Backward Anticipated Social Optima: Input Constraints and Partial Information

Abstract

A class of stochastic linear-quadratic (LQ) dynamic optimization problems involving a large population is investigated in this work. Here, the agents cooperate with each other to minimize certain social costs. Furthermore, differently from the classic social optima literature, the dynamics in this framework are driven by anticipated backward stochastic differential equations (ABSDE) in which the terminal instead of the initial condition is specified and the anticipated terms are involved. The individual admissible controls are constrained in closed convex subsets, and the common noise is considered. As a result, the related social cost is represented by a recursive functional in which the initial state is involved. By virtue of the so-called anticipated person-by-person optimality principle, a decentralized strategy can be derived. This is based on a class of new consistency condition systems, which are mean-field-type anticipated forward-backward stochastic differential delay equations (AFBSDDEs). The well-posedness of such a consistency condition system is obtained through a discounting decoupling method. Finally, the corresponding asymptotic social optimality is proved.

1. Introduction

Large population problems are widely investigated in many areas, such as engineering, biology, finance, economics, physics, etc. Due to a highly complicated coupling structure, it is infeasible and ineffective to obtain classical centralized strategies by combining the exact dynamics of all agents in a large population problem. As a substitute, it is more effective and tractable to investigate the corresponding decentralized strategies that only take into account their own individual dynamic (information) and some off-line quantities. In this research direction, many researchers have focus their efforts on mean field (MF) game studies. Interested readers can refer to [1,2,3,4,5,6,7] and the references therein.

Differently from the above works, the cooperative optimization problem has attracted much attention in the last ten years, including the so-called social optima problems. Readers can refer to [8,9,10,11,12,13,14,15,16] for related research. For more research and applications of MF social optima problems, readers can refer to [17,18,19].

It is known that both competitive behaviors and cooperative trades are influenced by the time factor. In many real cases, the evolution of a controlled system depends not only on the present state or decision policy at time t, but also on the past trajectories on a historical interval $[t-\delta ,t]$, where $\delta$ denotes a lagged index. As a result, a controlled system may contain a state delay, control delay, or both. Therefore, there are various practical problems where the systems are driven by stochastic differential delay equations (SDDEs).

In recent years, the study of SDDEs has attracted the attention of many researchers, see, e.g., [20,21,22], etc. The feature of backward dynamics makes our framework different from the existing works of the MF LQ team, wherein the individual states were driven by SDEs. Unlike SDEs, the terminal condition of a BSDE should be specified as a prior, instead of an initial condition. As a result, the BSDE will admit one adapted solution pair $(y_t,z_t)$. Linear BSDEs were first introduced in [23], and the general nonlinear BSDEs were first introduced in [24]. For more development of BSDEs, such as various applications in mathematical finance, readers can refer to [25,26,27], and to [28,29,30,31,32,33,34], etc., for optimal control and game theory.

A new type of BSDE called anticipated BSDEs (ABSDEs) was introduced in Peng and Yang [35] and a duality between SDDEs and ABSDEs was established. In an ABSDE, the generator not only contains the values of solutions for the present but also the future. This occurs in many phenomena in reality. Readers can refer to [36,37,38,39,40] and the references therein for more details of SDDE theory and its broad applications.

In this paper, we investigate an LQ mean-field game with an anticipated BSDE in the case of an input constraint with partial information and common noise. In all of the aforementioned papers concerning LQ control problems, the control was unconstrained and the (feedback) control was constructed through either a dynamic programming principle (DPP) or stochastic maximum principle (SMP), both of which are automatically admissible. However, if we impose constraints on the admissible control, the entire LQ approach fails to apply (see, e.g., [41,42]). For more applications in finance and economics, please refer to [4,43], etc. In addition, a partial information structure is introduced. Due to the common noise, the limiting process turns out to be stochastic.

The rest of this paper is organized as follows: We formulate the problem in Section 2. We derive the auxiliary control problem with an input constraint and partial information in Section 3. In Section 4, the consistency condition system and its well-posedness are established using the discounting method. Section 5 focuses on the asymptotic optimality of the decentralized strategy. In Section 6, we conclude the paper.

2. Problem Formulation

Consider a finite time horizon $[0,T]$ for fixed $T>0$. Assume that $(\Omega ,\mathcal{F},$${\left\{{\mathcal{F}}_t\right\}}_{0\le t\le T},\mathbb{P})$ is a complete filtered probability space satisfying the usual conditions and ${\{W_0\left(t\right),W_i\left(t\right),1\le i\le N\}}_{0\le t\le T}$ is a $d+N\times d$-dimensional Brownian motion in this space. Let ${\mathcal{F}}_t$ be the filtration generated by ${\{W_0\left(s\right),W_i\left(s\right),1\le i\le N\}}_{0\le s\le t}$ and augmented by ${\mathcal{N}}_{\mathbb{P}}$ (the class of all $\mathbb{P}$-null sets of $\mathcal{F}$). Let ${\mathcal{F}}_t^i$ be the augmentation of $\sigma \{W_0\left(s\right),W_i\left(s\right),0\le s\le t\}$ by ${\mathcal{N}}_{\mathbb{P}}$. Let ${\mathcal{F}}_t^0$ be the augmentation of $\sigma \{W_0\left(s\right),0\le s\le t\}$ by ${\mathcal{N}}_{\mathbb{P}}$. Let $\delta >0$ be a given fixed time horizon. $\mathbb{E}$ denotes the expectation under $\mathbb{P}$, and ${\mathbb{E}}^{{\mathcal{G}}_t}[·]:=\mathbb{E}[·|{\mathcal{G}}_t]$ denotes the conditional expectation. Define ${\mathcal{F}}_t:={\mathcal{F}}_0$ for $\forall t\in [-\delta ,0)$.

Let $\langle ·,·\rangle$ denote the standard Euclidean inner product. $x^{\top }$ denotes the transpose of a vector (or matrix) x. For a matrix $M\in {\mathbb{R}}^{n\times m}$, we define the norm $\left|M\right|:=\sqrt{\mathtt{tr}\left(M^{\top }M\right)}$. $M\in {\mathbb{S}}^n$ denotes the set of symmetric $n\times n$ matrices with real elements. $M>(\ge )0$ denotes that $M\in {\mathbb{S}}^n$, which is positive (semi)definite, while $M\gg 0$ denotes that, for some $\epsilon >0$, $M-\epsilon I\ge 0$. Let $\mathcal{H}$ be a given Hilbert space. Denote any times $t,t_1,t_2$ valued in $[0,T+\delta ]$ by

• $L_{\mathcal{F}}^2(\Omega ;\mathcal{H}):=\left\{\zeta :\Omega \to {\mathcal{H}\left|\zeta \mathrm{is}\mathcal{F}\text{-}\mathrm{measurable}\mathrm{such}\mathrm{that}\mathbb{E}\right|\zeta |}^2<\infty \right\}$
• $L_{\mathcal{F}}^2(\Omega ;C([t_1,t_2];\mathcal{H})):=\{\zeta (·):[t_1,t_2]\times \Omega \to \mathcal{H}|\zeta (·)$ is ${\mathcal{F}}_t$-adapted, continuous, such that $\mathbb{E}\left[\underset{s\in [t_1,t_2]}{sup}{\left|\zeta \left(s\right)\right|}^2\right]<\infty \}$
• $L_{\mathcal{F}}^2(t_1,t_2;\mathcal{H}):=\{\zeta (·):[t_1,t_2]\times \Omega \to \mathcal{H}|\zeta (·)\mathrm{is}\mathrm{an}{\mathcal{F}}_t\text{-}\mathrm{progressively}\mathrm{measurable}\mathrm{process}$
  $\mathrm{such}\mathrm{that}\mathbb{E}{\int }_{t_1}^{t_2}{\left|\zeta (·)\right|}^{2}dt<\infty \}$
• $L^2(t_1,t_2;\mathcal{H}):=\{\zeta (·):[t_1,t_2]\to \mathcal{H}|\zeta (·)\mathrm{is}\mathrm{a}\mathrm{deterministic}\mathrm{function}\mathrm{s}.\mathrm{t}.$${\int }_{t_1}^{t_2}{\left|\zeta (·)\right|}^{2}dt<\infty \}$
• $L^{\infty }(t_1,t_2;\mathcal{H}):=\left\{\zeta (·):[t_1,t_2]\to \mathcal{H}|\zeta (·)\mathrm{is}\mathrm{uniformly}\mathrm{bounded}\right\}$

In this work, we study an LQ large population system with K-type discrete heterogeneous agents $\{{\mathcal{A}}_i:1\le i\le N\}$. The dynamics of the agents are driven by a system of linear ABSDEs with mean-field coupling: that is, for $1\le i\le N,$

$$\left\{\begin{array}{cc}\hfill d{y}_i\left(t\right)=& -\left[A_{{\theta }_i}\left(t\right)y_i\left(t\right)+B\left(t\right)u_i\left(t\right)+C\left(t\right){\mathbb{E}}^{{\mathcal{F}}_t}\left[y^{\left(N\right)}(t+\delta )\right]\right]dt\hfill \\ \hfill & +z_i\left(t\right)d{W}_i\left(t\right)+\sum _{j=1,j\ne i}^N{z}_{ij}\left(t\right)d{W}_j\left(t\right)+z_{0i}\left(t\right)d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill y_i\left(t\right)=& {\xi }_i\left(t\right),z_i\left(t\right)={\eta }_i\left(t\right),z_{ij}\left(t\right)={\eta }_{ij}\left(t\right)(j\ne i),z_{0i}\left(t\right)={\eta }_{0i}\left(t\right),t\in [T,T+\delta ],\hfill \end{array}\right.$$

(1)

where $y^{\left(N\right)}(·)={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{y}_i(·)$ denotes the state-average of the agents. The number ${\theta }_i$ is a parameter of the agent ${\mathcal{A}}_i$ to model a heterogeneous population, and we assume that ${\theta }_i$ takes a value in a finite set $\Theta :=\{1,2,\cdots ,K\}$. We call ${\mathcal{A}}_i$ a k-type agent if ${\theta }_i=k\in \Theta$. In this paper, we are interested in the asymptotic behavior as N tends to infinity. For $1\le k\le K$, introduce

$${\mathcal{I}}_k=\left\{i\right|{\theta }_i=k,1\le i\le N\},N_k=\left|{\mathcal{I}}_k\right|,$$

where $N_k$ is the cardinality of index set ${\mathcal{I}}_k$. For $1\le k\le K$, let ${\pi }_k^{\left(N\right)}={\displaystyle \frac{N_k}{N}}$, then ${\pi }^{\left(N\right)}=({\pi }_1^{\left(N\right)},\cdots ,{\pi }_K^{\left(N\right)})$ is a probability vector representing the empirical distribution of ${\theta }_1,\cdots ,{\theta }_N.$ Let ${\Gamma }_{{\theta }_i}\subset {\mathbb{R}}^m$$(i=1,\cdots ,N)$ be nonempty closed convex sets. We introduce the following assumption:

(A1) 

There exists a probability mass vector $\pi =({\pi }_1,\cdots ,{\pi }_K)$ such that ${\displaystyle \underset{N\to +\infty }{lim}{\pi }^{\left(N\right)}=\pi }$, ${\displaystyle \underset{1\le k\le K}{min}{\pi }_k>0.}$

(A2) 

${\xi }_i(·)\in L_{{\mathcal{F}}^i}^2(\Omega ;C([T,T+\delta ];{\mathbb{R}}^n))$, ${\eta }_i(·),{\eta }_{0i}(·)\in L_{{\mathcal{F}}^i}^2(T,T+\delta ;{\mathbb{R}}^{n\times d})$, ${\eta }_{ij}(·)\in L_{\mathcal{F}}^2(T,T+\delta ;{\mathbb{R}}^{n\times d})(j\ne i)$, $i=1,\cdots ,N$. If ${\theta }_i={\theta }_j=k$, ${\xi }_i(·)$ and ${\xi }_j(·)$ are identically distributed and the common distribution is denoted by ${\xi }^{\left(k\right)}(·)$.

(A3) 

$A_{{\theta }_i}(·)\in L^{\infty }(0,T;{\mathbb{R}}^{n\times n})$ ($i=1,\cdots ,N$), $C(·)\in L^{\infty }(-\delta ,T;{\mathbb{R}}^{n\times n})$, $B(·)\in L^{\infty }(0,T;{\mathbb{R}}^{n\times m})$.

It follows that under (A1)–(A3), the state equation in (1) admits a unique solution for all $u_i\in {\mathcal{U}}_i.$ In fact, if we denote by

$$\left\{\begin{array}{cc}\hfill & \widehat{Y}=\left(\begin{array}{c}{y}_1\\ ⋮\\ y_N\end{array}\right),\widehat{Z}=\left(\begin{array}{ccccc}{z}_1& z_{12}& \cdots & z_{1,N-1}& z_{1N}\\ z_{21}& z_2& \cdots & z_{2,N-1}& z_{2N}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ z_{N1}& z_{N2}& \cdots & z_{N,N-1}& z_N\end{array}\right),{\widehat{Z}}_0=\left(\begin{array}{c}{z}_{01}\\ ⋮\\ z_{0N}\end{array}\right),\widehat{U}=\left(\begin{array}{c}{u}_1\\ ⋮\\ u_N\end{array}\right),\widehat{W}=\left(\begin{array}{c}{W}_1\\ ⋮\\ W_N\end{array}\right),\hfill \\ \hfill & \widehat{A}=\left(\begin{array}{ccc}{A}_{{\theta }_1}& & \\ & \ddots & \\ & & A_{{\theta }_N}\end{array}\right),\widehat{B}=\left(\begin{array}{ccc}B& & \\ & \ddots & \\ & & B\end{array}\right),\widehat{C}=\left(\begin{array}{ccc}C& & \\ & \ddots & \\ & & C\end{array}\right),J_N=\left(\begin{array}{ccc}1& \cdots & 1\\ ⋮& & ⋮\\ 1& \cdots & 1\end{array}\right),\hfill \\ \hfill & \widehat{\xi }=\left(\begin{array}{c}{\xi }_1\\ ⋮\\ {\xi }_N\end{array}\right),\widehat{\eta }=\left(\begin{array}{ccccc}{\eta }_1& {\eta }_{12}& \cdots & {\eta }_{1,N-1}& {\eta }_{1N}\\ {\eta }_{21}& {\eta }_2& \cdots & {\eta }_{2,N-1}& {\eta }_{2N}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\eta }_{N1}& {\eta }_{N2}& \cdots & {\eta }_{N,N-1}& {\eta }_N\end{array}\right),{\widehat{\eta }}_0=\left(\begin{array}{c}{\eta }_{01}\\ ⋮\\ {\eta }_{0N}\end{array}\right),\hfill \end{array}\right.$$

then, (1) can be rewritten as

$$\left\{\begin{array}{cc}\hfill & d\widehat{Y}\left(t\right)=-\left[\widehat{A}\left(t\right)\widehat{Y}\left(t\right)+\widehat{B}\left(t\right)\widehat{U}\left(t\right)+{\displaystyle \frac{1}{N}}\widehat{C}\left(t\right)J_N{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{Y}(t+\delta )\right]\right]dt+\widehat{Z}\left(t\right)d\widehat{W}\left(t\right)+{\widehat{Z}}_0\left(t\right)d{W}_0\left(t\right),\hfill \\ \hfill & t\in [0,T],\hfill \\ \hfill & \widehat{Y}\left(t\right)=\widehat{\xi }\left(t\right),\widehat{Z}\left(t\right)=\widehat{\eta }\left(t\right),{\widehat{Z}}_0\left(t\right)={\widehat{\eta }}_0\left(t\right),t\in [T,T+\delta ],\hfill \end{array}\right.$$

which is a linear ABSDE of vector value and admits a unique solution $(\widehat{Y},\widehat{Z},{\widehat{Z}}_0)\in L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{Nn})\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{Nn\times Nd})\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{Nn\times d})$ for $\widehat{U}\in L_{\mathcal{F}}^2(0,T;{\Gamma }_{{\theta }_1})\times \dots \times L_{\mathcal{F}}^2(0,T;{\Gamma }_{{\theta }_N})$, (see [4,35], etc.). Thus, for any $1\le i\le N$, the state equation (1) admits a unique solution $\left(y_i,z_i,z_{ij}(j\ne i),z_{0i}\right)\in L_{\mathcal{F}}^2(\Omega ;C([0,T+\delta ];{\mathbb{R}}^n))\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{n\times d})$ $\times \underset{N-1}{\underbrace{L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{n\times d})\times \cdots \times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{n\times d})}}\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{n\times d})$.

Let $u=(u_1,\cdots ,u_N)$ be the set of strategies of all N agents and $u_{-i}=(u_1,\cdots ,u_{i-1},u_{i+1},$$\cdots ,u_N)$, $1\le i\le N$. The cost functional for ${\mathcal{A}}_i$, $1\le i\le N$, is given by

$$\begin{array}{cc}\hfill {\mathcal{J}}_i(u_i(·),& u_{-i}(·))={\displaystyle \frac{1}{2}}\mathbb{E}\left\{{\int }_0^T\right[〈S\left(t\right)(y_i\left(t\right)-G\left(t\right)y^{\left(N\right)}\left(t\right)),y_i\left(t\right)-G\left(t\right)y^{\left(N\right)}\left(t\right)〉\hfill \\ \hfill & +〈\tilde{S}\left(t\right)\left({\mathbb{E}}^{{\mathcal{F}}_t}[y_i(t+\delta )-\tilde{G}\left(t\right)y^{\left(N\right)}(t+\delta )]\right),{\mathbb{E}}^{{\mathcal{F}}_t}[y_i(t+\delta )-\tilde{G}\left(t\right)y^{\left(N\right)}(t+\delta )]〉\hfill \\ \hfill & +〈R_{{\theta }_i}\left(t\right)u_i\left(t\right),u_i\left(t\right)〉]dt+〈Q(y_i\left(0\right)-H{y}^{\left(N\right)}\left(0\right)),y_i\left(0\right)-H{y}^{\left(N\right)}\left(0\right)〉\}.\hfill \end{array}$$

(2)

The aggregate team functional of N agents is

$${\mathcal{J}}_{soc}^{\left(N\right)}\left(u(·)\right)=\sum _{i=1}^N{\mathcal{J}}_i(u_i(·),u_{-i}(·)).$$

(3)

We impose the following assumptions on the coefficients of the cost functionals:

(A4) 

$S(·)\in L^{\infty }(0,T;{\mathbb{S}}^n)$, $\tilde{S}(·)\in L^{\infty }(-\delta ,T;{\mathbb{S}}^n)$, $S(·),\tilde{S}(·)\ge 0$, $G(·)\in L^{\infty }(0,T;{\mathbb{R}}^{n\times n})$, $\tilde{G}(·)\in L^{\infty }(-\delta ,T;{\mathbb{R}}^{n\times n})$, $R_{{\theta }_i}(·)\in L^{\infty }(0,T;{\mathbb{S}}^m)$, $R_{{\theta }_i}(·)\gg 0$ ($i=1,\cdots ,N$), $Q\in {\mathbb{S}}^{n\times n}$, $H\in {\mathbb{R}}^{n\times n}$, $Q\ge 0$.

For $i=1,\cdots ,N$, the centralized admissible strategy set for the $i^{th}$ agent is given by

$${\mathcal{U}}_i^c=\left\{u_i(·)|u_i(·)\in L_{\mathcal{F}}^2(0,T;{\Gamma }_{{\theta }_i})\right\}.$$

Correspondingly, the decentralized admissible strategy set for the $i^{th}$ agent is given by

$${\mathcal{U}}_i^d=\left\{u_i(·)|u_i(·)\in L_{{\mathcal{F}}^i}^2(0,T;{\Gamma }_{{\theta }_i})\right\}.$$

We propose the following optimal problem:

Problem (SO-IC-PI). Find a strategy set $\overline{u}=({\overline{u}}_1,\cdots ,{\overline{u}}_N)$ where ${\overline{u}}_i(·)\in {\mathcal{U}}_i^c$, $1\le i\le N$, such that

$${\mathcal{J}}_{soc}^{\left(N\right)}\left(\overline{u}(·)\right)=\underset{u_i\in {\mathcal{U}}_i^c,1\le i\le N}{inf}{\mathcal{J}}_{soc}^{\left(N\right)}(u_1(·),\cdots ,u_i(·),\cdots ,u_N(·)).$$

(4)

Definition 1.

A strategy ${\tilde{u}}_i(·)\in {\mathcal{U}}_i^d$, $i=1,\cdots ,N$ is an ε-social decentralized optimal strategy if there exists $\epsilon =\epsilon \left(N\right)>0$, ${lim}_{N\to \infty }\epsilon \left(N\right)=0$ such that

$${\displaystyle \frac{1}{N}}\left({\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}(·)\right)-\underset{u_i(·)\in {\mathcal{U}}_i^c,1\le i\le N}{inf}{\mathcal{J}}_{soc}^{\left(N\right)}\left(u(·)\right)\right)\le \epsilon .$$

Now, we briefly show the process of studying Problem (SO-IC-PI): Firstly, with the help of the anticipated person-by-person optimality principle and variational technique, we obtain an auxiliary LQ anticipated control problem. Then, the stochastic maximum principle ([35,36], etc.) is applied to derive the optimal control of auxiliary problem. Secondly, we establish and investigate the consistency condition system using the discounting method (e.g., [4,44]) to determine the frozen MF term and the off-line variables. Thirdly, by virtue of the standard estimations of the AFBSDDE ([35,36,39]), we verify that the decentralized strategy is the asymptotic optimality of the centralized strategy.

3. Stochastic Optimal Control Problem for the Agents ${\mathcal{A}}_{\mathit{i}}$

In this section, we try to solve the optimal control problem and derive the decentralized control.

3.1. Backward Person-by-Person Optimality

Let ${\{{\overline{u}}_i,{\overline{u}}_{-i}\in {\mathcal{U}}_i^c\}}_{i=1}^N$ be the centralized optimal strategy of all agents. Now, consider the perturbation that the agent ${\mathcal{A}}_i$ uses the strategy $u_i\in {\mathcal{U}}_i^c$ and all the other agents still apply the strategy ${\overline{u}}_{-i}=({\overline{u}}_1,\cdots ,{\overline{u}}_{i-1},{\overline{u}}_{i+1},\cdots ,{\overline{u}}_N)$. The realized states (1) corresponding to $(u_i,{\overline{u}}_{-i})$ and $({\overline{u}}_i,{\overline{u}}_{-i})$ are denoted by $((y_1,z_1,z_{1j},z_{01}),\cdots ,(y_N,z_N,z_{Nj},z_{0N}))$ and $(({\overline{y}}_1,{\overline{z}}_1,{\overline{z}}_{1j},{\overline{z}}_{01}),\cdots ,$$({\overline{y}}_N,{\overline{z}}_N,{\overline{z}}_{Nj},{\overline{z}}_{0N}))$, respectively. For $j=1,\cdots ,N$, denote the perturbation

$$\delta u_j=u_j-{\overline{u}}_j,\delta y_j=y_j-{\overline{y}}_j,\delta z_j=z_j-{\overline{z}}_j,\delta z_{jl}=z_{jl}-{\overline{z}}_{jl}(l\ne j),$$

$$\delta z_{0j}=z_{0j}-{\overline{z}}_{0j},\delta y^{\left(N\right)}=y^{\left(N\right)}-{\overline{y}}^{\left(N\right)},\delta {\mathcal{J}}_j={\mathcal{J}}_j(u_i,{\overline{u}}_{-i})-{\mathcal{J}}_j({\overline{u}}_i,{\overline{u}}_{-i}).$$

Therefore, the variation in the state for ${\mathcal{A}}_i$ is given by

$$\left\{\begin{array}{cc}\hfill & d\delta y_i=-\left[A_{{\theta }_i}\delta y_i+B\delta u_i+C{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y^{\left(N\right)}(t+\delta )\right]\right]dt+\delta z_i\left(t\right)d{W}_i\left(t\right)\hfill \\ \hfill & +\sum _{l=1,l\ne i}^N\delta z_{il}\left(t\right)d{W}_l\left(t\right)+\delta z_{0i}\left(t\right)d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & \delta y_i\left(t\right)=0,t\in [T,T+\delta ]\hfill \end{array}\right.$$

(5)

and for ${\mathcal{A}}_j$, $j\ne i$,

$$\left\{\begin{array}{cc}\hfill & d\delta y_j=-\left[A_{{\theta }_j}\delta y_j+C{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y^{\left(N\right)}(t+\delta )\right]\right]dt+\delta z_j\left(t\right)d{W}_j\left(t\right)\hfill \\ \hfill & +\sum _{l=1,l\ne j}^N\delta z_{jl}\left(t\right)d{W}_l\left(t\right)+\delta z_{0j}\left(t\right)d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & \delta y_j\left(t\right)=0,t\in [T,T+\delta ].\hfill \end{array}\right.$$

(6)

For $k=1,\cdots ,K$, define $\delta y_{\left(k\right)}={\sum }_{j\in {\mathcal{I}}_k,j\ne i}\delta y_j$, thus

$$\left\{\begin{array}{cc}\hfill & d\delta y_{\left(k\right)}=-\left[A_k\delta y_{\left(k\right)}+(N_k-I_{{\mathcal{I}}_k}\left(i\right))C{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y^{\left(N\right)}(t+\delta )\right]\right]dt+\sum _{j\in {\mathcal{I}}_k,j\ne i}\delta z_{j}d{W}_j\left(t\right)\hfill \\ \hfill & +\sum _{j\in {\mathcal{I}}_k,j\ne i}\sum _{l=1,l\ne j}^N\delta z_{jl}\left(t\right)d{W}_l\left(t\right)+\sum _{j\in {\mathcal{I}}_k,j\ne i}\delta z_{0j}d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & \delta y_{\left(k\right)}\left(t\right)=0,t\in [T,T+\delta ],\hfill \end{array}\right.$$

(7)

where $I_{{\mathcal{I}}_k}(·)$ denotes the indicative function, that is $I_{{\mathcal{I}}_k}\left(i\right)=\left\{\begin{array}{cc}1,& i\in {\mathcal{I}}_k\\ 0,& i\notin {\mathcal{I}}_k\end{array}\right.$. Through some elementary calculations, we can further obtain the variation in the cost functional of ${\mathcal{A}}_i$, as follows:

$$\begin{array}{cc}\hfill \Delta {\mathcal{J}}_i=& \mathbb{E}\left\{{\int }_0^T\right[〈S({\overline{y}}_i-G{\overline{y}}^{\left(N\right)}),\delta y_i-G\delta y^{\left(N\right)}〉+\langle \tilde{S}\left(t\right)\left({\mathbb{E}}^{{\mathcal{F}}_t}[{\overline{y}}_i(t+\delta )-\tilde{G}\left(t\right){\overline{y}}^{\left(N\right)}(t+\delta )]\right),\hfill \\ \hfill & {\mathbb{E}}^{{\mathcal{F}}_t}[\delta y_i(t+\delta )-\tilde{G}\left(t\right)\delta y^{\left(N\right)}(t+\delta )]\rangle +〈R_{{\theta }_i}{\overline{u}}_i,\delta u_i〉]dt\hfill \\ \hfill & +〈Q({\overline{y}}_i\left(0\right)-H{\overline{y}}^{\left(N\right)}\left(0\right)),\delta y_i\left(0\right)-H\delta y^{\left(N\right)}\left(0\right)〉\}.\hfill \end{array}$$

For $j\ne i$, the variation in the cost functional of ${\mathcal{A}}_j$ is given by

$$\begin{array}{cc}\hfill \Delta {\mathcal{J}}_j=& \mathbb{E}\left\{{\int }_0^T\right[〈S({\overline{y}}_j-G{\overline{y}}^{\left(N\right)}),\delta y_j-G\delta y^{\left(N\right)}〉+\langle \tilde{S}\left(t\right)\left({\mathbb{E}}^{{\mathcal{F}}_t}[{\overline{y}}_j(t+\delta )-\tilde{G}\left(t\right){\overline{y}}^{\left(N\right)}(t+\delta )]\right),\hfill \\ \hfill & {\mathbb{E}}^{{\mathcal{F}}_t}[\delta y_j(t+\delta )-\tilde{G}\left(t\right)\delta y^{\left(N\right)}(t+\delta )]\rangle ]dt+〈Q({\overline{y}}_j\left(0\right)-H{\overline{y}}^{\left(N\right)}\left(0\right)),\delta y_j\left(0\right)-H\delta y^{\left(N\right)}\left(0\right)〉\}.\hfill \end{array}$$

Therefore, by combining the above equalities, the variation in the social cost satisfies

$$\begin{array}{cc}\hfill \Delta {\mathcal{J}}_{soc}^{\left(N\right)}=& \mathbb{E}\left\{{\int }_0^T\right[〈S({\overline{y}}_i-G{\overline{y}}^{\left(N\right)}),\delta y_i-G\delta y^{\left(N\right)}〉+\sum _{j\ne i}〈S({\overline{y}}_j-G{\overline{y}}^{\left(N\right)}),\delta y_j-G\delta y^{\left(N\right)}〉\hfill \\ \hfill & +〈\tilde{S}\left(t\right)\left({\mathbb{E}}^{{\mathcal{F}}_t}[{\overline{y}}_i(t+\delta )-\tilde{G}\left(t\right){\overline{y}}^{\left(N\right)}(t+\delta )]\right),{\mathbb{E}}^{{\mathcal{F}}_t}[\delta y_i(t+\delta )-\tilde{G}\left(t\right)\delta y^{\left(N\right)}(t+\delta )]〉\hfill \\ \hfill & +\sum _{j\ne i}〈\tilde{S}\left(t\right)\left({\mathbb{E}}^{{\mathcal{F}}_t}[{\overline{y}}_j(t+\delta )-\tilde{G}\left(t\right){\overline{y}}^{\left(N\right)}(t+\delta )]\right),{\mathbb{E}}^{{\mathcal{F}}_t}[\delta y_j(t+\delta )-\tilde{G}\left(t\right)\delta y^{\left(N\right)}(t+\delta )]〉\hfill \\ \hfill & +〈R_{{\theta }_i}{\overline{u}}_i,\delta u_i〉]dt+\sum _{j=1}^N〈Q({\overline{y}}_j\left(0\right)-H{\overline{y}}^{\left(N\right)}\left(0\right)),\delta y_j\left(0\right)-H\delta y^{\left(N\right)}\left(0\right)〉\}.\hfill \end{array}$$

(8)

Step I: First, replacing ${\overline{y}}^{\left(N\right)}$ and ${\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}^{\left(N\right)}(t+\delta )\right]$ in (8) with the mean-field terms $\widehat{y}$ and ${\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right]$, which will be determined later,

$$\begin{array}{cc}\hfill \Delta {\mathcal{J}}_{soc}^{\left(N\right)}=& \mathbb{E}\{{\int }_0^T[\langle S{\overline{y}}_i,\delta y_i\rangle -\langle S_G\widehat{y},\delta y_i\rangle -\sum _{k=1}^K\langle S_G\widehat{y},\delta y_{\left(k\right)}\rangle +\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle S{\overline{y}}_j,N_k\delta y_j\rangle \hfill \\ \hfill & +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\langle {\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle {\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_{\left(k\right)}(t+\delta )\right]\rangle +\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}_j(t+\delta )\right],\hfill \\ \hfill & N_k{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_j(t+\delta )\right]\rangle +\langle R_{{\theta }_i}{\overline{u}}_i,\delta u_i\rangle ]dt+\langle Q{\overline{y}}_i\left(0\right),\delta y_i\left(0\right)\rangle -\langle Q_H\widehat{y}\left(0\right),\delta y_i\left(0\right)\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle Q_H\widehat{y}\left(0\right),\delta y_{\left(k\right)}\left(0\right)\rangle +\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle Q{\overline{y}}_j\left(0\right),N_k\delta y_j\left(0\right)\rangle \}+\sum _{l=1}^3{\epsilon }_l,\hfill \end{array}$$

where

$$\left\{\begin{array}{cc}\hfill & {\epsilon }_1=\mathbb{E}{\int }_0^T\langle S_G(\widehat{y}-{\overline{y}}^{\left(N\right)}),N\delta y^{\left(N\right)}\rangle dt,\hfill \\ \hfill & {\epsilon }_2=\mathbb{E}{\int }_0^T\langle {\tilde{S}}_{\tilde{G}}({\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right]-{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}^{\left(N\right)}(t+\delta )\right]),N{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y^{\left(N\right)}(t+\delta )\right]\rangle dt,\hfill \\ \hfill & {\epsilon }_3=\langle Q_H(\widehat{y}\left(0\right)-{\overline{y}}^{\left(N\right)}\left(0\right)),N\delta y^{\left(N\right)}\left(0\right)\rangle \hfill \end{array}\right.$$

and

$$\begin{array}{cc}\hfill & S_G:=SG+G^{\top }S-G^{\top }SG,{\tilde{S}}_{\tilde{G}}:=\tilde{S}\tilde{G}+{\tilde{G}}^{\top }\tilde{S}-{\tilde{G}}^{\top }\tilde{S}\tilde{G},Q_H:=QH+H^{\top }Q-H^{\top }QH.\hfill \end{array}$$

Step II: Next, for $j\in {\mathcal{I}}_k$, introduce the limit $(y_j^{*},z_j^{*},z_{jl}^{*}(l\ne j),z_{0j}^{*})$ to replace $(N_k\delta y_j,N_k\delta z_j,$$N_k\delta z_{jl}(l\ne j),N_k\delta z_{0j})$, and for $k=1,\cdots ,K$, introduce the limit $y_k^{**}$ to replace $\delta y_{\left(k\right)}$, where

$$\left\{\begin{array}{cc}\hfill & d{y}_j^{*}=-\left[A_k{y}_j^{*}+C{\pi }_k{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]+C{\pi }_k\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right]\right]dt+z_j^{*}d{W}_j\left(t\right)\hfill \\ \hfill & +\sum _{l=1,l\ne j}^N{z}_{jl}^{*}d{W}_l\left(t\right)+z_{0j}^{*}d{W}_0\left(t\right),\hfill \\ \hfill & d{y}_k^{**}=-\left[A_k{y}_k^{**}+C{\pi }_k{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]+C{\pi }_k\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right]\right]dt+\sum _{l=1}^N{z}_{kl}^{**}d{W}_l\left(t\right)\hfill \\ \hfill & +z_{0k}^{**}d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & y_j^{*}\left(t\right)=0,d{y}_k^{**}\left(t\right)=0,t\in [T,T+\delta ].\hfill \end{array}\right.$$

(9)

Therefore,

$$\begin{array}{cc}\hfill \Delta {\mathcal{J}}_{soc}^{\left(N\right)}=& \mathbb{E}\{{\int }_0^T[\langle S{\overline{y}}_i,\delta y_i\rangle -\langle S_G\widehat{y},\delta y_i\rangle -\sum _{k=1}^K\langle S_G\widehat{y},y_k^{**}\rangle +\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle S{\overline{y}}_j,y_j^{*}\rangle \hfill \\ \hfill & +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\langle {\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle {\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_k^{**}(t+\delta )\right]\rangle +\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}_j(t+\delta )\right],\hfill \\ \hfill & {\mathbb{E}}^{{\mathcal{F}}_t}\left[y_j^{*}(t+\delta )\right]\rangle +\langle R_{{\theta }_i}{\overline{u}}_i,\delta u_i\rangle ]dt+\langle Q{\overline{y}}_i\left(0\right),\delta y_i\left(0\right)\rangle -\langle Q_H\widehat{y}\left(0\right),\delta y_i\left(0\right)\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle Q_H\widehat{y}\left(0\right),y_k^{**}\left(0\right)\rangle +\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle Q{\overline{y}}_j\left(0\right),y_j^{*}\left(0\right)\rangle \}+\sum _{l=1}^8{\epsilon }_l,\hfill \end{array}$$

(10)

where

$$\left\{\begin{array}{cc}\hfill & {\epsilon }_4=\sum _{k=1}^K\mathbb{E}{\int }_0^T\langle S_G\widehat{y},y_k^{**}-\delta y_{\left(k\right)}\rangle dt,{\epsilon }_5=\sum _{k=1}^K\mathbb{E}{\int }_0^T{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle S{\overline{y}}_j,N_k\delta y_j-y_j^{*}\rangle dt,\hfill \\ \hfill & {\epsilon }_6=\sum _{k=1}^K\mathbb{E}{\int }_0^T\langle {\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}[y_k^{**}(t+\delta )-\delta y_{\left(k\right)}(t+\delta )]\rangle dt,\hfill \\ \hfill & {\epsilon }_7=\sum _{k=1}^K\mathbb{E}{\int }_0^T{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}_j(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}[N_k\delta y_j(t+\delta )-y_j^{*}(t+\delta )]\rangle dt,\hfill \\ \hfill & {\epsilon }_8=\sum _{k=1}^K\langle Q_H\widehat{y}\left(0\right),y_k^{**}\left(0\right)-\delta y_{\left(k\right)}\left(0\right)\rangle ,{\epsilon }_9=\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle Q{\overline{y}}_j\left(0\right),N_k\delta y_j\left(0\right)-y_j^{*}\left(0\right)\rangle .\hfill \end{array}\right.$$

Step III: Finally, we introduce the following adjoint equations $x_1^j$ and $x_2^k$ as

$$\left\{\begin{array}{cc}\hfill & d{x}_1^j={\alpha }_1^{j}dt,t\in [0,T],x_1^j\left(0\right)=-Q{\overline{y}}_j\left(0\right),x_1^j\left(t\right)=0,t\in [-\delta ,0),j=1,\cdots ,N,\hfill \\ \hfill & d{x}_2^k={\alpha }_2^{k}dt,t\in [0,T],x_2^k\left(0\right)=Q_H\widehat{y}\left(0\right),x_2^k\left(t\right)=0,t\in [-\delta ,0),k=1,\cdots ,K.\hfill \end{array}\right.$$

Applying Itô’s formula to $\langle x_1^j,y_j^{*}\rangle$, we have

$$\begin{array}{cc}\hfill d\langle x_1^j,y_j^{*}\rangle =& \left[\langle x_1^j,-(A_k{y}_j^{*}+C{\pi }_k{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]+C{\pi }_k\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right])\rangle +\langle {\alpha }_1^j,y_j^{*}\rangle \right]dt\hfill \\ \hfill & +\sum _{j=1}^N(\cdots )d{W}_j\left(t\right)+(\cdots )d{W}_0\left(t\right).\hfill \end{array}$$

For $j\in {\mathcal{I}}_k$, integrating from 0 to T and taking expectation, we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\langle Q{\overline{y}}_j\left(0\right),y_j^{*}\left(0\right)\rangle =\mathbb{E}\langle x_1^j\left(T\right),y_j^{*}\left(T\right)\rangle -\mathbb{E}\langle x_1^j\left(0\right),y_j^{*}\left(0\right)\rangle \hfill \\ \hfill =& \mathbb{E}{\int }_0^T\left[\langle x_1^j,-(A_k{y}_j^{*}+C{\pi }_k{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]+C{\pi }_k\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right])\rangle +\langle {\alpha }_1^j,y_j^{*}\rangle \right]dt\hfill \\ \hfill =& \mathbb{E}{\int }_0^T\left[\langle {\alpha }_1^j-A_k^{\top }{x}_1^j,y_j^{*}\rangle -\langle {\pi }_k{C}^{\top }{x}_1^j,{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\sum _{l=1}^K\langle {\pi }_k{C}^{\top }{x}_1^j,{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right]\rangle \right]dt.\hfill \end{array}$$

(11)

Similarly, we have

$$\begin{array}{cc}\hfill & -\mathbb{E}\langle Q_H\widehat{y}\left(0\right),y_k^{**}\left(0\right)\rangle =\mathbb{E}\langle x_2^k\left(T\right),y_k^{**}\left(T\right)\rangle -\mathbb{E}\langle x_2^k\left(0\right),y_k^{**}\left(0\right)\rangle \hfill \\ \hfill =& \mathbb{E}{\int }_0^T\left[\langle {\alpha }_2^k-A_k^{\top }{x}_2^k,y_k^{**}\rangle -\langle {\pi }_k{C}^{\top }{x}_2^k,{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\sum _{l=1}^K\langle {\pi }_k{C}^{\top }{x}_2^k,{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right]\rangle \right]dt.\hfill \end{array}$$

(12)

Recall that $x_1^j\left(t\right)=0$, for $t\in [-\delta ,0)$, then

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }{x}_1^j,{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle dt& =\mathbb{E}{\int }_{\delta }^{T+\delta }\langle {\pi }_k{C}^{\top }(t-\delta )x_1^j(t-\delta ),\delta y_i\left(t\right)\rangle dt\hfill \\ \hfill & =\mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }(t-\delta )x_1^j(t-\delta ),\delta y_i\left(t\right)\rangle dt.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }{x}_1^j,{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right]\rangle dt=\mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }(t-\delta )x_1^j(t-\delta ),y_l^{**}\left(t\right)\rangle dt,\hfill \\ \hfill & \mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }{x}_2^k,{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle dt=\mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta ),\delta y_i\left(t\right)\rangle dt,\hfill \\ \hfill & \mathbb{E}{\int }_0^T\sum _{k=1}^K\sum _{l=1}^K\langle {\pi }_k{C}^{\top }{x}_2^k,{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right]\rangle dt=\mathbb{E}{\int }_0^T\sum _{k=1}^K\sum _{l=1}^K\langle {\pi }_l{C}^{\top }(t-\delta )x_2^l(t-\delta ),y_k^{**}\left(t\right)\rangle dt.\hfill \end{array}$$

Letting

$$\left\{\begin{array}{cc}\hfill {\alpha }_1^j=& A_k^{\top }{x}_1^j-(S+\tilde{S}(t-\delta )){\overline{y}}_j,\hfill \\ \hfill {\alpha }_2^k=& A_k^{\top }{x}_2^k+(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ))\widehat{y}+\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )\mathbb{E}{x}_1^l(t-\delta )+\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )x_2^l(t-\delta ),\hfill \end{array}\right.$$

substituting (11) and (12) into (10), we have

$$\begin{array}{cc}\hfill \Delta {\mathcal{J}}_{soc}^{\left(N\right)}=& \mathbb{E}\left\{{\int }_0^T\right[\langle S{\overline{y}}_i,\delta y_i\rangle -\langle S_G\widehat{y},\delta y_i\rangle +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle \hfill \\ \hfill & -\langle {\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta ),\delta y_i\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta ),\delta y_i\rangle +\langle R_{{\theta }_i}{\overline{u}}_i,\delta u_i\rangle ]dt\hfill \\ \hfill & +\langle Q{\overline{y}}_i\left(0\right),\delta y_i\left(0\right)\rangle -\langle Q_H\widehat{y}\left(0\right),\delta y_i\left(0\right)\rangle \}+\sum _{l=1}^{11}{\epsilon }_l,\hfill \end{array}$$

where

$$\left\{\begin{array}{cc}\hfill & d{x}_1^j=\left[A_k^{\top }{x}_1^j-(S+\tilde{S}(t-\delta )){\overline{y}}_j\right]dt,t\in [0,T],x_1^j\left(0\right)=-Q{\overline{y}}_j\left(0\right),x_1^j\left(t\right)=0,t\in [-\delta ,0),\hfill \\ \hfill & d{x}_2^k=\left[A_k^{\top }{x}_2^k+(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ))\widehat{y}+\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )\mathbb{E}{x}_1^l(t-\delta )\right.\hfill \\ \hfill & \left.+\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )x_2^l(t-\delta )\right]dt,t\in [0,T],x_2^k\left(0\right)=Q_H\widehat{y}\left(0\right),x_2^k\left(t\right)=0,t\in [-\delta ,0),\hfill \\ \hfill & j=1,\cdots ,N,k=1,\cdots ,K.\hfill \end{array}\right.$$

(13)

and

$$\left\{\begin{array}{cc}\hfill & {\epsilon }_{10}=\sum _{k=1}^K\mathbb{E}{\int }_0^T〈\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_l(t-\delta )-\sum _{l=1}^K{\displaystyle \frac{{\pi }_l}{N_l}}\sum _{j\in {\mathcal{I}}_l,j\ne i}{C}^{\top }(t-\delta )x_1^j(t-\delta ),y_k^{**}〉dt,\hfill \\ \hfill & {\epsilon }_{11}=\sum _{k=1}^K\mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta )-{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{\pi }_k{C}^{\top }(t-\delta )x_1^j(t-\delta ),\delta y_i\rangle dt.\hfill \end{array}\right.$$

In the following, when considering the expectations, we will use ${\mathbf{x}}_k$ to denote the process $x_1^j$ defined in (13) of the representative agent of type k. Now, we introduce the decentralized auxiliary cost functional $J_i$ with perturbation as

$$\begin{array}{cc}\hfill \Delta J_i=& \mathbb{E}\left\{{\int }_0^T\right[\langle S{\overline{y}}_i,\delta y_i\rangle -\langle S_G\widehat{y},\delta y_i\rangle +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\overline{y}}_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle \hfill \\ \hfill & -\langle {\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta ),\delta y_i\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta ),\delta y_i\rangle +\langle R_{{\theta }_i}{\overline{u}}_i,\delta u_i\rangle ]dt\hfill \\ \hfill & +\langle Q{\overline{y}}_i\left(0\right),\delta y_i\left(0\right)\rangle -\langle Q_H\widehat{y}\left(0\right),\delta y_i\left(0\right)\rangle \}.\hfill \end{array}$$

(14)

3.2. Decentralized Strategy

Motivated by (14), we introduce the following auxiliary anticipated backward LQG control problem:

Problem (MFG-IC-PI). Minimize $J_i\left(u_i\right)$ over $u_i\in {\mathcal{U}}_i^d$ subject to

$$\left\{\begin{array}{cc}\hfill d{y}_i\left(t\right)=& -\left[A_{{\theta }_i}\left(t\right)y_i\left(t\right)+B\left(t\right)u_i\left(t\right)+C\left(t\right){\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right]\right]dt+z_i\left(t\right)d{W}_i\left(t\right)+z_{0i}\left(t\right)d{W}_0\left(t\right),\hfill \\ \hfill & t\in [0,T],\hfill \\ \hfill y_i\left(t\right)=& {\xi }_i\left(t\right),z_i\left(t\right)={\eta }_i\left(t\right),z_{0i}\left(t\right)={\eta }_{0i}\left(t\right),t\in [T,T+\delta ],\hfill \end{array}\right.$$

(15)

where

$$\begin{array}{cc}\hfill J_i\left(u_i\right)=& {\displaystyle \frac{1}{2}}\left\{\mathbb{E}{\int }_0^T\right[\langle S{y}_i,y_i\rangle +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_i(t+\delta )\right]\rangle -2\langle {\Theta }_1,y_i\rangle -2\langle {\Theta }_2,{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_i(t+\delta )\right]\rangle \hfill \\ \hfill & +\langle R_{{\theta }_i}{u}_i,u_i\rangle ]dt+\langle Q{y}_i\left(0\right),y_i\left(0\right)\rangle -2\langle {\Theta }_3,y_i\left(0\right)\rangle \},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\Theta }_1=S_G\widehat{y}+\sum _{k=1}^K{\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta )+\sum _{k=1}^K{\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta ),{\Theta }_2={\tilde{S}}_{\tilde{G}}{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right],\hfill \\ \hfill & {\Theta }_3=Q_H\widehat{y}\left(0\right),\hfill \end{array}$$

and $\widehat{y},x_2^k,{\widehat{x}}_k$ will be determined by the consistency condition in the following section.

Similarly to [29,35], we will apply a stochastic maximum principle to study Problem (MFG-IC-PI). First, introduce the following first order adjoint equation:

$$\left\{\begin{array}{cc}\hfill & d{p}_i\left(t\right)=\left[A_{{\theta }_i}^{\top }{p}_i+(S+\tilde{S}(t-\delta ))y_i-{\Theta }_1-{\Theta }_2(t-\delta )\right]dt,t\in [0,T],\hfill \\ \hfill & p_i\left(0\right)=Q{y}_i\left(0\right)-{\Theta }_3,p_i\left(t\right)=0,t\in [-\delta ,0).\hfill \end{array}\right.$$

The global stochastic maximum principle implies that

$${\overline{u}}_i\left(t\right)=-{\mathbf{P}}_{{\Gamma }_{{\theta }_i}}\left[R_{{\theta }_i}^{-1}\left(t\right)B^{\top }\left(t\right)p_i\left(t\right)\right],a.e.t\in [0,T],\mathbb{P}-a.s.,$$

where ${\mathbf{P}}_{{\Gamma }_{{\theta }_i}}[·]$ is the projection mapping from ${\mathbb{R}}^d$ to its closed convex subset ${\Gamma }_{{\theta }_i}$ under the norm ${\parallel ·\parallel }_{R_{{\theta }_i}}$. The related Hamiltonian system becomes

$$\left\{\begin{array}{cc}\hfill d{y}_i\left(t\right)=& -\left[A_{{\theta }_i}{y}_i-B{\mathbf{P}}_{{\Gamma }_{{\theta }_i}}\left[R_{{\theta }_i}^{-1}{B}^{\top }{p}_i\right]+C{\mathbb{E}}^{{\mathcal{F}}_t}\left[\widehat{y}(t+\delta )\right]\right]dt+z_i\left(t\right)d{W}_i\left(t\right)+z_{0i}\left(t\right)d{W}_0\left(t\right),\hfill \\ \hfill d{p}_i\left(t\right)=& \left[A_{{\theta }_i}^{\top }{p}_i+(S+\tilde{S}(t-\delta ))y_i-{\Theta }_1-{\Theta }_2(t-\delta )\right]dt,t\in [0,T],\hfill \\ \hfill y_i\left(t\right)=& {\xi }_i\left(t\right),z_i\left(t\right)={\eta }_i\left(t\right),z_{0i}\left(t\right)={\eta }_{0i}\left(t\right),t\in [T,T+\delta ],\hfill \\ \hfill p_i\left(0\right)=& Q{y}_i\left(0\right)-{\Theta }_3,p_i\left(t\right)=0,t\in [-\delta ,0).\hfill \end{array}\right.$$

4. Consistency Condition

Theorem 1.

Let (A1)–(A4) hold. The parameters in Problem (MFG-IC-PI) are determined by

$$\left(\widehat{y},{\widehat{x}}_k,x_2^k\right)=\left(\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right],\mathbb{E}{\check{x}}^k,\mathbb{E}{\check{x}}_2^k\right),$$

where $({\alpha }_k,{\beta }_k,{\gamma }_k,{\check{x}}^k,{\check{x}}_2^k)$ is the solution of the following MF-AFBSDDE, which is the so-called consistency condition system: for $k=1,\cdots ,K$,

$$\left\{\begin{array}{cc}\hfill & d{\alpha }_k\left(t\right)=-\left[A_k{\alpha }_k-B{\mathbf{P}}_{{\Gamma }_k}\left[R_k^{-1}{B}^{\top }{\gamma }_k\right]+C\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right]\right]dt+{\beta }_{k}d{W}^{\left(k\right)}\left(t\right)+{\beta }_{0k}d{W}_0\left(t\right),\hfill \\ \hfill & d{\gamma }_k\left(t\right)=[A_k^{\top }{\gamma }_k+(S+\tilde{S}(t-\delta )){\alpha }_k-(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ))\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right]\hfill \\ \hfill & -\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )\mathbb{E}{\check{x}}^l(t-\delta )-\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta ){\check{x}}_2^l(t-\delta )]dt,\hfill \\ \hfill & d{\check{x}}^k\left(t\right)=\left[A_k^{\top }{\check{x}}^k-(S+\tilde{S}(t-\delta )){\alpha }_k\right]dt,\hfill \\ \hfill & d{\check{x}}_2^k\left(t\right)=[A_k^{\top }{\check{x}}_2^k+(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ))\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right]+\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )\mathbb{E}{\check{x}}^l(t-\delta )\hfill \\ \hfill & +\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta ){\check{x}}_2^l(t-\delta )]dt,t\in [0,T],\hfill \\ \hfill & {\alpha }_k\left(t\right)={\xi }^{\left(k\right)}\left(t\right),{\beta }_k\left(t\right)={\eta }^{\left(k\right)}\left(t\right),{\beta }_{0k}\left(t\right)={\eta }_0^{\left(k\right)}\left(t\right),t\in [T,T+\delta ],\hfill \\ \hfill & {\gamma }_k\left(0\right)=Q{\alpha }_k\left(0\right)-Q_H\sum _{l=1}^K{\pi }_l{\alpha }_l\left(0\right),{\check{x}}^k\left(0\right)=-Q{\alpha }_k\left(0\right),\hfill \\ \hfill & {\check{x}}_2^k\left(0\right)=Q_H\sum _{l=1}^K{\pi }_l{\alpha }_l\left(0\right),{\gamma }_k\left(t\right)=0,{\check{x}}^k\left(t\right)=0,{\check{x}}_2^k\left(t\right)=0,t\in [-\delta ,0).\hfill \end{array}\right.$$

(17)

Define $Y={({\alpha }_1^{\top },\cdots ,{\alpha }_K^{\top })}^{\top }$, $X=({\gamma }_1^{\top },\cdots ,{\gamma }_K^{\top },{\left({\check{x}}^1\right)}^{\top },\cdots$, ${\left({\check{x}}^K\right)}^{\top },{\left({\check{x}}_2^1\right)}^{\top },\cdots ,{\left({\check{x}}_2^K\right)}^{\top }{)}^{\top }$, $W={({\left(W^{\left(1\right)}\right)}^{\top },\cdots ,{\left(W^{\left(K\right)}\right)}^{\top })}^{\top }$ and

$$Z=\left(\begin{array}{ccc}{\beta }_1& & \\ & \ddots & \\ & & {\beta }_K\end{array}\right),Z_0=\left(\begin{array}{c}{\beta }_{01}\\ ⋮\\ {\beta }_{0K}\end{array}\right),\Xi =\left(\begin{array}{c}{\xi }^{\left(1\right)}\\ ⋮\\ {\xi }^{\left(K\right)}\end{array}\right),\Gamma =\left(\begin{array}{c}{\eta }^{\left(1\right)}\\ ⋮\\ {\eta }^{\left(K\right)}\end{array}\right),$$

the MF-AFBSDDE (17), then take the following form:

$$\left\{\begin{array}{cc}\hfill & dY=-\left[{\mathbb{A}}_{1}Y+{\overline{\mathbb{A}}}_1{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[Y(t+\delta )\right]+{\mathbb{B}}_1\left(X\right)\right]dt+ZdW\left(t\right)+Z_{0}d{W}_0\left(t\right),\hfill \\ \hfill & dX=\left[{\mathbb{A}}_{2}X+{\tilde{\mathbb{A}}}_{2}X(t-\delta )+{\overline{\mathbb{A}}}_2\mathbb{E}\left[X(t-\delta )\right]+{\mathbb{B}}_{2}Y+{\overline{\mathbb{B}}}_2{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[Y\right]\right]dt,t\in [0,T],\hfill \\ \hfill & Y\left(t\right)=\Xi \left(t\right),Z\left(t\right)=\Gamma \left(t\right),t\in [T,T+\delta ],\hfill \\ \hfill & X\left(0\right)={\mathbb{H}}_{1}Y\left(0\right)+{\mathbb{H}}_2\mathbb{E}Y\left(0\right),X\left(t\right)=0,t\in [-\delta ,0),\hfill \end{array}\right.$$

(18)

where

$$\begin{array}{cc}\hfill & {\mathbb{A}}_1=\left(\begin{array}{ccc}{A}_1& & \\ & \ddots & \\ & & A_K\end{array}\right),{\overline{\mathbb{A}}}_1=\left(\begin{array}{ccc}C{\pi }_1& \cdots & C{\pi }_K\\ ⋮& \ddots & ⋮\\ C{\pi }_1& \cdots & C{\pi }_K\end{array}\right),{\mathbb{B}}_1\left(X\right)=\left(\begin{array}{ccccccccc}-B{\mathbf{P}}_{{\Gamma }_1}\left[R_1^{-1}{B}^{\top }{\gamma }_1\right]& & & 0& & & 0& & \\ & \ddots & & & \ddots & & & \ddots & \\ & & -B{\mathbf{P}}_{{\Gamma }_1}\left[R_K^{-1}{B}^{\top }{\gamma }_K\right]& & & 0& & & 0\end{array}\right),\hfill \\ \hfill & {\mathbb{A}}_2=\left(\begin{array}{ccccccccc}{A}_1^{\top }& & & 0& & & 0& & \\ & \ddots & & & \ddots & & & \ddots & \\ & & A_K^{\top }& & & 0& & & 0\\ 0& & & A_1^{\top }& & & 0& & \\ & \ddots & & & \ddots & & & \ddots & \\ & & 0& & & A_K^{\top }& & & 0\\ 0& & & 0& & & A_1^{\top }& & \\ & \ddots & & & \ddots & & & \ddots & \\ & & 0& & & 0& & & A_K^{\top }\end{array}\right),{\tilde{\mathbb{A}}}_2=\left(\begin{array}{ccccccccc}0& & & 0& & & -{\pi }_1{C}^{\top }(t-\delta )& \cdots & -{\pi }_K{C}^{\top }(t-\delta )\\ & \ddots & & & \ddots & & ⋮& \ddots & ⋮\\ & & 0& & & 0& -{\pi }_1{C}^{\top }(t-\delta )& \cdots & -{\pi }_K{C}^{\top }(t-\delta )\\ 0& & & 0& & & 0& & \\ & \ddots & & & \ddots & & & \ddots & \\ & & 0& & & 0& & & 0\\ 0& & & 0& & & {\pi }_1{C}^{\top }(t-\delta )& \cdots & {\pi }_K{C}^{\top }(t-\delta )\\ & \ddots & & & \ddots & & ⋮& \ddots & ⋮\\ & & 0& & & 0& {\pi }_1{C}^{\top }(t-\delta )& \cdots & {\pi }_K{C}^{\top }(t-\delta )\end{array}\right),\hfill \\ \hfill & {\overline{\mathbb{A}}}_2=\left(\begin{array}{ccccccccc}0& & & -{\pi }_1{C}^{\top }(t-\delta )& \cdots & -{\pi }_K{C}^{\top }(t-\delta )& 0& & \\ & \ddots & & ⋮& \ddots & ⋮& & \ddots & \\ & & 0& -{\pi }_1{C}^{\top }(t-\delta )& \cdots & -{\pi }_K{C}^{\top }(t-\delta )& & & 0\\ 0& & & 0& & & 0& & \\ & \ddots & & & \ddots & & & \ddots & \\ & & 0& & & 0& & & 0\\ 0& & & {\pi }_1{C}^{\top }(t-\delta )& \cdots & {\pi }_K{C}^{\top }(t-\delta )& 0& & \\ & \ddots & & ⋮& \ddots & ⋮& & \ddots & \\ & & 0& {\pi }_1{C}^{\top }(t-\delta )& \cdots & {\pi }_K{C}^{\top }(t-\delta )& & & 0\end{array}\right),{\mathbb{B}}_2=\left(\begin{array}{ccc}S+\tilde{S}(t-\delta )& & \\ & \ddots & \\ & & S+\tilde{S}(t-\delta )\\ -(S+\tilde{S}(t-\delta ))& & \\ & \ddots & \\ & & -(S+\tilde{S}(t-\delta ))\\ 0& & \\ & \ddots & \\ & & 0\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\overline{\mathbb{B}}}_2=\left(\begin{array}{ccc}-(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta )){\pi }_1& \cdots & -(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta )){\pi }_K\\ ⋮& \ddots & ⋮\\ -(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta )){\pi }_1& \cdots & -(S_G+{\tilde{S}}_{\tilde{G}}(t-\delta )){\pi }_K\\ 0& & \\ & \ddots & \\ & & 0\\ S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ){\pi }_1& \cdots & S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ){\pi }_K\\ ⋮& \ddots & ⋮\\ S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ){\pi }_1& \cdots & S_G+{\tilde{S}}_{\tilde{G}}(t-\delta ){\pi }_K\end{array}\right),{\mathbb{H}}_1=\left(\begin{array}{ccc}Q& & \\ & \ddots & \\ & & Q\\ -Q& & \\ & \ddots & \\ & & -Q\\ 0& & \\ & \ddots & \\ & & 0\end{array}\right),{\mathbb{H}}_2=\left(\begin{array}{ccc}-Q_H{\pi }_1& \cdots & -Q_H{\pi }_K\\ ⋮& \ddots & ⋮\\ -Q_H{\pi }_1& \cdots & -Q_H{\pi }_K\\ 0& & \\ & \ddots & \\ & & 0\\ Q_H{\pi }_1& \cdots & Q_H{\pi }_K\\ ⋮& \ddots & ⋮\\ Q_H{\pi }_1& \cdots & Q_H{\pi }_K\end{array}\right).\hfill \end{array}$$

In the following, we will use the discounting method of [4,44] to study the global solvability of MF-AFBSDDE (18). To start, we first give some results for the general nonlinear forward-backward system

$$\left\{\begin{array}{cc}\hfill & dX\left(t\right)=b\left(t,X\left(t\right),X(t-\delta ),\mathbb{E}X(t-\delta ),Y\left(t\right),{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[Y\left(t\right)\right]\right)dt,\hfill \\ \hfill & dY\left(t\right)=-f\left(t,X\left(t\right),Y\left(t\right),{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[Y(t+\delta )\right]\right)dt+Z\left(t\right)dW\left(t\right)+Z_0\left(t\right)d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & X\left(0\right)=\Phi \left(Y\right(0),\mathbb{E}Y(0\left)\right),X\left(t\right)=0,t\in [-\delta ,0),Y\left(t\right)=\Xi \left(t\right),Z\left(t\right)=\Delta \left(t\right),t\in [T,T+\delta ],\hfill \end{array}\right.$$

(19)

where the coefficients satisfy the following conditions

(A5) 

There exist ${\rho }_1,{\rho }_2\in \mathbb{R}$ and positive constants $k_i,i=1,\cdots ,10$ such that for t and all coefficients

• $\langle b(t,x_1,x^{\delta },{\overline{x}}^{\delta },y,\overline{y})-b(t,x_2,x^{\delta },{\overline{x}}^{\delta },y,\overline{y}),x_1-x_2\rangle \le {\rho }_1{|x_1-x_2|}^2,$
• $|b(t,x,x_1^{\delta },{\overline{x}}_1^{\delta },y_1,{\overline{y}}_1)-b(t,x,x_2^{\delta },{\overline{x}}_2^{\delta },y_2,{\overline{y}}_2)|\le k_1|x_1^{\delta }-x_2^{\delta }|+k_2|{\overline{x}}_1^{\delta }-{\overline{x}}_2^{\delta }|+k_3|y_1-y_2|+k_4|{\overline{y}}_1-{\overline{y}}_2|,$
• $|b(t,x,x^{\delta },{\overline{x}}^{\delta },y,\overline{y})|\le |b(t,0,x^{\delta },{\overline{x}}^{\delta },y,\overline{y})|+k_5(1+|x\left|\right),$
• $\langle f(t,x,y_1,{\overline{y}}^{\delta +})-f(t,x,y_2,{\overline{y}}^{\delta +}),y_1-y_2\rangle \le {\rho }_2{|y_1-y_2|}^2,$
• $|f(t,x_1,y,{\overline{y}}_1^{\delta +})-f(t,x_2,y,{\overline{y}}_2^{\delta +})|\le k_6|x_1-x_2|+k_7|{\overline{y}}_1^{\delta +}-{\overline{y}}_2^{\delta +}|,$
• $|f(t,x,y,{\overline{y}}^{\delta +})|\le |f(t,x,0,{\overline{y}}^{\delta +})|+k_8(1+|y\left|\right),$
• $|\Phi (y_1,{\overline{y}}_1)-\Phi (y_2,{\overline{y}}_2){|}^2\le k_9^2|y_1-y_2{|}^2+k_{10}^2{|{\overline{y}}_1-{\overline{y}}_2|}^2.$

(A6) 

$\mathbb{E}{\int }_0^T\left({\left|b(s,\mathbf{0})\right|}^2+{\left|f(s,\mathbf{0})\right|}^2\right)ds+\mathbb{E}{|\Phi (0,0)|}^2<+\infty .$

Let $\mathcal{H}$ be a Hilbert space. Recall that $L_{\mathcal{F}}^2(0,T;\mathcal{H})$ denotes the space of $\mathcal{H}-$valued $\left\{{\mathcal{F}}_s\right\}-$progressively measurable processes $\left\{v\right(s),s\in [0,T\left]\right\}$ such that ${\parallel v\parallel }^2:=\mathbb{E}{\int }_0^T{\left|v\left(s\right)\right|}^{2}ds<\infty$. Then, for $\rho \in \mathbb{R}$, we define an equivalent norm on $L_{\mathcal{F}}^2(0,T;\mathcal{H})$:

$${\parallel v\parallel }_{\rho }:={\left(\mathbb{E}{\int }_0^T{e}^{-\rho s}{\left|v\left(s\right)\right|}^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}.$$

Define

$$\begin{array}{c}\hfill L_{\rho ,k,\delta }:=2({\rho }_1+{\rho }_2)+k_1+k_2+k_7+(k_1+k_2)e^{-(2{\rho }_1+k_1+k_2)\delta }+k_7{e}^{-(2{\rho }_2+k_7)\delta }.\end{array}$$

Then, we have the following theorem:

Theorem 2.

Suppose that assumptions (A5) and (A6) hold. Then, there exists a ${ϵ}_0>0$, which depends on $k_i,{\rho }_1,{\rho }_2,T$, for $i=1,2,3,4,6,7$ such that when $k_j\in [0,{ϵ}_0),$ for $j=9,10$, there exists a unique adapted solution $(X,Y,Z,Z_0)\in L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^m)\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})$ to the MF-AFBSDDE (19). Further, if $L_{\rho ,k,\delta }<0$, there exists a ${ϵ}_1>0$, which depends on $k_i,{\rho }_1,{\rho }_2$, for $i=1,2,3,4,6,7$ and is independent of T, such that when $k_j\in [0,{ϵ}_1),$ for $j=9,10$, there exists a unique adapted solution $(X,Y,Z,Z_0)$ to the MF-AFBSDDE (19).

Before proving Theorem 2, we should carry out some preparative work. For any given $X(·)\in L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)$, the backward equation in the MF-AFBSDDE (19) admits a unique solution $(Y(·),Z(·),Z_0(·))\in L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^m)\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})$. Thus, we introduce a map ${\mathcal{M}}_1:L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)\to L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^m)\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})$, through

$$\left\{\begin{array}{cc}\hfill & Y\left(t\right)=\Xi \left(T\right)+{\int }_t^{T}f\left(s,X\left(s\right),Y\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y(s+\delta )\right]\right)ds-{\int }_t^{T}Z\left(s\right)dW\left(s\right)-{\int }_t^T{Z}_0\left(s\right)d{W}_0\left(s\right),\hfill \\ \hfill & t\in [0,T],\hfill \\ \hfill & Y\left(t\right)=\Xi \left(t\right),Z\left(t\right)=\Delta \left(t\right),t\in [T,T+\delta ].\hfill \end{array}\right.$$

(20)

The well-posedness of (20) under the assumptions (A5) and (A6) can be established by [35,45,46]. The proof is also omitted. Moreover, we have $\mathbb{E}{sup}_{t\in [0,T]}{\left|Y\left(t\right)\right|}^2<\infty$.

Lemma 1.

Let $(Y_i(·),Z_i(·),Z_{0i}(·))$ be the solution of (20) corresponding to $X(·)\in L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)$, $i=1,2$, respectively. Then, for all $\rho \in \mathbb{R}$ and the constants $l_1>0$, we have

$$\begin{array}{cc}\hfill & e^{-\rho t}\mathbb{E}|\widehat{Y}{\left(t\right)|}^2+{\overline{\rho }}_1\mathbb{E}{\int }_t^T{e}^{-\rho s}|\widehat{Y}{\left(s\right)|}^{2}ds+\mathbb{E}{\int }_t^T{e}^{-\rho s}\left(\right|\widehat{Z}{\left(s\right)|}^2+|{\widehat{Z}}_0{\left(s\right)|}^2)ds\le k_6{l}_1\mathbb{E}{\int }_t^T{e}^{-\rho s}{\left|\widehat{X}\left(s\right)\right|}^{2}ds,\hfill \end{array}$$

(21)

where ${\overline{\rho }}_1=-\rho -2{\rho }_1-k_6{l}_1^{-1}-k_7(1+e^{\rho \delta })$, and $\widehat{\psi }={\psi }_1-{\psi }_2,\psi =X,Y,Z,Z_0$. We also have that

$$\begin{array}{cc}\hfill & e^{-\rho t}\mathbb{E}|\widehat{Y}{\left(t\right)|}^2+\mathbb{E}{\int }_t^T{e}^{-{\tilde{\rho }}_1(s-t)-\rho s}\left(\right|\widehat{Z}{\left(s\right)|}^2+|{\widehat{Z}}_0{\left(s\right)|}^2)ds\le k_6{l}_1\mathbb{E}{\int }_t^T{e}^{-{\tilde{\rho }}_1(s-t)-\rho s}{\left|\widehat{X}\left(s\right)\right|}^{2}ds,\hfill \end{array}$$

(22)

where ${\tilde{\rho }}_1$ solves the equation ${\tilde{\rho }}_1=-\rho -2{\rho }_1-k_6{l}_1^{-1}-k_7(1+e^{(\rho +{\tilde{\rho }}_1)\delta })$. Moreover,

$$\begin{array}{cc}\hfill \parallel \widehat{Y}{\parallel }_{\rho }^2\le & {\displaystyle \frac{1-e^{-{\tilde{\rho }}_{1}T}}{{\tilde{\rho }}_1}}{k}_6{l}_1{\parallel \widehat{X}\parallel }_{\rho }^2,\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill |\widehat{Y}{\left(0\right)|}^2\le & (1\vee e^{-{\tilde{\rho }}_{1}T})k_6{l}_1{\parallel \widehat{X}\parallel }_{\rho }^2.\hfill \end{array}$$

(24)

Specifically, if ${\tilde{\rho }}_1>0$,

$$\begin{array}{cc}\hfill |\widehat{Y}{\left(0\right)|}^2\le & k_6{l}_1{\parallel \widehat{X}\parallel }_{\rho }^2.\hfill \end{array}$$

(25)

Proof. 

Denote $\widehat{f}\left(s\right):=f(s,X_1\left(s\right),Y_1\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1(s+\delta )\right])-f(s,X_2\left(s\right),Y_2\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_2(s+\delta )\right])$. Applying Itô’s formula to $e^{-\rho t}{\left|\widehat{Y}\left(t\right)\right|}^2$, we obtain

$$\begin{array}{cc}\hfill & e^{-\rho t}\mathbb{E}|\widehat{Y}{\left(t\right)|}^2-\rho \mathbb{E}{\int }_t^T{e}^{-\rho s}|\widehat{Y}{\left(s\right)|}^{2}ds+\mathbb{E}{\int }_t^T{e}^{-\rho s}\left(\right|\widehat{Z}{\left(s\right)|}^2+|{\widehat{Z}}_0\left(s\right){|}^2)ds\hfill \\ \hfill =& 2\mathbb{E}{\int }_t^T{e}^{-\rho s}\langle \widehat{Y}\left(s\right),\widehat{f}\left(s\right)\rangle ds.\hfill \end{array}$$

(26)

It follows that

$$\begin{array}{cc}\hfill 2\langle \widehat{Y}\left(s\right),\widehat{f}\left(s\right)\rangle =& 2\langle \widehat{Y}\left(s\right),f(s,X_1\left(s\right),Y_1\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1(s+\delta )\right])-f(s,X_1\left(s\right),Y_2\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1(s+\delta )\right])\rangle \hfill \\ \hfill & +2\langle \widehat{Y}\left(s\right),f(s,X_1\left(s\right),Y_2\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1(s+\delta )\right])-f(s,X_2\left(s\right),Y_2\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_2(s+\delta )\right])\rangle \hfill \\ \hfill \le & 2{\rho }_1|\widehat{Y}{\left(s\right)|}^2+2\left|\widehat{Y}\left(s\right)\right|\left(k_6|\widehat{X}\left(s\right)|+k_7\left|{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[\widehat{Y}(s+\delta )\right]\right|\right)\hfill \\ \hfill \le & (2{\rho }_1+k_6{l}_1^{-1}+k_7)|\widehat{Y}\left(s\right){|}^2+k_6{l}_1|\widehat{X}\left(s\right){|}^2+k_7{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[\right|\widehat{Y}(s+\delta ){|}^2].\hfill \end{array}$$

Notice that

$$\begin{array}{c}\hfill \mathbb{E}{\int }_t^T{e}^{-\rho s}\mathbb{E}|\widehat{Y}{(s+\delta )|}^{2}ds=\mathbb{E}{\int }_{t+\delta }^T{e}^{-\rho (s-\delta )}|\widehat{Y}{\left(s\right)|}^{2}ds\le e^{\rho \delta }\mathbb{E}{\int }_t^T{e}^{-\rho s}{\left|\widehat{Y}\left(s\right)\right|}^{2}ds.\end{array}$$

By noticing (26), we obtain (21).

Similarly, applying Itô’s formula to $e^{-{\tilde{\rho }}_1(s-t)-\rho s}{\left|\widehat{Y}\left(s\right)\right|}^2$ for $s\in [0,t]$, we have

$$\begin{array}{cc}\hfill & e^{-\rho t}\mathbb{E}|\widehat{Y}{\left(t\right)|}^2-(\rho +{\tilde{\rho }}_1)\mathbb{E}{\int }_t^T{e}^{-{\tilde{\rho }}_1(s-t)-\rho s}{\left|\widehat{Y}\left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\mathbb{E}{\int }_t^T{e}^{-{\tilde{\rho }}_1(s-t)-\rho s}\left(\right|\widehat{Z}{\left(s\right)|}^2+|{\widehat{Z}}_0\left(s\right){|}^2)ds=2\mathbb{E}{\int }_t^T{e}^{-{\tilde{\rho }}_1(s-t)-\rho s}\langle \widehat{Y}\left(s\right),\widehat{f}\left(s\right)\rangle ds.\hfill \end{array}$$

(27)

From the above estimates and (27), one can prove (22). We mention that there indeed exists a ${\tilde{\rho }}_1$ solving the equation ${\tilde{\rho }}_1=-\rho -2{\rho }_1-k_6{l}_1^{-1}-k_7(1+e^{(\rho +{\tilde{\rho }}_1)\delta })$ In fact, from the above equation, it follows that

$${\tilde{\rho }}_1+\rho +k_7{e}^{({\tilde{\rho }}_1+\rho )\delta }=-2{\rho }_1-k_6{l}_1^{-1}-k_7.$$

(28)

By noticing that $k_7$, $\delta$ are positive, which yields that $g_2\left(x\right):=x+k_7{e}^{x\delta }$ is an increasing continuous function satisfying $g_2(+\infty )=+\infty$ and $g_2(-\infty )=-\infty$. Therefore, for a given $\rho$ and $-2{\rho }_1-k_6{l}_1^{-1}-k_7$, there exists a unique ${\tilde{\rho }}_1$ such that (28) holds for intermediate value theorem.

Integrating from 0 to T on both sides of (22) and using ${\displaystyle \frac{1-e^{-{\tilde{\rho }}_{1}s}}{{\tilde{\rho }}_1}}\le {\displaystyle \frac{1-e^{-{\tilde{\rho }}_{1}T}}{{\tilde{\rho }}_1}},$ for all $s\in [0,T]$, we have (23). Letting $t=0$ in (22) and noticing $e^{-{\tilde{\rho }}_{1}s}\le 1\vee e^{-{\tilde{\rho }}_{1}T}$ for all $s\in [0,T]$, we obtain (24). If ${\tilde{\rho }}_1>0$, (25) is derived. □

Similarly, for a given $(Y(·),Z(·),Z_0(·))\in L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^m)\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})$, the forward equation in the MF-AFBSDDE (19) admits a unique solution $X(·)\in L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)$. Thus, we introduce a map ${\mathcal{M}}_2:L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^m)\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})\to L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)$, through

$$\left\{\begin{array}{cc}\hfill & X\left(t\right)=\Phi (Y\left(0\right),\mathbb{E}Y\left(0\right))+{\int }_0^{t}b\left(s,X\left(s\right),X(s-\delta ),\mathbb{E}X(s-\delta ),Y\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y\left(s\right)\right]\right)ds,t\in [0,T],\hfill \\ \hfill & X\left(t\right)=0,t\in [-\delta ,0).\hfill \end{array}\right.$$

(29)

Indeed, the well-posedness of (29) can be established by applying the contraction mapping method under assumptions (A5) and (A6), although the term $\mathbb{E}X(s-\delta )$ is involved. We omit the proof here. From Burkholder–Davis–Gundy inequality and the delay theory, it follows that $\mathbb{E}{sup}_{t\in [-\delta ,T]}{\left|X\left(t\right)\right|}^2<\infty$.

Lemma 2.

Let $X_i$ be the solution of (29) corresponding to $(Y_i(·),Z_i(·),Z_{0i}(·))\in L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^m)\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})\times L_{\mathcal{F}}^2(0,T+\delta ;{\mathbb{R}}^{m\times d})$, $i=1,2$, respectively. Then, for all $\rho \in \mathbb{R}$ and two constants $l_2,l_3>0$, we have

$$\begin{array}{cc}\hfill & e^{-\rho t}\mathbb{E}|\widehat{X}{\left(t\right)|}^2+{\overline{\rho }}_2\mathbb{E}{\int }_0^t{e}^{-\rho s}|\widehat{X}{\left(s\right)|}^{2}ds\le \left(k_9^2+k_{10}^2\right)|\widehat{Y}{\left(0\right)|}^2+\left(k_3{l}_2+k_4{l}_3\right)\mathbb{E}{\int }_0^t{e}^{-\rho s}{\left|\widehat{Y}\left(s\right)\right|}^{2}ds,\hfill \end{array}$$

(30)

where ${\overline{\rho }}_2=\rho -2{\rho }_2-(k_1+k_2)(1+e^{-\rho \delta })-k_3{l}_2^{-1}-k_4{l}_3^{-1}$, and $\widehat{\psi }={\psi }_1-{\psi }_2,\psi =X,Y$. We also have

$$\begin{array}{cc}\hfill & e^{-\rho t}\mathbb{E}|\widehat{X}{\left(t\right)|}^2\le e^{-{\tilde{\rho }}_{2}t}\left(k_9^2+k_{10}^2\right)|\widehat{Y}{\left(0\right)|}^2+\left(k_3{l}_2+k_4{l}_3\right)\mathbb{E}{\int }_0^t{e}^{-{\tilde{\rho }}_2(t-s)-\rho s}{\left|\widehat{Y}\left(s\right)\right|}^{2}ds,\hfill \end{array}$$

(31)

where ${\tilde{\rho }}_2$ solves the equation ${\tilde{\rho }}_2=\rho -2{\rho }_2-(k_1+k_2)(1+e^{-(\rho -{\tilde{\rho }}_2)\delta })-k_3{l}_2^{-1}-k_4{l}_3^{-1}.$ Moreover,

$$\begin{array}{c}\hfill \parallel \widehat{X}{\parallel }_{\rho }^2\le {\displaystyle \frac{1-e^{-{\tilde{\rho }}_{2}T}}{{\tilde{\rho }}_2}}\left[\left(k_9^2+k_{10}^2\right)|\widehat{Y}{\left(0\right)|}^2+\left(k_3{l}_2+k_4{l}_3\right){\parallel \widehat{Y}\parallel }_{\rho }^2\right].\end{array}$$

(32)

Proof. 

Denote $\widehat{b}\left(s\right):=b(s,X_1\left(s\right),X_1(s-\delta ),\mathbb{E}{X}_1(s-\delta ),Y_1\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1\left(s\right)\right])-b(s,X_2\left(s\right),X_2(s-\delta ),{\mathbb{E}}^{{\mathcal{F}}_s^0}[X_2(s-\delta ),Y_2\left(s\right),\mathbb{E}{Y}_2\left(s\right)])$. Applying Itô’s formula to $e^{-\rho s}{\left|\widehat{X}\left(s\right)\right|}^2$, we obtain

$$\begin{array}{c}\hfill e^{-\rho t}\mathbb{E}|\widehat{X}{\left(t\right)|}^2=|\widehat{X}{\left(0\right)|}^2-\rho \mathbb{E}{\int }_0^t{e}^{-\rho s}{\left|\widehat{X}\left(s\right)\right|}^{2}ds+2\mathbb{E}{\int }_0^t{e}^{-\rho s}\langle \widehat{X}\left(s\right),\widehat{b}\left(s\right)\rangle ds.\end{array}$$

(33)

It follows that

$$\begin{array}{cc}\hfill & 2\langle \widehat{X}\left(s\right),\widehat{b}\left(s\right)\rangle =2\langle \widehat{X}\left(s\right),b(s,X_1\left(s\right),X_1(s-\delta ),\mathbb{E}{X}_1(s-\delta ),Y_1\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1\left(s\right)\right])\hfill \\ \hfill & -b(s,X_2\left(s\right),X_1(s-\delta ),\mathbb{E}{X}_1(s-\delta ),Y_1\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1\left(s\right)\right])\rangle \hfill \\ \hfill & +2\langle \widehat{X}\left(s\right),b(s,X_2\left(s\right),X_1(s-\delta ),\mathbb{E}{X}_1(s-\delta ),Y_1\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_1\left(s\right)\right])\hfill \\ \hfill & -b(s,X_2\left(s\right),X_2(s-\delta ),\mathbb{E}{X}_2(s-\delta ),Y_2\left(s\right),{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[Y_2\left(s\right)\right])\rangle \hfill \\ \hfill \le & (2{\rho }_2+k_1+k_2+k_3{l}_2^{-1}+k_4{l}_3^{-1})|\widehat{X}\left(s\right){|}^2+k_1|\widehat{X}(s-\delta ){|}^2+k_2\mathbb{E}|\widehat{X}(s-\delta ){|}^2+k_3{l}_2|\widehat{Y}\left(s\right){|}^2+k_4{l}_3\mathbb{E}{\left|\widehat{Y}\left(s\right)\right|}^2.\hfill \end{array}$$

Notice that

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_0^t{e}^{-\rho s}\mathbb{E}{\left|\widehat{X}(s-\delta )\right|}^{2}ds& =\mathbb{E}{\int }_0^t{e}^{-\rho s}|\widehat{X}{(s-\delta )|}^{2}ds=\mathbb{E}{\int }_0^{t-\delta }{e}^{-\rho (s+\delta )}{\left|\widehat{X}\left(s\right)\right|}^{2}ds\hfill \\ \hfill & \le e^{-\rho \delta }\mathbb{E}{\int }_0^t{e}^{-\rho s}{\left|\widehat{X}\left(s\right)\right|}^{2}ds,\hfill \end{array}$$

and

$$|\widehat{X}{\left(0\right)|}^2=|\Phi (Y_1\left(0\right),\mathbb{E}{Y}_1\left(0\right))-\Phi (Y_2\left(0\right),\mathbb{E}{Y}_2\left(0\right)){|}^2\le k_9^2|\widehat{Y}{\left(0\right)|}^2+k_{10}^2|\mathbb{E}\widehat{Y}{\left(0\right)|}^2=(k_9^2+k_{10}^2){\left|\widehat{Y}\left(0\right)\right|}^2.$$

By noticing (33), we obtain (30).

Similarly, for ${\tilde{\rho }}_2$ which solves the equation ${\tilde{\rho }}_2=\rho -2{\rho }_2-(k_1+k_2)(1+e^{-(\rho -{\tilde{\rho }}_2)\delta })-k_3{l}_2^{-1}-k_4{l}_3^{-1}$, we apply Itô’s formula to $e^{-{\tilde{\rho }}_2(t-s)-\rho s}{\left|\widehat{X}\left(s\right)\right|}^2$ for $s\in [0,t]$, we have

$$\begin{array}{cc}\hfill e^{-\rho t}\mathbb{E}{\left|\widehat{X}\left(t\right)\right|}^2=& e^{-{\tilde{\rho }}_{2}t}|\widehat{X}{\left(0\right)|}^2-(\rho -{\tilde{\rho }}_2)\mathbb{E}{\int }_0^t{e}^{-{\tilde{\rho }}_2(t-s)-\rho s}{\left|\widehat{X}\left(s\right)\right|}^{2}ds\hfill \\ \hfill & +2\mathbb{E}{\int }_0^t{e}^{-{\tilde{\rho }}_2(t-s)-\rho s}\langle \widehat{X}\left(s\right),\widehat{b}\left(s\right)\rangle ds.\hfill \end{array}$$

(34)

From the above estimates and (34), one can prove (31). We note that there indeed exists a ${\tilde{\rho }}_2$ solving the equation ${\tilde{\rho }}_2=\rho -2{\rho }_2-(k_1+k_2)(1+e^{-(\rho -{\tilde{\rho }}_2)\delta })-k_3{l}_2^{-1}-k_4{l}_3^{-1}.$ In fact, from the above equation, it follows that

$${\tilde{\rho }}_2-\rho +(k_1+k_2)e^{({\tilde{\rho }}_2-\rho )\delta }=-2{\rho }_2-(k_1+k_2)-k_3{l}_2^{-1}-k_4{l}_3^{-1}.$$

(35)

By noticing that $k_1$, $k_2$, $\delta$ are positive which yields that $g_1\left(x\right):=x+(k_1+k_2)e^{x\delta }$ is an increasing continuous function satisfying $g_1(+\infty )=+\infty$ and $g_1(-\infty )=-\infty$. Therefore, for a given $\rho$ and $-2{\rho }_2-(k_1+k_2)-k_3{l}_2^{-1}-k_4{l}_3^{-1}$, there exists a unique ${\tilde{\rho }}_2$ such that (35) holds using intermediate value theorem.

Integrating from 0 to T on both sides of (31), we have

$$\begin{array}{cc}\hfill \parallel \widehat{X}{\parallel }_{\rho }^2\le & (k_9^2+k_{10}^2)|\widehat{Y}\left(0\right){|}^2{\int }_0^T{e}^{-{\tilde{\rho }}_{2}t}dt+\left(k_3{l}_2+k_4{l}_3\right)\mathbb{E}{\int }_0^T{\int }_0^t{e}^{-{\tilde{\rho }}_2(t-s)-\rho s}{\left|\widehat{Y}\left(s\right)\right|}^{2}dsdt\hfill \\ \hfill =& (k_9^2+k_{10}^2)|\widehat{Y}\left(0\right){|}^2{\displaystyle \frac{1-e^{-{\tilde{\rho }}_{2}T}}{{\tilde{\rho }}_2}}+\left(k_3{l}_2+k_4{l}_3\right)\mathbb{E}{\int }_0^T{\displaystyle \frac{1-e^{-{\tilde{\rho }}_2(T-s)}}{{\tilde{\rho }}_2}}{e}^{-\rho s}{\left|\widehat{Y}\left(s\right)\right|}^{2}ds.\hfill \end{array}$$

Noticing that for all $s\in [0,T]$, ${\displaystyle \frac{1-e^{-{\tilde{\rho }}_2(T-s)}}{{\tilde{\rho }}_2}}\le {\displaystyle \frac{1-e^{-{\tilde{\rho }}_{2}T}}{{\tilde{\rho }}_2}},$ then (32) follows. □

Now, we present the proof of Theorem 2.

Proof of Theorem 2.

Define $\mathcal{M}:={\mathcal{M}}_2\circ {\mathcal{M}}_1$. Since ${\mathcal{M}}_1$ is defined by (20) and ${\mathcal{M}}_2$ is defined by (29). Therefore, $\mathcal{M}$ maps $L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)$ onto itself. To prove the theorem, we only need to show that $\mathcal{M}$ is a contraction mapping for some equivalent norm ${\parallel ·\parallel }_{\rho }$. For $X_i\in L_{\mathcal{F}}^2(-\delta ,T;{\mathbb{R}}^n)$, let $(Y_i,Z_i):={\mathcal{M}}_1\left(X_i\right)$ and ${\overline{X}}_i:=\mathcal{M}\left(X_i\right)$; from (23), (24) and (32), we have

$$\begin{array}{cc}\hfill \parallel \widehat{X}{\parallel }_{\rho }^2\le & {\displaystyle \frac{1-e^{-{\tilde{\rho }}_{2}T}}{{\tilde{\rho }}_2}}\left[\left(k_9^2+k_{10}^2\right)|\widehat{Y}{\left(0\right)|}^2+\left(k_3{l}_2+k_4{l}_3\right){\parallel \widehat{Y}\parallel }_{\rho }^2\right]\hfill \\ \hfill \le & {\displaystyle \frac{1-e^{-{\tilde{\rho }}_{2}T}}{{\tilde{\rho }}_2}}\left[(1\vee e^{-{\tilde{\rho }}_{1}T})k_6{l}_1\left(k_9^2+k_{10}^2\right)\parallel \widehat{X}{\parallel }_{\rho }^2+{\displaystyle \frac{1-e^{-{\tilde{\rho }}_{1}T}}{{\tilde{\rho }}_1}}{k}_6{l}_1\left(k_3{l}_2+k_4{l}_3\right){\parallel \widehat{X}\parallel }_{\rho }^2\right]\hfill \\ \hfill =& {\displaystyle \frac{1-e^{-{\tilde{\rho }}_{2}T}}{{\tilde{\rho }}_2}}\left[(1\vee e^{-{\tilde{\rho }}_{1}T})k_6{l}_1\left(k_9^2+k_{10}^2\right)+{\displaystyle \frac{1-e^{-{\tilde{\rho }}_{1}T}}{{\tilde{\rho }}_1}}{k}_6{l}_1\left(k_3{l}_2+k_4{l}_3\right)\right]{\parallel \widehat{X}\parallel }_{\rho }^2.\hfill \end{array}$$

Recall that ${\tilde{\rho }}_1$ satisfies ${\tilde{\rho }}_1+\rho +k_7{e}^{({\tilde{\rho }}_1+\rho )\delta }=-2{\rho }_1-k_6{l}_1^{-1}-k_7$, and ${\tilde{\rho }}_2$ satisfies ${\tilde{\rho }}_2-\rho +(k_1+k_2)e^{({\tilde{\rho }}_2-\rho )\delta }=-2{\rho }_2-(k_1+k_2)-k_3{l}_2^{-1}-k_4{l}_3^{-1}$. By choosing a suitable $\rho$ (e.g., choosing $l_1=k_6,l_2=k_3,l_3=k_4$, and $\rho$ large enough such that ${\tilde{\rho }}_1\le 0$ and ${\tilde{\rho }}_2\ge 1$), the first assertion of Theorem 2 is obtained.

If $L_{\rho ,k,\delta }<0$, we can choose $\rho \in \mathbb{R}$ and a sufficiently large $l_j(j=1,2,3)$ such that

$${\tilde{\rho }}_1>0,{\tilde{\rho }}_2>0.$$

In fact, recall that ${\tilde{\rho }}_1$ satisfies ${\tilde{\rho }}_1+\rho +k_7{e}^{({\tilde{\rho }}_1+\rho )\delta }=-2{\rho }_1-k_6{l}_1^{-1}-k_7$, and ${\tilde{\rho }}_2$ satisfies ${\tilde{\rho }}_2-\rho +(k_1+k_2)e^{({\tilde{\rho }}_2-\rho )\delta }=-2{\rho }_2-(k_1+k_2)-k_3{l}_2^{-1}-k_4{l}_3^{-1}$. Denote ${\lambda }_1:={\tilde{\rho }}_1+\rho$ and ${\lambda }_2:={\tilde{\rho }}_2-\rho$, then

$${\lambda }_1+k_7{e}^{{\lambda }_1\delta }=-2{\rho }_1-k_6{l}_1^{-1}-k_7$$

and

$${\lambda }_2+(k_1+k_2)e^{{\lambda }_2\delta }=-2{\rho }_2-k_1-k_2-k_3{l}_2^{-1}-k_4{l}_3^{-1},$$

respectively (concerning the existence of ${\lambda }_1$, ${\lambda }_2$, see the proof of Lemmas 1 and 2). Then, it is obvious that ${\lambda }_1<-2{\rho }_1-k_6{l}_1^{-1}-k_7$ and ${\lambda }_2<-2{\rho }_2-k_1-k_2-k_3{l}_2^{-1}-k_4{l}_3^{-1}$. Thus,

$$\begin{array}{cc}\hfill -{\lambda }_1-{\lambda }_2=& 2({\rho }_1+{\rho }_2)+k_1+k_2+k_7+k_7{e}^{{\lambda }_1\delta }+(k_1+k_2)e^{{\lambda }_2\delta }+k_3{l}_2^{-1}+k_4{l}_3^{-1}+k_6{l}_1^{-1}\hfill \\ \hfill \le & 2({\rho }_1+{\rho }_2)+k_1+k_2+k_7+k_7{e}^{(-2{\rho }_1-k_6{l}_1^{-1}-k_7)\delta }+(k_1+k_2)e^{(-2{\rho }_2-k_1-k_2-k_3{l}_2^{-1}-k_4{l}_3^{-1})\delta }\hfill \\ & +k_3{l}_2^{-1}+k_4{l}_3^{-1}+k_6{l}_1^{-1},\hfill \end{array}$$

and by recalling that $L_{\rho ,k,\delta }<0$, one can choose sufficiently large $l_j(j=1,2,3)$ such that $-{\lambda }_2<{\lambda }_1$. Therefore, by recalling ${\tilde{\rho }}_1=-\rho +{\lambda }_1$ and ${\tilde{\rho }}_2=\rho +{\lambda }_2$, we can choose suitable $\rho$ (e.g., $\rho :={\displaystyle \frac{1}{2}}({\lambda }_1-{\lambda }_2)$) such that ${\tilde{\rho }}_1>0,{\tilde{\rho }}_2>0.$

Then, from (23), (25), (32), we have

$$\begin{array}{cc}\hfill \parallel \widehat{X}{\parallel }_{\rho }^2\le & {\displaystyle \frac{1}{{\tilde{\rho }}_2}}\left[\left(k_9^2+k_{10}^2\right)|\widehat{Y}{\left(0\right)|}^2+\left(k_3{l}_2+k_4{l}_3\right){\parallel \widehat{Y}\parallel }_{\rho }^2\right]\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\tilde{\rho }}_2}}\left[k_6{l}_1\left(k_9^2+k_{10}^2\right)+{\displaystyle \frac{1}{{\tilde{\rho }}_1}}{k}_6{l}_1\left(k_3{l}_2+k_4{l}_3\right)\right]{\parallel \widehat{X}\parallel }_{\rho }^2.\hfill \end{array}$$

This completes the second assertion of Theorem 2.

Let ${\rho }_1^{*}$ and ${\rho }_2^{*}$ be the largest eigenvalue of ${\displaystyle \frac{1}{2}}({\mathbb{A}}_2+{\mathbb{A}}_2^{\top })$ and ${\displaystyle \frac{1}{2}}({\mathbb{A}}_1+{\mathbb{A}}_1^{\top })$. Comparing (19) with (18), we can check that the coefficients of (A5) can be chosen as follows:

$$\begin{array}{cc}\hfill & {\rho }_1={\rho }_1^{*},{\rho }_2={\rho }_2^{*},k_1=\parallel {\tilde{\mathbb{A}}}_2\parallel ,k_2=\parallel {\overline{\mathbb{A}}}_2\parallel ,k_3=\parallel {\mathbb{B}}_2\parallel ,k_4=\parallel {\overline{\mathbb{B}}}_2\parallel ,k_5=\parallel {\mathbb{A}}_2\parallel ,\hfill \\ \hfill & k_6=\parallel {\mathbb{B}}_1\parallel ,k_7=\parallel {\overline{\mathbb{A}}}_1\parallel ,k_8=\parallel {\mathbb{A}}_1\parallel ,k_9=\parallel {\mathbb{H}}_1\parallel ,k_{10}=\parallel {\mathbb{H}}_2\parallel ,\hfill \\ \hfill & L_{\rho ,k,\delta }=2({\rho }_2^{*}+{\rho }_1^{*})+\parallel {\tilde{\mathbb{A}}}_2\parallel +\parallel {\overline{\mathbb{A}}}_1\parallel +\parallel {\overline{\mathbb{A}}}_2\parallel +(\parallel {\tilde{\mathbb{A}}}_2\parallel +\parallel {\overline{\mathbb{A}}}_2\parallel )e^{-(2{\rho }_2^{*}+\parallel {\tilde{\mathbb{A}}}_2\parallel +\parallel {\overline{\mathbb{A}}}_2\parallel )\delta }\hfill \\ \hfill & +\parallel {\overline{\mathbb{A}}}_1\parallel e^{-(2{\rho }_1^{*}+\parallel {\overline{\mathbb{A}}}_1\parallel )\delta }.\hfill \end{array}$$

Thus, by applying Theorem 2, we obtain the following global well-posedness of (18).

Theorem 3.

Suppose that $L_{\rho ,k,\delta }<0$, then there exists a ${ϵ}_1>0$, which depends on ${\rho }_1^{*},{\rho }_2^{*},\parallel {\tilde{\mathbb{A}}}_2\parallel$, $\parallel {\overline{\mathbb{A}}}_1\parallel ,\parallel {\overline{\mathbb{A}}}_2\parallel ,\parallel {\mathbb{B}}_1\parallel ,\parallel {\mathbb{B}}_2\parallel ,\parallel {\overline{\mathbb{B}}}_2\parallel$, and is independent of T, such that when $\parallel {\mathbb{H}}_1\parallel ,\parallel {\mathbb{H}}_2\parallel \in [0,{ϵ}_1)$, there exists a unique adapted solution $(\alpha ,\beta ,\gamma ,{\check{x}}^k,{\check{x}}_2^k)$ to consistency condition system (17).

5. Asymptotic $\mathit{\epsilon }$-Optimality

We start this section with some estimates, which play an important role in asymptotic optimality.

5.1. Agent ${\mathcal{A}}_i$ Perturbation

Let $\tilde{u}=({\tilde{u}}_1,\cdots ,{\tilde{u}}_N)$ be the decentralized strategy given by

$$\begin{array}{c}\hfill {\tilde{u}}_i\left(t\right)=-{\mathbf{P}}_{{\Gamma }_{{\theta }_i}}\left[R_{{\theta }_i}^{-1}\left(t\right)B^{\top }\left(t\right)p_i\left(t\right)\right],1\le i\le N,\end{array}$$

where

$$\left\{\begin{array}{cc}\hfill d{y}_i\left(t\right)=& -\left[A_{{\theta }_i}{y}_i-B{\mathbf{P}}_{{\Gamma }_{{\theta }_i}}\left[R_{{\theta }_i}^{-1}{B}^{\top }{p}_i\right]+C\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right]\right]dt+z_i\left(t\right)d{W}_i\left(t\right)+z_{0i}\left(t\right)d{W}_0\left(t\right),\hfill \\ \hfill d{p}_i\left(t\right)=& [A_{{\theta }_i}^{\top }{p}_i+(S+\tilde{S}(t-\delta ))y_i-S_G\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right]-\sum _{k=1}^K{\pi }_k{C}^{\top }(t-\delta ){\widehat{x}}_k(t-\delta )\hfill \\ \hfill & -\sum _{k=1}^K{\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta )-{\tilde{S}}_{\tilde{G}}(t-\delta )\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right]]dt,t\in [0,T],\hfill \\ \hfill y_i\left(t\right)=& {\xi }_i\left(t\right),z_i\left(t\right)={\eta }_i\left(t\right),z_{0i}\left(t\right)={\eta }_{0i}\left(t\right)t\in [T,T+\delta ],\hfill \\ \hfill p_i\left(0\right)=& Q{y}_i\left(0\right)-Q_H\sum _{l=1}^K{\pi }_l{\alpha }_l\left(0\right),p_i\left(t\right)=0,t\in [-\delta ,0)\hfill \end{array}\right.$$

with ${\alpha }_l,{\widehat{x}}_k,x_2^k$ given by Theorem 1.

Correspondingly, the realized state $({\tilde{x}}_1,\cdots ,{\tilde{x}}_N)$ under the decentralized strategy satisfies

$$\left\{\begin{array}{cc}\hfill d{\tilde{y}}_i\left(t\right)=& -\left[A_{{\theta }_i}{\tilde{y}}_i-B{\mathbf{P}}_{{\Gamma }_{{\theta }_i}}\left[R_{{\theta }_i}^{-1}{B}^{\top }{p}_i\right]+C{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\tilde{y}}^{\left(N\right)}(t+\delta )\right]\right]dt+{\tilde{z}}_i\left(t\right)d{W}_i\left(t\right)\hfill \\ \hfill & +\sum _{l=1,l\ne i}^N{\tilde{z}}_{il}d{W}_l\left(t\right)+{\tilde{z}}_{0i}\left(t\right)d{W}_0\left(t\right),\hfill \\ \hfill {\tilde{y}}_i\left(t\right)=& {\xi }_i\left(t\right),{\tilde{z}}_i\left(t\right)={\eta }_i\left(t\right),{\tilde{z}}_{0i}\left(t\right)={\eta }_{0i}\left(t\right),t\in [T,T+\delta ],\hfill \end{array}\right.$$

(36)

and ${\tilde{y}}^{\left(N\right)}(·)={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\tilde{y}}_i(·)$.

Let us consider the case that the agent ${\mathcal{A}}_i$ uses an alternative strategy $u_i$, while the other agents ${\mathcal{A}}_j,j\ne i$ use the strategy ${\tilde{u}}_{-i}$. The realized state with the i-th agent’s perturbation is

$$\left\{\begin{array}{cc}\hfill & d{\check{y}}_i=-\left[A_{{\theta }_i}{\check{y}}_i+B{u}_i+C{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\check{y}}^{\left(N\right)}(t+\delta )\right]\right]dt+{\check{z}}_{i}d{W}_i\left(t\right)+\sum _{l=1,l\ne i}^N{\check{z}}_{il}d{W}_l\left(t\right)+{\check{z}}_{0i}d{W}_0\left(t\right),\hfill \\ \hfill & d{\check{y}}_j=-\left[A_{{\theta }_j}{\check{y}}_j-B{\mathbf{P}}_{{\Gamma }_{{\theta }_j}}\left[R_{{\theta }_j}^{-1}{B}^{\top }{p}_j\right]+C{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\check{y}}^{\left(N\right)}(t+\delta )\right]\right]dt+{\check{z}}_{j}d{W}_j\left(t\right)\hfill \\ \hfill & +\sum _{l=1,l\ne j}^N{\check{z}}_{jl}d{W}_l\left(t\right)+{\check{z}}_{0j}d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & {\check{y}}_i\left(t\right)={\xi }_i\left(t\right),{\check{y}}_j\left(t\right)={\xi }_j\left(t\right),t\in [T,T+\delta ],1\le j\le N,j\ne i,\hfill \end{array}\right.$$

where ${\check{y}}^{\left(N\right)}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\check{y}}_i$. For $j=1,\cdots ,N$, denote the perturbation

$$\delta u_j=u_j-{\tilde{u}}_j,\delta y_j={\check{y}}_j-{\tilde{y}}_j,\delta {\mathcal{J}}_j={\mathcal{J}}_j(u_i,{\tilde{u}}_{-i})-{\mathcal{J}}_j({\tilde{u}}_i,{\tilde{u}}_{-i}).$$

Similarly to the computations in Section 3.1, we have

$$\begin{array}{cc}\hfill \Delta {\mathcal{J}}_{soc}^{\left(N\right)}=& \mathbb{E}\left\{{\int }_0^T\right[\langle S{\tilde{y}}_i,\delta y_i\rangle -\langle S_G\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right],\delta y_i\rangle +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\tilde{y}}_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle \hfill \\ \hfill & -\langle {\tilde{S}}_{\tilde{G}}\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta ),\delta y_i\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta ),\delta y_i\rangle +\langle R_{{\theta }_i}{\tilde{u}}_i,\delta u_i\rangle ]dt\hfill \\ \hfill & +\langle Q{\tilde{y}}_i\left(0\right),\delta y_i\left(0\right)\rangle -\langle Q_H\sum _{l=1}^K{\pi }_l\mathbb{E}{\alpha }_l\left(0\right),\delta y_i\left(0\right)\rangle \}+\sum _{l=1}^{11}{\epsilon }_l,\hfill \end{array}$$

(37)

where

$$\left\{\begin{array}{cc}\hfill & {\epsilon }_1=\mathbb{E}{\int }_0^T\langle S_G(\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right]-{\tilde{y}}^{\left(N\right)}),N\delta y^{\left(N\right)}\rangle dt,\hfill \\ \hfill & {\epsilon }_2=\mathbb{E}{\int }_0^T\langle {\tilde{S}}_{\tilde{G}}(\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right]-{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\tilde{y}}^{\left(N\right)}(t+\delta )\right]),N{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y^{\left(N\right)}(t+\delta )\right]\rangle dt,\hfill \\ \hfill & {\epsilon }_3=\langle Q_H(\sum _{l=1}^K{\pi }_l{\alpha }_l\left(0\right)-{\tilde{y}}^{\left(N\right)}\left(0\right)),N\delta y^{\left(N\right)}\left(0\right)\rangle ,\hfill \\ \hfill & {\epsilon }_4=\sum _{k=1}^K\mathbb{E}{\int }_0^T\langle S_G\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right],y_k^{**}-\delta y_{\left(k\right)}\rangle dt,{\epsilon }_5=\sum _{k=1}^K\mathbb{E}{\int }_0^T{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle S{\tilde{y}}_j,N_k\delta y_j-y_j^{*}\rangle dt,\hfill \\ \hfill & {\epsilon }_6=\sum _{k=1}^K\mathbb{E}{\int }_0^T\langle {\tilde{S}}_{\tilde{G}}\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}[y_k^{**}(t+\delta )-\delta y_{\left(k\right)}(t+\delta )]\rangle dt,\hfill \\ \hfill & {\epsilon }_7=\sum _{k=1}^K\mathbb{E}{\int }_0^T{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t^i}\left[{\tilde{y}}_j(t+\delta )\right],[N_k\delta y_j(t+\delta )-y_j^{*}(t+\delta )]\rangle dt,\hfill \\ \hfill & {\epsilon }_8=\sum _{k=1}^K\langle Q_H\sum _{l=1}^K{\pi }_l{\alpha }_l\left(0\right),y_k^{**}\left(0\right)-\delta y_{\left(k\right)}\left(0\right)\rangle ,{\epsilon }_9=\sum _{k=1}^K{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}\langle Q{\tilde{y}}_j\left(0\right),N_k\delta y_j\left(0\right)-y_j^{*}\left(0\right)\rangle ,\hfill \\ \hfill & {\epsilon }_{10}=\sum _{k=1}^K\mathbb{E}{\int }_0^T〈\sum _{l=1}^K{\pi }_l{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_l(t-\delta )-\sum _{l=1}^K{\displaystyle \frac{{\pi }_l}{N_l}}\sum _{j\in {\mathcal{I}}_l,j\ne i}{C}^{\top }(t-\delta )x_1^j(t-\delta ),y_k^{**}〉dt,\hfill \\ \hfill & {\epsilon }_{11}=\sum _{k=1}^K\mathbb{E}{\int }_0^T\langle {\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta )-{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{\pi }_k{C}^{\top }(t-\delta )x_1^j(t-\delta ),\delta y_i\rangle dt.\hfill \end{array}\right.$$

(38)

First, we need some estimations. In the proofs, L will denote a constant whose value may change from line to line. Similarly to the proof of Lemma 5.1 of [4], by virtue of estimations of the AFBSDDE, we derive

Lemma 3.

Let (A1)–(A4) hold. Then, there exists a constant L independent of N, such that

$$\begin{array}{cc}\hfill & \sum _{l=1}^K\mathbb{E}\underset{0\le t\le T}{sup}\left[|{\alpha }_l{\left(t\right)|}^2+|{\gamma }_l{\left(t\right)|}^2+|{\check{x}}^l{\left(t\right)|}^2+{\left|{\check{x}}_2^l\left(t\right)\right|}^2\right]+\underset{1\le i\le N}{sup}\mathbb{E}\underset{0\le t\le T}{sup}{\left|{\tilde{y}}_i\left(t\right)\right|}^2\hfill \\ \hfill & +\sum _{l=1}^K\mathbb{E}{\int }_0^T\left(\right|{\beta }_l{\left(t\right)|}^2+|{\beta }_{0l}\left(t\right){|}^2)dt\le L.\hfill \end{array}$$

(39)

Similarly to Lemma 3, using the $L^2$ boundness of $u_i,{\xi }_i$ and ${\xi }_j,p_j$$(1\le j\le N,j\ne i)$, we have

$$\underset{1\le i\le N}{sup}\mathbb{E}\underset{0\le t\le T}{sup}{\left|{\check{y}}_i\left(t\right)\right|}^2\le L.$$

(40)

where L is a constant independent of N.

Lemma 4.

Let (A1)–(A4) hold. Then, there exists a constant L independent of N such that

$$\mathbb{E}\underset{0\le t\le T}{sup}|{\tilde{y}}^{\left(N\right)}\left(t\right)-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\left(t\right)\right]{|}^2\le {\displaystyle \frac{L}{N}}+L{ϵ}_N^2,$$

(41)

where ${ϵ}_N={sup}_{1\le l\le K}|{\pi }_l^{\left(N\right)}-{\pi }_l|.$

Proof. 

For $1\le k\le K$, denote the k-type agent state average by ${\tilde{y}}^{\left(k\right)}:={\displaystyle \frac{1}{N_k}}{\sum }_{j\in {\mathcal{I}}_k}{\tilde{y}}_j$, and further ${\tilde{\xi }}^{\left(k\right)}:={\displaystyle \frac{1}{N_k}}{\sum }_{j\in {\mathcal{I}}_k}{\xi }_j$, thus

$$\left\{\begin{array}{cc}\hfill d{\tilde{y}}^{\left(k\right)}=& -\left[A_k{\tilde{y}}^{\left(k\right)}-{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{p}_j\right]+C{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\tilde{y}}^{\left(N\right)}(t+\delta )\right]\right]dt\hfill \\ \hfill & +{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}{\tilde{z}}_{j}d{W}_j\left(t\right)+{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}\sum _{l=1,l\ne j}^N{\tilde{z}}_{jl}d{W}_l\left(t\right)+{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}{\tilde{z}}_{0j}d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill {\tilde{y}}^{\left(k\right)}\left(t\right)=& {\tilde{\xi }}^{\left(k\right)}\left(t\right),t\in [T,T+\delta ].\hfill \end{array}\right.$$

Notice that

$$\left\{\begin{array}{cc}\hfill & d{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_k\right]=-\left[A_k{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_k\right]-{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{\mathbf{p}}_k\right]\right]+C\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right]\right]dt+{\beta }_{0k}d{W}_0\left(t\right),\hfill \\ & t\in [0,T],\hfill \\ \hfill & {\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_k\left(t\right)\right]={\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\xi }^{\left(k\right)}\left(t\right)\right],t\in [T,T+\delta ],\hfill \end{array}\right.$$

we have

$$\left\{\begin{array}{cc}\hfill & d\left({\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_k\right]\right)=-[A_k\left({\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_k\right]\right)-{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{p}_j\right]\hfill \\ \hfill & +{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{\mathbf{p}}_k\right]\right]+C\left({\mathbb{E}}^{{\mathcal{F}}_t}\left[{\tilde{y}}^{\left(N\right)}(t+\delta )\right]-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right]\right)]dt\hfill \\ \hfill & +{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}{\tilde{z}}_{j}d{W}_j\left(t\right)+{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}\sum _{l=1,l\ne j}^N{\tilde{z}}_{jl}d{W}_l\left(t\right)+{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}{\tilde{z}}_{0j}d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & {\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_k\right]\left(t\right)={\tilde{\xi }}^{\left(k\right)}\left(t\right)-{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\xi }^{\left(k\right)}\left(t\right)\right],t\in [T,T+\delta ].\hfill \end{array}\right.$$

Using the Cauchy–Schwartz inequality, Burkholder–Davis–Gundy inequality, and the estimates of ABSDE, we have

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}|{\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^2+\mathbb{E}{\int }_t^T\left({\displaystyle \frac{1}{N_k^2}}\sum _{j\in {\mathcal{I}}_k}\left(|{\tilde{z}}_j{|}^2+|{\tilde{z}}_{0j}{|}^2\right)+{\displaystyle \frac{1}{N_k^2}}\sum _{j\in {\mathcal{I}}_k}\sum _{l=1,l\ne j}^N|{\tilde{z}}_{jl}{|}^2\right)ds\hfill \\ \hfill \le & \mathbb{E}|{\tilde{\xi }}^{\left(k\right)}\left(T\right)-{\mathbb{E}}^{{\mathcal{F}}_T^0}\left[{\xi }^{\left(k\right)}\left(T\right)\right]{|}^2+L\mathbb{E}{\int }_t^T[|{\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^2+|{\tilde{y}}^{\left(N\right)}(s+\delta )\hfill \\ \hfill & -\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l(s+\delta )\right]{|}^2]ds+L\mathbb{E}{\int }_t^T|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{p}_j\right]\hfill \\ \hfill & -{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{\mathbf{p}}_k\right]\right]{|}^{2}ds.\hfill \end{array}$$

By (A2), for $1\le k\le K$, $\{{\xi }_j\left(T\right),j\in {\mathcal{I}}_k\}$ are independent identically distributed. Note that $p_j(·)\in {\mathcal{F}}_t^j$, thus $\{p_j,j\in {\mathcal{I}}_k\}$ are independent identically distributed. Then, we have

$$\begin{array}{cc}\hfill & \mathbb{E}|{\tilde{\xi }}^{\left(k\right)}\left(T\right)-{\mathbb{E}}^{{\mathcal{F}}_T^0}\left[{\xi }^{\left(k\right)}\left(T\right)\right]{|}^2=\mathbb{E}|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}({\xi }_j\left(T\right)-{\xi }^{\left(k\right)}\left(T\right)){|}^2={\displaystyle \frac{1}{N_k}}\mathbb{E}|{\xi }_j\left(T\right)-{\xi }^{\left(k\right)}\left(T\right){|}^2\le {\displaystyle \frac{L}{N_k}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_t^T|{\tilde{y}}^{\left(N\right)}(s+\delta )-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l(s+\delta )\right]{|}^{2}ds\hfill \\ \hfill & \le \mathbb{E}{\int }_t^T|{\tilde{y}}^{\left(N\right)}-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]{|}^{2}ds+\mathbb{E}{\int }_T^{T+\delta }|{\displaystyle \frac{1}{N}}\sum _{j=1}^N{\xi }_j-\sum _{l=1}^K{\pi }_l{\xi }^{\left(l\right)}{|}^{2}ds,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_t^T|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k}B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{p}_j\right]-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{\mathbf{p}}_k\right]\right]{|}^{2}ds\hfill \\ \hfill =& {\displaystyle \frac{1}{N_k}}\mathbb{E}{\int }_t^T|B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{p}_j\right]-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }{\mathbf{p}}_k\right]\right]{|}^{2}ds\le {\displaystyle \frac{L}{N_k}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}|{\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^2\le L\mathbb{E}{\int }_t^T|{\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^{2}ds\hfill \\ \hfill & +L\mathbb{E}{\int }_t^T|{\tilde{y}}^{\left(N\right)}-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]{|}^{2}ds+\mathbb{E}{\int }_T^{T+\delta }|{\displaystyle \frac{1}{N}}\sum _{j=1}^N{\xi }_j-\sum _{l=1}^K{\pi }_l{\xi }^{\left(l\right)}{|}^{2}ds+{\displaystyle \frac{L}{N_k}}.\hfill \end{array}$$

Using Gronwall inequality, we have

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}|{\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^2\hfill \\ \hfill \le & L\mathbb{E}{\int }_t^T|{\tilde{y}}^{\left(N\right)}-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]{|}^{2}ds+\mathbb{E}{\int }_T^{T+\delta }|{\displaystyle \frac{1}{N}}\sum _{j=1}^N{\xi }_j-\sum _{l=1}^K{\pi }_l{\xi }^{\left(l\right)}{|}^{2}ds+{\displaystyle \frac{L}{N_k}}.\hfill \end{array}$$

Since,

$$\begin{array}{cc}\hfill {\tilde{y}}^{\left(N\right)}-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]=& \sum _{l=1}^K({\pi }_l^{\left(N\right)}{\tilde{y}}^{\left(l\right)}-{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right])\hfill \\ \hfill =& \sum _{l=1}^K{\pi }_l^{\left(N\right)}({\tilde{y}}^{\left(l\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right])+\sum _{l=1}^K({\pi }_l^{\left(N\right)}-{\pi }_l){\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{N}}\sum _{j=1}^N{\xi }_j-\sum _{l=1}^K{\pi }_l{\xi }^{\left(l\right)}=\sum _{l=1}^K({\pi }_l^{\left(N\right)}{\tilde{\xi }}^{\left(l\right)}-{\pi }_l{\xi }^{\left(l\right)})=\sum _{l=1}^K{\pi }_l^{\left(N\right)}({\tilde{\xi }}^{\left(l\right)}-{\xi }^{\left(l\right)})+\sum _{l=1}^K({\pi }_l^{\left(N\right)}-{\pi }_l){\xi }^{\left(l\right)},\hfill \end{array}$$

we have

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_T^{T+\delta }|{\displaystyle \frac{1}{N}}\sum _{j=1}^N{\xi }_j-\sum _{l=1}^K{\pi }_l{\xi }^{\left(l\right)}{|}^{2}ds\le L\sum _{l=1}^K\mathbb{E}{\int }_T^{T+\delta }|{\tilde{\xi }}^{\left(l\right)}-{\xi }^{\left(l\right)}{|}^{2}ds+L{ϵ}_N^2\le {\displaystyle \frac{L}{N}}+L{ϵ}_N^2.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}|{\tilde{y}}^{\left(N\right)}-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]{|}^2\le L\sum _{l=1}^K\mathbb{E}\underset{t\le s\le T}{sup}|{\tilde{y}}^{\left(l\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]{|}^2+L{ϵ}_N^2\hfill \\ \hfill \le & L\mathbb{E}{\int }_t^T|{\tilde{y}}^{\left(N\right)}-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]{|}^{2}ds+{\displaystyle \frac{L}{N}}+L{ϵ}_N^2.\hfill \end{array}$$

Therefore, the result follows from Gronwall inequality. □

By Lemma 4, for $j=1,\cdots ,N$, we can derive

$$\mathbb{E}\underset{0\le t\le T}{sup}{|y_j\left(t\right)-{\tilde{y}}_j\left(t\right)|}^2\le {\displaystyle \frac{L}{N}}+L{ϵ}_N^2.$$

(42)

Lemma 5.

Let (A1)–(A4) hold. Then, there exists a constant L independent of N such that

$$\begin{array}{cc}\hfill & \underset{1\le j\le N,j\ne i}{sup}[\mathbb{E}\underset{0\le t\le T}{sup}{\left|\delta y_j\left(t\right)\right|}^2+\mathbb{E}{\int }_0^T\left(|\delta z_j\left(t\right){|}^2+|\delta z_{0j}\left(t\right){|}^2\right)dt\hfill \\ \hfill & +\mathbb{E}{\int }_0^T\sum _{l=1,l\ne j}^N|\delta z_{jl}\left(t\right){|}^{2}dt]\le {\displaystyle \frac{L}{N^2}}\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds.\hfill \end{array}$$

(43)

Proof. 

According to (5)–(7), this yields

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}{\left|\delta y_i\right|}^2+\mathbb{E}{\int }_t^T\left(|\delta z_i{|}^2+|\delta z_{0i}{|}^2\right)ds+\mathbb{E}{\int }_t^T\sum _{l=1,l\ne i}^N|\delta z_{il}{|}^{2}ds\hfill \\ \hfill & \le L\mathbb{E}{\int }_0^T|\delta u_i{|}^{2}ds+L\mathbb{E}{\int }_t^T|\delta y_i{|}^{2}ds+L\mathbb{E}{\int }_t^T{\left|{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y^{\left(N\right)}(s+\delta )\right]\right|}^{2}ds\hfill \\ \hfill & \le L\mathbb{E}{\int }_0^T|\delta u_i{|}^{2}ds+L\mathbb{E}{\int }_t^T|\delta y_i{|}^{2}ds+L\mathbb{E}{\int }_t^T{\left|\delta y^{\left(N\right)}\right|}^{2}ds,\hfill \end{array}$$

for $j\ne i$,

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}{\left|\delta y_j\right|}^2+\mathbb{E}{\int }_t^T\left(|\delta z_j{|}^2+|\delta z_{0j}{|}^2\right)ds+\mathbb{E}{\int }_t^T\sum _{l=1,l\ne j}^N|\delta z_{jl}{|}^{2}ds\hfill \\ \hfill & \le L\mathbb{E}{\int }_t^T|\delta y_j{|}^{2}ds+L\mathbb{E}{\int }_t^T{\left|\delta y^{\left(N\right)}\right|}^{2}ds,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}{\left|\delta y_{\left(k\right)}\right|}^2+\mathbb{E}{\int }_t^T\sum _{j\in {\mathcal{I}}_k,j\ne i}\left(|\delta z_j{|}^2+|\delta z_{0j}{|}^2\right)ds+\mathbb{E}{\int }_t^T\sum _{j\in {\mathcal{I}}_k,j\ne i}\sum _{l=1,l\ne j}^N|\delta z_{jl}{|}^{2}ds\hfill \\ \hfill \le & L\mathbb{E}{\int }_t^T|\delta y_{\left(k\right)}{|}^{2}ds+L{N}^2\mathbb{E}{\int }_t^T{\left|\delta y^{\left(N\right)}\right|}^{2}ds.\hfill \end{array}$$

Noticing that

$$\delta y^{\left(N\right)}={\displaystyle \frac{1}{N}}\delta y_i+{\displaystyle \frac{1}{N}}\sum _{l=1}^K\delta y_{\left(l\right)},$$

we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}{\left|\delta y_i\right|}^2+\mathbb{E}{\int }_t^T\left(|\delta z_i{|}^2+|\delta z_{0i}{|}^2\right)ds+\mathbb{E}{\int }_t^T\sum _{l=1,l\ne i}^N|\delta z_{il}{|}^{2}ds\hfill \\ \hfill & \le L\mathbb{E}{\int }_0^T|\delta u_i{|}^{2}ds+L\mathbb{E}{\int }_t^T|\delta y_i{|}^{2}ds+{\displaystyle \frac{L}{N^2}}\sum _{l=1}^K\mathbb{E}{\int }_t^T{\left|\delta y_{\left(l\right)}\right|}^{2}ds,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}{\left|\delta y_{\left(k\right)}\right|}^2+\mathbb{E}{\int }_t^T\sum _{j\in {\mathcal{I}}_k,j\ne i}\left(|\delta z_j{|}^2+|\delta z_{0j}{|}^2\right)ds+\mathbb{E}{\int }_t^T\sum _{j\in {\mathcal{I}}_k,j\ne i}\sum _{l=1,l\ne j}^N|\delta z_{jl}{|}^{2}ds\hfill \\ \hfill \le & L\mathbb{E}{\int }_t^T|\delta y_{\left(k\right)}{|}^{2}ds+L\mathbb{E}{\int }_t^T|\delta y_i{|}^{2}ds+L\sum _{l=1}^K\mathbb{E}{\int }_t^T{\left|\delta y_{\left(l\right)}\right|}^{2}ds.\hfill \end{array}$$

Therefore, it follows from the Gronwall inequality that

$$\begin{array}{c}\hfill \mathbb{E}\underset{t\le s\le T}{sup}|\delta y_i{|}^2+\sum _{l=1}^K\mathbb{E}\underset{t\le s\le T}{sup}|\delta y_{\left(l\right)}{|}^2\le L\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds.\end{array}$$

Thus,

$$\mathbb{E}\underset{t\le s\le T}{sup}|\delta y^{\left(N\right)}{|}^2\le {\displaystyle \frac{L}{N^2}}\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds.$$

By the Gronwall inequality again, we have (43). □

Remark 1.

Note that in (43), the upper bound depends on $\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds$. However, when studying the asymptotic optimality, we only need to consider the perturbations satisfying (47). Hence, in Section 5, when applying Lemma 5, a similar estimation still holds while the upper bound is ${\displaystyle \frac{L}{N^2}}$ and L is a general constant.

Lemma 6.

Let (A1)–(A4) hold. Then, there exist constants L independent of N such that

$$\sum _{l=1}^K\mathbb{E}\underset{0\le t\le T}{sup}|y_l^{**}\left(t\right)-\delta y_{\left(l\right)}{\left(t\right)|}^2\le \left({\displaystyle \frac{L}{N^2}}+L{ϵ}_N^2\right)\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds,$$

(44)

and for $j\in {\mathcal{I}}_k$, $1\le k\le K$,

$$\mathbb{E}\underset{0\le t\le T}{sup}|N_k\delta y_j\left(t\right)-y_j^{*}{\left(t\right)|}^2\le \left({\displaystyle \frac{L}{N^2}}+L{ϵ}_N^2\right)\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds.$$

(45)

Proof. 

First,

$$\left\{\begin{array}{cc}\hfill & d(y_k^{**}-\delta y_{\left(k\right)})=-[A_k(y_k^{**}-\delta y_{\left(k\right)})+C\left({\pi }_k-{\displaystyle \frac{N_k-I_{{\mathcal{I}}_k}\left(i\right)}{N}}\right){\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\hfill \\ \hfill & +C{\pi }_k\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}[y_l^{**}(t+\delta )-\delta y_{\left(l\right)}(t+\delta )]+C\left({\pi }_k-{\displaystyle \frac{N_k-I_{{\mathcal{I}}_k}\left(i\right)}{N}}\right)\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_{\left(l\right)}(t+\delta )\right]]dt\hfill \\ \hfill & +\sum _{l=1}^N{z}_{kl}^{**}d{W}_l\left(t\right)-\sum _{j\in {\mathcal{I}}_k,j\ne i}\delta z_{j}d{W}_j\left(t\right)-\sum _{j\in {\mathcal{I}}_k,j\ne i}\sum _{l=1,l\ne j}^N\delta z_{jl}d{W}_l\left(t\right)\hfill \\ \hfill & +\left(z_{0k}^{**}-\sum _{j\in {\mathcal{I}}_k,j\ne i}\delta z_{0j}\right)d{W}_0\left(t\right),t\in [0,T],\hfill \\ \hfill & (y_k^{**}-\delta y_{\left(k\right)})\left(t\right)=0,t\in [T,T+\delta ],\hfill \end{array}\right.$$

and for $j\in {\mathcal{I}}_k$,

$$\left\{\begin{array}{cc}\hfill & d(y_j^{*}-N_k\delta y_j)=-[A_k(y_j^{*}-N_k\delta y_j)+C({\pi }_k-{\pi }_k^{\left(N\right)}){\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\hfill \\ \hfill & +C({\pi }_k-{\pi }_k^{\left(N\right)})\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}\left[y_l^{**}(t+\delta )\right]+C{\pi }_k^{\left(N\right)}\sum _{l=1}^K{\mathbb{E}}^{{\mathcal{F}}_t}[y_l^{**}(t+\delta )-\delta y_{\left(l\right)}(t+\delta )]]dt\hfill \\ \hfill & +(z_j^{*}-N_k\delta z_j)d{W}_j\left(t\right)+\sum _{l=1,l\ne j}^N(z_{jl}^{*}-N_k\delta z_{jl})d{W}_l\left(t\right)+(z_{0j}^{*}-N_k\delta z_{0j})d{W}_0\left(t\right),\hfill \\ \hfill & (y_j^{*}-N_k\delta y_j)\left(t\right)=0,t\in [T,T+\delta ].\hfill \end{array}\right.$$

Therefore, it follows from the Burkholder–Davis–Gundy inequality that

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{t\le s\le T}{sup}{|y_k^{**}-\delta y_{\left(k\right)}|}^2\hfill \\ \hfill \le & L\mathbb{E}{\int }_t^T|y_k^{**}-\delta y_{\left(k\right)}{|}^{2}ds+L\sum _{l=1}^K\mathbb{E}{\int }_t^T{\left|{\mathbb{E}}^{{\mathcal{F}}_s}[y_l^{**}(s+\delta )-\delta y_{\left(l\right)}(s+\delta )]\right|}^{2}ds\hfill \\ \hfill & +\left({\displaystyle \frac{L}{N^2}}+L{ϵ}_N^2\right)\mathbb{E}{\int }_t^T\left[|{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]{|}^2+\sum _{l=1}^K{\left|{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_{\left(l\right)}(s+\delta )\right]\right|}^2\right]ds\hfill \\ \hfill \le & L\mathbb{E}{\int }_t^T|y_k^{**}-\delta y_{\left(k\right)}{|}^{2}ds+L\sum _{l=1}^K\mathbb{E}{\int }_t^T|y_l^{**}-\delta y_{\left(l\right)}{|}^{2}ds+\left({\displaystyle \frac{L}{N^2}}+L{ϵ}_N^2\right)\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \sum _{l=1}^K\mathbb{E}\underset{t\le s\le T}{sup}|y_l^{**}-\delta y_{\left(l\right)}{|}^2\le L\sum _{l=1}^K\mathbb{E}{\int }_t^T|y_l^{**}-\delta y_{\left(l\right)}{|}^2+\left({\displaystyle \frac{L}{N^2}}+L{ϵ}_N^2\right)\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds.\end{array}$$

It then follows from the Gronwall inequality that

$$\sum _{l=1}^K\mathbb{E}\underset{0\le t\le T}{sup}|y_l^{**}\left(t\right)-\delta y_{\left(l\right)}{\left(t\right)|}^2\le \left({\displaystyle \frac{L}{N^2}}+L{ϵ}_N^2\right)\mathbb{E}{\int }_0^T{\left|\delta u_i\right|}^{2}ds.$$

Similarly, we have (45). □

Applying the above estimations, using the standard estimations of the AFBSDDE, (13) and (36), we can obtain the following result.

Lemma 7.

Let (A1)–(A4) hold. Then, there exists a constant L independent of N such that

$$\sum _{k=1}^K\mathbb{E}\underset{0\le t\le T+\delta }{sup}{|\mathbb{E}{\mathbf{x}}_k(t-\delta )-{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j(t-\delta )|}^2\le {\displaystyle \frac{L}{N}}+L{ϵ}_N^2.$$

(46)

Proof. 

It follows from (13) that

$$\left\{\begin{array}{cc}\hfill & d\left({\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j\right)=\left[{\displaystyle \frac{A_k^{\top }}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j-{\displaystyle \frac{S+\tilde{S}(t-\delta )}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{\tilde{y}}_j\right]dt,t\in [0,T],\hfill \\ \hfill & {\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j\left(0\right)=-{\displaystyle \frac{Q}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{\tilde{y}}_j\left(0\right),{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j\left(t\right)=0,t\in [-\delta ,0).\hfill \end{array}\right.$$

By the definition of ${\mathbf{x}}_k$, we have

$$\left\{\begin{array}{cc}\hfill & d\mathbb{E}{\mathbf{x}}_k=\left[A_k^{\top }\mathbb{E}{\mathbf{x}}_k-(S+\tilde{S}(t-\delta ))\mathbb{E}{\tilde{y}}_k\right]dt,t\in [0,T],\hfill \\ \hfill & \mathbb{E}{\mathbf{x}}_k\left(0\right)=-Q\mathbb{E}{\tilde{y}}_k\left(0\right),\mathbb{E}{\mathbf{x}}_k\left(t\right)=0,t\in [-\delta ,0),k=1,\cdots ,K,\hfill \end{array}\right.$$

where ${\tilde{y}}_k$ denotes the optimal state of $k$-type corresponding to (36) and $\mathbb{E}{\tilde{y}}_k$ satisfies

$$\left\{\begin{array}{cc}\hfill & d\mathbb{E}{\tilde{y}}_k=-\left[A_k\mathbb{E}{\tilde{y}}_k-B{\mathbf{P}}_{{\Gamma }_{{\theta }_k}}\left[R_k^{-1}{B}^{\top }\mathbb{E}{\mathbf{p}}_k\right]+C\mathbb{E}{\tilde{y}}^{\left(N\right)}(t+\delta )\right]dt,t\in [0,T],\hfill \\ \hfill & \mathbb{E}{\tilde{y}}_k\left(t\right)=\mathbb{E}{\xi }^{\left(k\right)}\left(t\right),t\in [T,T+\delta ].\hfill \end{array}\right.$$

Recall the notation ${\tilde{y}}^{\left(k\right)}$ defined in proof of Lemma 4. Noticing

$${\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{\tilde{y}}_j={\tilde{y}}^{\left(k\right)}-{\displaystyle \frac{I_{{\mathcal{I}}_k}\left(i\right)}{N_k}}{\tilde{y}}_i,$$

we have

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le s\le t}{sup}|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j-\mathbb{E}{\mathbf{x}}_k{|}^2\le L\mathbb{E}|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{\tilde{y}}_j\left(0\right)-\mathbb{E}{\tilde{y}}_k\left(0\right){|}^2\hfill \\ \hfill & +L\mathbb{E}{\int }_t^T|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j-\mathbb{E}{\mathbf{x}}_k{|}^{2}ds+L\mathbb{E}{\int }_t^T|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{\tilde{y}}_j-\mathbb{E}{\tilde{y}}_k{|}^{2}ds\hfill \\ \hfill & \le L\mathbb{E}{\int }_t^T|{\displaystyle \frac{1}{N_k}}\sum _{j\in {\mathcal{I}}_k,j\ne i}{x}_1^j-\mathbb{E}{\mathbf{x}}_k{|}^{2}ds+L\mathbb{E}\left(\right|{\tilde{y}}^{\left(k\right)}\left(0\right)-\mathbb{E}{\alpha }_k{\left(0\right)|}^2+{|\mathbb{E}{\tilde{y}}_k\left(0\right)-\mathbb{E}{\alpha }_k\left(0\right)|}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{N_k^2}}{\left|{\tilde{y}}_i\left(0\right)\right|}^2)+L\mathbb{E}{\int }_t^T\left(|{\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^2+|\mathbb{E}{\tilde{y}}_k-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^2+{\displaystyle \frac{1}{N_k^2}}{\left|{\tilde{y}}_i\right|}^2\right)ds.\hfill \end{array}$$

Notice

$$\begin{array}{cc}\hfill \underset{t\le s\le T}{sup}|\mathbb{E}{\tilde{y}}_k-{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_k\right]{|}^2& \le L\mathbb{E}{\int }_t^T|\mathbb{E}{\tilde{y}}^{\left(N\right)}(s+\delta )-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l(s+\delta )\right]{|}^{2}ds\hfill \\ \hfill & \le L\mathbb{E}{\int }_t^T|{\tilde{y}}^{\left(N\right)}-\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_s^0}\left[{\alpha }_l\right]{|}^{2}ds+{\displaystyle \frac{L}{N}}.\hfill \end{array}$$

With the help of the proof of Lemma 4, one obtains

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le t\le T}{sup}|{\tilde{y}}^{\left(k\right)}-{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_k\right]{|}^2\le {\displaystyle \frac{L}{N}}+L{ϵ}_N^2.\hfill \end{array}$$

Then, (46) is obtained based on the above inequalities, $L^2$ boundness of ${\tilde{y}}_i$, Lemma 3, Lemma 4, and the Gronwall inequality. □

5.2. Asymptotic Optimality

In order to prove asymptotic optimality, it suffices to consider the perturbations $u_i\in {\mathcal{U}}_i^c,1\le i\le N$ such that ${\mathcal{J}}_{soc}^{\left(N\right)}(u_1,\cdots ,u_N)\le {\mathcal{J}}_{soc}^{\left(N\right)}({\tilde{u}}_1,$$\cdots ,{\tilde{u}}_N).$ It is easy to check that

$${\mathcal{J}}_{soc}^{\left(N\right)}({\tilde{u}}_1,\cdots ,{\tilde{u}}_N)\le LN,$$

where $L\ge 0$ is a constant independent of N. Therefore, in the following, we only consider the perturbations $u_i\in {\mathcal{U}}_i^c$ satisfying

$$\sum _{i=1}^N\mathbb{E}{\int }_0^T{\left|u_i\right|}^{2}dt\le LN.$$

(47)

Let $\delta u_i=u_i-{\tilde{u}}_i,1\le i\le N$, and consider the perturbation $u=\tilde{u}+(\delta u_1,\cdots ,\delta u_N):=\tilde{u}+\delta u$.

Theorem 4.

Let (A1)–(A4) hold. Then, $\tilde{u}=({\tilde{u}}_1,\cdots ,{\tilde{u}}_N)$ is a $\left({\displaystyle \frac{1}{\sqrt{N}}}+{ϵ}_N\right)$-optimal strategy for the agents.

Proof. 

We denote ${\displaystyle \frac{\partial {\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)}{\partial {\tilde{u}}_i}}\left(\delta u_i\right)$ as the variation in ${\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)$ on the i-th direction. From (37), we know that, ${\displaystyle \frac{\partial {\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)}{\partial {\tilde{u}}_i}}\left(\delta u_i\right)$ exists and is continuous, for each $i=1,2,\dots ,N$. Thus,

$$\begin{array}{cc}\hfill & {\mathcal{J}}_{soc}^{\left(N\right)}(\tilde{u}+\delta u)-{\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)=\sum _{i=1}^N{\displaystyle \frac{\partial {\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)}{\partial {\tilde{u}}_i}}\left(\delta u_i\right)+o(\parallel \delta u\parallel ).\hfill \end{array}$$

Here, $o(\parallel \delta u\parallel )$ is the higher order infinitesimal of $\parallel \delta u\parallel$ and the norm of $\delta u$ is given by

$${\parallel \delta u\parallel }^2:=\sum _{i=1}^N\mathbb{E}{\int }_0^T{\left|u_i\right|}^{2}dt.$$

From (47), we know that $o(\parallel \delta u\parallel )=o\left(\sqrt{N}\right)$, which yields that ${\displaystyle \frac{1}{N}}o(\parallel \delta u\parallel )=o\left({\displaystyle \frac{1}{\sqrt{N}}}\right)$.

Therefore, in order to prove asymptotic optimality, we only need to show that

$${\displaystyle \frac{\partial {\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)}{\partial {\tilde{u}}_i}}\left(\delta u_i\right)=O\left({\displaystyle \frac{1}{\sqrt{N}}}+{ϵ}_N\right).$$

According to Section 5.1, we derive

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)}{\partial {\tilde{u}}_i}}\left(\delta u_i\right)=\mathbb{E}\left\{{\int }_0^T\right[\langle S{\tilde{y}}_i,\delta y_i\rangle -\langle S_G\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right],\delta y_i\rangle +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\tilde{y}}_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle \hfill \\ \hfill & -\langle {\tilde{S}}_{\tilde{G}}\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta ),\delta y_i\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta ),\delta y_i\rangle +\langle R_{{\theta }_i}{\tilde{u}}_i,\delta u_i\rangle ]dt+\langle Q{\tilde{y}}_i\left(0\right),\delta y_i\left(0\right)\rangle -\langle Q_H\sum _{l=1}^K{\pi }_l{\alpha }_l\left(0\right),\delta y_i\left(0\right)\rangle \}\hfill \\ \hfill & +\sum _{l=1}^{11}{\epsilon }_l.\hfill \end{array}$$

It follows from the optimality of $\tilde{u}$ that

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{{\int }_0^T\right[\langle S{\tilde{y}}_i,\delta y_i\rangle -\langle S_G\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l\right],\delta y_i\rangle +\langle \tilde{S}{\mathbb{E}}^{{\mathcal{F}}_t}\left[{\tilde{y}}_i(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle \hfill \\ \hfill & -\langle {\tilde{S}}_{\tilde{G}}\sum _{l=1}^K{\pi }_l{\mathbb{E}}^{{\mathcal{F}}_t^0}\left[{\alpha }_l(t+\delta )\right],{\mathbb{E}}^{{\mathcal{F}}_t}\left[\delta y_i(t+\delta )\right]\rangle -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )\mathbb{E}{\mathbf{x}}_k(t-\delta ),\delta y_i\rangle \hfill \\ \hfill & -\sum _{k=1}^K\langle {\pi }_k{C}^{\top }(t-\delta )x_2^k(t-\delta ),\delta y_i\rangle +\langle R_{{\theta }_i}{\tilde{u}}_i,\delta u_i\rangle ]dt+\langle Q{\tilde{y}}_i\left(0\right),\delta y_i\left(0\right)\rangle -\langle Q_H\sum _{l=1}^K{\pi }_l{\alpha }_l\left(0\right),\delta y_i\left(0\right)\rangle \}=0.\hfill \end{array}$$

Moreover, by Lemmas 4–7 (recall that when calculating ${\displaystyle \frac{\partial {\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)}{\partial {\tilde{u}}_i}}\left(\delta u_i\right)$, it only needs to take perturbation for one component of $\tilde{u}$), we have

$$\sum _{l=1}^{11}{\epsilon }_l=O\left({\displaystyle \frac{1}{\sqrt{N}}}+{ϵ}_N\right).$$

Therefore,

$${\displaystyle \frac{\partial {\mathcal{J}}_{soc}^{\left(N\right)}\left(\tilde{u}\right)}{\partial {\tilde{u}}_i}}\left(\delta u_i\right)=O\left({\displaystyle \frac{1}{\sqrt{N}}}+{ϵ}_N\right).$$

□

6. Conclusions

A class of stochastic LQ dynamic optimization problems involving a large population was researched in the work. Unlike the well-studied MF game, these agents cooperate to minimize a social cost, while the dynamics are evolved by the ABSDE in the case of an input constraint and partial information. By virtue of the so-called anticipated person-by-person optimality, we solved an auxiliary control problem and derived a decentralized social strategy based on an MF-type consistency condition system. We also established the well-posedness of this MF-AFBSDDE using the discounting decoupling method. Finally, the related asymptotic social optimality was verified.