On distribution dependent stochastic differential equations driven by $G$-Brownian motion

TL;DR

This paper investigates the existence and uniqueness of solutions for distribution dependent stochastic differential equations driven by $G$-Brownian motion within the $G$-expectation framework, extending classical results to this more general setting.

Contribution

It introduces a new formulation and proper distance for distribution dependent $G$-SDEs, establishing well-posedness under Lipschitz conditions using fixed point arguments.

Findings

Proved existence and uniqueness of solutions for distribution dependent $G$-SDEs.

Developed a new formulation and distance measure for these equations.

Derived estimates for the solutions.

Abstract

Distribution dependent stochastic differential equations have been a very hot subject with extensive studies. On the other hand, under the $G$-expectation framework, stochastic differential equations driven by $G$-Brownian motion (in short form, $G$-SDEs) have received increasing attentions, and the existence and uniqueness of solutions to $G$-SDEs under Lipschitz and non-Lipschitz conditions have been obtained. Based on these studies, it is very natural and also important to investigate the $G$-SDEs which are also distribution dependent. In this paper, we are concerned with the well-posedness of the distribution dependent $G$-SDEs. To this end, we first introduce a proper distance of the involved distribution functions and propose a new formulation of the distribution dependent $G$-SDEs. Then, by utilising fix point argument, we establish existence and uniqueness of the solutions of distributed dependent $G$-SDEs under Lipschitz condition. Finally, we derive certain estimates for the solutions of the distribution dependent $G$-SDEs.

Equations

$$X_t=x_0+{\int }_0^t\hat{\mathbb{E}}[b(s,x,X_s)]{|}_{x=X_s}ds+{\int }_0^t\hat{\mathbb{E}}[{\sigma }_j(s,x,X_s)]{|}_{x=X_s}d{B}_s+{\int }_0^t\hat{\mathbb{E}}[h_{ij}(s,x,X_s)]{|}_{x=X_s}d⟨B^i,B^j{⟩}_s$$

$${\mathbb{F}}_X:\phi \in C_{Lip}\left({\mathbb{R}}^n\right)\mapsto {\mathbb{F}}_X\left(\phi \right):=\mathbb{E}\left[\phi \left(X\right)\right]\in \mathbb{R}$$

$${\mathbb{F}}_t^X:\phi \in C_{Lip}\left({\mathbb{R}}^n\right)\mapsto {\mathbb{F}}_t^X\left(\phi \right):={\mathbb{F}}_{X_t}\left(\phi \right)=\mathbb{E}\left[\phi \left(X_t\right)\right]\in \mathbb{R}$$

$$\rho \left({\omega }^1,{\omega }^2\right):=\sum _{i=1}^{\infty }{2}^{-i}\left[\left(\underset{t\in [0,i]}{\max }\left|{\omega }_t^1-{\omega }_t^2\right|\right)\wedge 1\right],\text{ for }{\omega }^1,{\omega }^2\in \Omega .$$

$$Lip\left({\Omega }_T\right):=\left\{\phi \left(B_{t_1},\dots ,B_{t_n}\right):t_1,\dots ,t_n\in [0,T],\phi \in C_{b.Lip}\left({\mathbb{R}}^{d\times n}\right),n\ge 1\right\},$$

$$G(A):=\frac{1}{2}\hat{\mathbb{E}}[⟨A{B}_1,B_1⟩],A\in \mathbb{S}(d),$$

$$\mu \left({\pi }_T^N\right):=\max \left\{\left|t_{i+1}-t_i\right|:i=0,1,\cdots ,N-1\right\}.$$

$${\eta }_t:=\sum _{k=0}^{N-1}{\xi }_k{1}_{\left[t_k,t_{k+1}\right)}(t).$$

$${\Vert \eta \Vert }_{M_G^p(0,T)}:={\left(\hat{\mathbb{E}}\left[{\int }_0^T{\left|{\eta }_t\right|}^{p}dt\right]\right)}^{\frac{1}{p}}.$$

$${\Vert \eta \Vert }_{{\bar{M}}_G^p(0,T)}:={\left({\int }_0^T\hat{\mathbb{E}}\left[{\left|{\eta }_t\right|}^p\right]dt\right)}^{\frac{1}{p}}.$$

$$I(\eta )={\int }_0^T{\eta }_{t}d{B}_t^{\mathbf{a}}:=\sum _{k=0}^{N-1}{\xi }_k\left(B_{t_{k+1}}^{\mathbf{a}}-B_{t_k}^{\mathbf{a}}\right).$$

$${\int }_0^T{\eta }_{t}d{B}_t^{\mathbf{a}}:=I(\eta ).$$

$$\hat{\mathbb{E}}\left[{\int }_0^T{\eta }_{t}d{B}_t^{\mathbf{a}}\right]=0,$$

$$\hat{\mathbb{E}}\left[{\left({\int }_0^T{\eta }_{t}d{B}_t^{\mathbf{a}}\right)}^2\right]\le {\sigma }_{\mathbf{a}{\mathbf{a}}^T}^2\hat{\mathbb{E}}\left[{\int }_0^T{\eta }_t^{2}dt\right].$$

$${⟨B^{\mathbf{a}}⟩}_t:=\underset{\mu \left({\pi }_T^N\right)\to 0}{\lim }\sum _{k=0}^{N-1}{\left(B_{t_{k+1}^N}^{\mathbf{a}}-B_{t_k^N}^{\mathbf{a}}\right)}^2={\left(B_t^{\mathbf{a}}\right)}^2-2{\int }_0^t{B}_s^{\mathbf{a}}d{B}_s^{\mathbf{a}},$$

$${\mathcal{Q}}_{0,T}(\eta )={\int }_0^T{\eta }_{t}d{⟨B^{\mathbf{a}}⟩}_t:=\sum _{k=0}^{N-1}{\xi }_k\left({⟨B^{\mathbf{a}}⟩}_{t_{k+1}}-{⟨B^{\mathbf{a}}⟩}_{t_k}\right)$$

$${⟨B^{\mathbf{a}},B^{\overline{\mathbf{a}}}⟩}_t:=\frac{1}{4}\left[{⟨B^{\mathbf{a}+\overline{\mathbf{a}}}⟩}_t-{⟨B^{\mathbf{a}-\overline{\mathbf{a}}}⟩}_t\right].$$

$${\int }_0^T{\eta }_{t}d{⟨B^{\mathbf{a}},B^{\overline{\mathbf{a}}}⟩}_t:=\frac{1}{4}\left({\int }_0^T{\eta }_{t}d{⟨B^{\mathbf{a}+\overline{\mathbf{a}}}⟩}_t-{\int }_0^T{\eta }_{t}d{⟨B^{\mathbf{a}-\overline{\mathbf{a}}}⟩}_t\right).$$

$$\hat{\mathbb{E}}\left[\underset{s\le u\le t}{\sup }{\left|{\int }_s^u{\eta }_{r}d{B}_r^{\mathbf{a}}\right|}^p\right]$$

$$\le C_p{\sigma }_{\mathbf{a}{\mathbf{a}}^{\mathrm{T}}}^p|t-s{|}^{\frac{p}{2}-1}\left({\int }_s^t\hat{\mathbb{E}}\left[{\left|{\eta }_u\right|}^p\right]du\right),$$

$$\hat{\mathbb{E}}\left[\underset{s\le u\le t}{\sup }{\left|{\int }_s^u{\eta }_{r}d{⟨B^{\mathbf{a}},B^{\overline{\mathbf{a}}}⟩}_r\right|}^p\right]$$

$$\le {\left(\frac{{\sigma }_{(\mathbf{a}+\bar{\mathbf{a}}){(\mathbf{a}+\bar{\mathbf{a}})}^T}^2+{\sigma }_{(\mathbf{a}-\bar{\mathbf{a}}){(\mathbf{a}-\bar{\mathbf{a}})}^T}^2}{4}\right)}^p(t-s{)}^{p-1}{\int }_s^t\hat{\mathbb{E}}\left[{\left|{\eta }_u\right|}^p\right]du.$$

$$\underset{C_{\phi }\le r}{\sup }\left|f(\phi )-{\mathbb{F}}_0(\phi )\right|=r\underset{C_{\frac{\phi }{r}}\le 1}{\sup }\left|f(\frac{\phi }{r})-{\mathbb{F}}_0(\frac{\phi }{r})\right|\le r\underset{C_{\phi }\le 1}{\sup }\left|f(\phi )-{\mathbb{F}}_0(\phi )\right|<\infty ,$$

$$\underset{C_{\phi }\le 1}{\sup }\left|f(\phi )-{\mathbb{F}}_0(\phi )\right|=\frac{1}{r}\underset{C_{r\phi }\le r}{\sup }\left|f(r\phi )-{\mathbb{F}}_0(r\phi )\right|\le \frac{1}{r}\underset{C_{\phi }\le r}{\sup }\left|f(\phi )-{\mathbb{F}}_0(\phi )\right|<\infty ,$$

$$\left|F_t^n(\phi )-F_t^m(\phi )\right|\le d_1^T(F^n,F^m)<\epsilon .$$

$${\tilde{F}}_t\left(\phi \right):=\underset{n\to \infty }{\lim }{F}_t^n(\phi ).$$

$$F_t(\psi ):=C_{\psi }{\tilde{F}}_t\left(\frac{\psi }{C_{\psi }}\right)=\underset{n\to \infty }{\lim }{F}_t^n(\psi ).$$

$$\left|F_t^n(\phi )-F_t(\phi )\right|<\epsilon .$$

$$\left|F_t^N\left(\phi \right)-\phi \left(0\right)\right|\le k.$$

$$\left|F_t\left(\phi \right)-\phi \left(0\right)\right|\le \left|F_t^N\left(\phi \right)-\phi \left(0\right)\right|+\left|F_t\left(\phi \right)-F_t^N\left(\phi \right)\right|\le k+\epsilon .$$

$$\underset{t\in [0,T]}{\sup }\underset{C_{\phi }\le 1}{\sup }\left|F_t\left(\phi \right)-\phi \left(0\right)\right|<\infty .$$

Full text

On distribution dependent stochastic differential equations driven by $G$-Brownian motion

De Suna, Jiang-Lun Wub and Panyu Wu*c,*111Corresponding author. E-mail: wupanyu@sdu.edu.cn  

a School of Mathematics, Shandong University, Jinan 250100, China

b Department of Mathematics, Computational Foundry, Swansea University, Swansea SA1 8EN, UK

c Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China

(Emails: sunde@mail.sdu.edu.cn; j.l.wu@swansea.ac.uk; wupanyu@sdu.edu.cn)

Abstract: Distribution dependent stochastic differential equations have been a very hot subject with extensive studies. On the other hand, under the $G$-expectation framework, stochastic differential equations driven by $G$-Brownian motion (in short form, $G$-SDEs) have received increasing attentions, and the existence and uniqueness of solutions to $G$-SDEs under Lipschitz and non-Lipschitz conditions have been obtained. Based on these studies, it is very natural and also important to investigate the $G$-SDEs which are also distribution dependent. In this paper, we are concerned with the well-posedness of the distribution dependent $G$-SDEs. To this end, we first introduce a proper distance of the involved distribution functions and propose a new formulation of the distribution dependent $G$-SDEs. Then, by utilising fix point argument, we establish existence and uniqueness of the solutions of distributed dependent $G$-SDEs under Lipschitz condition. Finally, we derive certain estimates for the solutions of the distribution dependent $G$-SDEs.

MSC (2020): 60H10; 60H30

Keywords: Distribution dependent; Stochastic differential equations; $G$-Brownian motion; Existence and uniqueness

1 Introduction

We start with a brief account of the history of distribution dependent stochastic differential equations. The prototype model equation was first proposed in 1938 by Vlasov (see the reprint [22]) with the named of mean field interaction in mathematical physics. Furthermore, inspired by Kac’s profound foundations of kinetic theory [10], in the seminal work [14], McKean had formulated the relevant SDEs with coefficients involving distributions of the solutions, hereafter names as McKean-Vlasov equations (also called mean-field SDEs in some literature). It was Sznitman who first established the existence and uniqueness of solutions of McKean-Vlasov SDEs with bounded drift coefficients and constant diffusion coefficients [21]. Furthermore, with the help of Wasserstein distance, Méléard [15] proved the existence and uniqueness of solutions of McKean-Vlasov SDEs under Lipschitz condition by using the fixed point theorem. Moreover, Buckdahn et al [2] studied the relation of mean-field SDEs to non-linear PDEs. Moreover, in Wang [23], by iterating in distributions, strong solutions of McKean-Vlasov SDEs were constructed. In Huang and Yang [9], the existence and uniqueness of the distribution-dependent SDEs with the Hölder continuous drift coefficients driven by a $\alpha$-stable process were obtained. Furthermore, Röckner and Zhang [18] studied the situation in which the diffusion coefficients are uniformly non degenerate, bounded, Hölder continuous and the drift coefficients are integrable, they established the existence and uniqueness of strong solutions as well as weak solutions of McKean-Vlasov SDEs.

On the other hand, motivated by uncertainty problems, risk measures and super-hedging in finance, Peng [16][17] invented a framework of time consistent sublinear expectation, called $G$-expectation. Under the $G$-expectation framework, a new notion of $G$-normal distribution was initiated, which plays the same important role in the theory of sublinear expectation as that of normal distribution in the classical probability theory. Based on the $G$-normal distribution, a new type of $G$-Brownian motion and the related stochastic calculus of Itô’s type have been developed. Furthermore, stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs) have been considered by Gao [4] and Peng [17], wherein the solvability of $G$-SDEs under Lipschitz conditions has been obtained by the contraction mapping theorem. Along this line, Lin [13] obtained a pathwise uniqueness result for non-Lipschitz $G$-SDEs with bounded coefficients. Bai and Lin [1] further studied the existence and uniqueness of solutions to $G$-SDEs with integral-Lipschitz coefficients. More investigation on $G$-SDEs can also be found in [5][6][8][12][24] (and references therein), just mention a few.

Based on the above studies, it is natural to consider the situation that the $G$-SDEs are also distribution dependent. The preliminary question is then the proper formulation of the distribution dependent $G$-SDEs. It was Sun [19] first considered this topic. Sun introduced the following mean-field $G$-SDE

[TABLE]

and showed the existence and uniqueness of solutions under Lipschitz condition, see [19] for details, wherein Sun also introduced the mean-field backward SDE driven by $G$-Brownian motion and established the existence and uniqueness theorem for the equation under Lipschitz condition. Further in Sun [20], the equation with uniformly continuous coefficients was considered. In the present paper, we want to study the equation (1.1) with distribution dependent coefficients. This motives us to establish a new formulation of distribution dependent $G$-SDEs in $G$-expectation space. To this end, we first construct a new distance for the distribution functions, which turns to be equivalent to the 1-Wasserstein distance in probability space (see our remark 3.3 below), then we formulate the well-posedness of these equations, and by utilising fix point argument, we establish the existence and uniqueness of the solutions of our distribution dependent $G$-SDEs with Lipschitz coefficients. We ends our paper by deriving certain estimates for the solutions to distribution dependent $G$-SDEs.

The rest of our paper is organised as follows. Section 2 presents the necessary preliminaries on sublinear expectation spaces and the $G$-framework. In Section 3, we introduce several complete metric spaces of sublinear functionals. Section 4 is devoted to establishing the existence and uniqueness theorem for distribution dependent $G$-SDEs with Lipschitz coefficients, deriving certain estimates for the solutions and providing an example to support our obtained results.

2 Notations and preliminaries

In this section, we will recall some definitions and results in sublinear expectation spaces and the $G$-framework. The readers may refer to Denis et al [3], Gao [4], Hu et al [7] and Peng [16], [17] for more details. Let $\Omega$ be a given nonempty set and $\cal{H}$ be a linear space of real functions defined on $\Omega$ such that if ${X_{1}},\cdots,{X_{n}}\in\cal{H}$, then $\varphi\left({{X_{1}},\cdots,{X_{n}}}\right)\in\cal{H}$ for each $\varphi\in{C_{Lip}}({{\mathbb{R}^{n}}})$, where ${C_{Lip}}({{\mathbb{R}^{n}}})$ denotes the linear space of functions $\varphi$ satisfying the following Lipschitz condition:

$\left|{\varphi(x)-\varphi(y)}\right|\leq{C_{\varphi}}\left|{x-y}\right|$, for $x,y\in{{\mathbb{R}^{n}}},$

where ${C_{\varphi}}>0$ is the minimal Lipschitz constant for the Lipschtian function$\varphi$.

Definition 2.1

A functional $\mathbb{E}[\cdot]:\mathcal{H}\rightarrow\mathbb{R}$ is called a sublinear expectation on $\mathcal{H}$, denoted by $\mathbb{E}[\cdot]$, if for any $X,Y\in\mathcal{H}$, the following four conditions are fulfilled

(1) (Monotonicity) if $X\geq Y$, then $\mathbb{E}[X]\geq\mathbb{E}[Y]$;

(2) (Constants preserving) $\mathbb{E}[c]=c$, for $c\in\mathbb{R}$;

(3) (Sub-additivity) $\mathbb{E}[X+Y]\leq\mathbb{E}[X]+\mathbb{E}[Y]$;

(4) (Positive homogeneity) $\mathbb{E}[\lambda X]=\lambda\mathbb{E}[X]$, for $\lambda\in\mathbb{R}^{+}$.

The triple $(\Omega,\mathcal{H},\mathbb{E})$ is called a sublinear expectation space.

Definition 2.2

Let $n\in\mathbb{N}$ and $X=\left(X_{1},\ldots,X_{n}\right)$ be a given $n$-dimensional random vector on a sublinear expectation space $\left({\Omega,\cal{H},\mathbb{E}}\right)$, the sublinear functional ${\mathbb{F}}_{X}$ defined by

[TABLE]

is called the $n$-dimensional distribution of $X$ under $\left({\Omega,\cal{H},\mathbb{E}}\right)$.

Remark 2.1

The triple $(\mathbb{R}^{n},{C_{Lip}}(\mathbb{R}^{n}),{{\mathbb{F}}_{X}})$ forms a sublinear expectation space. In other words, ${{\mathbb{F}}_{X}}$ satisfies monotonicity, constants preserving, sub-additivity and positive homogeneity.

Next, we introduce the notion of distributions of stochastic processes on the sublinear expectation space $\left({\Omega,\cal{H},\mathbb{E}}\right)$.

Definition 2.3

Let $\left(X_{t}\right)_{t\in[0,T]}$ be an $\mathbb{R}^{n}$-valued stochastic process on the sublinear expectation space $\left({\Omega,\cal{H},\mathbb{E}}\right)$, the functional process $\left({\mathbb{F}}_{t}^{X}\right)_{t\in[0,T]}$ defined by

[TABLE]

is called the distribution of $\left(X_{t}\right)_{t\in[0,T]}$ on $\left({\Omega,\cal{H},\mathbb{E}}\right)$.

In the rest of the paper, for arbitrarily fixed $d\in\mathbb{N}$, we let $\Omega=C_{0}({{\mathbb{R}}^{+};\mathbb{R}^{d}})$ be the space of all ${\mathbb{R}}^{d}$-valued continuous paths ${({\omega_{t}})_{t\in{{\mathbb{R}}^{+}}}}$ on ${\mathbb{R}}^{+}$ with ${\omega_{0}}={\bf 0}\in\mathbb{R}^{d}$, equipped with the following distance

[TABLE]

For each fixed $T\in[0,\infty)$, set ${\Omega_{T}}:=\left\{{{\omega_{\cdot\wedge T}}:\omega\in\Omega}\right\}$ and let ${B_{t}}(\omega)={\omega_{t}}$ be the canonical process and

[TABLE]

where ${C_{b.Lip}}\left({{\mathbb{R}^{d\times n}}}\right)$ stands for the linear space of bounded functions in ${C_{Lip}}\left({{\mathbb{R}^{d\times n}}}\right)$.

Peng [16] constructed a consistent sublinear expectation space $\left({\Omega,Lip({\Omega_{T}}),\hat{\mathbb{E}}}\right)$, called the $G$-expectation space and the canonical process $(B_{t})_{t\in[0,t]}$ is called a $G$-Brownian motion. The monotonic and sublinear function $G:\ \mathbb{S}(d)\to\mathbb{R}$ is defined by

[TABLE]

where $\langle\,,\,\rangle$ is the scalar product on $\mathbb{R}^{d}$ and $\mathbb{S}(d)$ denotes the collection of $d\times d$ symmetric matrices.

For each given $p\geq 1$, define $\left\|X\right\|_{L_{G}^{p}}=(\hat{\mathbb{E}}[|X|^{p}])^{1/p}$ for $X\in Lip({\Omega_{T}})$, and denote by $L_{G}^{p}({{\Omega_{T}}})$ the completion of $Lip({\Omega_{T}})$ under the norm $\left\|\cdot\right\|_{L_{G}^{p}}$. Then $\hat{\mathbb{E}}$ can be extended continuously to $L_{G}^{p}({{\Omega_{T}}})$. Next, recall that a partition ${\pi_{T}^{N}}$ of $[0,T]$ is a finite, ordered subset ${\pi_{T}}=\left\{{{t_{0}},{t_{1}},\cdots{t_{N}}}\right\}$ such that $0={t_{0}}<{t_{1}}<\cdots<{t_{N}}=T$. We further set

[TABLE]

For a given partition ${\pi_{T}^{N}}$ of $[0,T]$ and any given ${\xi_{k}}\in{Lip}\left({{\Omega_{{t_{k}}}}}\right),k=0,1,2.\cdots,N-1$, we define the simple process

[TABLE]

The totality of all simple processes is denoted by $M_{G}^{p,0}(0,T)$. Furthermore, we let $M_{G}^{p}(0,T)$ be the completion of $M_{G}^{p,0}(0,T)$ under the norm

[TABLE]

Denote by $\bar{M}_{G}^{p}\left(0,T\right)$ the completion of $M_{G}^{p,0}\left(0,T\right)$ under the norm

[TABLE]

Since $\hat{\mathbb{E}}$ is sub-additive, it is clear that $\bar{M}_{G}^{p}\left(0,T\right)\subset M_{G}^{p}\left(0,T\right)$.

For each fixed $\mathbf{a}\in{\mathbb{R}}^{d}$, let $B_{t}^{\mathbf{a}}:=\left\langle{{\mathbf{a}},{B_{t}}}\right\rangle$. Then $(B_{t}^{\mathbf{a}})_{t\geq 0}$ is a $1$-dimensional $G_{\mathbf{a}}$-Brownian motion with ${G_{\mathbf{a}}}\left(\alpha\right)=\frac{1}{2}\left({\sigma_{{\mathbf{a}}{{\mathbf{a}}^{T}}}^{2}{\alpha^{+}}-\sigma_{-{\mathbf{a}}{{\mathbf{a}}^{T}}}^{2}{\alpha^{-}}}\right)$, where ${\sigma_{\mathbf{a}{\mathbf{a}^{T}}}^{2}}=2G(\mathbf{a}\mathbf{a}^{T})={\mathbb{E}}\left[{{{\left\langle{{\bf{a}},{B_{1}}}\right\rangle}^{2}}}\right]$ and $\sigma_{-{\mathbf{a}}{{\mathbf{a}}^{T}}}^{2}=-2G(-\mathbf{a}\mathbf{a}^{T})=-{\mathbb{E}}\left[{-{{\left\langle{{\bf{a}},{B_{1}}}\right\rangle}^{2}}}\right]$.

Definition 2.4

For each $\eta\in M_{G}^{2,0}(0,T)$ of the form $\eta_{t}=\sum_{k=0}^{N-1}\xi_{k}\mathbf{1}_{\left[t_{k},t_{k+1}\right)}(t),$ we define the stochastic integral

[TABLE]

This can be continuously extended to $I:M_{G}^{2}([0,T])\rightarrow L_{G}^{2}\left(\Omega_{T}\right)$. Then for each $\eta\in M_{G}^{2}(0,T)$, we define the stochastic integral

[TABLE]

The following Lemma was established in [17] (see Lemma 3.3.4 therein).

Lemma 2.1

For each $\eta\in M_{G}^{2}(0,T)$,

[TABLE]

[TABLE]

The quadratic variation process of $B^{\mathbf{a}}$ is defined by

[TABLE]

which is not always a deterministic process as in the classical theory.

Definition 2.5

We specify the mapping $\mathcal{Q}_{0,T}:M_{G}^{1,0}([0,T])\rightarrow L_{G}^{1}\left(\Omega_{T}\right)$ via

[TABLE]

and $\mathcal{Q}_{0,T}$ can be uniquely extended to $M_{G}^{1}([0,T])\rightarrow L_{G}^{1}\left(\Omega_{T}\right)$. We still denote this mapping by

$\int_{0}^{T}\eta_{s}d\left\langle B^{\mathbf{a}}\right\rangle_{s}:=\mathcal{Q}_{0,T}(\eta)$, for each $\eta\in M_{G}^{1}(0,T).$

Let $\mathbf{a}$ and $\overline{\mathbf{a}}$ be two given vectors in $\mathbb{R}^{d}$. The mutual variation process of $B^{\mathbf{a}}$ and $B^{\overline{\mathbf{a}}}$ is defined by

[TABLE]

Then, for each $\eta\in M_{G}^{1}(0,T)$,

[TABLE]

The following BDG type inequalities can be found in Gao [4] (see Theorem 2.1 and 2.2 therein).

Lemma 2.2

Let $p\geq 2$, $\eta\in M_{G}^{p}(0,T)$ and $\mathbf{a}\in\mathbb{R}^{d}$. For $0\leq s\leq t\leq T$,

[TABLE]

where $C_{p}>0$ is a constant independent of $\mathbf{a},\eta$.

Lemma 2.3

Let $p\geq 1$, $\eta\in M_{G}^{p}(0,T)$ and ${\bf{a}},{\bf{\bar{a}}}\in\mathbb{R}^{d}$ and $0\leq s\leq t\leq T$, then

[TABLE]

Remark 2.2

We would like to pointed out that Gao [4] (see Theorem 2.1 and 2.2 therein) proved Lemmas 2.2 and 2.3 for $\eta\in\bar{M}_{G}^{p}(0,T)$. While Lemmas 2.2 and 2.3 can be verified, respectively, for $\eta\in M_{G}^{p}(0,T)$ by utilising similar arguments, therefore we omit their proofs here.

In the remaining of this paper, we write by $B^{i}=B^{{\mathbf{e}}_{i}}$ for the $i$-th coordinate of the $G$-Brownian motion $B$, under a given orthonormal basis $\left(\mathbf{e}_{1},\dots\mathbf{e}_{d}\right)$ in the space ${\mathbb{R}}^{d}$. For $1\leq i\leq j\leq d$, we denote $\left\langle B\right\rangle_{t}^{ij}:={\left\langle B^{i},B^{j}\right\rangle}_{t}.$

3 Complete metric spaces

For any $r>0$, let ${{\cal{D}}_{r}}$ denote the space of functionals $f$ on ${C_{Lip}}\left({{\mathbb{R}^{n}}}\right)$ satisfying monotonicity, constants preserving, sub-additivity and positive homogeneity, and further fulfilling the following

$\mathop{\sup}\limits_{{C_{\varphi}}\leq r}\left|{f\left(\varphi\right)-{{\mathbb{F}}_{\bf{0}}}}(\varphi)\right|=\mathop{\sup}\limits_{{C_{\varphi}}\leq r}\left|{f\left(\varphi\right)-\varphi\left({\bf{0}}\right)}\right|<\infty,$

where ${\mathbb{F}}_{\bf{0}}(\varphi):=\varphi({\bf{0}})$ and ${\bf{0}}$ is the zero vector in $\mathbb{R}^{n}$. We define the following metric $d_{r}$ on ${\cal{D}}_{r}$

$d_{r}\left({{f^{1}},{f^{2}}}\right):=\mathop{\sup}\limits_{\scriptstyle{C_{\varphi}}\leq{r}\hfill}\left|{{f^{1}}\left(\varphi\right)-{f^{2}}\left(\varphi\right)}\right|$, for $f^{1},f^{2}\in{\cal{D}}_{r}$.

Similarly, let ${\cal{D}}_{r}^{T}$ denote the space of functional processes $F=(F_{t})_{t\in[0,T]}$ on $\left[{0,T}\right]\times{C_{Lip}}\left({{{\mathbb{R}}^{n}}}\right)$ satisfying that $F_{t}\in\mathcal{D}_{r}$ and

$\mathop{\sup}\limits_{t\in[0,T]}\mathop{\sup}\limits_{{C_{\varphi}}\leq r}\left|{F_{t}\left(\varphi\right)-{{{\mathbb{F}}_{\bf{0}}}}}\right|=\mathop{\sup}\limits_{t\in[0,T]}\mathop{\sup}\limits_{{C_{\varphi}}\leq r}\left|{F_{t}\left(\varphi\right)-\varphi\left({\bf{0}}\right)}\right|<\infty.$

Then, we can introduce the following metric $d_{r}^{T}$ on ${{\cal{D}}_{r}^{T}}$

$d_{r}^{T}\left({{F^{1}},{F^{2}}}\right):=\mathop{\sup}\limits_{t\in[0,T]}\mathop{\sup}\limits_{\scriptstyle{C_{\varphi}}\leq{r}\hfill}\left|{{F_{t}^{1}}(\varphi)}-{{F_{t}^{2}}(\varphi)}\right|$, for $F^{1},F^{2}\in{\cal{D}}_{r}^{T}$.

Proposition 3.1

The spaces ${{\cal{D}}_{r}}$ and ${{\cal{D}}^{T}_{r}}$ are independent of $r>0$ so that they can be denoted by ${\cal{D}}$ and ${\cal{D}}^{T}$, respectively. Moreover, for any $r>0$, we have ${d_{r}}=r{d_{1}}$ and ${d_{r}^{T}}=r{d_{1}^{T}}$.

Proof. We only prove the first assertion for ${\cal{D}}$, the proofs of other assertions are similar, so are omitted. In fact, for any $f\in\mathcal{D}_{1}$, we have for arbitrarily fixed $r>0$

[TABLE]

thus, $f\in\mathcal{D}_{r}$ for $r>0$.

On the other hand, let us arbitrarily fix $r>0$. Then, for any $f\in\mathcal{D}_{r}$, we have

[TABLE]

thus, $f\in\mathcal{D}_{1}$. Hence, $\mathcal{D}_{r}\equiv\mathcal{D}_{1}=:\mathcal{D}$. The proof of Proposition 3.1 is completed. $\square$

Remark 3.1

It follows from Proposition 3.1 that for all $r>0$, $d_{r}$ are equivalent distances on $\mathcal{D}$, so we will just consider the metric $d_{1}$ on $\mathcal{D}$ for simplicity. Similarly, we will only consider the metric $d_{1}^{T}$ on the space ${{{\cal{D}}}^{T}}$.

Remark 3.2

If $f\in\mathcal{D}$ is the distribution of $X$ on the sublinear expectation space $(\Omega,\mathcal{H},\mathbb{E})$, then it is easy to check that

$\mathop{\sup}\limits_{{C_{\varphi}}\leq 1}\left|{f\left(\varphi\right)-{{\mathbb{F}}_{\bf{0}}}}(\varphi)\right|=d_{1}(\mathbb{F}_{X},{\mathbb{F}}_{\bf{0}})=\mathbb{E}[|X|].$**

Remark 3.3

It is worthwhile to point out that in case the sublinear expectation space $(\Omega,\mathcal{H},\mathbb{E})$ degenerates into a usual single probability space $(\Omega,\mathcal{F},P)$, for $\mu,\nu$ being the corresponding probability measures on $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ induced by the random vectors $X$ and $Y$, respectively, then the 1-Wasserstein distance of $\mu,\nu$ equals to $d_{1}(\mathbb{F}_{X},\mathbb{F}_{Y})$. This can be verified by the duality theorem of Kantorovich and Rubinstein [11].

Proposition 3.2

Both metric spaces $(\mathcal{D},d_{1})$ and $(\mathcal{D}^{T},d_{1}^{T})$ are complete metric spaces.

Proof. We will only show that $(\mathcal{D}^{T},d_{1}^{T})$ is complete, the other can be proved similarly. The proof will be divided into three steps.

**Step 1. **If $\left\{{{F^{n}}}\right\}_{n=1}^{\infty}$ is a Cauchy sequence in ${\cal{D}}^{T}$, then for any $\varepsilon>0$, there exists $N>0$ such that for all $n,m\geq N$, $t\in[0,T]$, $\varphi\in{C_{Lip}}\left({{{\mathbb{R}}^{n}}}\right)$ satisfying ${{C_{\varphi}}}\leq 1$, we have

[TABLE]

Therefore, for any fixed $t\in[0,T]$, $\left\{{F_{t}^{n}(\varphi)}\right\}_{n=1}^{\infty}$ is a Cauchy sequence in $\mathbb{R}$, and it definitely converges. Let ${\tilde{F}_{t}}\left(\varphi\right)$ denote the limit of $\left\{{F_{t}^{n}\left(\varphi\right)}\right\}_{n=1}^{\infty}$, that is

[TABLE]

For all $t\in[0,T]$, $\psi\in{C_{Lip}}\left({{{\mathbb{R}}^{n}}}\right)$, we have ${{C_{\frac{\psi}{C_{\psi}}}}}\leq 1$. Define

[TABLE]

Clearly, $F$ is a functional process on ${C_{Lip}}\left({{{\mathbb{R}}^{n}}}\right)\times\left[{0,T}\right]$.

**Step 2. ** We want to verify that $F\in{\cal{D}}^{T}$. Taking $m\to\infty$ in inequality (3.1), we obtain for any $\varepsilon>0$, there exists $N>0$ such that for all $n\geq N$, $t\in[0,T]$, $\varphi\in{C_{Lip}}\left({{{\mathbb{R}}^{n}}}\right)$ satisfying $C_{\varphi}\leq 1$, that the following holds

[TABLE]

Since $F^{N}\in{\cal{D}}^{T}$, there exists $k>0$ such that for all $t\in[0,T]$, $\varphi\in{C_{Lip}}\left({{{\mathbb{R}}^{n}}}\right)$ satisfying $C_{\varphi}\leq 1$,

[TABLE]

Therefore, for all $t\in[0,T]$, $\varphi\in{C_{Lip}}\left({{{\mathbb{R}}^{n}}}\right)$ satisfying $C_{\varphi}\leq 1$, we have

[TABLE]

Hence

[TABLE]

It is straightforward that $F_{t}$ is monotonic, constants preserved, sub-additive and positive homogeneous, thus $F\in{\cal{D}}^{T}$.

**Step 3. ** From inequality (3.2), we get

[TABLE]

Therefore, ${\cal{D}}^{T}$ is a complete metric space with respect to $d_{1}^{T}$. The proof of Proposition 3.2 is then completed. $\square$

4 Existence and uniqueness of distribution dependent SDEs driven by $G$-Brownian motion

In this final section, we are concerned with the following distribution dependent SDE driven by $G$-Brownian motion (in short, distribution dependent $G$-SDE)

[TABLE]

Here we are using Einstein summation convention, that is, the repeated indices $i$ and $j$ in the above equation means tacitly the summation over those indices. The initial data $x_{0}\in{\mathbb{R}^{n}}$ is a constant vector, $\left({\mathbb{F}}_{t}^{X}\right)_{t\in[0,T]}$ is the distribution of $\left(X_{t}\right)_{t\in[0,T]}$, $b,h_{ij},\sigma_{j}:[0,T]\times{\mathbb{R}^{n}}\times{\cal{D}}\rightarrow{\mathbb{R}^{n}}$ are functionals satisfying the following assumptions (H1) and (H2):

(H1) there exists $K>0$ such that for all $t\in[0,T],\ x,y\in\mathbb{R}^{n},f^{1},f^{2}\in\cal{D}$, we have

[TABLE]

(H2) ${b(\cdot,x,F_{\cdot}),{h_{ij}}(\cdot,x,F_{\cdot}),{\sigma_{j}}(\cdot,x,F_{\cdot})\in L^{2}(0,T;{\mathbb{R}^{n}})}$, for each $x\in{\mathbb{R}^{n}},F\in{\cal{D}}^{T}$.

By a solution, it is meant to be a stochastic process $(X_{t})_{t\in[0,T]}\in\bar{M}^{2}_{G}(0,T;\mathbb{R}^{n})$ fulfilling the distribution dependent $G$-SDE (4.1).

To ensure that (4.1) is well-posed, primarily the integrands with respect to $dB_{s}^{j}$ should be in $M_{G}^{2}\left(0,T;{\mathbb{R}^{n}}\right)$ and the integrands with respect to $d\langle B\rangle_{s}^{ij}$ or $ds$ should be in $M_{G}^{1}\left(0,T;{\mathbb{R}^{n}}\right)$. Thus, we need the following lemma.

Lemma 4.1

Fix any $p\geq 1$, let $\zeta:[0,T]\times{\mathbb{R}^{n}}\times{\cal{D}}\rightarrow{\mathbb{R}^{n}}$ be a functional such that $\zeta(\cdot,x,{{\mathbb{F}}^{x}})\in L^{p}(0,T;{{\mathbb{R}}^{n}})$ for each $x\in{{\mathbb{R}}^{n}}$. If $\zeta$ satisfies the Lipschitz condition in the sense that for all $t\in[0,T]$, $x,y\in{{\mathbb{R}}^{n}}$, ${f^{1}},{f^{2}}\in{\cal D}$,

[TABLE]

then, we claim that $\zeta(\cdot,{X_{\cdot}},{\mathbb{F}}_{\cdot}^{X})$ is an element in $M_{G}^{p}(0,T;{{\mathbb{R}}^{n}})$ for any $X_{\cdot}\in\bar{M}_{G}^{p}(0,T;{{\mathbb{R}}^{n}})$.

Proof. The proof is motivated by Lemma 5.1 in Bai and Lin [1]. For readers convenience, we present our proof here. Without loss of generality, we only give the proof for the one dimensional case. For $X_{\cdot}\in\bar{M}_{G}^{p}(0,T)$, choose ${\{{X^{N}_{\cdot}}\}_{N=1}^{\infty}}\subset M_{G}^{p,0}(0,T)$ such that ${\left\|{X_{\cdot}^{N}-{X_{\cdot}}}\right\|_{\bar{M}_{G}^{p}(0,T)}}\to 0$ as $N\to\infty$, where $X^{N}_{\cdot}$ has the following simple form

[TABLE]

Then, by the Lipschitz condition (4.2) and Hölder inequality of $\hat{\mathbb{E}}$ (Proposition 1.4.2 in Peng [17]), we get

[TABLE]

Hence, it is suffices to prove that $\zeta(\cdot,X_{\cdot}^{N},\mathbb{F}_{\cdot}^{{X^{N}}})\in M_{G}^{p}(0,T)$, that is, $\zeta(\cdot,{\xi_{k}},{\mathbb{F}^{{\xi_{k}}}}){I_{[{t_{k}},{t_{k+1}})}}(\cdot)\in M_{G}^{p}(0,T)$ for each $k\in\mathbb{N}$. For simplicity of the notation, we want to make a new assertion which is equivalent to the one stated above: for fixed $T\geq 1$, if $\eta\in{Lip}\left({{\Omega_{1}}}\right)$, then $\zeta(\cdot,\eta,{\mathbb{F}^{\eta}}){I_{[1,T)}}(\cdot)\in M_{G}^{p}(0,T)$. Before we proceed further, let us prove the assertion.

Since $\eta\in{Lip}\left({{\Omega_{1}}}\right)$, there exists an $M>0$ such that $\eta\in\left[{-M,M}\right]$. For each $n\in\mathbb{N}$, there is an open cover ${\left\{{{G_{i}}}\right\}_{i\in I}}$ of $\mathbb{R}$ with the Lebesgue measure $\lambda(G_{i})<\frac{1}{n}$ for each $i\in I$. By the partition of unity theorem, there exists a family $\{\phi_{i}^{n}\}_{i\in I}$ of $C_{0}^{\infty}(\mathbb{R})$-valued functions such that for each $i\in I$, supp$\left({\phi_{i}^{n}}\right)\in{G_{i}}$, $0\leq\phi_{i}^{n}\leq 1$, and for each $x\in\mathbb{R}$, $\sum\limits_{i\in I}{\phi_{i}^{n}\left(x\right)}=1$. Moreover, there exists a finite sub-family of $\{\phi_{i}^{n}\}_{i\in I}$, denoted by $\{\phi_{i}^{n}\}_{1\leq i\leq N(n)}$, such that for $x\in\left[{-M,M}\right]$, $\sum\limits_{i=1}^{N(n)}{\phi_{i}^{n}\left(x\right)}=1$. Choosing, for each $i=1,\cdots,N(n)$, a point $x_{i}^{n}$ such that $\phi_{i}^{n}\left({x_{i}^{n}}\right)>0$. Then let

[TABLE]

We have

[TABLE]

which implies that ${\zeta^{n}}\left({\cdot,\eta,{F^{\eta}}}\right){I_{[1,T)}}(\cdot)$ converges to $\zeta\left({\cdot,\eta,{F^{\eta}}}\right){I_{[1,T)}}(\cdot)$ in $M_{G}^{p}\left({0,T}\right)$. Therefore, it suffices to prove that ${\zeta^{n}}\left({\cdot,\eta,{F^{\eta}}}\right){I_{[1,T)}}(\cdot)$ belongs to $M_{G}^{p}\left({0,T}\right)$, that is, $\zeta\left({\cdot,x_{i}^{n},{{\mathbb{F}}^{x_{i}^{n}}}}\right)\phi_{i}^{n}\left(\eta\right){I_{[1,T)}}(\cdot)\in M_{G}^{p}\left({0,T}\right)$, $i=1,\cdots,N(n)$, which can be deduced from Lemma 5.2 in Bai and Lin [1]. The proof of Lemma 4.1 is thus completed. $\square$

Remark 4.1

When $p=2$, all the coefficients in the distribution dependent $G$-SDE (4.1) satisfy the conditions of Lemma 4.1 under the assumptions (H1) and (H2). Therefore, the $G$-stochastic integrals in the $G$-SDE (4.1) are well defined for any solution $\left(X_{t}\right)_{t\in[0,T]}\in\bar{M}_{G}^{2}\left(0,T;{\mathbb{R}^{n}}\right)$.

Our main result can be then formulated as follows.

Theorem 4.1

Suppose the assumptions (H1) and (H2) hold. Then there exists a unique solution $\left(X_{t}\right)_{t\in[0,T]}\in\bar{M}_{G}^{2}\left(0,T;{\mathbb{R}^{n}}\right)$ to the distribution dependent $G$-SDE (4.1).

Our proof of Theorem 4.1 is inspired by [15] and [21] and is based on a fixed point theorem. One considers the mapping $U:{\cal{D}}^{T}\to{\cal{D}}^{T}$ which associates with $F$ the distribution of $X^{F}$, that is $U(F)={\mathbb{F}}^{{X^{F}}}$, where $X^{F}$ fulfils the following

[TABLE]

We need show the following three lemmas before we present our proof of Theorem 4.1.

Lemma 4.2

Assume that (H1) and (H2) hold. Then, the mapping $U(F)={\mathbb{F}}^{{X^{F}}}:{\cal{D}}^{T}\to{\cal{D}}^{T}$ is well-defined.

Proof. Fixing $F\in{\cal{D}}^{T}$, the equation (4.3) then becomes a SDE driven by $G$-Brownian motion. Thanks to the assumptions (H1) and (H2), it can be deduced from Theorem 5.1.3 in [17] that there is a unique solution $\left(X_{t}^{F}\right)_{t\in[0,T]}\in\bar{M}_{G}^{2}\left(0,T;{\mathbb{R}^{n}}\right)$ to the equation (4.3). We then only need to prove that $U(F)\in{\cal{D}}^{T}$. In fact, it follows from the sub-additivity of $\hat{\mathbb{E}}$ that

[TABLE]

where the last inequality is due to the estimate for the solution of $G$-SDE (see Proposition 3.3 in Lin [12]). Thus, the mapping $U:{\cal{D}}^{T}\to{\cal{D}}^{T}$ is well-defined. The proof of Lemma 4.2 is completed. $\square$

Lemma 4.3

For $0\leq t\leq T$ and ${F^{1}},{F^{2}}\in{\cal{D}}^{T}$, we have

[TABLE]

where $C_{T}$ is a constant depending only on the constants $K$ and $T$.

Proof. For $F^{1},F^{2}\in{\cal{D}}_{T}$, we notice that $U({F^{1}})$ and $U({F^{2}})$ are the distributions of ${X^{1}}$, ${X^{2}}$ respectively, where ${X^{1}}$ and ${X^{2}}$ are determined by the following

[TABLE]

[TABLE]

for $t\in[0,T]$. Then by Hölder inequality of $\hat{\mathbb{E}}$ (see Proposition 1.4.2 in Peng [17]), we get

[TABLE]

Meanwhile, for $0\leq t\leq T^{\prime}\leq T$, it follows from Lemma 2.1 and Lemma 2.3 that

[TABLE]

Next by Gronwall inequality, we have

[TABLE]

Let $t=T^{\prime}$, we obtain the following

[TABLE]

where $C_{T}=C{e^{CT}}$, $C$ is a constant depending only on $K$ and $T$, and may vary line by line. The proof of Lemma 4.3 is completed. $\square$

Lemma 4.4

For $k\geq 1$, $F\in{\cal{D}}^{T}$, $0\leq t\leq T$, we have

[TABLE]

where $C_{T}$ is the same constant as in Lemma 4.3.

Proof. For any $F\in{\cal{D}}^{T}$, $k\geq 1$, let ${U^{(k)}}\left(F\right)=U\left({{U^{(k-1)}}\left(F\right)}\right)$ and ${U^{(0)}}\left(F\right)=F$. Then ${{U^{(k)}}\left(F\right)}$ is the distribution of ${X^{(k)}}$, where $X^{(k)}$ for each $k\geq 1$ is defined by the following

[TABLE]

By Lemma 4.3, we have

[TABLE]

We next use mathematical induction argument to show the inequality (4.4). Suppose the inequality (4.4) holds for $k-1$, namely

[TABLE]

Similar to the proof of Lemma 4.3, we have

[TABLE]

The proof of Lemma 4.4 is completed. $\square$

Proof of Theorem 4.1. First, we notice that if there exist ${F^{1}},{F^{2}}\in{\cal{D}}^{T}$ such that $U({F^{1}})={F^{1}},U({F^{2}})={F^{2}}$, then it follows from Lemma 4.3 that for $0\leq t\leq T$,

[TABLE]

Using Gronwall inequality, we have

${d_{1}^{t}}\left({{F^{1}},{F^{2}}}\right)=0$, for $0\leq t\leq T.$

Therefore, we conclude that the mapping $U:{\cal{D}}^{T}\to{\cal{D}}^{T}$ has at most one fixed point.

Second, it follows from Proposition 3.3 of Lin [12] that $\mathop{\sup}\limits_{t\in[0,T]}\hat{\mathbb{E}}\left[{\left|{X_{t}^{(1)}}\right|}\right]<\infty$, then due to the fact that $F\in{\cal{D}}^{T}$, we have

[TABLE]

From Lemma 4.4, one gets that for $m>n\geq 0$

[TABLE]

Let $n\to\infty$, we have $d_{1}^{T}\left({{U^{(m)}}\left(F\right),{U^{(n)}}\left(F\right)}\right)\to 0$. Thus ${\left\{{{U^{(k)}}\left(F\right)}\right\}_{k=1}^{\infty}}$ is a Cauchy sequence in ${\cal{D}}^{T}$. Since ${\cal{D}}^{T}$ is a complete metric space with respect to the metric $d_{r}^{T}$ (see our Proposition 3.2), then ${\left\{{{U^{(k)}}\left(F\right)}\right\}_{k=1}^{\infty}}$ converges. That is, there exists ${F^{*}}\in{\cal{D}}^{T}$ such that

[TABLE]

From Lemma 4.3 and ${U^{(k)}}\left(F\right)=U\left({{U^{(k-1)}}\left(F\right)}\right)$, we have

[TABLE]

Let $k\to\infty$, we have $d_{1}^{T}\left({U\left({{F^{*}}}\right),{F^{*}}}\right)=0$. Hence $F^{*}$ is the fixed point of $U$.

If $\left(X_{t}\right)_{t\in[0,T]}$ is the solution of the distribution dependent $G$-SDE (4.1), then the distribution of $\left(X_{t}\right)_{t\in[0,T]}$ is the fixed point of $U:{\cal{D}}^{T}\to{\cal{D}}^{T}$ and vice versa. Namely, the existence and uniqueness of the fixed point of $U:{\cal{D}}^{T}\to{\cal{D}}^{T}$ is equivalent to the existence and uniqueness of the solution $\left(X_{t}\right)_{t\in[0,T]}\in\bar{M}_{G}^{2}\left(0,T;{\mathbb{R}^{n}}\right)$ of the distribution dependent $G$-SDE (4.1). The proof of Theorem 4.1 is completed. $\hfill\Box$

Theorem 4.2

For any $p\geq 2$, assume that (H1) and (H2) hold and $\phi(\cdot,{\bf{0}},\mathbb{F}_{\bf{0}})\in L^{p}(0,T;\mathbb{R}^{n})$, that is there exists a positive constant $M$ such that $\int_{0}^{T}\left|\phi(s,{\bf{0}},\mathbb{F}_{\bf{0}})\right|^{p}ds\leq M$, where $\phi=b,{h_{ij}},{\sigma_{j}}$, respectively. Then, we have the following estimate for the solution ${\left({X_{t}^{x}}\right)_{t\in[0,T]}}$ of the distribution dependent $G$-SDE (4.1) with initial condition $x_{0}=x$

[TABLE]

where

[TABLE]

and $C_{p}$ is the constant as in Lemma 2.2.

Proof. For $p\geq 2$, we have

[TABLE]

It follows from Hölder inequality, the subadditivity of $G$-expectation $\hat{\mathbb{E}}$ and Lemma 2.2, Lemma 2.3 that

[TABLE]

where ${{C_{\sigma}}}={\left({\frac{{\sigma_{({{\bf{e}}_{i}}+{{\bf{e}}_{j}}){{({{\bf{e}}_{i}}+{{\bf{e}}_{j}})}^{T}}}^{2}+\sigma_{({{\bf{e}}_{i}}-{{\bf{e}}_{j}}){{({{\bf{e}}_{i}}-{{\bf{e}}_{j}})}^{T}}}^{2}}}{4}}\right)^{p}}$ and $C_{p}$ is the constant in Lemma 2.2. From Assumption (H1), we get

[TABLE]

then summarily we have

[TABLE]

for $\phi=b,{h_{ij}},{\sigma_{j}}$, respectively. Thus,

[TABLE]

Using Gronwall inequality, we have $\hat{\mathbb{E}}\left[{\mathop{\sup}\limits_{s\in[0,t]}{{\left|{X_{s}^{x}}\right|}^{p}}}\right]\leq{C_{1}}{e^{{C_{2}}t}}.$ $\square$

Theorem 4.3

For any $p\geq 2$, assume that (H1) and (H2) hold, then there exists a positive constant $C_{3}$ such that for all $x,y\in{\mathbb{R}^{n}},~{}t\in\left[{0,T}\right]$,

[TABLE]

where $C_{3}={4^{p-1}}\exp\left\{{{2^{3p-2}}{K^{p}}\left({{T^{p-1}}+{C_{\sigma}}{T^{p-1}}+{C_{p}}{T^{\frac{p}{2}-1}}}\right)t}\right\}.$

Proof. For $p\geq 2$, we have

[TABLE]

It follows from Hölder inequality, the subadditivity of $G$-expectation $\hat{\mathbb{E}}$ and Lemma 2.2, Lemma 2.3 that

[TABLE]

By Assumption (H2), we get

[TABLE]

thus summarily

[TABLE]

for $\phi=b,{h_{ij}},{\sigma_{j}}$, respectively. Hence, we obtain

[TABLE]

where ${C}={2^{3p-2}}{K^{p}}\left({{T^{p-1}}+{C_{\sigma}}{T^{p-1}}+{C_{p}}\sigma_{{\mathbf{e_{j}}}{\mathbf{e_{j}}}^{T}}^{p}{T^{\frac{p}{2}-1}}}\right).$ Furthermore, by Gronwall inequality, we derive that

[TABLE]

where $C_{3}={4^{p-1}}{e^{{C}t}}$. The proof of Theorem 4.3 is thus completed. $\square$

Example 4.1

*Let $b^{\prime}(t,x,y),h^{\prime}_{ij}(t,x,y),\sigma^{\prime}_{j}(t,x,y):[0,T]\times{\mathbb{R}^{n}}\times{\mathbb{R}^{n}}\rightarrow{\mathbb{R}^{n}}$ satisfy the following assumptions (A1) and (A2):

(A1) there exists $K>0$ such that for all $t\in[0,T],\ x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}^{n}$, we have*

[TABLE]

(A2) for any fixed $(t,x)$, $b^{\prime}(t,x,\cdot),h^{\prime}_{ij}(t,x,\cdot),\sigma^{\prime}_{j}(t,x,\cdot)$ are bounded functions on $\mathbb{R}^{n}$.

Specify $b,h_{ij},\sigma_{j}:[0,T]\times{\mathbb{R}^{n}}\times{\cal{D}}\rightarrow{\mathbb{R}^{n}}$, respectively, by the following

[TABLE]

It can be checked that $b,h_{ij},\sigma_{j}$ fulfil Assumptions (H1) and (H2). Hence, the equation (4.1) becomes the following

[TABLE]

which coincides with the mean-field $G$-SDE (1.1) introduced by Sun in [19].

Acknowledgement

The work is supported by the National Key Research and Development Program of China (No. 2018YFA0703900) and the Natural Science Foundation of Shandong Province (No. ZR2021MA098 and ZR2019ZD41).