General Time-Symmetric Mean-Field Forward-Backward Doubly Stochastic Differential Equations

Abstract

In this paper, a general class of time-symmetric mean-field stochastic systems, namely the so-called mean-field forward-backward doubly stochastic differential equations (mean-field FBDSDEs, in short) are studied, where coefficients depend not only on the solution processes but also on their law. We first verify the existence and uniqueness of solutions for the forward equation of general mean-field FBDSDEs under Lipschitz conditions, and we obtain the associated comparison theorem; similarly, we also verify those results about the backward equation. As the above two comparison theorems’ application, we prove the existence of the maximal solution for general mean-field FBDSDEs under some much weaker monotone continuity conditions. Furthermore, under appropriate assumptions we prove the uniqueness of the solution for the equations. Finally, we also obtain a comparison theorem for coupled general mean-field FBDSDEs.

1. Introduction

In our paper, we propose a new class of time-symmetric forward–backward stochastic differential equations (FBSDEs), which we call the fully coupled general mean-field forward–backward doubly stochastic differential equations (mean-field FBDSDEs), and the specific form is as follows: for $\eta \in L^2(\mathsf{\Omega },{\mathcal{F}}_0,P;\mathbb{R})$, $t\in [0,T]$,

$$\left\{\begin{array}{c}{X}_t=\eta +{\int }_0^{t}f(s,P_{(X_s,Y_s)},X_s,Q_s,Y_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,\hfill \\ Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{(X_s,Y_s)},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s\hfill \\ -{\int }_t^T{Z}_{s}d{W}_s.\hfill \end{array}\right.$$

(1)

The above equation involves both a standard (forward) stochastic integral driven by a Brownian motion W and a backward stochastic integral governed by a Brownian motion B, which are two mutually independent standard Brownian motions. We first study the existence and uniqueness of the solution for (1) and obtained the associated comparison theorem.

It is worth emphasizing that when (1) does not depend on the process Q and $G=0$, it will degenerate into the FBSDEs, whose development can be traced back to the end of the 20th century. In 1990, Pardoux and Peng [1] studied nonlinear backward stochastic differential equations (BSDEs), and the BSDEs theory has been widely studied and applied, especially in stochastic control, stochastic differential games, financial mathematics, partial differential equations (PDEs), and so on. In order to meet the needs of control theory research, fully coupled FBSDEs were first provided by Antonelli [2]. Then, many researchers extended this result. For example, Hu and Peng [3] proved the existence and uniqueness of the solution for FBSDEs under a certain ‘monotonicity’ condition. Antonelli and Hamadene [4] obtained the existence of solutions for FBSDEs with continuous monotonic coefficients. On the other hand, Peng [5] has also shown that this type of BSDE can provide a probabilistic representation for a large class of systems of quasilinear parabolic and elliptic PDEs, generalizing the classical Feynman–Kac formula for linear parabolic and elliptic PDEs. In order to further generalize the Feynman–Kac formula and propose a class of systems of quasilinear parabolic stochastic partial differential equations (SPDEs), Pardoux and Peng [6] proposed a class of backward doubly stochastic differential equations (BDSDEs) in 1994, which contain two kinds of stochastic integrals in opposite directions, resulting in two kinds of opposite information flows possessing more complex measurability. On the basis of the above literature, we further study a class of FBDSDEs with two opposite directions in this paper. In more detail, we prove the existence and uniqueness of solutions for both forward Equation (2)

$$X_t=\eta +{\int }_0^{t}f(s,P_{(X_s,Q_s)},X_s,Q_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,t\in [0,T]$$

(2)

and backward Equation (3)

$$Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{(Y_s,Z_s)},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s$$

(3)

of general mean-field FBDSDEs under Lipschitz conditions, respectively, (see Lemma 1 and Lemma 2). We also obtain the associated comparison theorems of (4)

$$X_t=\eta +{\int }_0^{t}f(s,P_{X_s},X_s,Q_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s$$

(4)

and (5)

$$Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{Y_s},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s.$$

(5)

(See Lemma 3 and Lemma 4).

The above work is different from the existing results on time-symmetric FBDSDEs (see [7,8,9,10]); we not only consider the backward stochastic integral term, but also introduce the mean field term in our work. Mean-field problems have been widely applied in various fields, such as statistical mechanics, quantum mechanics, quantum chemistry and so on. In order to simplify the study of complex problems, many scholars have applied mean-field theory to reduce high-dimensional problems to low-dimensional problems. In 2007, Lasry and Lions [11] formally introduced the concept of mean-field games. Inspired by them, Buckdahn et al. [12,13] studied a special class of mean-field problems completely by using stochastic methods and derived a new type of BSDEs, called mean-field backward stochastic differential equations (mean-field BSDEs). Since then, an increasing number of scholars have devoted their energy to the study of mean-field problems. Buckdahn et al. [14] proposed a general class of mean-field SDEs, in which the coefficients of the stochastic system depend not only on the solution processes, but also on their law, and detailed the existence and uniqueness theorem of the solution. This result elevated the research of mean-field problems to a new height. Subsequently, Li and Xing [15] studied the existence of solutions for the general mean-field BDSDEs with continuous coefficients, and Li et al. [16] studied a class of general coupled mean-field reflected FBSDEs and proved the existence and uniqueness of the solution. In this paper, we use the method of [17] to study a class of general mean-field FBDSDEs. This kind of equation depends not only on the solution processes but also on their law. In [17], the authors introduced a very important and technical lemma (Lemma 3.1) which plays a key role in proving the existence of solutions for general mean-field BSDEs with continuous coefficients. Inspired by this work, we also want to prove the existence of solutions for our proposed equation under the continuous condition. Compared to their equations, we add the backward stochastic integral term and consider a fully coupled system of forward and backward equations. We first construct a sequence of equations for $n\ge 1$, $\eta \in L^2(\mathsf{\Omega },{\mathcal{F}}_0,P;\mathbb{R})$,

$$\left\{\begin{array}{c}{X}_t^n=\eta +{\int }_0^{t}f(s,P_{(X_s^n,Y_s^n)},X_s^n,Q_s^n,Y_s^n)ds+{\int }_0^{t}g(s,X_s^n,Q_s^n)d{W}_s-{\int }_0^t{Q}_s^{n}d{\overleftarrow{B}}_s,\hfill \\ Y_t^n=\mathsf{\Phi }(P_{X_T^{n-1}},X_T^{n-1})+{\int }_t^{T}F(s,P_{(X_s^{n-1},Y_s^n)},X_s^{n-1},Y_s^n,Z_s^n)ds\hfill \\ -{\int }_t^{T}G(s,Y_s^n,Z_s^n)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{n}d{W}_s,0\le t\le T.\hfill \end{array}\right.$$

(6)

From Lemma 1 and Lemma 2, we obtain the unique solution $(X^n,Q^n,Y^n,Z^n)$ of (6), and we prove that $(X^n,Q^n,Y^n,Z^n)$ is monotonic and its limit verifies (1); therefore, the existence and uniqueness of the solution for (1) can be obtained. For details, readers can refer to Theorem 1.

In addition to the above results, in this paper we also prove that with some additional appropriate conditions, we can find the unique solution of the general mean-field FBDSDEs (1), which is shown by numerical simulations in Example 1. We also propose the comparison theorem for fully coupled general mean-field FBDSDEs (see Theorem 3), which is distinguished from the comparison theorems in Section 2. This type of comparison theorem for fully coupled equations has not been previously studied in the literature.

Although we have relatively rich theoretical results, we still want to explore the applications of these theories in finance, stochastic control and other fields in the future. For example, inspired by [18], we can further study the controlled mean-field forward backward doubly stochastic systems in the future, and explore the system controllability problems, so that a series of issues in the stock market can be studied such as asset pricing, risk assessment, and so on.

Our paper is organized as follows: In Section 2, we provide some preliminaries of general mean-field FBDSDEs. We also verify the existence and uniqueness of the solution for the general mean-field FBDSDEs under Lipschitz conditions and obtain the associated comparison theorems. Section 3 is devoted to proving the existence of the maximal solution for general mean-field FBDSDEs with continuous coefficients. Furthermore, under appropriate assumptions, we prove the uniqueness of solutions for the equations in Section 4. Finally, we obtain comparison theorem for coupled general mean-field FBDSDEs in Section 5.

2. Preliminaries

Now, we begin by introducing some necessary notations and concepts.

Let $(\mathsf{\Omega },\mathcal{F},P)$ be a complete probability space, and $T>0$ be an arbitrarily fixed time horizon throughout this paper. Let $\{W_t;0\le t\le T\}$ and $\{B_t;0\le t\le T\}$ be two mutually independent standard Brownian Motions with values respectively in ${\mathbb{R}}^d$ and ${\mathbb{R}}^l$, defined on $(\mathsf{\Omega },\mathcal{F},P)$. Let $\mathcal{N}$ denote the class of P-null sets of $\mathcal{F}$, and ${\mathcal{P}}_2({\mathbb{R}}^k)$ denote the set of the probability measures $\mu$ over $({\mathbb{R}}^k,\mathcal{B}({\mathbb{R}}^k))$ with a finite second moment, that is, ${\int }_{{\mathbb{R}}^k}{|x|}^2\mu (dx)<\infty$. Here, $\mathcal{B}({\mathbb{R}}^k)$ denotes the Borel $\sigma$-field over ${\mathbb{R}}^k$, and the probability space $(\mathsf{\Omega },\mathcal{F},P)$ needs to be rich, so we assume that there is a sub-$\sigma$-field $\mathcal{N}\subset {\mathcal{F}}^0\subset \mathcal{F}$ such that

(i)

The two dimensional Brownian motion $(B,W)$ is independent of ${\mathcal{F}}^0$;

(ii)

${\mathcal{F}}^0$ is ‘rich enough’, that is, for every $\mu \in {\mathcal{P}}_2({\mathbb{R}}^k)$ there is a random variable $\xi \in L^2(\mathsf{\Omega },\mathcal{F},P;{\mathbb{R}}^k)$ such that $P_{\xi }=\mu$.

The space ${\mathcal{P}}_2({\mathbb{R}}^k)$ is endowed with the 2-Wasserstein metric: for $\mu ,\nu \in {\mathcal{P}}_2({\mathbb{R}}^k),$

$$\begin{array}{cc}\hfill W_2(\mu ,\nu ):=\inf \{({\int }_{{\mathbb{R}}^{2k}}& {|x-y|}^2\rho (dxdy){)}^{\frac{1}{2}}:\rho \in {\mathcal{P}}_2({\mathbb{R}}^{2k}),\hfill \\ \hfill & \rho (·\times {\mathbb{R}}^k)=\mu ,\rho ({\mathbb{R}}^k\times ·)=\nu \}.\hfill \end{array}$$

For each $t\in [0,T]$, we define ${\mathcal{F}}_t:={\mathcal{F}}_t^W\vee {\mathcal{F}}_{t,T}^B\vee {\mathcal{F}}^0,$ where for any process $\left\{{\zeta }_t\right\}$, ${\mathcal{F}}_{s,t}^{\zeta }=\sigma \{{\zeta }_r-{\zeta }_s; s\le r\le t\}$, ${\mathcal{F}}_t^{\zeta }={\mathcal{F}}_{0,t}^{\zeta }$. Note that ${\left\{{\mathcal{F}}_t\right\}}_{t\in [0,T]}$ is neither increasing nor decreasing, and it does not constitute a filtration.

We will work with the following spaces of stochastic processes:

• $L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R}):=\{\mathbb{R}$-value $\mathcal{F}$-measurable random variables $\xi$: ${\parallel \xi \parallel }_{L^2}:={(E|\xi |}^2{)}^{\frac{1}{2}}<\infty \}$;
• ${\mathcal{S}}^2(0,T;\mathbb{R}):=\{\mathbb{R}$-value continuous processes $\zeta$: for any $t\in [0,T]$, ${\zeta }_t$ is ${\mathcal{F}}_t$-measurable with

  $${\parallel \zeta \parallel }_{{\mathcal{S}}^2}:={\left(E[\underset{0\le s\le T}{\sup }|\zeta (s){|}^2]\right)}^{\frac{1}{2}}<\infty \};$$
• ${\mathcal{H}}^2(0,T;{\mathbb{R}}^k):=\{{\mathbb{R}}^k$-value processes $\zeta$: for any $t\in [0,T]$, ${\zeta }_t$ is ${\mathcal{F}}_t$-measurable with

  $${\parallel \zeta \parallel }_{{\mathcal{H}}^2}:={\left(E[{\int }_0^t|\zeta (s){|}^{2}ds]\right)}^{\frac{1}{2}}<\infty \}.$$

Let $f:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2({\mathbb{R}}^{1+l})\times \mathbb{R}\times {\mathbb{R}}^l\to \mathbb{R}$, $g:[0,T]\times \mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^l\to {\mathbb{R}}^d$, $F:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2({\mathbb{R}}^{1+d})\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d\to \mathbb{R}$, $G:[0,T]\times \mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^d\to {\mathbb{R}}^l$, $\mathsf{\Phi }:{\mathcal{P}}_2(\mathbb{R})\times \mathbb{R}\to \mathbb{R}$ be jointly measurable and satisfy the following assumptions:

(A1) 

$f(t,\omega ,{\delta }_0^{(1)},0,0)\in {\mathcal{H}}^2(0,T;\mathbb{R})$, $g(t,\omega ,0,0)\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$, where ${\delta }_0^{(1)}$ denotes throughout the paper the Dirac measure with mass at $0\in {\mathbb{R}}^{1+l}$;

(A2) 

g is Lipschitz in $(x,q)$: there are constants $c_0>0, 0<\gamma <1$ such that, for all $x,x^{\prime }\in \mathbb{R},q,q^{\prime }\in {\mathbb{R}}^l$,

$|g(t,x,q)-g(t,x^{\prime },q^{\prime }){|}^2\le c_0|x-x^{\prime }{|}^2+\gamma {|q-q^{\prime }|}^2;$

(A3) 

f is Lipschitz in $(\mu ,x,q)$: there is a constant $c_0>0$ such that, for all $\mu ,{\mu }^{\prime }\in {\mathcal{P}}_2({\mathbb{R}}^{1+l})$, $x,x^{\prime }\in \mathbb{R},q,q^{\prime }\in {\mathbb{R}}^l$,

$|f(t,\mu ,x,q)-f(t,{\mu }^{\prime },x^{\prime },q^{\prime })|\le c_0(W_2(\mu ,{\mu }^{\prime })+|x-x^{\prime }|+|q-q^{\prime }|)$;

(A4) 

$G(t,\omega ,0,0)\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)$, and for fixed $X\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$, $F(t,\omega ,{\delta }_0^{(2)},X,0,0)\in {\mathcal{H}}^2(0,T;\mathbb{R})$, where ${\delta }_0^{(2)}$ denotes throughout the paper the Dirac measure with mass at $0\in {\mathbb{R}}^{1+d}$;

(A5) 

G is Lipschitz in $(y,z)$: there are constants $c_0>0, 0<\gamma <1$ such that, for all $y,y^{\prime }\in \mathbb{R},z,z^{\prime }\in {\mathbb{R}}^d$,

$|G(t,y,z)-G(t,y^{\prime },z^{\prime }){|}^2\le c_0|y-y^{\prime }{|}^2+\gamma {|z-z^{\prime }|}^2;$

(A6) 

For fixed $X\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$, F is Lipschitz in $(\mu ,y,z)$: there is a constant $c_0>0$ such that, for all $\mu ,{\mu }^{\prime }\in {\mathcal{P}}_2({\mathbb{R}}^{1+d})$, $y,y^{\prime }\in \mathbb{R},z,z^{\prime }\in {\mathbb{R}}^d$,

$|F(t,\mu ,X,y,z)-F(t,{\mu }^{\prime },X,y^{\prime },z^{\prime })|\le c_0(W_2(\mu ,{\mu }^{\prime })+|y-y^{\prime }|+|z-z^{\prime }|)$;

(A7) 

$\mathsf{\Phi }({\delta }_0^{(3)},0)\in {\mathcal{H}}^2(0,T;\mathbb{R})$, where ${\delta }_0^{(3)}$ denotes throughout the paper the Dirac measure with mass at $0\in \mathbb{R}$;

(A8) 

$\mathsf{\Phi }$ is Lipschitz in $(\mu ,x)$: there is a constant $c_0>0$ such that, for all $\mu ,{\mu }^{\prime }\in {\mathcal{P}}_2(\mathbb{R})$, $x,x^{\prime }\in \mathbb{R}$, $|\mathsf{\Phi }(\mu ,x)-\mathsf{\Phi }({\mu }^{\prime },x^{\prime })|\le c_0(W_2(\mu ,{\mu }^{\prime })+|x-x^{\prime }|)$.

We now consider the following general mean-field FBDSDE: for $\eta \in L^2(\mathsf{\Omega },{\mathcal{F}}_0,P;\mathbb{R})$,

$$\left\{\begin{array}{c}{X}_t=\eta +{\int }_0^{t}f(s,P_{(X_s,Q_s)},X_s,Q_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,\hfill \\ Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{(Y_s,Z_s)},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s\hfill \\ -{\int }_t^T{Z}_{s}d{W}_s,t\in [0,T].\hfill \end{array}\right.$$

(7)

Lemma 1.

Under the assumptions (A1)–(A3), the following forward equation of general mean-field FBDSDE (7)

$$X_t=\eta +{\int }_0^{t}f(s,P_{(X_s,Q_s)},X_s,Q_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,t\in [0,T],$$

(8)

has an unique solution $(X,Q)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)$ and there is a constant ${\kappa }_1$ that depends only on $c_0{,\gamma ,T,x,E[|\eta |}^2],E[{\int }_0^T|f(s,{\delta }_0^{(1)},0,0){|}^{2}ds],E[{\int }_0^T|g(s,0,0){|}^{2}ds]$ such that

$${\parallel X\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}+{\parallel Q\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^l)}\le {\kappa }_1.$$

(9)

Proof of Lemma 1.

We first introduce a norm which is equivalent to the canonical norm on the space ${\mathcal{H}}^2(0,T;{\mathbb{R}}^{1+l})$

$${\parallel (u,v)\parallel }_{\beta ,a}=(E[{\int }_0^T{e}^{-\beta s}(|a{u}_s{|}^2+|v_s{|}^2{)ds])}^{\frac{1}{2}},$$

where $\beta ,a>0$ will be specified later. For arbitrary $(x,q)\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^{1+l})$, we consider the following general mean-field double SDE, for $t\in [0,T]$,

$$X_t=\eta +{\int }_0^{t}f(s,P_{(x_s,q_s)},X_s,Q_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s.$$

Then, from a classical result for BDSDEs (see [6]), we obtain the existence and uniqueness of the solution $(X,Q)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)$.

Let us define the mapping $I(x,q):=(X,Q):{\mathcal{H}}^2(0,T;{\mathbb{R}}^{1+l})\to {\mathcal{H}}^2(0,T;{\mathbb{R}}^{1+l}).$ For arbitrary $(x^i,q^i)\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^{1+l}),$ we denote $(X^i,Q^i)=I(x^i,q^i),$$i=1,2$. Let $(\widehat{X},\widehat{Q}):=(X^1-X^2,Q^1-Q^2)$.

For any $\beta >0$, applying It$\widehat{\mathrm{o}}$’s formula to $e^{-\beta t}{|{\widehat{X}}_t|}^2$, we obtain

$$\begin{array}{cc}\hfill & E[e^{-\beta t}|{\widehat{X}}_t{|}^2]+E[{\int }_0^t\beta e^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+E[{\int }_0^t{e}^{-\beta s}|{\widehat{Q}}_s{|}^{2}ds]\hfill \\ \hfill =& 2E[{\int }_0^t{e}^{-\beta s}{\widehat{X}}_s{\widehat{f}}_{s}ds]+E[{\int }_0^t{e}^{-\beta s}|{\widehat{g}}_s{|}^{2}ds]:={\Delta }_1+{\Delta }_2,\hfill \end{array}$$

(10)

where ${\widehat{f}}_s=f(s,P_{(X_s^1,Q_s^1)},X_s^1,Q_s^1)-f(s,P_{(X_s^2,Q_s^2)},X_s^2,Q_s^2)$, ${\widehat{g}}_s=g(s,X_s^1,Q_s^1)-g(s,X_s^2,Q_s^2)$, ${\Delta }_1$ is the first integral, and ${\Delta }_2$ is the second one. From $2ab\le \frac{1}{\delta }{a}^2+\delta b^2,\delta >0$, there exist constants ${\delta }_1=1,{\delta }_2=\frac{1-\gamma }{4c_0},{\delta }_3=\frac{1-\gamma }{2c_0}>0$, such that

$$\begin{array}{cc}\hfill {\Delta }_1\le & 2c_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s|W_2(P_{(x_s^1,q_s^1)},P_{(x_s^2,q_s^2)})ds]\hfill \\ \hfill & +2c_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+2c_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s||{\widehat{Q}}_s|ds]\hfill \\ \hfill \le & \frac{c_0}{{\delta }_1}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+{\delta }_1{c}_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{x}}_s{|}^{2}ds]\hfill \\ \hfill & +\frac{c_0}{{\delta }_2}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+{\delta }_2{c}_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{q}}_s{|}^{2}ds]+2c_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]\hfill \\ \hfill & +\frac{c_0}{{\delta }_3}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+{\delta }_3{c}_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{Q}}_s{|}^{2}ds]\hfill \\ \hfill \le & (3c_0+\frac{6c_0^2}{1-\gamma })E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+\frac{1-\gamma }{2}E[{\int }_0^t{e}^{-\beta s}|{\widehat{Q}}_s{|}^{2}ds]\hfill \\ \hfill & +c_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{x}}_s{|}^{2}ds]+\frac{1-\gamma }{4}E[{\int }_0^t{e}^{-\beta s}|{\widehat{q}}_s{|}^{2}ds],\hfill \\ \hfill {\Delta }_2\le & c_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+\gamma E[{\int }_0^t{e}^{-\beta s}|{\widehat{Q}}_s{|}^{2}ds].\hfill \end{array}$$

Substituting the estimates of ${\Delta }_1$ and ${\Delta }_2$ into (10), we have

$$\begin{array}{cc}\hfill & (\beta -4c_0-\frac{6c_0^2}{1-\gamma })E[{\int }_0^t{e}^{-\beta s}|{\widehat{X}}_s{|}^{2}ds]+\frac{1-\gamma }{2}E[{\int }_0^t{e}^{-\beta s}|{\widehat{Q}}_s{|}^{2}ds]\hfill \\ \hfill \le & c_{0}E[{\int }_0^t{e}^{-\beta s}|{\widehat{x}}_s{|}^{2}ds]+\frac{1-\gamma }{4}E[{\int }_0^t{e}^{-\beta s}|{\widehat{q}}_s{|}^{2}ds].\hfill \end{array}$$

Let $a^2=\frac{4c_0}{1-\gamma }$ and take $\beta =4c_0+\frac{6c_0^2}{1-\gamma }+a^2·\frac{1-\gamma }{2}$; therefore, $\parallel (\widehat{X},\widehat{Q}){\parallel }_{\beta ,a}\le \sqrt{\frac{1}{2}}\parallel (\widehat{x},\widehat{q})\parallel$. Thus, we obtain that I is a contraction map on ${\mathcal{H}}^2(0,T;{\mathbb{R}}^{1+l})$ endowed with the norm ${\parallel ·\parallel }_{\beta ,a}$, and from the contraction mapping theorem we know that there is a unique fixed point $(X,Q)\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^{1+l})$ such that $I(X,Q)=(X,Q)$; thus, there is a unique pair $(X,Q)$ which solves the Equation (8).

Next, we prove that the estimate (9) holds.

For any $\beta >0$, applying It$\widehat{\mathrm{o}}$’s formula to $e^{-\beta t}{|X_t|}^2$, we obtain

$$\begin{array}{cc}\hfill & e^{-\beta t}|X_t{|}^2+\beta {\int }_0^t{e}^{-\beta s}|X_s{|}^{2}ds+{\int }_0^t{e}^{-\beta s}{|Q_s|}^{2}ds\hfill \\ \hfill =& {|\eta |}^2+2{\int }_0^t{e}^{-\beta s}{X}_{s}f(s,P_{(X_s,Q_s)},X_s,Q_s)ds+2{\int }_0^t{e}^{-\beta s}{X}_{s}g(s,X_s,Q_s)d{W}_s\hfill \\ \hfill & -2{\int }_0^t{e}^{-\beta s}{X}_s{Q}_{s}d{\overleftarrow{B}}_s+{\int }_0^t{e}^{-\beta s}{|g(s,X_s,Q_s)|}^{2}ds.\hfill \end{array}$$

(11)

On the one hand, there exist ${\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4,{\delta }_5>0$ such that

$$\begin{array}{cc}\hfill & E[e^{-\beta t}|X_t{|}^2]+\beta E[{\int }_0^t{e}^{-\beta s}|X_s{|}^{2}ds]+E[{\int }_0^t{e}^{-\beta s}|Q_s{|}^{2}ds]\hfill \\ \hfill =& {E[|\eta |}^2]+2E[{\int }_0^t{e}^{-\beta s}{X}_{s}f(s,P_{(X_s,Q_s)},X_s,Q_s)ds]+E[{\int }_0^t{e}^{-\beta s}|g(s,X_s,Q_s){|}^{2}ds]\hfill \\ \hfill \le & {E[|\eta |}^2]+2c_{0}E[{\int }_0^t{e}^{-\beta s}|X_s|(E[|X_s{|}^2{])}^{\frac{1}{2}}ds]+2c_{0}E[{\int }_0^t{e}^{-\beta s}|X_s|(E[|Q_s{|}^2{])}^{\frac{1}{2}}ds]\hfill \\ \hfill & +2c_{0}E[{\int }_0^t{e}^{-\beta s}|X_s{|}^{2}ds]+2c_{0}E[{\int }_0^t{e}^{-\beta s}|X_s||Q_s|ds]+2E[{\int }_0^t{e}^{-\beta s}|X_s||f(s,{\delta }_0^{(1)},0,0)|ds]\hfill \\ \hfill & +E[{\int }_0^t{e}^{-\beta s}|g(s,X_s,Q_s)-g(s,0,0){|}^{2}ds]+E[{\int }_0^t{e}^{-\beta s}|g(s,0,0){|}^{2}ds]\hfill \\ \hfill & +2E[{\int }_0^t{e}^{-\beta s}|g(s,X_s,Q_s)-g(s,0,0)||g(s,0,0)|ds]\hfill \\ \hfill \le & {E[|\eta |}^2]+(\frac{c_0}{{\delta }_1}+{\delta }_1{c}_0+\frac{c_0}{{\delta }_2}+2c_0+\frac{c_0}{{\delta }_3}+\frac{1}{{\delta }_4}+c_0+\frac{c_0}{{\delta }_5})E[{\int }_0^t{e}^{-\beta s}|X_s{|}^{2}ds]\hfill \\ \hfill & +{\delta }_{4}E[{\int }_0^t{e}^{-\beta s}|f(s,{\delta }_0^{(1)},0,0){|}^{2}ds]+({\delta }_2{c}_0+{\delta }_3{c}_0+(1+\frac{1}{{\delta }_5})\gamma )E[{\int }_0^t{e}^{-\beta s}|Q_s{|}^{2}ds]\hfill \\ \hfill & +(1+{\delta }_5)E[{\int }_0^t{e}^{-\beta s}|g(s,0,0){|}^{2}ds].\hfill \end{array}$$

Let ${\delta }_1={\delta }_4=1,{\delta }_2={\delta }_3=\frac{1-\gamma }{4c_0},{\delta }_5=\frac{4\gamma }{1-\gamma }$ and $\beta =1+\frac{c_0}{{\delta }_1}+{\delta }_1{c}_0+\frac{c_0}{{\delta }_2}+2c_0+\frac{c_0}{{\delta }_3}+\frac{1}{{\delta }_4}+c_0+\frac{c_0}{{\delta }_5},$ we obtain

$$\begin{array}{cc}\hfill & E[|X_t{|}^2]+E[{\int }_0^t|X_s{|}^{2}ds]+\frac{1-\gamma }{4}E[{\int }_0^t|Q_s{|}^{2}ds]\hfill \\ \hfill \le & e^{\beta T}(E[e^{-\beta t}|X_t{|}^2]+E[{\int }_0^t{e}^{-\beta s}|X_s{|}^{2}ds]+\frac{1-\gamma }{4}E[{\int }_0^t{e}^{-\beta s}|Q_s{|}^{2}ds])\hfill \\ \hfill \le & e^{\beta T}{(E[|\eta |}^2]+E[{\int }_0^t|f(s,{\delta }_0^{(1)},0,0){|}^{2}ds]+\frac{1+3\gamma }{1-\gamma }E[{\int }_0^t|g(s,0,0){|}^{2}ds])\le {\kappa }_1,\hfill \end{array}$$

then,

$$\begin{array}{c}\hfill E[|X_t{|}^2]+E[{\int }_0^t|X_s{|}^{2}ds]+E[{\int }_0^t|Q_s{|}^{2}ds]\le {\kappa }_1,\end{array}$$

(12)

where ${\kappa }_1$ depends on $c_0{,\gamma ,T,E[|\eta |}^2],E[{\int }_0^t|f(s,{\delta }_0^{(1)},0,0){|}^{2}ds]$ and $E[{\int }_0^t|g(s,0,0){|}^{2}ds]$.

On the other hand, from (11) and the Burkholder–Davis–Gundy inequality, for all $t\in [0,T]$, there exist ${\delta }_6,{\delta }_7>0$, such that

$$\begin{array}{cc}\hfill & E[\underset{r\in [0,t]}{\sup }{e}^{-\beta r}|X_r{|}^2]\hfill \\ \hfill \le & {E[|\eta |}^2]+2c_{0}E[{\int }_0^t{e}^{-\beta s}|X_s|(W_2(P_{(X_s,Q_s)},{\delta }_0^{(1)})+|X_s|+|Q_s|)ds]\hfill \\ \hfill & +2E[{\int }_0^t{e}^{-\beta s}|X_s||f(s,{\delta }_0^{(1)},0,0)|ds]+2E[\underset{r\in [0,t]}{\sup }{e}^{-\frac{\beta r}{2}}|X_r|\underset{r\in [0,t]}{\sup }|{\int }_0^r{e}^{-\frac{\beta s}{2}}g(s,X_s,Q_s)d{W}_s|]\hfill \\ \hfill & +2E[\underset{r\in [0,t]}{\sup }{e}^{-\frac{\beta r}{2}}|X_r|\underset{r\in [0,t]}{\sup }|{\int }_0^r{e}^{-\frac{\beta s}{2}}{Q}_{s}d{\overleftarrow{B}}_s|]+2E[{\int }_0^t{e}^{-\beta s}(c_0|X_s{|}^2+\gamma |Q_s{|}^2)ds]\hfill \\ \hfill & +2E[{\int }_0^t{e}^{-\beta s}|g(s,0,0){|}^{2}ds]\hfill \\ \hfill \le & {E[|\eta |}^2]+(\frac{1}{{\delta }_6}+\frac{1}{{\delta }_7})E[\underset{r\in [0,t]}{\sup }{e}^{-\beta r}|X_r{|}^2]+[(8+2{\delta }_6{\tilde{C}}_1)c_0+1]E[{\int }_0^t{e}^{-\beta s}|X_s{|}^{2}ds]\hfill \\ \hfill & +(2c_0+2{\delta }_6{\tilde{C}}_1\gamma +{\delta }_7{\tilde{C}}_2+2\gamma )E[{\int }_0^t{e}^{-\beta s}|Q_s{|}^{2}ds]+E[{\int }_0^t{e}^{-\beta s}|f(s,{\delta }_0^{(1)},0,0){|}^{2}ds]\hfill \\ \hfill & +(2{\delta }_6{\tilde{C}}_1+2)E[{\int }_0^t{e}^{-\beta s}|g(s,0,0){|}^{2}ds].\hfill \end{array}$$

Taking ${\delta }_6={\delta }_7=4,{\tilde{C}}_1={\tilde{C}}_2=1$, then we have

$$\begin{array}{cc}\hfill & E[\underset{r\in [0,t]}{\sup }|X_r{|}^2]\le e^{\beta T}E[\underset{r\in [0,t]}{\sup }{e}^{-\beta r}|X_r{|}^2]\hfill \\ \hfill \le & e^{\beta T}{\left\{2E\right[|\eta |}^2]+(32c_0+2)E[{\int }_0^t{e}^{-\beta s}|X_s{|}^{2}ds]+(4c_0+20\gamma +8)E[{\int }_0^t{e}^{-\beta s}|Q_s{|}^{2}ds]\hfill \\ \hfill & +2E[{\int }_0^t{e}^{-\beta s}|f(s,{\delta }_0^{(1)},0,0){|}^{2}ds]+20E[{\int }_0^t{e}^{-\beta s}|g(s,0,0){|}^{2}ds]\}\hfill \\ \hfill \le & e^{\beta T}{\left\{2E\right[|\eta |}^2]+(32c_0+2)E[{\int }_0^t|X_s{|}^{2}ds]+(4c_0+20\gamma +8)E[{\int }_0^t|Q_s{|}^{2}ds]\hfill \\ \hfill & +2E[{\int }_0^t|f(s,{\delta }_0^{(1)},0,0){|}^{2}ds]+20E[{\int }_0^t|g(s,0,0){|}^{2}ds]\}\le {\kappa }_1,\mathrm{for}\mathrm{all}t\in [0,T].\hfill \end{array}$$

(13)

By combining (12) and (13), we obtain the estimate (9).  □

Lemma 2.

Under the assumptions (A4)–(A8), the backward equation of general mean-field FBDSDE (7)

$$Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{(Y_s,Z_s)},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s,$$

has an unique solution $(Y,Z)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$ and there is a constant ${\kappa }_2$ that depends only on $c_0,\gamma ,T,{\kappa }_1,E[|\mathsf{\Phi }({\delta }_0^{(3)},0){|}^2],E[{\int }_0^T|F(s,{\delta }_0^{(2)},X,0,0){|}^{2}ds],E[{\int }_0^T|G(s,0,0){|}^{2}ds]$, such that

$${\parallel Y\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}+{\parallel Z\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^d)}\le {\kappa }_2.$$

Proof of Lemma 2.

From Lemma 1, we first deduce the existence and uniqueness of the process $(X,Q)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)$ for the forward equation of general mean-field FBDSDE (7). Once we know $(X,Q)$, the backward equation of general mean-field FBDSDE (7) becomes a classical equation with the coefficients $\tilde{F}(s,\mu ,y,z)=F(s,\mu ,X,y,z)$, $G(s,y,z)$ and the terminal condition $\mathsf{\Phi }(P_X,X)$. Therefore, according to the proof in [15], we know that the backward equation of general mean-field FBDSDE (7) has a unique solution under the assumptions (A4)–(A8).  □

Next, we would like to prove another important result, the comparison theorem. From Example 3.1 and 3.2 in [13] and Example 3.1 and 3.2 in [15] we know that, for the forward equation of coupled general mean-field FBDSDEs, if the coefficient f depends on the law of Q or is non-increasing in the law of X, or the coefficient g depends on the law of X or the law of Q, we usually do not have the comparison theorem; similarly, for the backward equation, if the coefficient F depends on the law of Z or is non-increasing in the law of Y, or the coefficient G depends on the law of Y or the law of Z, we also do not have the comparison theorem. Therefore, we now consider the following general mean-field FBDSDE:

$$\left\{\begin{array}{c}{X}_t=\eta +{\int }_0^{t}f(s,P_{X_s},X_s,Q_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,\hfill \\ Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{Y_s},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s.\hfill \end{array}\right.$$

(14)

In addition, we assume that $f:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2(\mathbb{R})\times \mathbb{R}\times {\mathbb{R}}^l\to \mathbb{R}$ and $F:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2(\mathbb{R})\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d\to \mathbb{R}$ satisfy the following conditions:

(A9) 

For any $X_1,X_2\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$ and $(s,x,q)\in [0,T]\times \mathbb{R}\times {\mathbb{R}}^l$, there exists a constant $L>0$ such that $f(s,P_{X_1},x,q)-f(s,P_{X_2},x,q)\le L(E[|{(X_1-X_2)}^{+}{|}^2{])}^{\frac{1}{2}}$;

(A10) 

Fixed $X\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$, for any $Y_1,Y_2\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$ and $(s,y,z)\in [0,T]\times \mathbb{R}\times {\mathbb{R}}^d$, there exists a constant $L>0$ such that $F(s,P_{Y_1},X,y,z)-F(s,P_{Y_2},X,y,z)\le L(E[|{(Y_1-Y_2)}^{+}{|}^2{])}^{\frac{1}{2}}$;

(A11) 

For any $X_1,X_2\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$ and $x\in \mathbb{R}$, there exists a constant $L>0$ such that $\mathsf{\Phi }(P_{X_1},x)-\mathsf{\Phi }(P_{X_2},x)\le L(E[|{(X_1-X_2)}^{+}{|}^2{])}^{\frac{1}{2}}$, and for all $\mu \in {\mathcal{P}}_2(\mathbb{R})$, $x,x^{\prime }\in \mathbb{R}$, there is $\mathsf{\Phi }(\mu ,x)\le \mathsf{\Phi }(\mu ,x^{\prime })$, as $x\le x^{\prime }$.

Lemma 3.

Let $g=g(s,\omega ,x,q)$ satisfy (A1) and (A2), and $f^i=f^i(s,\omega ,\mu ,x,q),i=1,2$ be two drivers satisfying (A1). Moreover, we assume that

• (i) 

  One of the drivers $f^i$ satisfies assumption (A3);

  (ii) 

  One of the drivers $f^i$ satisfies assumption (A9).

Denote by $(X^1,Q^1)$ and $(X^2,Q^2)$ the solutions of the forward equation of general mean-field FBDSDE (14) with data $({\eta }^1,f^1,g)$ and $({\eta }^2,f^2,g)$, respectively:

$$X_t=\eta +{\int }_0^{t}f(s,P_{X_s},X_s,Q_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s.$$

Then, if for all $\mu ,x,q$, ${\eta }^1\le {\eta }^2$, $f^1(s,\mu ,x,q)\le f^2(s,\mu ,x,q)$, dsdP-a.e., it holds that also $X_s^1\le X_s^2,$ for all $s\in [0,T],$ P-a.s.

Proof of Lemma 3.

For notational simplicity, we assume $l=d=1$. Without loss of generality, we assume that the driver $f^1$ satisfies (A3) and (A9). Let us define $\widehat{\eta }:={\eta }^1-{\eta }^2$, $({\widehat{X}}_t,{\widehat{Q}}_t):=(X_t^1-X_t^2,Q_t^1-Q_t^2),t\in [0,T]$. Because of ${\eta }^1\le {\eta }^2$, we have ${\widehat{\eta }}^{+}=0$. From It$\widehat{\mathrm{o}}$’s formula and the inequality $2ab\le \frac{1}{\lambda }{a}^2+\lambda b^2,\lambda >0$, there is

$$\begin{array}{cc}\hfill & E[{({\widehat{X}}_t^{+})}^2]+E[{\int }_0^t{I}_{\{{\widehat{X}}_s\ge 0\}}|{\widehat{Q}}_s{|}^{2}ds]\hfill \\ \hfill =& E[{({\widehat{\eta }}^{+})}^2]+2E[{\int }_0^t{\widehat{X}}_s^{+}(f^1(s,P_{X_s^1},X_s^1,Q_s^1)-f^2(s,P_{X_s^2},X_s^2,Q_s^2))ds]\hfill \\ \hfill & +E[{\int }_0^t{I}_{\{{\widehat{X}}_s\ge 0\}}|g(s,X_s^1,Q_s^1)-g(s,X_s^2,Q_s^2){|}^{2}ds]\hfill \\ \hfill \le & 2E[{\int }_0^t{\widehat{X}}_s^{+}(f^1(s,P_{X_s^1},X_s^1,Q_s^1)-f^1(s,P_{X_s^2},X_s^1,Q_s^1)+f^1(s,P_{X_s^2},X_s^1,Q_s^1)-f^1(s,P_{X_s^2},X_s^2,Q_s^2)\hfill \\ \hfill & +f^1(s,P_{X_s^2},X_s^2,Q_s^2)-f^2(s,P_{X_s^2},X_s^2,Q_s^2))ds]+E[{\int }_0^t{I}_{\{{\widehat{X}}_s\ge 0\}}(c_0|{\widehat{X}}_s{|}^2+\gamma |{\widehat{Q}}_s{|}^2)ds]\hfill \\ \hfill \le & 2E[{\int }_0^t{\widehat{X}}_s^{+}(L(E[|{\widehat{X}}_s^{+}{|}^2{])}^{\frac{1}{2}}+c_0|{\widehat{X}}_s|+c_0|{\widehat{Q}}_s|)ds]+E[{\int }_0^t{I}_{\{{\widehat{X}}_s\ge 0\}}(c_0|{\widehat{X}}_s{|}^2+\gamma |{\widehat{Q}}_s{|}^2)ds]\hfill \\ \hfill \le & (2L+3c_0+\frac{2c_0^2}{1-\gamma })E[{\int }_0^t{({\widehat{X}}_s^{+})}^{2}ds]+\frac{1+\gamma }{2}E[{\int }_0^t{I}_{\{{\widehat{X}}_s\ge 0\}}|{\widehat{Q}}_s{|}^{2}ds].\hfill \end{array}$$

From Gronwall’s inequality, $E[{({\widehat{X}}_t^{+})}^2]=0,t\in [0,T],$ i.e., $X_t^1\le X_t^2$, P-a.s., $t\in [0,T]$. Hence, $X_t^1\le X_t^2,t\in [0,T],$ P-a.s.  □

Lemma 4.

Denote by $(X,Q)$ the solution of the forward equation of general mean-field FBDSDE (14). Let $G=G(s,\omega ,y,z)$ satisfy (A4) and (A5), $F^i=F^i(s,\omega ,\mu ,X,y,z),i=1,2$ be two drivers satisfying (A4), and Φ satisfy (A7), (A8) and (A11). Moreover, we assume that

• (i) 

  One of the drivers $F^i$ satisfies assumption (A6);

  (ii) 

  One of the drivers $F^i$ satisfies assumption (A10).

Denote by $(X^1,Q^1)$ and $(X^2,Q^2)$ the solutions of the forward equation of general mean-field FBDSDE (14) with data $({\eta }^1,f^1,g)$ and $({\eta }^2,f^2,g)$, $(Y^1,Z^1)$ and $(Y^2,Z^2)$ the solutions of the backward equation of general mean-field FBDSDE (14) with data $(\mathsf{\Phi },F^1,G)$ and $(\mathsf{\Phi },F^2,G)$, respectively:

$$Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{Y_s},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s.$$

Then, if $X_T^1\le X_T^2,$ P-a.s., and for all $\mu ,y,z,$$F^1(s,\mu ,X,y,z)\le F^2(s,\mu ,X,y,z)$, dsdP-a.e., it also holds that $Y_s^1\le Y_s^2,$ for all $s\in [0,T],$ P-a.s.

Proof of Lemma 4.

For notational simplicity, we assume $l=d=1$. Without loss of generality, we assume that the driver $F^1$ satisfies (A6) and (A10). Let us define $({\widehat{Y}}_t,{\widehat{Z}}_t):=(Y_t^1-Y_t^2,Z_t^1-Z_t^2),t\in [0,T]$. From It$\widehat{\mathrm{o}}$’s formula, we have

$$\begin{array}{cc}\hfill & E[{({\widehat{Y}}_t^{+})}^2]+E[{\int }_t^T{I}_{\{{\widehat{Y}}_s\ge 0\}}|{\widehat{Z}}_s{|}^{2}ds]\hfill \\ \hfill =& 2E[{\int }_t^T{\widehat{Y}}_s^{+}(F^1(s,P_{Y_s^1},X_s,Y_s^1,Z_s^1)-F^2(s,P_{Y_s^2},X_s,Y_s^2,Z_s^2))ds]\hfill \\ \hfill & +E[{\int }_t^T{I}_{\{{\widehat{Y}}_s\ge 0\}}|G(s,Y_s^1,Z_s^1)-G(s,Y_s^2,Z_s^2){|}^{2}ds]\hfill \end{array}$$

Notice that, since $F^1$ and G are Lipschitz continuous and $F^1(s,\mu ,X,y,z)\le F^2(s,\mu ,X,y,z),$ dsdP-a.e., for all $\mu ,y,z$, and by the inequality $2ab\le \frac{1}{\lambda }{a}^2+\lambda b^2,\lambda >0$, there is

$$\begin{array}{cc}\hfill & E[{({\widehat{Y}}_t^{+})}^2]+E[{\int }_t^T{I}_{\{{\widehat{Y}}_s\ge 0\}}|{\widehat{Z}}_s{|}^{2}ds]\hfill \\ \hfill \le & 2E[{\int }_t^T{\widehat{Y}}_s^{+}(F^1(s,P_{Y_s^1},X_s,Y_s^1,Z_s^1)-F^1(s,P_{Y_s^2},X_s,Y_s^1,Z_s^1)+F^1(s,P_{Y_s^2},X_s,Y_s^1,Z_s^1)\hfill \\ \hfill & -F^1(s,P_{Y_s^2},X_s,Y_s^2,Z_s^2)+F^1(s,P_{Y_s^2},X_s,Y_s^2,Z_s^2)-F^2(s,P_{Y_s^2},X_s,Y_s^2,Z_s^2))ds]\hfill \\ \hfill & +E[{\int }_t^T{I}_{\{{\widehat{Y}}_s\ge 0\}}(c_0|{\widehat{Z}}_s{|}^2+\gamma |{\widehat{Z}}_s{|}^2)ds]\hfill \\ \hfill \le & 2E[{\int }_t^T{\widehat{Y}}_s^{+}(L(E[|{\widehat{Y}}_s^{+}{|}^2{])}^{\frac{1}{2}}+c_0|{\widehat{Y}}_s|+c_0|{\widehat{Z}}_s|)ds]+E[{\int }_t^T{I}_{\{{\widehat{Y}}_s\ge 0\}}(c_0|{\widehat{Z}}_s{|}^2+\gamma |{\widehat{Z}}_s{|}^2)ds]\hfill \\ \hfill \le & (2L+3c_0+\frac{2c_0^2}{1-\gamma })E[{\int }_t^T{({\widehat{Y}}_s^{+})}^{2}ds]+\frac{1+\gamma }{2}E[{\int }_t^T{I}_{\{{\widehat{Y}}_s\ge 0\}}|{\widehat{Z}}_s{|}^{2}ds],t\in [0,T].\hfill \end{array}$$

From Gronwall’s inequality, $E[{({\widehat{Y}}_t^{+})}^2]=0,t\in [0,T],$ i.e., $Y_t^1\le Y_t^2,$ P-a.s., $t\in [0,T]$. Hence, $Y_t^1\le Y_t^2,t\in [0,T],$ P-a.s.  □

3. General Mean-Field FBDSDEs with Continuous Coefficients

In this section, we prove the existence of the maximal solution for general mean-field FBDSDEs under some much weaker monotone continuity conditions, which can be regarded as the application of the above two comparison theorems. We consider the general mean-field FBDSDE as follows: for $\eta \in L^2(\mathsf{\Omega },{\mathcal{F}}_0,P;\mathbb{R})$, $t\in [0,T]$,

$$\left\{\begin{array}{c}{X}_t=\eta +{\int }_0^{t}f(s,P_{(X_s,Y_s)},X_s,Q_s,Y_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,\hfill \\ Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{(X_s,Y_s)},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s\hfill \\ -{\int }_t^T{Z}_{s}d{W}_s,\hfill \end{array}\right.$$

(15)

where $f:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\to \mathbb{R}$, $g:[0,T]\times \mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^l\to {\mathbb{R}}^d$, $F:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d\to \mathbb{R}$, $G:[0,T]\times \mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^d\to {\mathbb{R}}^l$, $\mathsf{\Phi }:\mathsf{\Omega }\times {\mathcal{P}}_2(\mathbb{R})\times \mathbb{R}\to \mathbb{R}$ are jointly measurable and satisfy the following assumptions:

(B1) 

$f(s,\omega ,{\delta }_0^{(4)},0,0,0)\in {\mathcal{H}}^2(0,T;\mathbb{R})$, $g(s,\omega ,0,0)\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$, $F(s,\omega ,{\delta }_0^{(4)},0,0,0)\in {\mathcal{H}}^2(0,$$T;\mathbb{R})$, $G(s,\omega ,0,0)\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)$, $\mathsf{\Phi }({\delta }_0^{(3)},0)\in {\mathcal{H}}^2(0,T;\mathbb{R})$, where ${\delta }_0^{(4)}$ denotes throughout the Dirac measure with mass at $0\in {\mathbb{R}}^2$;

(B2) 

There are constants $K>0, 0<\gamma <1$ such that, for all $s\in [0,T]$, $x,x^{\prime },y,y^{\prime }\in \mathbb{R},q,q^{\prime }\in {\mathbb{R}}^l,z,z^{\prime }\in {\mathbb{R}}^d$, $X,Y\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R}),$

$|f(s,P_{(X,Y)},x,q,y)|\le K(1+W_2(P_X,{\delta }_0^{(3)})+|x|+|q|)$,

$|g(s,x,q)-g(s,x^{\prime },q^{\prime }){|}^2\le K|x-x^{\prime }{|}^2+\gamma {|q-q^{\prime }|}^2,$

$|F(s,P_{(X,Y)},x,y,z)|\le K(1+W_2(P_{(X,Y)},{\delta }_0^{(4)})+|x|+|y|+|z|),$

$|G(s,y,z)-G(s,y^{\prime },z^{\prime }){|}^2\le K|y-y^{\prime }{|}^2+\gamma {|z-z^{\prime }|}^2,$

$|\mathsf{\Phi }(P_X,x)|\le K(1+W_2(P_X,{\delta }_0^{(3)})+|x|)$;

(B3) 

For all $\theta ,{\theta }^{\prime }\in L^2(\mathsf{\Omega },\mathcal{F},P;{\mathbb{R}}^2),X,X^{\prime }\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R}),$$s\in [0,T]$, $x,y\in \mathbb{R},q\in {\mathbb{R}}^l,z\in {\mathbb{R}}^d$,

$f(s,P_{\theta },x,q,y)\le f(s,P_{{\theta }^{\prime }},x,q,y)$, dsdP-a.e., $\theta \le {\theta }^{\prime }$, P-a.s.,

$F(s,P_{\theta },x,y,z)\le F(s,P_{{\theta }^{\prime }},x,y,z)$, dsdP-a.e., $\theta \le {\theta }^{\prime }$, P-a.s.,

$\mathsf{\Phi }(P_X,x)\le \mathsf{\Phi }(P_{X^{\prime }},x)$, dsdP-a.e., $X\le X^{\prime }$, P-a.s.,

where ${\theta }_i$ is the ith component of $\theta$, $\theta \le {\theta }^{\prime }$ means ${\theta }_i\le {\theta }_i^{\prime },i=1,2$;

(B4) 

For almost every $(s,\omega )\in [0,T]\times \mathsf{\Omega },$$f(s,\omega ,·,·,·,·),F(s,\omega ,·,·,·,·)$ and $\mathsf{\Phi }(\omega ,·,·)$ are continuous with a continuity modulus $\rho :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ for $\mu$, for all ${\mu }_1,{\mu }_1^{\prime }\in {\mathcal{P}}_2({\mathbb{R}}^2),{\mu }_2,{\mu }_2^{\prime }$$\in {\mathcal{P}}_2(\mathbb{R}),$$(s,\omega ,x,q,y,z)\in [0,T]\times \mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times {\mathbb{R}}^d,$

$|f(s,\omega ,{\mu }_1,x,q,y)-f(s,\omega ,{\mu }_1^{\prime },x,q,y)|\le \rho (W_2({\mu }_1,{\mu }_1^{\prime })),$

$|F(s,\omega ,{\mu }_1,x,y,z)-F(s,\omega ,{\mu }_1^{\prime },x,y,z)|\le \rho (W_2({\mu }_1,{\mu }_1^{\prime })),$

$|\mathsf{\Phi }(\omega ,{\mu }_2,x)-\mathsf{\Phi }(\omega ,{\mu }_2^{\prime },x)|\le \rho (W_2({\mu }_2,{\mu }_2^{\prime })),$

where $\rho$ is supposed to be nondecreasing such that $\rho (0+)=0$;

(B5) 

Quasimonotonous growth. For all $(s,\mu ,\nu ,x,x^{\prime },q,y,y^{\prime },z)\in [0,T]\times {\mathcal{P}}_2({\mathbb{R}}^2)\times {\mathcal{P}}_2(\mathbb{R})\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d$, there is $f(s,\mu ,x,q,y)\le f(s,\mu ,x,q,y^{\prime })$, as $y\le y^{\prime }$; $F(s,\mu ,x,y,z)$$\le F(s,\mu ,x^{\prime },y,z)$, as $x\le x^{\prime }$; and $\mathsf{\Phi }(\nu ,x)\le \mathsf{\Phi }(\nu ,x^{\prime })$, as $x\le x^{\prime }$.

Before we come to our main results, let us introduce a technical lemma with the following notations: for $\mu ,\nu \in {\mathcal{P}}_2({\mathbb{R}}^2)$,

$$\begin{array}{c}\hfill W_{2,-}(\mu ,\nu )=\inf \left\{{\left({\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}{|{(x-y)}^{-}|}^2\rho (dxdy)\right)}^{\frac{1}{2}},\rho \in {\mathcal{P}}_2({\mathbb{R}}^4),\rho (.\times {\mathbb{R}}^2)=\mu ,\rho ({\mathbb{R}}^2\times .)=\nu \right\}.\end{array}$$

Lemma 5.

Let $f:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\to \mathbb{R},$$F:[0,T]\times \mathsf{\Omega }\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d\to \mathbb{R}$ be functions satisfying assumptions (B1)–(B5), then the sequence of functions

$$\begin{array}{cc}\hfill {\overline{f}}^n(s,\omega ,\mu ,x,q,y)=\underset{(\nu ,a,b,e)\in {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}}{\mathrm{esssup}}\{& f(s,\omega ,\nu ,a,b,e)-n{W}_{2,-}(\mu ,\nu )-n|x-a|\hfill \\ \hfill & -n|q-b|-n|y-e|\},\hfill \\ \hfill {\overline{F}}^n(s,\omega ,\mu ,x,y,z)=\underset{(\nu ,a,e,p)\in {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d}{\mathrm{esssup}}\{& F(s,\omega ,\nu ,a,e,p)-n{W}_{2,-}(\mu ,\nu )-n|x-a|\hfill \\ \hfill & -n|y-e|-n|z-p|\},\hfill \end{array}$$

are well defined for $n\ge K$ and have the following properties:

• (i) 

  Linear growth: For all $(s,\mu ,x,q,y,z)\in [0,T]\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times {\mathbb{R}}^d,$

$$\begin{array}{cc}\hfill -K(1+W_2(P_X,{\delta }_0^{(3)})+|x|+|q|)\le & f(s,\mu ,x,q,y)\le {\overline{f}}^n(s,\mu ,x,q,y)\hfill \\ \hfill \le & K(1+W_2(P_X,{\delta }_0^{(3)})+|x|+|q|),\mathit{\text{P-a.s.}}\hfill \\ \hfill -K(1+W_2(P_{(X,Y)},{\delta }_0^{(4)})+|x|+|y|+|z|)\le & F(s,\mu ,x,y,z)\le {\overline{F}}^n(s,\mu ,x,y,z)\hfill \\ \hfill \le & K(1+W_2(P_{(X,Y)},{\delta }_0^{(4)})+|x|+|y|+|z|),\mathit{\text{P-a.s.}}\hfill \end{array}$$

• (ii) 

  Monotonicity in μ: For all $(x,q,y,z)\in \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times {\mathbb{R}}^d$, and $\theta ,{\theta }^{\prime }\in L^2(\mathsf{\Omega },\mathcal{F},P;{\mathbb{R}}^2)$ with $\theta \le {\theta }^{\prime }$, P-a.s.,

$$\begin{array}{c}\hfill {\overline{f}}^n(s,P_{\theta },x,q,y)\le {\overline{f}}^n(s,P_{{\theta }^{\prime }},x,q,y),\mathit{dsd}\mathit{\text{P-a.e.}}\\ \hfill {\overline{F}}^n(s,P_{\theta },x,y,z)\le {\overline{F}}^n(s,P_{{\theta }^{\prime }},x,y,z),\mathit{dsd}\mathit{\text{P-a.e.}}\end{array}$$

• (iii) 

  Monotonicity in n: For all $(s,\mu ,x,q,y,z)\in [0,T]\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times {\mathbb{R}}^d,$

$$\begin{array}{cc}\hfill & {\overline{f}}^{n+1}(s,\mu ,x,q,y)\le {\overline{f}}^n(s,\mu ,x,q,y),\mathit{\text{P-a.s.}}\hfill \\ \hfill & {\overline{F}}^{n+1}(s,\mu ,x,y,z)\le {\overline{F}}^n(s,\mu ,x,y,z),\mathit{\text{P-a.s.}}\hfill \end{array}$$

• (iv) 

  Lipschitz condition: For all $(s,\mu ,{\mu }^{\prime },x,x^{\prime },q,q^{\prime },y,y^{\prime },z,z^{\prime })\in [0,T]\times {\mathcal{P}}_2({\mathbb{R}}^2)\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^l\times {\mathbb{R}}^l\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d\times {\mathbb{R}}^d$,

$$\begin{array}{cc}\hfill & |{\overline{f}}^n(s,\mu ,x,q,y)-{\overline{f}}^n(s,{\mu }^{\prime },x^{\prime },q^{\prime },y^{\prime })|\hfill \\ \hfill \le & n(W_2(\mu ,{\mu }^{\prime })+|x-x^{\prime }|+|q-q^{\prime }|+|y-y^{\prime }|),\mathit{\text{P-a.s.}}\hfill \\ \hfill & |{\overline{F}}^n(s,\mu ,x,y,z)-{\overline{F}}^n(s,{\mu }^{\prime },x^{\prime },y^{\prime },z^{\prime })|\hfill \\ \hfill \le & n(W_2(\mu ,{\mu }^{\prime })+|x-x^{\prime }|+|y-y^{\prime }|+|z-z^{\prime }|),\mathit{\text{P-a.s.}}\hfill \end{array}$$

• (v) 

  Quasimonotonous growth: For all $(s,\mu ,x,x^{\prime },q,y,y^{\prime },z)\in [0,T]\times {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d$, there are ${\overline{f}}^n(s,\mu ,x,q,y)\le {\overline{f}}^n(s,\mu ,x,q,y^{\prime })$, P-a.s., as $y\le y^{\prime }$, and ${\overline{F}}^n(s,\mu ,x,y,z)\le {\overline{F}}^n(s,\mu ,x^{\prime },y,z)$, P-a.s., as $x\le x^{\prime }$.

  (vi) 

  Strong convergence: If $({\mu }_n,x_n,q_n,y_n,z_n)\to (\mu ,x,q,y,z)$ in ${\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times {\mathbb{R}}^d$ when $n\to \infty$, then ${\overline{f}}^n(s,{\mu }_n,x_n,q_n,y_n)$$\to f(s,\mu ,x,q,y)$, ${\overline{F}}^n(s,{\mu }_n,x_n,y_n,z_n)$$\to F(s,\mu ,x,y,z)$ when $n\to \infty$.

The proof of Lemma 5 is similar to that of Li et al. [17,19].

Remark 1.

Let ${\overline{f}}^n$ and ${\overline{F}}^n$ be monotonic in μ and Lipschitz (Lipschitz constant n), then, for all $\theta ,{\theta }^{\prime }\in L^2(\mathsf{\Omega },\mathcal{F},P;{\mathbb{R}}^2)$ and $(s,x,q,y,z)\in [0,T]\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}\times {\mathbb{R}}^d$, there is $|{\overline{f}}^n(s,P_{\theta },x,q,y)-{\overline{f}}^n(s,P_{{\theta }^{\prime }},x,q,y)|\le n(E[|{(\theta -{\theta }^{\prime })}^{+}{|}^2{])}^{\frac{1}{2}}$, P-a.s., and $|{\overline{F}}^n(s,P_{\theta },x,y,z)-{\overline{F}}^n(s,P_{{\theta }^{\prime }},x,y,z)|\le n(E[|{(\theta -{\theta }^{\prime })}^{+}{|}^2{])}^{\frac{1}{2}},$ P-a.s.

We now consider $h_1(s,\omega ,\mu ,x,q)=-K(1+W_2(\mu ,{\delta }_0^{(3)})+|x|+|q|),h_2(s,\omega ,\mu ,x,q)=K(1+W_2(\mu ,{\delta }_0^{(3)})+|x|+|q|)$, $H_1(s,\omega ,\mu ,{\mu }^{\prime },x,x^{\prime },y,z)=-K(1+W_2(\mu ,{\delta }_0^{(4)})+W_2({\mu }^{\prime },{\delta }_0^{(4)})+|x|+|x^{\prime }|+|y|+|z|)$, $H_2(s,\omega ,\mu ,{\mu }^{\prime },x,x^{\prime },y,z)=K(1+W_2(\mu ,{\delta }_0^{(4)})+W_2({\mu }^{\prime },{\delta }_0^{(4)})+|x|+|x^{\prime }|+|y|+|z|)$. We notice that $h_1,h_2,H_1,H_2$ are jointly measurable, and $h_1,h_2$ are Lipschitz in $(\mu ,x,q)$, $H_1,H_2$ are Lipschitz in $(\mu ,{\mu }^{\prime },x,x^{\prime },y,z)$, uniformly in $(s,\omega )$.

Theorem 1.

Under the assumptions (B1)–(B5), the general mean-field FBDSDE (15) has a maximal solution $(X,Q,Y,Z)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)\times {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$.

Proof of Theorem 1.

In order to construct a maximal solution of (15), our basic idea is to consider the following equation: for all $n\ge 1$, $\eta \in L^2(\mathsf{\Omega },{\mathcal{F}}_0,P;\mathbb{R})$,

$$\left\{\begin{array}{c}{X}_t^n=\eta +{\int }_0^{t}f(s,P_{(X_s^n,Y_s^n)},X_s^n,Q_s^n,Y_s^n)ds+{\int }_0^{t}g(s,X_s^n,Q_s^n)d{W}_s-{\int }_0^t{Q}_s^{n}d{\overleftarrow{B}}_s,\hfill \\ Y_t^n=\mathsf{\Phi }(P_{X_T^{n-1}},X_T^{n-1})+{\int }_t^{T}F(s,P_{(X_s^{n-1},Y_s^n)},X_s^{n-1},Y_s^n,Z_s^n)ds\hfill \\ -{\int }_t^{T}G(s,Y_s^n,Z_s^n)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{n}d{W}_s,0\le t\le T.\hfill \end{array}\right.$$

(16)

We will prove that $(X^n,Q^n,Y^n,Z^n)$ is monotonic and its limit verifies (15). Now, we divide the proof into four steps.

Step 1: Construction of the starting point.

Consider the following two general mean-field doubly SDEs:

$$\begin{array}{cc}\hfill & {\tilde{X}}_t^0=\eta +{\int }_0^t{h}_1(s,P_{{\tilde{X}}_s^0},{\tilde{X}}_s^0,{\tilde{Q}}_s^0)ds+{\int }_0^{t}g(s,{\tilde{X}}_s^0,{\tilde{Q}}_s^0)d{W}_s-{\int }_0^t{\tilde{Q}}_s^{0}d{\overleftarrow{B}}_s,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & X_t^0=\eta +{\int }_0^t{h}_2(s,P_{X_s^0},X_s^0,Q_s^0)ds+{\int }_0^{t}g(s,X_s^0,Q_s^0)d{W}_s-{\int }_0^t{Q}_s^{0}d{\overleftarrow{B}}_s.\hfill \end{array}$$

(18)

We obtain from Lemma 1 that the above two SDEs (17) and (18) have unique solutions $({\tilde{X}}^0,{\tilde{Q}}^0)$ and $(X^0,Q^0),$ respectively, which satisfy $\parallel {\tilde{X}}^0{\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}+{\parallel {\tilde{Q}}^0\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^l)}\le {\kappa }_1,$$\parallel X^0{\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}+{\parallel Q^0\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^l)}\le {\kappa }_1$, and then we obtain from Lemma 3 that ${\tilde{X}}^0\le X^0,$ P-a.s.

Similarly, from Lemma 2, it follows that the following general mean-field BDSDEs (19) and (20) have unique solutions $({\tilde{Y}}^0,{\tilde{Z}}^0)$ and $(Y^0,Z^0)$, respectively, and the solutions satisfy $\parallel {\tilde{Y}}^0{\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}+{\parallel {\tilde{Z}}^0\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^d)}\le {\kappa }_2$ and $\parallel Y^0{\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}+{\parallel Z^0\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^d)}\le {\kappa }_2$:

$$\begin{array}{cc}\hfill {\tilde{Y}}_t^0=& \mathsf{\Phi }(P_{{\tilde{X}}_T^0},{\tilde{X}}_T^0)+{\int }_t^T{H}_1(s,P_{(X_s^0,{\tilde{Y}}_s^0)},P_{({\tilde{X}}_s^0,{\tilde{Y}}_s^0)},X_s^0,{\tilde{X}}_s^0,{\tilde{Y}}_s^0,{\tilde{Z}}_s^0)ds\hfill \\ \hfill & -{\int }_t^{T}G(s,{\tilde{Y}}_s^0,{\tilde{Z}}_s^0)d{\overleftarrow{B}}_s-{\int }_t^T{\tilde{Z}}_s^{0}d{W}_s,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill Y_t^0=& \mathsf{\Phi }(P_{X_T^0},X_T^0)+{\int }_t^T{H}_2(s,P_{(X_s^0,Y_s^0)},P_{({\tilde{X}}_s^0,Y_s^0)},X_s^0,{\tilde{X}}_s^0,Y_s^0,Z_s^0)ds\hfill \\ \hfill & -{\int }_t^{T}G(s,Y_s^0,Z_s^0)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{0}d{W}_s.\hfill \end{array}$$

(20)

Step 2: Construction of $(X^1,Q^1,Y^1,Z^1)$.

From Step 1 we know that $(X^0,Q^0)$ is the solution of (18), so we will prove that the following general mean-field BDSDE has a solution $(Y^1,Z^1)$:

$$\begin{array}{cc}\hfill Y_t^1=& \mathsf{\Phi }(P_{X_T^0},X_T^0)+{\int }_t^{T}F(s,P_{(X_s^0,Y_s^1)},X_s^0,Y_s^1,Z_s^1)ds\hfill \\ \hfill & -{\int }_t^{T}G(s,Y_s^1,Z_s^1)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{1}d{W}_s,t\in [0,T].\hfill \end{array}$$

(21)

Now, for $m\ge K$, let us define

$$\begin{array}{cc}\hfill F^{1,m}(s,\omega ,P_Y,y,z):=& {\overline{F}}^m(s,\omega ,P_{(X_s^0,Y)},X_s^0,y,z)\hfill \\ \hfill :=& \underset{(\nu ,a,e,p)\in {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d}{\mathrm{esssup}}\{F(s,\omega ,\nu ,a,e,p)-m{W}_{2,-}(P_{(X_s^0,Y)},\nu )\hfill \\ \hfill & -m|X_s^0-a|-m|y-e|-m|z-p|\},\hfill \end{array}$$

then, from Lemma 5 the sequence $F^{1,m}$ satisfies the following properties: for all $y,y^{\prime }\in \mathbb{R},$$z,z^{\prime }\in {\mathbb{R}}^d,$$\vartheta ,{\vartheta }^{\prime }\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$

$$\begin{array}{cc}\hfill & (\mathrm{i})|F^{1,m}(s,P_{\vartheta },y,z)|\le K(1+W_2(P_{(X_s^0,\vartheta )},{\delta }_0^{(4)})+|X_s^0|+|y|+|z|),P\text{-a.s.};\hfill \\ \hfill & (\mathrm{ii})|F^{1,m}(s,P_{\vartheta },y,z)-F^{1,m}(s,P_{{\vartheta }^{\prime }},y^{\prime },z^{\prime })|\le m(W_2(P_{(X_s^0,\vartheta )},P_{(X_s^0,{\vartheta }^{\prime })})+|y-y^{\prime }|+|z-z^{\prime }|),P\text{-a.s.};\hfill \\ \hfill & (\mathrm{iii})F^{1,m}(s,P_{\vartheta },y,z)\le F^{1,m}(s,P_{{\vartheta }^{\prime }},y,z),\mathit{dsdP}\text{-a.e.},\mathrm{when}\vartheta \le {\vartheta }^{\prime },P\text{-a.s.};\hfill \\ \hfill & (\mathrm{iv})F(s,P_{(X_s^0,\vartheta )},X_s^0,y,z)\le F^{1,m+1}(s,P_{\vartheta },y,z)\le F^{1,m}(s,P_{\vartheta },y,z),P\text{-a.s.};\hfill \\ \hfill & (\mathrm{v})F^{1,m}(s,\mu ,x,y,z)\le F^{1,m}(s,\mu ,x^{\prime },y,z),P\text{-a.s.},\mathrm{when}x\le x^{\prime };\hfill \\ \hfill & (\mathrm{vi})F^{1,m}(s,{\mu }_m,y_m,z_m)\to F(s,\mu ,X_s^0,y,z),\mathrm{when}({\mu }_m,y_m,z_m)\to (\mu ,y,z),\mathrm{where}\mu =P_{(X_s^0,\vartheta )}.\hfill \end{array}$$

(22)

From Lemma 2, the following general mean-field BDSDE has an unique solution $(Y^{1,m},Z^{1,m})$$\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$:

$$\begin{array}{cc}\hfill Y_t^{1,m}=& \mathsf{\Phi }(P_{X_T^0},X_T^0)+{\int }_t^T{F}^{1,m}(s,P_{Y_s^{1,m}},Y_s^{1,m},Z_s^{1,m})ds\hfill \\ \hfill & -{\int }_t^{T}G(s,Y_s^{1,m},Z_s^{1,m})d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{1,m}d{W}_s,t\in [0,T].\hfill \end{array}$$

(23)

From (22)—(i), (iv) and Lemma 4, it follows that

$${\tilde{Y}}_t^0\le Y_t^{1,m+1}\le Y_t^{1,m}\le Y_t^0,t\in [0,T],P\text{-a.s.}$$

(24)

Apply It$\widehat{\mathrm{o}}$’s formula to $|Y_t^{1,m}{|}^2$ we have

$$\begin{array}{cc}\hfill & |Y_t^{1,m}{|}^2+{\int }_t^T|Z_s^{1,m}{|}^{2}ds={|\mathsf{\Phi }(P_{X_T^0},X_T^0)|}^2+2{\int }_t^T{Y}_s^{1,m}{F}^{1,m}(s,P_{Y_s^{1,m}},Y_s^{1,m},Z_s^{1,m})ds\hfill \\ \hfill & -2{\int }_t^T{Y}_s^{1,m}{Z}_s^{1,m}d{W}_s-2{\int }_t^T{Y}_s^{1,m}G(s,Y_s^{1,m},Z_s^{1,m})d{\overleftarrow{B}}_s+{\int }_t^T{|G(s,Y_s^{1,m},Z_s^{1,m})|}^{2}ds.\hfill \end{array}$$

(25)

Taking expectations on both sides of the above Equation (25), then by the inequality $2ab\le \frac{1}{\lambda }{a}^2+\lambda b^2,\lambda >0$, there exist constants ${\lambda }_1,{\lambda }_2>0$ such that

$$\begin{array}{cc}\hfill & E[|Y_t^{1,m}{|}^2]+E[{\int }_t^T|Z_s^{1,m}{|}^{2}ds]\hfill \\ \hfill =& E[|\mathsf{\Phi }(P_{X_T^0},X_T^0){|}^2]+2E[{\int }_t^T{Y}_s^{1,m}{F}^{1,m}(s,P_{Y_s^{1,m}},Y_s^{1,m},Z_s^{1,m})ds]+E[{\int }_t^T|G(s,Y_s^{1,m},Z_s^{1,m}){|}^{2}ds]\hfill \\ \hfill \le & K^{2}E[(1+W_2(P_{X_T^0},{\delta }_0^{(3)})+|X_T^0{|)}^2]+2KE[{\int }_t^T|Y_s^{1,m}|(1+W_2(P_{(X_s^0,Y_s^{1,m})},{\delta }_0^{(4)})+|X_s^0|\hfill \\ \hfill & +|Y_s^{1,m}|+|Z_s^{1,m}|)ds]+E[{\int }_t^T|G(s,Y_s^{1,m},Z_s^{1,m})-G(s,0,0){|}^{2}ds]+E[{\int }_t^T|G(s,0,0){|}^{2}ds]\hfill \\ \hfill & +2E[{\int }_t^T|G(s,Y_s^{1,m},Z_s^{1,m})-G(s,0,0)||G(s,0,0)|ds]\hfill \\ \hfill \le & K^{2}E[{\left(1+W_2(P_{X_T^0},{\delta }_0^{(3)})+|X_T^0|\right)}^2]+{\lambda }_{1}KE[{\int }_t^T(1+W_2(P_{(X_s^0,Y_s^{1,m})},{\delta }_0^{(4)})+|X_s^0|+|Y_s^{1,m}|\hfill \\ \hfill & +|Z_s^{1,m}|{)}^{2}ds]+\frac{K}{{\lambda }_1}E[{\int }_t^T|Y_s^{1,m}{|}^{2}ds]+(1+\frac{1}{{\lambda }_2})E[{\int }_t^T|G(s,Y_s^{1,m},Z_s^{1,m})-G(s,0,0){|}^{2}ds]\hfill \\ \hfill & +(1+{\lambda }_2)E[{\int }_t^T|G(s,0,0){|}^{2}ds]\hfill \\ \hfill \le & C_1(1+E[{\int }_t^T|Y_s^{1,m}{|}^{2}ds])+(5{\lambda }_{1}K+\gamma +\frac{\gamma }{{\lambda }_2})E[{\int }_t^T|Z_s^{1,m}{|}^{2}ds],\hfill \end{array}$$

where $C_1$ depends on $K,T,{\kappa }_1$ and $E[{\int }_0^T|G(s,0,0){|}^{2}ds]$. Taking ${\lambda }_1=\frac{1-\gamma }{20K}$, ${\lambda }_2=\frac{4\gamma }{1-\gamma }$, then we have

$$E[|Y_t^{1,m}{|}^2]+\frac{1-\gamma }{2}E[{\int }_t^T|Z_s^{1,m}{|}^{2}ds]\le C_1(1+E[{\int }_t^T|Y_s^{1,m}{|}^{2}ds]),\mathrm{for}\mathrm{all}t\in [0,T].$$

According to Gronwall’s inequality, we obtain

$$\underset{t\in [0,T]}{\sup }E[|Y_t^{1,m}{|}^2]\le C_1,E[{\int }_0^T|Z_s^{1,m}{|}^{2}ds]\le C_1,$$

(26)

then, from $\parallel X^0{\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}+{\parallel Q^0\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^l)}\le {\kappa }_1$, (22)—(i) and (26) we have

$$\begin{array}{cc}\hfill & E[{\int }_0^T|F^{1,m}(s,P_{Y_s^{1,m}},Y_s^{1,m},Z_s^{1,m}){|}^{2}ds]\hfill \\ \hfill \le & K^{2}E[{\int }_0^T(1+W_2(P_{(X_s^0,Y_s^{1,m})},{\delta }_0^{(4)})+|X_s^0|+|Y_s^{1,m}|+|Z_s^{1,m}{|)}^{2}ds]\le C_1.\hfill \end{array}$$

(27)

On the other hand, let us take ${\sup }_{t\in [0,T]}$ first and then take the expectation for (25), by using the Burkholder–Davis–Gundy inequality; there exist constants ${\lambda }_1={\lambda }_2={\lambda }_3={\lambda }_6=1,{\lambda }_4={\lambda }_5=4,{\tilde{D}}_1={\tilde{D}}_2=1$ such that

$$\begin{array}{cc}\hfill & E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]\hfill \\ \hfill \le & K^{2}E[(1+W_2(P_{X_T^0},{\delta }_0^{(3)})+|X_T^0{|)}^2]+2KE[{\int }_0^T|Y_s^{1,m}|(1+W_2(P_{(X_s^0,Y_s^{1,m})},{\delta }_0^{(4)})+|X_s^0|+|Y_s^{1,m}|\hfill \\ \hfill & +|Z_s^{1,m}|)ds]+2{\left(E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]\right)}^{\frac{1}{2}}{\left(E[\underset{t\in [0,T]}{\sup }|{\int }_t^T{Z}_s^{1,m}d{W}_s{|}^2]\right)}^{\frac{1}{2}}+2{\left(E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]\right)}^{\frac{1}{2}}\hfill \\ \hfill & ·{\left(E[\underset{t\in [0,T]}{\sup }|{\int }_t^{T}G(s,Y_s^{1,m},Z_s^{1,m})d{\overleftarrow{B}}_s{|}^2]\right)}^{\frac{1}{2}}+E[{\int }_0^T|G(s,Y_s^{1,m},Z_s^{1,m}){|}^{2}ds]\hfill \\ \hfill \le & K^{2}E[(1+W_2(P_{X_T^0},{\delta }_0^{(3)})+|X_T^0{|)}^2]+\frac{K}{{\lambda }_3}E[{\int }_0^T|Y_s^{1,m}{|}^{2}ds]+{\lambda }_{3}KE[{\int }_0^T(1+W_2(P_{(X_s^0,Y_s^{1,m})},{\delta }_0^{(4)})\hfill \\ \hfill & +|X_s^0|+|Y_s^{1,m}|+|Z_s^{1,m}|{)}^{2}ds]+\frac{1}{{\lambda }_4}E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]+{\lambda }_4{\tilde{D}}_{1}E[{\int }_0^T|Z_s^{1,m}{|}^{2}ds]\hfill \\ \hfill & +\frac{1}{{\lambda }_5}E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]+({\lambda }_5{\tilde{D}}_2+1)E[{\int }_0^T|G(s,0,0){|}^{2}ds]+({\lambda }_5{\tilde{D}}_2+1)E[{\int }_0^T|G(s,Y_s^{1,m},Z_s^{1,m})\hfill \\ \hfill & {-G(s,0,0)|}^{2}ds]+2({\lambda }_5{\tilde{D}}_2+1)E[{\int }_0^T|G(s,Y_s^{1,m},Z_s^{1,m})-G(s,0,0)||G(s,0,0)|ds]\hfill \\ \hfill \le & 3K^2+6K^{2}E[|X_T^0{|}^2]+5{\lambda }_{3}KT+10{\lambda }_{3}KE[{\int }_0^T|X_s^0{|}^{2}ds]+(1+{\lambda }_6)({\lambda }_5{\tilde{D}}_2+1)E[{\int }_0^T|G(s,0,0){|}^{2}ds]\hfill \\ \hfill & +(5{\lambda }_{3}K+{\lambda }_4{\tilde{D}}_1+(1+\frac{1}{{\lambda }_6})({\lambda }_5{\tilde{D}}_2+1)\gamma )E[{\int }_0^T|Z_s^{1,m}{|}^{2}ds]+(\frac{1}{{\lambda }_4}+\frac{1}{{\lambda }_5})E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]\hfill \\ \hfill & +(\frac{K}{{\lambda }_3}+10{\lambda }_{3}K+(1+\frac{1}{{\lambda }_6})({\lambda }_5{\tilde{D}}_2+1)K)E[{\int }_0^T|Y_s^{1,m}{|}^{2}ds]\hfill \\ \hfill \le & \frac{1}{2}E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^{2}ds]+21KE[{\int }_0^T|Y_s^{1,m}{|}^{2}ds]+C_1.\hfill \end{array}$$

Then,

$$E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]\le 42KE[{\int }_0^T|Y_s^{1,m}{|}^{2}ds]+C_1.$$

From (26) we have

$$E[\underset{t\in [0,T]}{\sup }|Y_t^{1,m}{|}^2]\le C_1.$$

(28)

Combining (24) and (26), from the monotone bounded convergence theorem we obtain that ${\left\{Y^{1,m}\right\}}_{m\ge 1}$ has a limit, denoted by $Y^1=\underset{m\to \infty }{lim}{Y}^{1,m}$. Then, from Fatou’s Lemma we know that $E[{\sup }_{t\in [0,T]}|Y_t^1{|}^2]\le C_1.$ By using the dominated convergence theorem and Dini’s theorem, we obtain $E[{\sup }_{t\in [0,T]}|Y_t^{1,m}-Y_t^1{|}^2]\to 0,m\to \infty .$

Applying It$\widehat{\mathrm{o}}$’s formula to $|Y_t^{1,i}-Y_t^{1,m}{|}^2$ and then Hölder’s inequality and (27) we obtain

$$\begin{array}{cc}\hfill & E[|Y_0^{1,i}-Y_0^{1,m}{|}^2]+E[{\int }_0^T|Z_t^{1,i}-Z_t^{1,m}{|}^{2}dt]\hfill \\ \hfill \le & 2E[{\int }_0^T|Y_t^{1,i}-Y_t^{1,m}||F^{1,i}(t,P_{Y_t^{1,i}},Y_t^{1,i},Z_t^{1,i})-F^{1,m}(t,P_{Y_t^{1,m}},Y_t^{1,m},Z_t^{1,m})|dt]\hfill \\ \hfill & +E[{\int }_0^T|G(t,Y_t^{1,i},Z_t^{1,i})-G(t,Y_t^{1,m},Z_t^{1,m}){|}^{2}dt]\hfill \\ \hfill \le & 2(E[{\int }_0^T|Y_t^{1,i}-Y_t^{1,m}{|}^2{dt])}^{\frac{1}{2}}(E[{\int }_0^T|F^{1,i}(t,P_{Y_t^{1,i}},Y_t^{1,i},Z_t^{1,i})-F^{1,m}(t,P_{Y_t^{1,m}},Y_t^{1,m},Z_t^{1,m}){|}^{2}dt{])}^{\frac{1}{2}}\hfill \\ \hfill & +KE[{\int }_0^T|Y_t^{1,i}-Y_t^{1,m}{|}^{2}dt]+\gamma E[{\int }_0^T|Z_t^{1,i}-Z_t^{1,m}{|}^{2}dt]\hfill \\ \hfill \le & 4C_1(E[{\int }_0^T|Y_t^{1,i}-Y_t^{1,m}{|}^2{dt])}^{\frac{1}{2}}+KE[{\int }_0^T|Y_t^{1,i}-Y_t^{1,m}{|}^{2}dt]+\gamma E[{\int }_0^T|Z_t^{1,i}-Z_t^{1,m}{|}^{2}dt].\hfill \end{array}$$

Hence ${\left\{Z^{1,m}\right\}}_{m\ge 1}$ is a Cauchy sequence in ${\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$ and we have

$$E[{\int }_0^T(|Y_t^{1,i}-Y_t^{1,m}{|}^2+|Z_t^{1,i}-Z_t^{1,m}{|}^2)dt]\to 0,\mathrm{as}i,m\to \infty ,$$

(29)

that is, ${\left\{Z^{1,m}\right\}}_{m\ge 1}$ has a limit $Z^1\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$, denoted by $Z^1=\underset{m\to \infty }{lim}{Z}^{1,m}$.

Now let $\tau \in [0,T]$ be a stopping time. For all $m\ge K$ we have

$$\begin{array}{c}\hfill Y_{\tau }^{1,m}=Y_0^{1,m}-{\int }_0^{\tau }{F}^{1,m}(s,P_{Y_s^{1,m}},Y_s^{1,m},Z_s^{1,m})ds+{\int }_0^{\tau }G(s,Y_s^{1,m},Z_s^{1,m})d{\overleftarrow{B}}_s+{\int }_0^{\tau }{Z}_s^{1,m}d{W}_s.\end{array}$$

Notice that ${\int }_0^T{W}_2^2(P_{(X_t^0,Y_t^{1,i})},P_{(X_t^0,Y_t^{1,m})})ds\le E[{\int }_0^T|Y_t^{1,i}-Y_t^{1,m}{|}^{2}dt]\to 0,\mathrm{as}i,m\to \infty ,$ and then from (22), (26), (28) and (29), there exist ${\tilde{D}}_3,{\tilde{D}}_4>0$ such that

$$\begin{array}{cc}\hfill & E[{\int }_0^T|F^{1,m}(s,P_{Y_s^{1,m}},Y_s^{1,m},Z_s^{1,m})-F(s,P_{(X_s^0,Y_s^1)},X_s^0,Y_s^1,Z_s^1){|}^{2}ds]\to 0,\mathrm{as}m\to \infty ,\hfill \\ \hfill & E[\underset{t\in [0,T]}{\sup }|{\int }_0^t(G(s,Y_s^{1,m},Z_s^{1,m})-G(s,Y_s^1,Z_s^1))d{\overleftarrow{B}}_s{|}^2]\hfill \\ \hfill \le & {\tilde{D}}_{3}E[{\int }_0^T|G(s,Y_s^{1,m},Z_s^{1,m})-G(s,Y_s^1,Z_s^1){|}^{2}ds]\hfill \\ \hfill \le & {\tilde{D}}_{3}KE[{\int }_0^T|Y_s^{1,m}-Y_s^1{|}^{2}ds]+{\tilde{D}}_3\gamma E[{\int }_0^T|Z_s^{1,m}-Z_s^1{|}^{2}ds]\to 0,\mathrm{as}m\to \infty ,\hfill \\ \hfill & E[\underset{t\in [0,T]}{\sup }|{\int }_0^t{Z}_s^{1,m}d{W}_s-{\int }_0^t{Z}_s^{1}d{W}_s{|}^2]\le {\tilde{D}}_{4}E[{\int }_0^T|Z_s^{1,m}-Z_s^1{|}^{2}ds]\to 0,\mathrm{as}m\to \infty .\hfill \end{array}$$

It implies that for any stopping time $\tau$ we have

$$\begin{array}{c}\hfill Y_{\tau }^1=Y_0^1-{\int }_0^{\tau }F(s,P_{(X_s^0,Y_s^1)},X_s^0,Y_s^1,Z_s^1)ds+{\int }_0^{\tau }G(s,Y_s^1,Z_s^1)d{\overleftarrow{B}}_s+{\int }_0^{\tau }{Z}_s^{1}d{W}_s.\end{array}$$

Since $Y^1$ and the right-hand side are optional process, then apply the optional section theorem we have P-a.s. for any $t\in [0,T]$,

$$\begin{array}{c}\hfill Y_t^1=Y_0^1-{\int }_0^{t}F(s,P_{(X_s^0,Y_s^1)},X_s^0,Y_s^1,Z_s^1)ds+{\int }_0^{t}G(s,Y_s^1,Z_s^1)d{\overleftarrow{B}}_s+{\int }_0^t{Z}_s^{1}d{W}_s.\end{array}$$

As $Y_T^1=\mathsf{\Phi }(P_{X_T^0},X_T^0)$, then ${(Y_t^1,Z_t^1)}_{t\in [0,T]}$ is a solution of the general mean-field BDSDE (21).

Next, using the same method as that on $X^0$, we can construct $X^1$ based on $Y^1$. Now for $m\ge K$, we let

$$\begin{array}{cc}\hfill f^{1,m}(s,\omega ,P_X,x,q):=& {\overline{f}}^m(s,\omega ,P_{(X,Y_s^1)},x,q,Y_s^1)\hfill \\ \hfill =& \underset{(\nu ,a,b,e)\in {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times {\mathbb{R}}^l\times \mathbb{R}}{\mathrm{esssup}}\{f(s,\omega ,\nu ,a,b,e)-m{W}_{2,-}(P_{(X,Y_s^1)},\nu )\hfill \\ \hfill & -m|x-a|-m|q-b|-m|Y_s^1-e|\},\hfill \end{array}$$

then from Lemma 5 the sequence $f^{1,m}$ satisfies the following properties: for all $x,x^{\prime }\in \mathbb{R},$$q,q^{\prime }\in {\mathbb{R}}^l,$$\varsigma ,{\varsigma }^{\prime }\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$, $s\in [0,T]$,

$$\begin{array}{cc}\hfill & (\mathrm{i})|f^{1,m}(s,P_{\varsigma },x,q)|\le K(1+W_2(P_{(\varsigma ,Y_s^1)},{\delta }_0^{(4)})+|x|+|q|),P\text{-a.s.};\hfill \\ \hfill & (\mathrm{ii})|f^{1,m}(s,P_{\varsigma },x,q)-f^{1,m}(s,P_{{\varsigma }^{\prime }},x^{\prime },q^{\prime })|\le m(W_2(P_{(\varsigma ,Y_s^1)},P_{({\varsigma }^{\prime },Y_s^1)})+|x-x^{\prime }|+|q-q^{\prime }|),P\text{-a.s.};\hfill \\ \hfill & (\mathrm{iii})f^{1,m}(s,P_{\varsigma },x,q)\le f^{1,m}(s,P_{{\varsigma }^{\prime }},x,q),\mathit{dsdP}\text{-a.e.},\mathrm{when}\varsigma \le {\varsigma }^{\prime },P\text{-a.s.};\hfill \\ \hfill & (\mathrm{iv})f(s,P_{(\varsigma ,Y_s^1)},x,q,Y_s^1)\le f^{1,m+1}(s,P_{\varsigma },x,q)\le f^{1,m}(s,P_{\varsigma },x,q),P\text{-a.s.};\hfill \\ \hfill & (\mathrm{v})f^{1,m}(s,\mu ,x,q,y)\le f^{1,m}(s,\mu ,x,q,y^{\prime }),P\text{-a.s.},\mathrm{when}y\le y^{\prime };\hfill \\ \hfill & (\mathrm{vi})f^{1,m}(s,{\mu }_m,x_m,q_m)\to f(s,\mu ,x,q,Y_s^1),\mathrm{when}({\mu }_m,x_m,q_m)\to (\mu ,x,q),\mathrm{where}\mu =P_{(\varsigma ,Y_s^1)}.\hfill \end{array}$$

(30)

We consider the following SDE: for $\eta \in L^2(\mathsf{\Omega },{\mathcal{F}}_0,P;\mathbb{R})$,

$$X_t^{1,m}=\eta +{\int }_0^t{f}^{1,m}(s,P_{X_s^{1,m}},X_s^{1,m},Q_s^{1,m})ds+{\int }_0^{t}g(s,X_s^{1,m},Q_s^{1,m})d{W}_s-{\int }_0^t{Q}_s^{1,m}d{\overleftarrow{B}}_s,0\le t\le T.$$

(31)

We know from Lemma 1 that there is a unique solution $(X^{1,m},Q^{1,m})$ of Equation (31). From (30) and Lemma 3, there is ${\tilde{X}}_t^0\le X_t^{1,m+1}\le X_t^{1,m}\le X_t^0,$$t\in [0,T]$, P-a.s. So, there exist a upper semi-continuous process $X^1$ and a process $Q^1$ such that for all $t\in [0,T]$,

$$\begin{array}{cc}\hfill & \parallel X_t^{1,m}-X_t^1{\parallel }_{{\mathcal{S}}^2(0,T;\mathbb{R})}\to 0,\mathrm{as}m\to \infty ;\hfill \\ \hfill & \parallel Q_t^{1,m}-Q_t^1{\parallel }_{{\mathcal{H}}^2(0,T;{\mathbb{R}}^l)}\to 0,\mathrm{as}m\to \infty ;\hfill \\ \hfill & {\tilde{X}}_t^0\le X_t^1\le X_t^0,P\text{-a.s.}\hfill \end{array}$$

(32)

Now, let $\tau \in [0,T]$ be a stopping time. For all $m\ge K$, we obtain

$$X_{\tau }^{1,m}=\eta +{\int }_0^{\tau }{f}^{1,m}(s,P_{X_s^{1,m}},X_s^{1,m},Q_s^{1,m})ds+{\int }_0^{\tau }g(s,X_s^{1,m},Q_s^{1,m})d{W}_s-{\int }_0^{\tau }{Q}_s^{1,m}d{\overleftarrow{B}}_s.$$

(33)

From the dominated convergence theorem and (30), we obtain

$$E[\underset{t\in [0,T]}{\sup }{\int }_t^T|f^{1,m}(s,P_{X_s^{1,m}},X_s^{1,m},Q_s^{1,m})-f(s,P_{(X_s^1,Y_s^1)},X_s^1,Q_s^1,Y_s^1){|}^{2}ds]\to 0,\mathrm{as}m\to \infty .$$

From the Burkholder–Davis–Gundy inequality and (32), there exist ${\tilde{D}}_5,{\tilde{D}}_6>0$ such that

$$\begin{array}{cc}\hfill & E[\underset{t\in [0,T]}{\sup }|{\int }_t^T(g(s,X_s^{1,m},Q_s^{1,m})-g(s,X_s^1,Q_s^1))d{W}_s{|}^2]\hfill \\ \hfill \le & {\tilde{D}}_{5}KE[{\int }_0^T|X_s^{1,m}-X_s^1{|}^{2}ds]+{\tilde{D}}_5\gamma E[{\int }_0^T|Q_s^{1,m}-Q_s^1{|}^{2}ds]\to 0,\mathrm{as}m\to \infty ,\hfill \\ \hfill & E[\underset{t\in [0,T]}{\sup }|{\int }_t^T{Q}_s^{1,m}d{\overleftarrow{B}}_s-{\int }_t^T{Q}_s^{1}d{\overleftarrow{B}}_s{|}^2]\le {\tilde{D}}_{6}E[{\int }_0^T|Q_s^{1,m}-Q_s^1{|}^{2}ds]\to 0,\mathrm{as}m\to \infty .\hfill \end{array}$$

Taking the limit in (33), it follows that

$$\begin{array}{c}\hfill X_{\tau }^1=\eta +{\int }_0^{\tau }f(s,P_{(X_s^1,Y_s^1)},X_s^1,Q_s^1,Y_s^1)ds+{\int }_0^{\tau }g(s,X_s^1,Q_s^1)d{W}_s-{\int }_0^{\tau }{Q}_s^{1}d{\overleftarrow{B}}_s,\end{array}$$

and by applying the section theorem we find that $(X^1,Q^1)$ is the solution for the following equation

$$\begin{array}{c}\hfill X_t^1=\eta +{\int }_0^{t}f(s,P_{(X_s^1,Y_s^1)},X_s^1,Q_s^1,Y_s^1)ds+{\int }_0^{t}g(s,X_s^1,Q_s^1)d{W}_s-{\int }_0^t{Q}_s^{1}d{\overleftarrow{B}}_s,0\le t\le T,\end{array}$$

that is, $X^1$ is continuous, and similarly, $E[{\sup }_{t\in [0,T]}|X_t^{1,m}-X_t^1{|}^2]\to 0,m\to \infty .$

Step 3: Construction of $(X^n,Q^n,Y^n,Z^n)$.

Now, for $m\ge K$, we let

$$\begin{array}{cc}\hfill F^{2,m}(s,\omega ,P_Y,y,z):=& {\overline{F}}^m(s,\omega ,P_{(X_s^1,Y)},X_s^1,y,z)\hfill \\ \hfill :=& \underset{(\nu ,a,b,e)\in {\mathcal{P}}_2({\mathbb{R}}^2)\times \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^d}{\mathrm{esssup}}\{F(s,\omega ,\nu ,a,b,e)-m{W}_{2,-}(P_{(X_s^1,Y)},\nu )\hfill \\ \hfill & -m|X_s^1-a|-m|y-b|-m|z-e|\},\hfill \end{array}$$

It is obvious that the following general mean-field BDSDE has a unique solution $(Y^{2,m},Z^{2,m})\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$:

$$\begin{array}{cc}\hfill Y_t^{2,m}=& \mathsf{\Phi }(P_{X_T^1},X_T^1)+{\int }_t^T{F}^{2,m}(s,P_{Y_s^{2,m}},Y_s^{2,m},Z_s^{2,m})ds\hfill \\ \hfill & -{\int }_t^{T}G(s,Y_s^{2,m},Z_s^{2,m})d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{2,m}d{W}_s,t\in [0,T].\hfill \end{array}$$

From (32), we know $X_t^1\le X_t^0,$$t\in [0,T],$ P-a.s. From assumptions (B3) and (B5), it follows that $\mathsf{\Phi }(P_{X_T^1},X_T^1)\le \mathsf{\Phi }(P_{X_T^0},X_T^0)$. Moreover, from Lemma 5—(ii) and Lemma 5—(v), we have that for all $Y\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R}),y\in \mathbb{R},z\in {\mathbb{R}}^d$, $t\in [0,T]$, there is ${\overline{F}}^m(t,\omega ,P_{(X_t^1,Y)},X_t^1,y,z)\le {\overline{F}}^m(t,\omega ,P_{(X_t^0,Y)},X_t^0,y,z)$, P-a.s.; therefore, $F^{2,m}(t,\omega ,P_Y,y,z)\le F^{1,m}(t,\omega ,P_Y,y,z),$$t\in [0,T],$ P-a.s. By Lemma 4, it follows that $Y_t^{2,m}\le Y_t^{1,m}$, for all $t\in [0,T],$ P-a.s. Then, as $m\to \infty$, $Y_t^2\le Y_t^1,$ for all $t\in [0,T]$, P-a.s.

• Similarly, we prove that $(X^2,Q^2)$ is the solution of the SDE

  $$\begin{array}{c}\hfill X_t^2=\eta +{\int }_0^{t}f(s,P_{(X_s^2,Y_s^2)},X_s^2,Q_s^2,Y_s^2)ds+{\int }_0^{t}g(s,X_s^2,Q_s^2)d{W}_s-{\int }_0^t{Q}_s^{2}d{\overleftarrow{B}}_s,0\le t\le T,\end{array}$$

  and ${\tilde{X}}_t^0\le X_t^2\le X_t^1\le X_t^0,$ for all $t\in [0,T]$, P-a.s.

Repeating the same procedure, we obtain the existence of a sequence $(X^n,Q^n,Y^n,Z^n)$ which satisfies (16) and for all $t\in [0,T]$,

$$\begin{array}{c}\hfill {\tilde{X}}_t^0\le ···\le X_t^{n+1}\le X_t^n\le ···\le X_t^2\le X_t^1\le X_t^0,P\text{-a.s.};\\ \hfill {\tilde{Y}}_t^0\le ···\le Y_t^{n+1}\le Y_t^n\le ···\le Y_t^2\le Y_t^1\le Y_t^0,P\text{-a.s.}\end{array}$$

Therefore, there are two lower semi-continuous processes X and Y such that for all $t\in [0,T]$,

$$\begin{array}{c}\hfill X_t=\underset{n\to \infty }{lim}{X}_t^n,Y_t=\underset{n\to \infty }{lim}{Y}_t^n,P\text{-a.s.}\end{array}$$

Now, let $\tau \in [0,T]$ be a stopping time. For all $n\ge 1$, we have

$$\begin{array}{c}\hfill X_{\tau }^n=\eta +{\int }_0^{\tau }f(s,P_{(X_s^n,Y_s^n)},X_s^n,Q_s^n,Y_s^n)ds+{\int }_0^{\tau }g(s,X_s^n,Q_s^n)d{W}_s-{\int }_0^{\tau }{Q}_s^{n}d{\overleftarrow{B}}_s.\end{array}$$

Using the same method as that use for $X^1$, it is obvious that

$$\begin{array}{cc}\hfill & E[{\int }_0^T|f(s,P_{(X_s^n,Y_s^n)},X_s^n,Q_s^n,Y_s^n)-f(s,P_{(X_s,Y_s)},X_s,Q_s,Y_s){|}^{2}ds]+E[\underset{t\in [0,T]}{\sup }|{\int }_t^T(g(s,X_s^n,Q_s^n)\hfill \\ \hfill & -g(s,X_s,Q_s))d{W}_s{|}^2]+E[\underset{t\in [0,T]}{\sup }|{\int }_t^T{Q}_s^{n}d{\overleftarrow{B}}_s-{\int }_t^T{Q}_{s}d{\overleftarrow{B}}_s{|}^2]\to 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

Using the section theorem once again, it follows that

$$\begin{array}{c}\hfill X_t=\eta +{\int }_0^{t}f(s,P_{(X_s,Y_s)},X_s,Q_s,Y_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,t\in [0,T].\end{array}$$

To finish the proof, let us focus on the backward equation

$$\begin{array}{cc}\hfill Y_t^n=& \mathsf{\Phi }(P_{X_T^{n-1}},X_T^{n-1})+{\int }_t^{T}F(s,P_{(X_s^{n-1},Y_s^n)},X_s^{n-1},Y_s^n,Z_s^n)ds\hfill \\ \hfill & -{\int }_t^{T}G(s,Y_s^n,Z_s^n)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{n}d{W}_s,0\le t\le T.\hfill \end{array}$$

Applying It$\widehat{\mathrm{o}}$’s formula to $|Y_t^n-Y_t^m{|}^2$, we obtain

$$\begin{array}{cc}\hfill & E[|Y_0^n-Y_0^m{|}^2]+E[{\int }_0^T|Z_t^n-Z_t^m{|}^{2}dt]\hfill \\ \hfill \le & 2E[{\int }_0^T|Y_t^n-Y_t^m||F(t,P_{(X_t^{n-1},Y_t^n)},X_t^{n-1},Y_s^n,Z_t^n)-F(t,P_{(X_t^{m-1},Y_t^m)},X_t^{m-1},Y_t^m,Z_t^m)|dt]\hfill \\ \hfill & +E[{\int }_0^T|G(t,Y_t^n,Z_t^n)-G(t,Y_t^m,Z_t^m){|}^{2}dt]\hfill \\ \hfill \le & 2(E[{\int }_0^T|F(t,P_{(X_t^{n-1},Y_t^n)},X_t^{n-1},Y_t^n,Z_t^n)-F(t,P_{(X_t^{m-1},Y_t^m)},X_t^{m-1},Y_t^m,Z_t^m){|}^{2}dt{])}^{\frac{1}{2}}\hfill \\ \hfill & ·(E[{\int }_0^T|Y_t^n-Y_t^m{|}^2{dt])}^{\frac{1}{2}}+KE[{\int }_0^T|Y_t^n-Y_t^m{|}^{2}dt]+\gamma E[{\int }_0^T|Z_t^n-Z_t^m{|}^{2}dt]\hfill \\ \hfill \le & 4C_1(E[{\int }_0^T|Y_t^n-Y_t^m{|}^2{dt])}^{\frac{1}{2}}+KE[{\int }_0^T|Y_t^n-Y_t^m{|}^{2}dt]+\gamma E[{\int }_0^T|Z_t^n-Z_t^m{|}^{2}dt].\hfill \end{array}$$

Hence, ${\left\{Z^n\right\}}_{n\ge 1}$ is a Cauchy sequence in ${\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$ and we have

$$E[{\int }_0^T(|Y_t^n-Y_t^m{|}^2+|Z_t^n-Z_t^m{|}^2)dt]\to 0,\mathrm{as}n,m\to \infty ,$$

(34)

that is, ${\left\{Z^n\right\}}_{n\ge 1}$ has a limit $Z\in {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$, denoted by $Z=\underset{n\to \infty }{lim}{Z}^n$.

Finally, we conclude that $(X,Q,Y,Z)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)\times {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$ is a solution of general mean-field FBDSDE (15).

Step 4: Construction of the maximal solution.

Let us assume that $(\overline{X},\overline{Q},\overline{Y},\overline{Z})\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)\times {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$ is an arbitrary solution of (15). Therefore, from assumption (B2) we have

$$\begin{array}{cc}\hfill & -K(1+W_2(P_X,{\delta }_0^{(3)})+|x|+|q|)\hfill \\ \hfill \le & f(s,P_{(X,\overline{Y})},x,q,\overline{Y})\le K(1+W_2(P_X,{\delta }_0^{(3)})+|x|+|q|),\mathrm{for}\mathrm{all}s\in [0,T],x\in \mathbb{R},q\in {\mathbb{R}}^l.\hfill \end{array}$$

From Lemma 3, we know that ${\tilde{X}}_t^0\le {\overline{X}}_t\le X_t^0,t\in [0,T]$, P-a.s.

On the other hand, we have for all $s\in [0,T]$, $y\in \mathbb{R}$, $z\in {\mathbb{R}}^d$,

$$\begin{array}{cc}\hfill & -K(1+W_2(P_{(\overline{X},Y)},{\delta }_0^{(4)})+|\overline{X}|+|y|+|z|)\le F(s,P_{(\overline{X},Y)},\overline{X},y,z)\hfill \\ \hfill \le & K(1+W_2(P_{(\overline{X},Y)},{\delta }_0^{(4)})+|\overline{X}|+|y|+|z|),P\text{-a.s.}\hfill \end{array}$$

From assumptions (B3) and (B5), we find that $\mathsf{\Phi }(P_{{\overline{X}}_T},{\overline{X}}_T)\le \mathsf{\Phi }(P_{X_T^0},X_T^0)$, and then using Lemma 4 we find immediately that ${\tilde{Y}}_t^0\le {\overline{Y}}_t\le Y_t^0,t\in [0,T],$ P-a.s. Moreover, it follows that $F(s,P_{(\overline{X},Y)},\overline{X},y,z)\le F(s,P_{(X^0,Y)},X^0,y,z),\mathrm{for}\mathrm{all}s\in [0,T],y\in \mathbb{R},z\in {\mathbb{R}}^d$. Again, according to Lemma 4, we conclude that ${\overline{Y}}_t\le Y_t^1,t\in [0,T],$ P-a.s. Repeating the same procedure, we obtain, for all $n\in \mathbb{N}$,

$$\begin{array}{c}\hfill {\tilde{X}}_t^0\le {\overline{X}}_t\le \cdots \le X_t^n\cdots \le X_t^1\le X_t^0,t\in [0,T],P\text{-a.s.};\\ \hfill {\tilde{Y}}_t^0\le {\overline{Y}}_t\le \cdots \le Y_t^n\le \cdots \le Y_t^1\le Y_t^0,t\in [0,T],P\text{-a.s.}\end{array}$$

It implies that $\overline{Y}\le Y$, P-a.s., and $\overline{X}\le X$, P-a.s., that is, (15) has a maximal solution $(X,Q,Y,Z)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)\times {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$.  □

4. Unique Solvability

We will continue to consider the following general mean-field FBDSDE for $t\in [0,T],$

$$\left\{\begin{array}{c}{X}_t=\eta +{\int }_0^{t}f(s,P_{(X_s,Y_s)},X_s,Q_s,Y_s)ds+{\int }_0^{t}g(s,X_s,Q_s)d{W}_s-{\int }_0^t{Q}_{s}d{\overleftarrow{B}}_s,\hfill \\ Y_t=\mathsf{\Phi }(P_{X_T},X_T)+{\int }_t^{T}F(s,P_{(X_s,Y_s)},X_s,Y_s,Z_s)ds-{\int }_t^{T}G(s,Y_s,Z_s)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s.\hfill \end{array}\right.$$

(35)

In addition to the (B1)–(B5) conditions, the following assumptions are now assumed to be satisfied

(C1) 

$f,F$ are Lipschitz in $\mu$: there is a constant $c_1>0$ such that, for all $\mu ,{\mu }^{\prime }\in {\mathcal{P}}_2({\mathbb{R}}^2)$, $x,y\in \mathbb{R},q\in {\mathbb{R}}^l,z\in {\mathbb{R}}^d$,

$|f(t,\mu ,x,q,y)-f(t,{\mu }^{\prime },x,q,y){|}^2\le c_1{W}_2^2(\mu ,{\mu }^{\prime })$,

$|F(t,\mu ,x,y,z)-F(t,{\mu }^{\prime },x,y,z){|}^2\le c_1{W}_2^2(\mu ,{\mu }^{\prime })$;

(C2) 

$\mathsf{\Phi }$ is Lipschitz in $\mu$: there is a constant $c_2>0$ such that, for all $\mu ,{\mu }^{\prime }\in {\mathcal{P}}_2(\mathbb{R}),x\in \mathbb{R}$;

$$\begin{array}{c}\hfill |\mathsf{\Phi }(\mu ,x)-\mathsf{\Phi }({\mu }^{\prime },x)|\le c_2{W}_2(\mu ,{\mu }^{\prime });\end{array}$$

(C3) 

There are constants ${\beta }_1,{\beta }_2,{\beta }_3>0$ such that, for all $\mu \in {\mathcal{P}}_2({\mathbb{R}}^2),x,x^{\prime },y,y^{\prime }\in \mathbb{R},q,q^{\prime }\in {\mathbb{R}}^l,z,z^{\prime }\in {\mathbb{R}}^d$, $X\in L^2(\mathsf{\Omega },\mathcal{F},P;\mathbb{R})$,

$$\begin{array}{cc}\hfill & \langle A(t,\mu ,x,q,y,z)-A(t,\mu ,x^{\prime },q^{\prime },y^{\prime },z^{\prime }),(x,y)-(x^{\prime },y^{\prime })\rangle \hfill \\ \hfill \le & -{\beta }_1(|x-x^{\prime }{|}^2+|q-q^{\prime }{|}^2)-{\beta }_2(|y-y^{\prime }{|}^2+|z-z^{\prime }{|}^2),\hfill \\ \hfill & (\mathsf{\Phi }(P_X,x)-\mathsf{\Phi }(P_X,x^{\prime }))(x-x^{\prime })\ge {\beta }_3{|x-x^{\prime }|}^2,\hfill \end{array}$$

where $A(t,\mu ,x,q,y,z)=(-F(t,\mu ,x,y,z),f(t,\mu ,x,q,y))$;

(C4) 

${\beta }_3>c_2$ and there is a constant $k>0$ such that ${\beta }_1,{\beta }_2>B$, where $B=\frac{1}{2k}+k{c}_1+\frac{kK}{2}+\frac{k\gamma }{2}$.

Theorem 2.

Under the assumptions (B1)–(B5) and (C1)–(C4), the general mean-field FBDSDE (35) has a unique solution $(X,Q,Y,Z)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)\times {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$.

Proof of Theorem 2.

Let $(X^1,Q^1,Y^1,Z^1)$ and $(X^2,Q^2,Y^2,Z^2)$ be two solutions of general mean-field FBDSDE (35). We set $(\widehat{X},\widehat{Q},\widehat{Y},\widehat{Z}):=(X^1-X^2,Q^1-Q^2,Y^1-Y^2,Z^1-Z^2)$. Apply It$\widehat{\mathrm{o}}$’s formula to ${\widehat{X}}_t{\widehat{Y}}_t$, by using assumptions (B1)–(B5), (C1)–(C4) and the inequality $ab\le \frac{1}{2\delta }{a}^2+\frac{\delta }{2}{b}^2,\delta >0$, there exists a constant $k>0$ such that

$$\begin{array}{cc}\hfill & E[{\widehat{X}}_T(\mathsf{\Phi }(P_{X_T^1},X_T^1)-\mathsf{\Phi }(P_{X_T^2},X_T^2))]=E[{\int }_0^T({\widehat{Y}}_t{\widehat{f}}_t+\langle {\widehat{Z}}_t,{\widehat{g}}_t\rangle -{\widehat{X}}_t{\widehat{F}}_t+\langle {\widehat{Q}}_t,{\widehat{G}}_t\rangle )dt]\hfill \\ \hfill \le & E[{\int }_0^T(〈({\widehat{X}}_t,{\widehat{Y}}_t),A(t,P_{(X_t^1,Y_t^1)},X_t^1,Q_t^1,Y_t^1,Z_t^1)-A(t,P_{(X_t^1,Y_t^1)},X_t^2,Q_t^2,Y_t^2,Z_t^2)〉\hfill \\ \hfill & +{\widehat{Y}}_t\left(f(t,P_{(X_t^1,Y_t^1)},X_t^2,Q_t^2,Y_t^2)-f(t,P_{(X_t^2,Y_t^2)},X_t^2,Q_t^2,Y_t^2)\right)+{\widehat{X}}_t(F(t,P_{(X_t^1,Y_t^1)},X_t^2,Y_t^2,Z_t^2)\hfill \\ \hfill & -F(t,P_{(X_t^2,Y_t^2)},X_t^2,Y_t^2,Z_t^2)))dt]+E[{\int }_0^T|{\widehat{g}}_t||{\widehat{Z}}_t|dt]+E[{\int }_0^T|{\widehat{Q}}_t||{\widehat{G}}_t|dt]\hfill \\ \hfill \le & -{\beta }_{1}E[{\int }_0^T|{\widehat{X}}_t{|}^{2}dt]-{\beta }_{1}E[{\int }_0^T|{\widehat{Q}}_t{|}^{2}dt]-{\beta }_{2}E[{\int }_0^T|{\widehat{Y}}_t{|}^{2}dt]-{\beta }_{2}E[{\int }_0^T|{\widehat{Z}}_t{|}^{2}dt]\hfill \\ \hfill & +\frac{1}{2k}E[{\int }_0^T|{\widehat{Y}}_t{|}^{2}dt]+\frac{k}{2}E[{\int }_0^T|f(t,P_{(X_t^1,Y_t^1)},X_t^2,Q_t^2,Y_t^2)-f(t,P_{(X_t^2,Y_t^2)},X_t^2,Q_t^2,Y_t^2){|}^{2}dt]\hfill \\ \hfill & +\frac{1}{2k}E[{\int }_0^T|{\widehat{X}}_t{|}^{2}dt]+\frac{k}{2}E[{\int }_0^T|F(t,P_{(X_t^1,Y_t^1)},X_t^2,Y_t^2,Z_t^2)-F(t,P_{(X_t^2,Y_t^2)},X_t^2,Y_t^2,Z_t^2){|}^{2}dt]\hfill \\ \hfill & +\frac{1}{2k}E[{\int }_0^T|{\widehat{Z}}_t{|}^{2}dt]+\frac{k}{2}E[{\int }_0^T|{\widehat{g}}_t{|}^{2}dt]+\frac{1}{2k}E[{\int }_0^T|{\widehat{Q}}_t{|}^{2}dt]+\frac{k}{2}E[{\int }_0^T|{\widehat{G}}_t{|}^{2}dt]\hfill \\ \hfill \le & (-{\beta }_1+\frac{1}{2k}+k{c}_1+\frac{kK}{2})E[{\int }_0^T|{\widehat{X}}_t{|}^{2}dt]+(-{\beta }_1+\frac{1}{2k}+\frac{k\gamma }{2})E[{\int }_0^T|{\widehat{Q}}_t{|}^{2}dt]\hfill \\ \hfill & +(-{\beta }_2+\frac{1}{2k}+k{c}_1+\frac{kK}{2})E[{\int }_0^T|{\widehat{Y}}_t{|}^{2}dt]+(-{\beta }_2+\frac{1}{2k}+\frac{k\gamma }{2})E[{\int }_0^T|{\widehat{Z}}_t{|}^{2}dt],\hfill \end{array}$$

(36)

where

$$\begin{array}{cc}\hfill & {\widehat{f}}_s=f(s,P_{(X_s^1,Y_s^1)},X_s^1,Q_s^1,Y_s^1)-f(s,P_{(X_s^2,Y_s^2)},X_s^2,Q_s^2,Y_s^2),\hfill \\ \hfill & {\widehat{F}}_s=F(s,P_{(X_s^1,Y_s^1)},X_s^1,Y_s^1,Z_s^1)-F(s,P_{(X_s^2,Y_s^2)},X_s^2,Y_s^2,Z_s^2),\hfill \\ \hfill & {\widehat{g}}_s=g(s,X_s^1,Q_s^1)-g(s,X_s^2,Q_s^2),{\widehat{G}}_s=G(s,Y_s^1,Z_s^1)-G(s,Y_s^2,Z_s^2).\hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill & E[{\widehat{X}}_T(\mathsf{\Phi }(P_{X_T^1},X_T^1)-\mathsf{\Phi }(P_{X_T^2},X_T^2))]\hfill \\ \hfill =& E[{\widehat{X}}_T(\mathsf{\Phi }(P_{X_T^1},X_T^1)-\mathsf{\Phi }(P_{X_T^1},X_T^2)+\mathsf{\Phi }(P_{X_T^1},X_T^2)-\mathsf{\Phi }(P_{X_T^2},X_T^2))]\hfill \\ \hfill \ge & ({\beta }_3-c_2)E[|{\widehat{X}}_T{|}^2]\ge 0.\hfill \end{array}$$

(37)

Combine (36) and (37) we have

$$\begin{array}{cc}\hfill & ({\beta }_1-\frac{1}{2k}-k{c}_1-\frac{kK}{2})E[{\int }_0^T|{\widehat{X}}_t{|}^{2}dt]+({\beta }_1-\frac{1}{2k}-\frac{k\gamma }{2})E[{\int }_0^T|{\widehat{Q}}_t{|}^{2}dt]\hfill \\ \hfill & +({\beta }_2-\frac{1}{2k}-k{c}_1-\frac{kK}{2})E[{\int }_0^T|{\widehat{Y}}_t{|}^{2}dt]+({\beta }_2-\frac{1}{2k}-\frac{k\gamma }{2})E[{\int }_0^T|{\widehat{Z}}_t{|}^{2}dt]\le 0.\hfill \end{array}$$

Then from (C4) we have $E[{\int }_0^T|(X_t^1,Q_t^1,Y_t^1,Z_t^1)-(X_t^2,Q_t^2,Y_t^2,Z_t^2){|}^{2}dt]=0,$ that is, $(X^1,Q^1,$$Y^1,Z^1)=(X^2,Q^2,Y^2,Z^2)$, P-a.s.  □

Example 1.

Let us consider the following mean-field FBDSDE:

$$\left\{\begin{array}{c}{X}_t=\frac{1}{2}+{\int }_0^t\frac{1}{5}d{W}_s,\hfill \\ Y_t=(\frac{11}{2}{X}_T+\frac{1}{10}E{X}_T)\vee (-1)\wedge 1+{\int }_t^T\left((6Y_s+6Z_s+\frac{1}{10}E{Y}_s)\vee (-1)\wedge 1\right)ds\hfill \\ -{\int }_t^T\left((\frac{1}{4}{Y}_s+\frac{1}{4}{Z}_s)\vee (-1)\wedge 1\right)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s,t\in [0,T].\hfill \end{array}\right.$$

Taking $K=6$, $k=\frac{1}{2}$, $\gamma =\frac{1}{2}$, $c_1=c_2=1$, ${\beta }_1={\beta }_2=\frac{11}{2}$, ${\beta }_3=5$, the above mean-field FBDSDE satisfies the assumptions (B1)–(B5) and (C1)–(C4), from Theorem 2 we know that the equation group has a unique solution $(X,Q,Y,Z)\in {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^l)\times {\mathcal{S}}^2(0,T;\mathbb{R})\times {\mathcal{H}}^2(0,T;{\mathbb{R}}^d)$. In order for readers to understand this result more intuitively, we used the numerical calculation method provided in [20] to find out the solution processes of the above equation. See Figure 1, Figure 2, Figure 3 and Figure 4 for details.

5. Comparison Theorem

In Section 2, we only proved the comparison theorem for single equation; in this section, we will study the comparison theorem for coupled general mean-field FBDSDEs. Consider the following general mean-field FBDSDEs: for ${\eta }^i\in L^2(\mathsf{\Omega },{\mathcal{F}}_0,P;\mathbb{R})$, $i=1,2$, $t\in [0,T]$,

$$\left\{\begin{array}{c}{X}_t^i={\eta }^i+{\int }_0^t{f}^i(s,P_{(X_s^i,Y_s^i)},X_s^i,Q_s^i,Y_s^i)ds+{\int }_0^{t}g(s,X_s^i,Q_s^i)d{W}_s-{\int }_0^t{Q}_s^{i}d{\overleftarrow{B}}_s,\hfill \\ Y_t^i=\mathsf{\Phi }(P_{X_T^i},X_T^i)+{\int }_t^T{F}^i(s,P_{(X_s^i,Y_s^i)},X_s^i,Y_s^i,Z_s^i)ds-{\int }_t^{T}G(s,Y_s^i,Z_s^i)d{\overleftarrow{B}}_s\hfill \\ -{\int }_t^T{Z}_s^{i}d{W}_s.\hfill \end{array}\right.$$

(38)

Theorem 3.

Let $f^i=f^i(s,\omega ,\mu ,x,q,y)$, $F^i=F^i(s,\omega ,\mu ,x,y,z)$, $i=1,2$, $g=g(s,\omega ,x,q)$, $G=G(s,\omega ,y,z)$ and $\mathsf{\Phi }(\omega ,\mu ,x)$ satisfy the assumptions (B1)–(B5). Denote by $(X^1,Q^1,Y^1,Z^1)$ and $(X^2,Q^2,Y^2,$$Z^2)$ the solutions of the general mean-field FBDSDEs (38) with data $({\eta }^1,f^1,g,\mathsf{\Phi },F^1,G)$ and $({\eta }^2,f^2,g,\mathsf{\Phi },F^2,G)$, respectively. Then, if ${\eta }^1\le {\eta }^2$, and for all $\mu ,x,q,y,z$, $f^1(s,\mu ,x,q,y)\le f^2(s,\mu ,x,q,y)$, $F^1(s,\mu ,x,y,z)\le F^2(s,\mu ,x,y,z)$, $ds\otimes dP$-a.e., we have $X_t^1\le X_t^2,$ and $Y_t^1\le Y_t^2$, for all $t\in [0,T]$, P-a.s.

Proof of Theorem 3.

Consider the following equations for $t\in [0,T]$, $i=1,2$, $m\ge 1$,

$$\left\{\begin{array}{c}{X}_t^{i,m}={\eta }^i+{\int }_0^t{f}^i(s,P_{(X_s^{i,m},Y_s^{i,m})},X_s^{i,m},Q_s^{i,m},Y_s^{i,m})ds+{\int }_0^{t}g(s,X_s^{i,m},Q_s^{i,m})d{W}_s\hfill \\ -{\int }_0^t{Q}_s^{i,m}d{\overleftarrow{B}}_s,\hfill \\ Y_t^{i,m}=\mathsf{\Phi }(P_{X_T^{i,m-1}},X_T^{i,m-1})+{\int }_t^T{F}^i(s,P_{(X_s^{i,m-1},Y_s^{i,m})},X_s^{i,m-1},Y_s^{i,m},Z_s^{i,m})ds\hfill \\ -{\int }_t^{T}G(s,Y_s^{i,m},Z_s^{i,m})d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{i,m}d{W}_s.\hfill \end{array}\right.$$

(39)

From the proof of Theorem 1, we obtain $X^{i,m}\to X^i$, $Q^{i,m}\to Q^i$, $Y^{i,m}\to Y^i$, $Z^{i,m}\to Z^i$, as $m\to \infty$, $i=1,2$; that is, $(X^{i,m},Y^{i,m},Q^{i,m},Z^{i,m})$ (the solutions of (39)) converge to $(X^i,Y^i,Q^i,Z^i)$ (the solutions of (38)).

Now, let us construct the starting point. Consider the following equations: for $t\in [0,T]$, $i=1,2$,

$$X_t^{i,0}={\eta }^i+{\int }_0^{t}K(1+W_2(P_{X_s^{i,0}},{\delta }_0)+|X_s^{i,0}|+|Q_s^{i,0}|)ds+{\int }_0^{t}g(s,X_s^{i,0},Q_s^{i,0})d{W}_s-{\int }_0^t{Q}_s^{i,0}d{\overleftarrow{B}}_s.$$

(40)

From ${\eta }^1\le {\eta }^2$ and Lemma 3, we know $X_t^{1,0}\le X_t^{2,0}$, for all $t\in [0,T]$, P-a.s.

Next, let $m=1$ to construct $Y^{1,1}$ and $Y^{2,1}$. We have the following general mean-field BDSDEs: for $t\in [0,T]$, $i=1,2$,

$$\begin{array}{cc}\hfill Y_t^{i,1}=& \mathsf{\Phi }(P_{X_T^{i,0}},X_T^{i,0})+{\int }_t^T{F}^i(s,P_{(X_s^{i,0},Y_s^{i,1})},X_s^{i,0},Y_s^{i,1},Z_s^{i,1})ds\hfill \\ \hfill & -{\int }_t^{T}G(s,Y_s^{i,1},Z_s^{i,1})d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{i,1}d{W}_s.\hfill \end{array}$$

(41)

Let $k\ge K$ and define

$$\begin{array}{cc}\hfill & F^{i,1,k}(s,\omega ,P_Y,y,z):={\overline{F}}^{i,k}(s,\omega ,P_{(X_s^{i,0},Y)},X_s^{i,0},y,z)\hfill \\ \hfill :=& \underset{(\nu ,a,e,p)\in {\mathcal{P}}_2({\mathbb{R}}^2)\times {\mathbb{R}}^{d+2}}{\mathrm{esssup}}\left\{F(s,\omega ,\nu ,a,e,p)-k{W}_{2,-}(P_{(X_s^{i,0},Y)},\nu )-k|X_s^{i,0}-a|-k|y-e|-k|z-p|\right\}.\hfill \end{array}$$

Consider the following general mean-field BDSDEs: for $t\in [0,T]$, $i=1,2$,

$$\begin{array}{cc}\hfill Y_t^{i,1,k}=& \mathsf{\Phi }(P_{X_T^{i,0}},X_T^{i,0})+{\int }_t^T{F}^{i,1,k}(s,P_{Y_s^{i,1,k}},Y_s^{i,1,k},Z_s^{i,1,k})ds\hfill \\ \hfill & -{\int }_t^{T}G(s,Y_s^{i,1,k},Z_s^{i,1,k})d{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^{i,1,k}d{W}_s,\hfill \end{array}$$

(42)

and it follows that as $k\to \infty$, $(Y^{i,1,k},Z^{i,1,k})$ (the solutions of (42)) converge to $(Y^{i,1},Z^{i,1})$ (the solutions of (41)). Because of $X_t^{1,0}\le X_t^{2,0}$, for all $t\in [0,T]$, P-a.s., and assumptions (B3) and (B5), we have $\mathsf{\Phi }(P_{X_T^{1,0}},X_T^{1,0})\le \mathsf{\Phi }(P_{X_T^{2,0}},X_T^{2,0})$; moreover, from Lemma 5—(ii) and (v), we obtain

$$\begin{array}{cc}\hfill & F^{1,1,k}(s,\omega ,P_Y,y,z)={\overline{F}}^{1,k}(s,\omega ,P_{(X_s^{1,0},Y)},X_s^{1,0},y,z)\hfill \\ \hfill \le & {\overline{F}}^{2,k}(s,\omega ,P_{(X_s^{2,0},Y)},X_s^{2,0},y,z)=F^{2,1,k}(s,\omega ,P_Y,y,z),P\text{-a.s.}\hfill \end{array}$$

Therefore, from Lemma 4, $Y_t^{1,1,k}\le Y_t^{2,1,k}$, for all $t\in [0,T]$, P-a.s. Because $(Y^{i,1,k},Z^{i,1,k})$ converge to $(Y^{i,1},Z^{i,1})$, as $k\to \infty$, we can obtain that $Y_t^{1,1}\le Y_t^{2,1}$, for all $t\in [0,T]$, P-a.s.

Similarly, let $m=1$ to construct $X^{1,1}$ and $X^{2,1}$. We have the following equations: for $t\in [0,T]$, $i=1,2$,

$$X_t^{i,1}={\eta }^i+{\int }_0^t{f}^i(s,P_{(X_s^{i,1},Y_s^{i,1})},X_s^{i,1},Q_s^{i,1},Y_s^{i,1})ds+{\int }_0^{t}g(s,X_s^{i,1},Q_s^{i,1})d{W}_s-{\int }_0^t{Q}_s^{i,1}d{\overleftarrow{B}}_s.$$

(43)

Let $k\ge K$ and define

$$\begin{array}{cc}\hfill & f^{i,1,k}(s,\omega ,P_X,x,q):={\overline{f}}^{i,k}(s,\omega ,P_{(X,Y_s^{i,1})},x,q,Y_s^{i,1})\hfill \\ \hfill :=& \underset{(\nu ,a,b,e)\in {\mathcal{P}}_2({\mathbb{R}}^2)\times {\mathbb{R}}^{l+2}}{\mathrm{esssup}}\left\{f(s,\omega ,\nu ,a,b,e)-k{W}_{2,-}(P_{(X,Y_s^{i,1})},\nu )-k|x-a|-k|q-b|-k|Y_s^{i,1}-e|\right\}.\hfill \end{array}$$

Consider the following equations for $t\in [0,T]$, $i=1,2$,

$$\begin{array}{cc}{X}_t^{i,1,k}=\hfill & {\eta }^i+{\int }_0^t{f}^{i,1,k}(s,P_{(X_s^{i,1,k},Y_s^{i,1})},X_s^{i,1,k},Q_s^{i,1,k},Y_s^{i,1})ds\hfill \\ \hfill & +{\int }_0^{t}g(s,X_s^{i,1,k},Q_s^{i,1,k})d{W}_s-{\int }_0^t{Q}_s^{i,1,k}d{\overleftarrow{B}}_s,\hfill \end{array}$$

(44)

and it follows that as $k\to \infty$, $(X^{i,1,k},Q^{i,1,k})$ (the solutions of (44)) converge to $(X^{i,1},Q^{i,1})$ (the solutions of (43)). We know that ${\eta }^1\le {\eta }^2$. Because of Lemma 5—(ii) and (v), we have

$$\begin{array}{cc}\hfill & f^{1,1,k}(s,\omega ,P_X,x,q)={\overline{f}}^{1,k}(s,\omega ,P_{(X,Y_s^{1,1})},x,q,Y_s^{1,1})\hfill \\ \hfill \le & {\overline{f}}^{2,k}(s,\omega ,P_{(X,Y_s^{2,1})},x,q,Y_s^{2,1})=f^{2,1,k}(s,\omega ,P_X,x,q),P\text{-a.s.}\hfill \end{array}$$

Therefore, using Lemma 3, we can obtain $X_t^{1,1,k}\le X_t^{2,1,k}$, for all $t\in [0,T]$, P-a.s. Because $(X^{i,1,k},Q^{i,1,k})$ converge to $(X^{i,1},Q^{i,1})$, as $k\to \infty$, we can also obtain $X_t^{1,1}\le X_t^{2,1}$, for all $t\in [0,T]$, P-a.s.

Repeating the same procedure we find, for all $m\in \mathbb{N}$, that

$$X_t^{1,m}\le X_t^{2,m},Y_t^{1,m}\le Y_t^{2,m},0\le t\le T,P\text{-a.s.}$$

From the proof of Theorem 1, it follows that $X^{1,m}\to X^1,Q^{1,m}\to Q^1,Y^{1,m}\to Y^1,Z^{1,m}\to Z^1$, as $m\to \infty$, that is, $(X^{1,m},Q^{1,m},Y^{1,m},Z^{1,m})$ converges to $(X^1,Q^1,Y^1,Z^1)$, as $m\to \infty$. Therefore,

$$X_t^1\le X_t^2,Y_t^1\le Y_t^2,0\le t\le T,P\text{-a.s.}$$

□

6. Conclusions

Mean-field problems have been widely applied in various fields, such as statistical mechanics, quantum mechanics, quantum chemistry and so on. In order to simplify the study of complex problems, many scholars have applied mean-field theory to reduce high-dimensional problems to low-dimensional problems. In this paper, a general class of mean-field forward–backward doubly stochastic differential equations (mean-field FBDSDEs, in short) are studied, where coefficients depend not only on the solution processes but also on their law. Firstly, we begin our studies with the above mean-field FBDSDE (7), where we solve the existence and uniqueness of the solution for the forward equation of Equation (7) under Lipschitz conditions (and the backward equation is in the similar process), and we obtain the associated comparison theorem. Then, with the help of the above two comparison theorems, we obtain the existence of the maximal solution for Equation (15) under some much weaker monotone continuity conditions. Furthermore, under appropriate assumptions we prove the uniqueness of the solution for Equation (15). Finally, we obtain the comparison theorem for coupled general mean-field FBDSDEs.