mathcal{P}$ being a weakly compact set that represents $\widehat{\mathbb{E}}$, we define the capacity

$$c(A):=\sup_{P\in\mathcal{P}}P(A),\qquad\forall A\in\mathcal{B}(\Omega_{T}).$$

A set $A\in\mathcal{B}(\Omega_{T})$ is called polar if $c(A)=0$. A property holds $``quasi$-$surely"$ (q.s.) if it holds outside a polar set. In the sequel, we do not distinguish two random variables $X$ and $Y$ if $X=Y$, q.s..

Denote by $\langle B\rangle$ the quadratic variation process of the $G$-Brownian motion $B$. For two processes $\xi\in M_{G}^{1}(0,T)$ and $\eta\in M_{G}^{2}(0,T)$, the $G$-Itô integrals $(\int^{t}_{0}\xi_{s}d\langle B\rangle_{s})_{0\leq t\leq T}$ and $(\int^{t}_{0}\eta_{s}dB_{s})_{0\leq t\leq T}$ are well defined, see Li and Peng, (2011) and Peng, (2019). The following proposition can be regarded as the Burkholder–Davis–Gundy inequality under the $G$-expectation framework.

Let

$$S_{G}^{0}(0,T):=\Big{\{}h(t,B_{t_{1}\wedge t},\ldots,B_{t_{n}\wedge t}):t_{1},\ldots,t_{n}\in[0,T],\;h\in C_{b,Lip}(\mathbb{R}^{n+1})\Big{\}}.$$

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

$$\|\eta\|_{S_{G}^{p}}:=\left\{\widehat{\mathbb{E}}\sup_{t\in[0,T]}|\eta_{t}|^{p}\right\}^{1/p}.$$

Denote by $S_{G}^{p}(0,T)$ the completion of $S_{G}^{0}(0,T)$ under the norm $\|\cdot\|_{S_{G}^{p}}$. We have the following uniform continuity property for the processes in $S_{G}^{p}(0,T)$.

For $\xi\in L_{ip}(\Omega_{T})$, let

$$\mathcal{E}(\xi):=\widehat{\mathbb{E}}\left[\sup_{t\in[0,T]}\widehat{\mathbb{E}}_{t}[\xi]\right].$$

For $p\geq 1$ and $\xi\in L_{ip}(\Omega_{T})$, define

$$\|\xi\|_{p,\mathcal{E}}:=[\mathcal{E}(|\xi|^{p})]^{1/p}$$

and denote by $L_{\mathcal{E}}^{p}(\Omega_{T})$ the completion of $L_{ip}(\Omega_{T})$ under $\|\cdot\|_{p,\mathcal{E}}$. The following theorem can be regarded as the Doob’s maximal inequality under the $G$-expectation.

We can see that unlike the classical case, the order of the right-hand side is strictly greater than that of the left-hand side under the $G$-expectation.

We review some fundamental results about $G$-BSDEs. The solution of $G$-BSDE with terminal value $\xi$ and generators $f,g$, is a triple of processes $(Y,Z,A)$ evolve according to the following equation:

$$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})d\langle B\rangle_{s}-\int_{t}^{T}Z_{s}dB_{s}-(K_{T}-K_{t}),$$

(2.2)

where $Y\in S_{G}^{\alpha}(0,T)$, $Z\in H_{G}^{\alpha}(0,T)$, and $K$ is a non-increasing $G$-martingale such that $K_{0}=0$ and $K_{T}\in L_{G}^{\alpha}(\Omega_{T})$. To establish the well-posedness of the $G$-BSDE (2.2), consider the generators

$$f(t,\omega,y,z),\ g(t,\omega,y,z):[0,T]\times\Omega_{T}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$$

satisfy the following properties:

(H1)

There exists some $\beta>1$, such that for any $y,z\in\mathbb{R}$, $f(\cdot,\cdot,y,z),\ g(\cdot,\cdot,y,z)\in M_{G}^{\beta}(0,T)$;

(H2)

There exists some $\kappa>0$, such that

$$|f(t,\omega,y,z)-f(t,\omega,y^{\prime},z^{\prime})|+|g(t,\omega,y,z)-g(t,\omega,y^{\prime},z^{\prime})|\leq\kappa(|y-y^{\prime}|+|z-z^{\prime}|);$$

(H3)

The terminal value $\xi\in L_{G}^{\beta}(\Omega_{T})$.

The following results will be needed in our proofs. Note that $(Y,Z,K)$ in Theorem 2.7 is not the solution to the $G$-BSDE (2.2).

Similar to the classical case, the comparison theorem for $G$-BSDEs still holds.

In contrast to classical BSDEs, the inclusion of the additional non-increasing $G$-martingale $K$ in $G$-BSDEs introduces model uncertainty and complicates the analysis. Song, (2019) demonstrated that the non-increasing $G$-martingale cannot be expressed in the form ${\int_{0}^{t}\eta_{s}dt}$ or ${\int_{0}^{t}\gamma_{s}d\langle B\rangle_{s}}$, where $\eta,\gamma\in M_{G}^{1}(0,T)$. Specifically, the author established the following result.

We call the following process $u$ a generalized $G$-Itô process:

$$u_{t}=u_{0}+\int_{0}^{t}\eta_{s}ds+\int_{0}^{t}\zeta_{s}dB_{s}+K_{t},$$

where $\eta\in M_{G}^{1}(0,T)$, $\zeta\in H_{G}^{1}(0,T)$, and $K$ is a non-increasing $G$-martingale such that $K_{0}=0$. By Theorem 2.9, the decomposition of the generalized $G$-Itô process is unique.

Now we review the reflected $G$-BSDE with a lower obstacle studied in Li et al., 2018b . Their parameters consist of a terminal value $\xi$, generators $f,g$, and an obstacle $L$, where $L$ satisfies the following condition:

(H4)

$L\in S_{G}^{\beta}(0,T)$ is bounded from above by a generalized $G$-Itô process $L^{\prime}$ of the following form:

$$L^{\prime}_{t}=L^{\prime}_{0}+\int_{0}^{t}b^{\prime}(s)ds+\int_{0}^{t}\sigma^{\prime}(s)dB_{s}+K^{\prime}_{t},$$

where $b^{\prime}\in M_{G}^{\beta}(0,T)$, $\sigma^{\prime}\in H_{G}^{\beta}(0,T)$, and $K^{\prime}\in S_{G}^{\beta}(0,T)$ is a non-increasing $G$-martingale such that $K^{\prime}_{0}=0$ and $\beta>2$. Additionally, $\xi\geq L_{T}$ q.s.

A triple of processes $(Y,Z,A)$ for some $1<\alpha<\beta$, is called a solution of the reflected $G$-BSDE with a lower obstacle with parameters $(\xi,f,g,L)$, if

$\displaystyle\begin{cases}Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})d\langle B\rangle_{s}-\int_{t}^{T}Z_{s}dB_{s}+(A_{T}-A_{t}),\vspace{0.2cm}\\ Y_{t}\geq L_{t},\quad 0\leq t\leq T,\vspace{0.2cm}\\ \big{\{}-\int_{0}^{t}(Y_{s}-L_{s})dA_{s}\big{\}}_{t\in[0,T]}\textrm{ is a non-increasing $G$-martingale},\end{cases}$

(2.3)

where $Y\in S_{G}^{\alpha}(0,T)$, $Z\in H_{G}^{\alpha}(0,T)$, and $A$ is a continuous non-decreasing process such that $A_{0}=0$ and $A\in S_{G}^{\alpha}(0,T)$. The following theorem provides the well-posedness of the reflected $G$-BSDE (2.3).

The following theorem provides the comparison theorem for the reflected $G$-BSDE (2.3).

A triple of processes $(Y,Z,A)$, with $Y,A\in S_{G}^{\alpha}(0,T)$ and $Z\in H_{G}^{\alpha}(0,T)$ for some $2\leq\alpha<\beta$, is called a solution to the doubly reflected $G$-BSDE with the parameters $(\xi,f,g,L,U)$, if

$\displaystyle\begin{cases}Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})d\langle B\rangle_{s}-\int_{t}^{T}Z_{s}dB_{s}+(A_{T}-A_{t}),\vspace{0.2cm}\\ L_{t}\leq Y_{t}\leq U_{t},\quad 0\leq t\leq T,\vspace{0.2cm}\\ (Y,A)\textrm{ satisfies the }\textmd{ASC}_{\alpha}.\end{cases}$

(3.1)

A pair of processes $(Y,A)$ with $Y,A\in S_{G}^{\alpha}(0,T)$ is said to satisfy the $\textmd{ASC}_{\alpha}$, if there exist non-decreasing processes $\{A^{n,+}\}_{n\in\mathbb{N}}$, $\{A^{n,-}\}_{n\in\mathbb{N}}$, and non-increasing $G$-martingales $\{K^{n}\}_{n\in\mathbb{N}}$, such that

• •

  $\widehat{\mathbb{E}}\Big{[}|A_{T}^{n,+}|^{\alpha}+|A_{T}^{n,-}|^{\alpha}+|K^{n}_{T}|^{\alpha}\Big{]}\leq C$, where $C$ is independent of $n$;

• •

  $\widehat{\mathbb{E}}\sup\limits_{t\in[0,T]}\Big{|}A_{t}-(A_{t}^{n,+}-A_{t}^{n,-}-K_{t}^{n})\Big{|}^{\alpha}\rightarrow 0$, as $n\rightarrow\infty$;

• •

  $\lim\limits_{n\rightarrow\infty}\widehat{\mathbb{E}}\left|\int_{0}^{T}(Y_{s}-L_{s})dA_{s}^{n,+}\right|^{\alpha/2}=0$;

• •

  $\lim\limits_{n\rightarrow\infty}\widehat{\mathbb{E}}\left|\int_{0}^{T}(U_{s}-Y_{s})dA_{s}^{n,-}\right|^{\alpha/2}=0$.

We call $\{A^{n,+}\}_{n\in\mathbb{N}}$, $\{A^{n,-}\}_{n\in\mathbb{N}}$, and $\{K^{n}\}_{n\in\mathbb{N}}$ the approximate sequences for $(Y,A)$ with order $\alpha$ w.r.t. the lower obstacle $L$ and the upper obstacle $U$.

Consider the parameters of the doubly reflected $G$-BSDE (3.1), namely the terminal value $\xi$, the generators $f,g$, and the obstacles $L,U$, satisfy (H2), (H3) and the following assumptions:

(A1)

There exists some $\beta>2$, such that for any $y,z\in\mathbb{R}$, $f(\cdot,\cdot,y,z)$, $g(\cdot,\cdot,y,z)\in S_{G}^{\beta}(0,T)$;

(A2)

$L$, $U\in S_{G}^{\beta}(0,T)$. There exists some $I\in S_{G}^{\beta}(0,T)$ satisfying the following representation:

$$I_{t}=I_{0}+A^{I,-}_{t}-A_{t}^{I,+}+\int_{0}^{t}\sigma^{I}(s)dB_{s}.$$

Here, $A^{I,+}$, $A^{I,-}\in S_{G}^{\beta}(0,T)$ are two non-decreasing processes such that $A^{I,+}_{0}=A^{I,-}_{0}=0$; $\sigma^{I}\in S_{G}^{\beta}(0,T)$ satisfies $L_{t}\leq I_{t}\leq U_{t}$; $U+A^{I,+}$ is a generalized $G$-Itô process evolves according to

$$U_{t}+A^{I,+}_{t}=U_{0}+\int_{0}^{t}b(s)ds+\int_{0}^{t}\sigma(s)dB_{s}+K^{u}_{t},$$

(3.2)

where $b,\sigma\in S_{G}^{\beta}(0,T)$, and $K^{u}\in S_{G}^{\beta}(0,T)$ is a non-increasing $G$-martingale such that $K^{u}_{0}=0$. Additionally, $L_{T}\leq\xi\leq U_{T}$, q.s..