# Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

Authors: Hui Min; Ying Peng; Yongli Qin
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.


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## Body

## 1. Introduction

Pardoux and Peng [1] in 1990 first introduced nonlinear classical backward stochastic differential equations (BSDEs). They proved the uniqueness and the existence of the solutions of nonlinear BSDEs under Lipschitz assumption. Since then the theory of BSDEs developed very fast and had found many applications, for example, in the stochastic control and partial differential equations. On the other hand, those stochastic Hamilton systems, derived from the stochastic maximum principle of stochastic optimal control problems, are forward-backward stochastic differential equations (FBSDEs).The theory of fully coupled FBSDEs develops also very dynamically. There are many works on the existence and the uniqueness of solutions of fully coupled FBSDEs. Antonelli [2] first proved the existence and the uniqueness of solutions of fully coupled FBSDEs driven by Brownian motion on a small time interval with the fixed point theorem. There are also many other methods to study fully coupled FBSDEs on an arbitrarily given time interval, mainly three methods. One is “four-step scheme” approach (see Ma et al. [3]) which combines PDE methods and probability methods. The authors proved the existence and the uniqueness for fully coupled FBSDEs on an arbitrarily given time interval, but they required the diffusion coefficients to be nondegenerate and deterministic. Another one is purely probabilistic continuation method; refer to Hu and Peng [4], Pardoux and Tang [5], Peng and Wu [6], Yong [7], and so on. Another method is inspired by the numerical approaches for some linear FBSDEs (see Delarue and Menozzi [8] and Zhang [9]). There are also other methods; see Ma et al. [10]. For more details about fully coupled FBSDEs, the readers also refer to Ma and Yong [11] or Yong [7] and the references therein.On the other hand, the theory of the modern optimal control has been developed widely since Pontryagin et al.’s work [12] about the maximum principle and Bellman’s work [13] on the dynamic programming approach. Later there have been a lot of works on the stochastic maximum principle; see, for example, Kushner [14, 15], Bensoussan [16], Haussmann [17], Peng [18], Wu [19], and so on. Wu [19] discussed the stochastic maximum principle for the fully coupled FBSDEs. Recently the methods of mean-field are used in various fields, such as in Finance, Chemistry, and Game Theory. The mean-field backward stochastic differential equations (mean-field BSDEs) were introduced by [20]; for more properties about mean-field BSDEs we refer to [21]. There are also many works on stochastic maximum principle for SDEs of mean-field type; see Andersson and Djehiche [22], Buckdahn et al. [23], Li [24], Bensoussan et al. [25], and so on. For more details we may refer to Yong [7].In this paper, we consider the following fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs in short):(1)dx(t)=E′[f(t,χ(t))]dt+E′[σ(t,χ(t))]dBt,-dy(t)=E′[g(t,χ(t))]dt-z(t)dBt,x(0)=a,y(T)=E′[Φ(x(T),(x(T))′)],
where χ(t):=(x(t),y(t), z(t),(x(t))′, (y(t))′,(z(t))′), (x(·),y(·),z(·)) take values in Rn×Rm×Rm×d; f,σ,g,Φ are mappings with appropriate dimensions which are 𝔽--progressively measurable. The time duration T≥0 is an arbitrarily fixed number. Our aim is first to find a triplet 𝔽-adapted processes (x(·),y(·),z(·)) satisfying (1) and then study the stochastic maximum principle of mean-field FBSDEs with controls. For more works we refer to Qin [26].In Section2, we introduce the mean-field BSDEs. In Section 3, we prove the existence and the uniqueness of solution of mean-field FBSDE by the continuation method. In Section 4, we give the continuity of solutions of mean-field FBSDE with respect to the parameters and also give an example to show that our monotonicity conditions are necessary. In Section 5 we study the stochastic maximum principle for mean-field FBSDEs with controls and obtain the necessary condition of the stochastic maximum principle. In Section 6 we discuss mean-field backward stochastic linear quadratic optimal control problem as an example.


## 2. Preliminaries

Let(Ω,ℱ,P) be a complete probability space with a standard d-dimensional Brownian motion B=(Bt)t≥0, and let ℱt be the natural filtration generated by B and augmented by all P-null sets (i.e., ℱt=σ{Bs:0≤s≤t}⋁𝒩P,t≥0, where 𝒩P is the set of all P-null subsets). T>0 is the fixed time horizon. 𝔽={ℱt,t≥0}.Let(Ω-,ℱ-,P-)=(Ω×Ω,ℱ⊗ℱ,P⊗P) be the (noncompleted) product of (Ω,ℱ,P) with itself. This product space is endowed with the filtration 𝔽-={ℱ-t=ℱ⊗ℱt,0≤t≤T}. A random variable ξ∈L0(Ω,ℱ,P;ℝn) originally defined on Ω is extended canonically to Ω-:ξ′(ω′,ω)=ξ(ω′),(ω′,ω)∈Ω-=Ω×Ω. For any θ∈L1(Ω-,ℱ-,P-) the variable θ(·,ω):Ω→ℝ is in L1(Ω,ℱ,P),P(dω)-a.s., and its expectation is denoted by
(2)E′[θ(·,ω)]=∫Ω‍θ(ω′,ω)P(dω′).
We notice that E′[θ]=E′[θ(·,ω)]∈L1(Ω,ℱ,P) and
(3)E-[θ](=∫Ω-‍θdP-=∫Ω‍E′[θ(·,ω)]P(dω))=E[E′[θ]].The generator of our mean-field BSDE is a mapping:(4)f=f(t,ω,ω′,y,z,y~,z~):[0,T]×Ω-×ℝm×ℝm×d×ℝm×ℝm×d⟶ℝm,
which is 𝔽--progressively measurable, for all (y,z,y~,z~), and satisfies the following assumptions.We assume the following.(H2.1)
(i)
f
(
t
,
y
,
z
,
y
~
,
z
~
)is uniformly Lipschitz with respect to y,z,y~,z~;
(ii)
f
(
·
,
0,0
,
0,0
)
∈
M
𝔽
-
2
(
0
,
T
;
ℝ
m
); that is,f(·,0,0,0,0) is ℝm-valued𝔽--progressively measurable andE-[∫0T‍|f(t,0,0,0,0)|2dt]<+∞.Lemma 1.
Let (H2.1) hold, for any random variableξ∈L2(Ω,ℱT,P;ℝm); the mean-field BSDE
(5)Yt=ξ+∫tT‍E′[f(s,Ys,Zs,(Ys)′,(Zs)′)]ds-∫tT‍ZsdBs,0≤t≤T
has a unique solution (Y,Z)∈S𝔽2(0,T;ℝm)×M𝔽2(0,T;ℝm×d); that is, Y is ℝm-valued 𝔽-adapted continuous process and E[sup⁡0≤t≤T⁡|Yt|2]<∞; Z is ℝm×d-valued 𝔽-progressively measurable process and E[∫0T‍|Zt|2dt]<∞.For the proof, the readers may refer to [20].Remark 2.
From the above notions, the generator of the above mean-feld BSDE has to be understood as follows:(6)E′[f(s,Ys,Zs,(Ys)′,(Zs)′)](ω)=E′[f(s,Ys(ω),Zs(ω),(Ys)′,(Zs)′)]=∫Ω‍f(ω′,ω,s,Ys(ω),Zs(ω),Ys(ω′),Zs(ω′))P(dω′),ω∈Ω.Remark 3.
If we assume that(i)
b(·,x,x~) and σ(·,x,x~) are 𝔽--progressively measurable continuous processes, for all x,x~∈ℝn and there exists some constant C>0 such that |b(t,x,x~)|+|σ(t,x,x~)|≤C(1+|x|+|x~|),a.s.,0≤t≤T,x,x~∈ℝn,(ii)
b and σ are Lipschitz in x,x~,then, for any random variable ζ∈L2(Ω,ℱT,P;ℝn), the following mean-field SDE which is also the Mckean-Vlasov SDE has a unique adapted solution X∈S𝔽2(0,T;ℝn):
(7)Xt=ζ+∫0t‍E′[b(s,Xs,(Xs)′)]ds+∫0t‍E′[σ(s,Xs,(Xs)′)]dBs,0≤t≤T.
For more details, the reader may refer to, for example, [20] or [24].


## 3. Mean-Field FBSDE: Existence and Uniqueness

We consider the following fully coupled mean-field forward-backward stochastic differential equations:(8)dx(t)=E′[f(t,χ(t))]dt+E′[σ(t,χ(t))]dBt,-dy(t)=E′[g(t,χ(t))]dt-z(t)dBt,x(0)=a,y(T)=E′[Φ(x(T),(x(T))′)],
where
(9)χ(t)=(x(t),y(t),z(t),(x(t))′,(y(t))′,(z(t))′),f:[0,T]×Ω-×ℝn×ℝm×ℝm×d×ℝn×ℝm×ℝm×d⟶ℝn,σ:[0,T]×Ω-×ℝn×ℝm×ℝm×d×ℝn×ℝm×ℝm×d⟶ℝn×d,g:[0,T]×Ω-×ℝn×ℝm×ℝm×d×ℝn×ℝm×ℝm×d⟶ℝm,Φ:Ω-×ℝn×ℝn⟶ℝm.Remark 4.
In Li [24], the author studied the stochastic maximum principle in mean-field controls; the related feedback control system takes a special case of the mean-field FBSDE (8).Given anm×n full-rank matrix G. We use the following notations:
(10)λ=(xyz),λ~=(x~y~z~),A(t,λ,λ~)=(-GTgGfGσ)(t,λ,λ~),
where Gσ=(Gσ1,…,Gσd). We use the standard inner product and Euclidean norm in ℝm×d.Definition 5.
A triple of processes(X,Y,Z) is called an adapted solution of mean-field FBSDE (8), if (X,Y,Z)∈M𝔽2(0,T;ℝn×ℝm×ℝm×d), and satisfies mean-field FBSDE (8).We assume the following.(H3.1)
(i)
A
(
t
,
λ
,
λ
~
)is uniformly Lipschitz with respect to λ,λ~;
(ii)
for eachλ,λ~,A(·,λ,λ~) is in M𝔽-2(0,T);
(iii)
Φ
(
x
,
x
~
) is uniformly Lipchitz with respect to x,x~∈ℝn;
(iv)
for eachx,x~,Φ(x,x~)is inL2(Ω-,ℱ-T,P-);
(v)
the coefficients(f,σ,g)are uniformly Lipschitz to (x,y,z,x~,y~,z~).We also need the following monotonicity assumptions.(H3.2)
(i)
〈
A
(
t
,
λ
,
λ
~
)
-
A
(
t
,
λ
-
,
λ
~
)
,
λ
-
λ
-
〉
≤
-
β
1
|
x
^
|
2
-
β
2
(
|
y
^
|
2
+
|
z
^
|
2
);
(ii)
〈
Φ
(
x
,
x
~
)
-
Φ
(
x
-
,
x
~
)
,
G
(
x
-
x
-
)
〉
≥
μ
1
|
x
^
|
2;
(11)∀λ=(x,y,z),λ~=(x~,y~,z~),λ-=(x-,y-,z-),l^=l-l-,wherel=x,y,z,x~,y~,z~,respectively.
β
1, β2, and μ1 are given nonnegative constants with β1-LA≥0,β2-LA≥0 (the equalities cannot be established at the same time), and μ1-LΦλ1>0; LA, LΦ are the Lipschitz constants of A,Φ with respect to λ~, x~, respectively; and λ1 satisfies |Gl^(T)|≤λ1|l^(T)|.Or we need the following.(H3.3)
(i)
〈
A
(
t
,
λ
,
λ
~
)
-
A
(
t
,
λ
-
,
λ
~
)
,
λ
-
λ
-
〉
≥
β
1
|
x
^
|
2
+
β
2
(
|
y
^
|
2
+
|
z
^
|
2
);
(ii)
〈
Φ
(
x
,
x
~
)
-
Φ
(
x
-
,
x
~
)
,
G
(
x
-
x
-
)
〉
≤
-
μ
1
|
x
^
|
2;
(12)∀λ=(x,y,z),λ~=(x~,y~,z~),λ-=(x-,y-,z-),l^=l-l-,wherel=x,y,z,x~,y~,z~,respectively.
β
1, β2, and μ1 are given nonnegative constants with β1-LA≥0,β2-LA≥0 (the equalities cannot be established at the same time), and μ1-LΦλ1>0.Then we have the following two main results in this section.Theorem 6.
One assumes that (H3.1) and (H3.2) hold; then mean-field FBSDE (8) has a unique adapted solution (X,Y,Z).Remark 7.
Similarly, if (H3.1) and (H3.3) hold, then mean-field FBSDE (8) has a unique adapted solution (X,Y,Z).Proof.
We first prove the uniqueness. Letλ(t)=(x(t),y(t),z(t)) and λ-(t)=(x-(t),y-(t),z-(t)) be two solutions of (8). We set l^=l-l-, where l=x,y,z,x~,y~,z~, respectively. Applying Itô's formula to 〈Gx^(s),y^(s)〉, we get
(13)E〈(E′[Φ(x(T),(x(T))′)]-E′[Φ(x-(T),(x-(T))′)]),G(x(T)-x-(T))〉=E∫0T‍〈(-E′[A(t,λ-(t),(λ-(t))′)])E′[A(t,λ(t),(λ(t))′)]-E′[A(t,λ-(t),(λ-(t))′)]),λ(t)-λ-(t)〉dt.
From (H3.2) the monotonicity assumptions of Φ and A, we get
(14)(μ1-LΦλ1)E[|x^(T)|2]≤-E∫0T‍[β1|x^(t)|2+β2(|y^(t)|2+|z^(t)|2)]dt+LAE∫0T‍[|x^(t)|2+|y^(t)|2+|z^(t)|2]dt.
When β1-LA>0,β2-LA≥0,μ1-LΦλ1>0,|x^(t)|2=0,ds dP-a.e. In this case we have x^(t)=0,P-a.s., for all t∈[0,T]. Thus, Φ(x(T),(x(T))′)=Φ(x-(T),(x-(T))′), P--a.s. Therefore, from Lemma 1 it follows that y(t)=y-(t),P-a.s. and z(t)=z-(t),P-a.s.a.e. When β1-LA=0,β2-LA>0,μ1-LΦλ1>0, thus y(t)=y-(t),z(t)=z-(t),x(T)=x-(T),P-a.s.a.e. From the uniqueness of solutions of McKean-Vlasov equations (refer to [20] or Remark 2), we get x(t)=x-(t),P-a.s., for all t∈[0,T].For the existence, we need to combine the above techniques and an a priori estimate to construct a contraction mapping. For this we first prove the following lemma.Forβ1-LA≥0,β2-LA≥0,μ1-LΦλ1>0 (the equalities cannot be established at the same time). We consider the following mean-field FBSDEs parameterized by α∈[0,1]:
(15)dxα(t)=[αE′[f(t,χα(t))]+E′[ϕ(t)]]dt+[αE′[σ(t,χα(t))]+E′[ψ(t)]]dBt,-dyα(t)=[(1-α)β1Gxα(t)+αE′[g(t,χα(t))]+E′[γ(t)]]dt-zα(t)dBt,xα(0)=a,yα(T)=αE′[Φ(xα(T),(xα(T))′)]+(1-α)Gxα(T)+ξ,
where χα(t)=(xα(t),yα(t),zα(t),(xα(t))′,(yα(t))′,(zα(t))′) and ϕ,ψ, and γ are given processes in M𝔽-2(0,T) with values in ℝn,ℝn×d, and ℝm, respectively. ξ∈L2(Ω,ℱT,P). Obviously, when α=1, the existence of (15) implies that of (8). From the existence and the uniqueness of Mckean-Vlasov equation and mean-field BSDE, (15) has a unique solution when α=0. The following lemma is needed.Lemma 8.
One assumes that (H3.1) and (H3.2) hold. If for anα0∈[0,1) there exists a solution (xα0,yα0,zα0) of (15), then there exists a positive constant δ0 such that for each δ∈[0,δ0] there exists a solution (xα0+δ,yα0+δ,zα0+δ) of mean-field FBSDE (15) for α=α0+δ.Proof.
Since for everyϕ∈M𝔽-2(0,T;ℝn),γ∈M𝔽-2(0,T;ℝm),ψ∈M𝔽-2(0,T,ℝn×d),α0∈[0,1), there exists a (unique) solution of (15); for each x(T)∈L2(Ω,ℱT,P) and a triple (λ(t))0≤t≤T=(x(t),y(t),z(t))0≤t≤T∈M𝔽2(0,T;ℝn+m+m×d) there exists a unique triple
(16)(Λ(t))0≤t≤T=(X(t),Y(t),Z(t))0≤t≤T∈M𝔽2(0,T;ℝn+m+m×d)
satisfying the following mean-field FBSDEs:(17)dX(t)=[α0E′[f(t,Λ(t),(Λ(t))′)]+δE′[f(t,λ(t),(λ(t))′)]+E′[ϕ(t)]]dt+[α0E′[σ(t,Λ(t),(Λ(t))′)]+δE′[σ(t,λ(t),(λ(t))′)]+E′[ψ(t)]]dBt,-dY(t)=[(1-α0)β1GX(t)+α0E′[g(t,Λ(t),(Λ(t))′)]+δ(-β1Gx(t)+E′[g(t,λ(t),(λ(t))′)])+E′[γ(t)]]dt-Z(t)dBt,X(0)=a,Y(T)=α0E′[Φ(X(T),(X(T))′)]+(1-α0)GX(T)+δ(E′[Φ(x(T),(x(T))′)]-Gx(T))+ξ.
We want to prove that if δ is small enough, the mapping defined by Iα0+δ(λ×x(T))=Λ×X(T):M𝔽2(0,T;ℝn+m+m×d)×L2(Ω,ℱT,P)↦M𝔽2(0,T;ℝn+m+m×d)×L2(Ω,ℱT,P) is a contraction. Let λ-=(x-,y-,z-)∈M𝔽2(0,T;ℝn+m+m×d) and Λ-×X-(T)=Iα0+δ(λ-×x-(T)). We define λ^=(x^,y^,z^)=(x-x-,y-y-,z-z-),λ^′=(x^′,y^′)=(x′-x-′,y′-y-′,z′-z-′),Λ^=(X^,Y^,Z^)=(X-X-,Y-Y-,Z-Z-),Λ^′=(X^′,Y^′,Z^′)=(X′-X-′,Y′-Y-′,Z′-Z-′).
Applying Itô’s formula to〈GX^(t),Y^(t)〉, it yields(18)E〈α0E′[Φ(X(T),(X(T))′)-Φ(X-(T),(X-(T))′)]+(1-α0)GX^(T)+δ(E′[Φ(x(T),(x(T))′)-Φ(x-(T),(x-(T))′)]-Gx^(T)),GX^(T)[Φ(X(T),(X(T))′)-Φ(X-(T),(X-(T))′)]〉=E∫0T{〈α0E′[A(t,Λ-(t),(Λ-(t))′)A(t,Λ(t),(Λ(t))′)-A(t,Λ-(t),(Λ-(t))′)],Λ^(t)〉-(1-α0)β1〈GX^(t),GX^(t)〉+δE′[β1〈GX^(t),Gx^(t)〉+〈GX^(t),g^(t)〉+〈GTY^(t),f^(t)〉+〈GTZ^(t),σ^(t)〉〈α0E′[A(t,Λ(t),(Λ(t))′)]}dt,
where f^(t)=f(t,λ(t),(λ(t))′)-f(t,λ-(t),(λ-(t))′),g^(t)=-g(t,λ(t),(λ(t))′)+g(t,λ-(t),(λ-(t))′),σ^(t)=σ(t,λ(t),(λ(t))′)-σ(t,λ-(t),(λ-(t))′).
From (H3.1) and (H3.2), we know that ifβ1-LA=0,μ1-LΦλ1>0, then β2-LA>0. Then, we have
(19)E[∫0T‍(|Y^(t)|2+|Z^(t)|2)dt]≤δC2E{∫0T‍|Λ^(t)|2dt+|X^(T)|2+∫0T‍|λ^(t)|2dt+|x^(T)|2}.
On the other hand, from standard technique to the forward equation for X^(t)=X(t)-X-(t), we get
(20)sup⁡0≤t≤T⁡E[|X^(t)|2]≤δC2E[∫0T‍|λ^(t)|2dt]+C2E[∫0T‍[|Y^(t)|2+|Z^(t)|2]dt].
From the above two estimates, we have
(21)E{∫0T‍|Λ^(t)|2dt+|X^(T)|2}≤C-δE{∫0T‍|λ^(t)|dt+|x^(T)|2}.
Here the constant C- depends on the Lipschitz constants, λ1,β1,β2, and T.
Ifβ1-LA>0,μ1-LΦλ1>0, then β2-LA≥0. Then, we have
(22)E[|X^(T)|2]+E[∫0T‍|X^(t)|2dt]≤δC1E{∫0T‍|Λ^(t)|2dt+∫0T‍|λ^(t)|2dt+|x^(T)|2+|X^(T)|2}.
Then from the standard estimate of the mean-field BSDE part, we get
(23)E[∫0T‍(|Y^(t)|2+|Z^(t)|2)dt]≤C1δE{∫0T‍|λ^(t)|2dt+|x^(T)|2}+C1{E∫0T‍|X^(t)|2dt+E|X^(T)|2}.
Here the constant C1 depends on the Lipschitz constants, λ1,β1,μ1,α0, and T.
From the above two estimates and the standard estimate ofX^(t), it follows that, for the sufficiently small δ>0,
(24)E{∫0T‍|Λ^(t)|2dt+|X^(T)|2}≤C-δE{∫0T‍|λ^(t)|dt+|x^(T)|2}.
Here the constant C- depends only on the Lipschitz constants, λ1,β1,μ1,α0, and T.
From above all, we now chooseδ0=1/2C-. Obviously, for every fixed δ∈[0,δ0], the mapping Iα0+δ is a contraction in the sense that
(25)E{∫0T‍|Λ^(t)|2dt+|X^(T)|2}≤12E{∫0T‍|λ^(t)|dt+|x^(T)|2}.
It means immediately that this mapping has a unique fixed point:
(26)Λα0+δ=(Xα0+δ,Yα0+δ,Zα0+δ),
which is the solution of (15) for α=α0+δ.Now we can give the proof of the existence of the solution of mean-field FBSDE (8).Proof (continued).
Whenα=0, (15) has a unique solution. Then from Lemma 8, there exists a positive constant δ0 depending on Lipschitz constants, β1,μ1,λ1,λ2, and T, such that, for every δ∈[0,δ0], (15) for α=δ has a unique solution. We can repeat this process N times where 1≤Nδ0≤1+δ0. It means that, in particular, mean-field FBSDE (15) for α=1 has a unique solution; that is, (8) has a unique solution.
The proof is complete.Example 9.
We consider(27)dx(t)=E′[-y′(t)-2y(t)]dt+E′[-z′(t)-2z(t)]dBt,t∈[0,T],-dy(t)=E′[x′(t)+2x(t)]dt-z(t)dBt,t∈[0,T],x(0)=1,y(T)=E′[x′(T)+2x(T)].
The above FBSDE satisfes (H3.1) and (H3.2), form Theorem 6, we know it has a unique solution.Remark 10.
The proof of Remark7 is similar. Notice that (15) should be changed into the following form:
(28)dxα(t)=[αE′[f(t,χα(t))]+E′[ϕ(t)]]dt+[αE′[σ(t,χα(t))]+E′[ψ(t)]]dBt,-dyα(t)=[E′[f(t,χα(t))]-(1-α)β1Gxα(t)+αE′[g(t,χα(t))]+E′[γ(t)]]dt-zα(t)dBt,xα(0)=a,yα(T)=αE′[Φ(xα(T),(xα(T))′)]-(1-α)Gxα(T)+ξ.Remark 11.
WhenΦ does not depend on x,x~, that is, Φ(x,x~)=ξ∈L2(Ω,ℱT,P) is given, for the existence and the uniqueness of the solution of mean-field FBSDE (8), the monotonicity assumption (H3.2) can be weakened as
(29)〈A(t,λ,λ~)-A(t,λ-,λ~),λ-λ-〉≤-β1|x^|2-β2|y^|2;
similarly, (H3.3) can be weakened as
(30)〈A(t,λ,λ~)-A(t,λ-,λ~),λ-λ-〉≥β1|x^|2+β2|y^|2,
where β1, β2, and μ1 are given nonnegative constants with β1-LA≥0,β2-LA≥0, and μ1-LΦλ1>0, where the equalities cannot be established at the same time; LA,LΦ are the Lipchitz constants of A,Φ with respect to λ~, x~, respectively; and λ1 satisfies |Gl^(T)|≤λ1|l^(T)|.Lemma 12.
Whenσ does not depend on z,z′, the mean-field FBSDE (8) also has a unique adapted solution, but the monotonicity (H3.2) should be weakened as(i)
〈A(t,λ,λ~)-A(t,λ-,λ~),λ-λ-〉≤-β1|x^|2;(ii)
〈Φ(x,x~)-Φ(x-,x~),G(x-x-)〉≥μ1|x^|2;similarly, (H3.3) can be weakened as(i)
〈A(t,λ,λ~)-A(t,λ-,λ~),λ-λ-〉≥β1|x^|2;(ii)
〈Φ(x,x~)-Φ(x-,x~),G(x-x-)〉≤-μ1|x^|2, where β1 and μ are given nonnegative constants. Moreover, one has β1>LA+2LACLg,T, μ1>LΦλ1+8CLg,TLΦ2LA, and CLg,T=exp{(4Lg+12Lg2+1)T}.The proof of this lemma is similar to that of Theorem6; we now only give the proof of the uniqueness.Proof.
Letλ(t)=(x(t),y(t),z(t)) and λ-(t)=(x-(t),y-(t),z-(t)) be two solutions of (8). We set l^=l-l-, where l=x,y,z,x~,y~,z~, respectively. Applying Itô’s formula to 〈Gx^(s),y^(s)〉, we get
(31)E〈(E′[Φ(x(T),(x(T))′)]-E′[Φ(x-(T),(x-(T))′)]),G(x(T)-x-(T))[Φ(x(T),(x(T))′)]〉=E∫0T‍〈(E′[A(t,λ-(t),(λ-(t))′)]E′[A(t,λ(t),(λ(t))′)]-E′[A(t,λ-(t),(λ-(t))′)]),λ(t)-λ-(t)(E′[A(t,λ-(t),(λ-(t))′)]E′[A(t,λ(t),(λ(t))′)]〉dt.
From (H3.2) the monotonicity assumptions of Φ and A, we get
(32)(μ1-LΦλ1)E[|x^(T)|2]≤-(β1-LA)E∫0T‍|x^(t)|2dt+LAE∫0T‍(|y^(t)|2+|z^(t)|2)dt.
Applying Itô’s formula to eβs|y^(s)|2, we get
(33)deβs|y^(s)|2=βeβs|y^(s)|2ds+2eβsy^(s)dy^(s)+eβs(dy^(s))2=βeβs|y^(s)|2ds-2eβsy^(s)E′[g^]ds+2eβsy^(s)z^(s)dBs+eβs|z^(s)|2ds,
where g^(s)=g(s,λ(s),λ′(s))-g(s,λ-(s),(λ-(s))′).
Then we get(34)E∫0T‍βeβs|y^(s)|2ds+E∫0T‍eβs|z^(s)|2ds=E[eβT|y^(T)|2]+E∫0T‍2eβsy^(s)E′[g^]ds≤4eβTLΦ2E|x^(T)|2+E∫0T‍eβs|x^(s)|2ds+12E∫0T‍eβs|z^(s)|2ds+(4Lg+12Lg2)E∫0T‍eβs|y^(s)|2ds.
Hence, we have
(35)E∫0T‍(β-4Lg-12Lg2)eβs|y^(s)|2ds+12E∫0T‍eβs|z^(s)|2ds≤4eβTLΦ2E|x^(T)|2+eβTE∫0T‍|x^(s)|2ds.
Thus, taking β=4Lg+12Lg2+1, we get
(36)E∫0T‍(|y^(s)|2+|z^(s)|2)ds≤8CLg,TLΦ2E|x^(T)|2+2CLg,TE∫0T‍|x^(s)|2ds,
where CLg,T=exp{(4Lg+12Lg2+1)T}.
Combining with (32), we have
(37)(μ1-LΦλ1-8CLg,TLΦ2LA)E[|x^(T)|2]≤-(β1-LA-2CLg,TLA)E∫0T‍|x^(t)|2dt.
When β1>LA+2LACLg,T, μ1>LΦλ1+8CLg,TLΦ2LA, we have |x^(t)|2=0dsdP-a.e. In this case we have x^(t)=0,P-a.s., for all t∈[0,T]. Thus, Φ(x(T),(x(T))′)=Φ(x-(T),(x-(T))′), P--a.s. Therefore, from Lemma 1 it follows that y(t)=y-(t),P-a.s. and z(t)=z-(t),P-a.s.


## 4. Continuity Property on the Parameters

In this section we will discuss the continuity of the solution of (8) depending on parameters. We consider the following mean-field FBSDEs with coefficients (fα,σα,gα,Φα),α∈ℝ:
(38)dxα(t)=E′[fα(t,χα(t))]dt+E′[σα(t,χα(t))]dBt,-dyα(t)=E′[gα(t,χα(t))]dt-zα(t)dBt,xα(0)=a,yα(T)=E′[Φα(xα(T),(xα(T))′)],
where χα(t)=(xα(t),yα(t),zα(t), (xα(t))′,(yα(t))′, (zα(t))′) and fα,σα,gα,Φα,α∈ℝ, satisfy (H3.1) and (H3.2) for each α∈ℝ. Then, from Theorem 6 we know that mean-field FBSDE (38) has a unique solution (xα,yα,zα) for each α∈ℝ.Let us give some more assumptions.(H4.1)
(i)
The coefficients(fα,σα,gα,Φα),α∈ℝ, are uniformly Lipschitz to (x,y,z,x~,y~,z~);
(ii)
the mappingsα↦(fα,σα,gα,Φα),α∈ℝ, are continuous, respectively.Then we have the following continuity property.Theorem 13.
Let the coefficients(fα,σα,gα,Φα),α∈ℝ, satisfy (H3.1), (H3.2), and (H4.1), and the associated solution of mean-field FBSDE (38) is denoted by (xα,yα,zα). Then, the mappings
(39)α⟼(xα,yα,zα,xα(T)):ℝ⟼M𝔽2×(0,T;ℝn×ℝm×ℝm×d)×L2(Ω,ℱT,P;ℝn)
are continuous.Proof.
For simplicity of notations, we only prove the continuity of the solutions(xα,yα,zα,xα(T)) of mean-field FBSDE (38) at α=0. We want to prove that (xα,yα,zα,xα(T)) converges to (x0,y0,z0,x0(T)) in M𝔽2(0,T;ℝn×ℝm×ℝm×d)×L2(Ω,ℱT,P;ℝn) as α tends to 0. We set λα(t)=(xα(t),yα(t),zα(t)) and λ^(t)=(λα(t)-λ0(t))=(x^(t),y^(t),z^(t))=(xα(t)-x0(t),yα(t)-y0(t),zα(t)-z0(t)); then from (38) we know that
(40)dx^(t)=E′[(t,λ0(t),(λ0(t))′)fα(t,λα(t),(λα(t))′)-f0(t,λ0(t),(λ0(t))′)]dt+E′[(t,λ0(t),(λ0(t))′)σα(t,λα(t),(λα(t))′)-σ0(t,λ0(t),(λ0(t))′)]dBt,-dy^(t)=E′[(t,λ0(t),(λ0(t))′)gα(t,λα(t),(λα(t))′)-g0(t,λ0(t),(λ0(t))′)]dt-z^(t)dBt,x^(0)=0,y^(T)=E′[(t,λ0(t),(λ0(t))′)Φα(xα(T),(xα(T))′)-Φ0(x0(T),(x0(T))′)].
From assumptions (H3.1), (H3.2), and (H4.1) and standard estimates of x^(t) and (y^(t),z^(t)), we get
(41)sup⁡0≤t≤T⁡E|x^(t)|2≤C1E∫0T‍(|y^(t)|2+|z^(t)|2)dt+C1E-∫0T‍[|f^(t)|2+|σ^(t)|2]dt;(42)E∫0T‍(|y^(t)|2+|z^(t)|2)dt≤C1{E∫0T‍|x^(t)|2dt+E|x^(T)|2+E-∫0T‍|g^(t)|2dt+E-[|Φ^(T)|2]},
where C1 depends on the Lipchitz constants and T, where
(43)f^(t)=fα(t,λ0(t),(λ0(t))′)-f0(t,λ0(t),(λ0(t))′),σ^(t)=σα(t,λ0(t),(λ0(t))′)-σ0(t,λ0(t),(λ0(t))′),g^(t)=-gα(t,λ0(t),(λ0(t))′)+g0(t,λ0(t),(λ0(t))′),Φ^(T)=Φα(x0(T),(x0(T))′)-Φ0(x0(T),(x0(T))′).
Applying Itô’s formula to 〈Gx^(t),y^(t)〉, it yields
(44)E〈E′[(x0(T),(x0(T))′)Φα(xα(T),(xα(T))′)-Φα(x0(T),(x0(T))′)],Gx^(T)〉+E〈E′[Φα(x0(T),(x0(T))′)-Φ0(x0(T),(x0(T))′)],Gx^(T)〉=E∫0T‍E′〈(t,λ0(t),(λ0(t))′)Aα(t,λα(t),(λα(t))′)-Aα(t,λ0(t),(λ0(t))′),λ^(t)〉dt+E∫0T‍E′[〈Gx^(t),g^(t)〉+〈GTy^(t),f^(t)〉+〈GTz^(t),σ^(t)〉]dt,
where
(45)Aα(t,λ,λ′)=(-GTgαGfαGσα)(t,λ,λ′).
Then, we have
(46)(β1-LA)E∫0T|x^(t)|2dt+(μ1-LΦλ1)E|x^(T)|2+(β2-LA)E×∫0T‍(|y^(t)|2+|z^(t)|2)dt≤C2E[∫0TE′(|f^(t)|2+|g^(t)|2+|σ^(t)|2)dtE′|Φ^(T)|2+∫0TE′(|f^(t)|2+|g^(t)|2+|σ^(t)|2)dt]+δ[E|x^(T)|2+E∫0T(|x^(t)|2+|y^(t)|2+|z^(t)|2)dt].
From (H3.2) we know if β1-LA>0,μ1-LΦλ1>0, then β2-LA≥0. Then, from (46) we have
(47)(β1-LA)E∫0T|x^(t)|2dt+(μ1-LΦλ1)E|x^(T)|2≤C2E[+∫0TE′(|f^(t)|2+|g^(t)|2+|σ^(t)|2)dtE′|Φ^(T)|2+∫0TE′(|f^(t)|2+|g^(t)|2+|σ^(t)|2)dt]+δ[E|x^(T)|2+E∫0T(|x^(t)|2+|y^(t)|2+|z^(t)|2)dt].
With the help of (42) and (47) we can take sufficiently small δ such that
(48)E|x^(T)|2+E∫0T‍|λ^(t)|2dt≤CE-[|Φ^(T)|2∫0T‍|f^(t)|2+∫0T‍(|f^(t)|2+|g^(t)|2+|σ^(t)|2)dt],
where the constant C only depends on C1,C2,β1,μ1,LA,LΦ.
Similarly, ifβ1-LA=0,μ1-LΦλ1>0, then β2-LA>0. Then, from (46) we have
(49)(β2-LA)E∫0T‍(|y^(t)|2+|z^(t)|2)dt≤C2E[∫0T‍E′|Φ^(T)|2+∫0T‍E′(|f^(t)|2+|g^(t)|2+|σ^(t)|2)dt]+δ[∫0T‍E|x^(T)|2+E∫0T‍(|x^(t)|2+|y^(t)|2+|z^(t)|2)dt].
With the help of (41) and (49) we can take sufficiently small δ such that
(50)E|x^(T)|2+E∫0T‍|λ^(t)|2dt≤CE-[|Φ^(T)|2+∫0T‍(|f^(t)|2+|g^(t)|2+|σ^(t)|2)dt],
where the constant C only depends on C1,C2,β2,LA,LΦ.
Hence, we have that(xα,yα,zα,xα(T)) converges to (x0,y0,z0,x0(T)) in M𝔽2(0,T;ℝn×ℝm×ℝm×d)×L2(Ω,ℱT,P;ℝn) as α tends to 0.Now we will give an example to explain that (H3.2) (or (H3.3)) is necessary; that is, if the coefficients do not satisfy (H3.2) (or (H3.3)), then (8) may not have a solution. We take m=n=d=1 here. We consider
(51)dx(t)=E[y(t)]dt+dBt,t∈[0,34π],-dy(t)=E[x(t)]dt-z(t)dBt,t∈[0,34π],x(0)=1,y(34π)=-E[x(34π)],t∈[0,34π].
It is easy to check that this equation does not satisfy (H3.2) (or (H3.3)); we point out that, there is also no adapted solution. In fact, if (x,y,z)0≤t≤(3/4)π is the solution of mean-field FBSDE (51), then (E[x(t)],E[y(t)]) is the solution of the following ordinary differential equation:
(52)X˙=Y,Y˙=-X,X(0)=1,Y(34π)=-X(34π),t∈[0,34π].
But we know this ODE has no solution; therefore, there is no adapted solution of (51).


## 5. Maximum Principle for the Controlled Mean-Field FBSDEs

We consider the following controlled mean-field fully coupled forward-backward SDEs:(53)dx(t)=E′[f(t,χ(t),v(t))]dt+E′[σ(t,χ(t),v(t))]dBt,-dy(t)=E′[g(t,χ(t),v(t))]dt-z(t)dBt,x(0)=a,y(T)=E′[Φ(x(T),(x(T))′)],
where χ(t)=(x(t),y(t),z(t),(x(t))′,(y(t))′,(z(t))′), (x,y,z) takes value in ℝn×ℝm×ℝm×d. Let U be a nonempty convex subset of ℝk(54)𝒰ad={v(·)∈M𝔽-2(0,T;ℝk)∣v(t)∈U,0≤t≤T,P--a.s.}.
An element v of 𝒰ad is called an admissible control. We define the following cost functional:
(55)J(v(·))=E[∫0TE′[L(χ(t),v(t))]dt+E′[φ(x(T),(x(T))′)]+h(y(0))∫0TE′[L(χ(t),v(t))]dt],
where
(56)f:[0,T]×ℝn×ℝm×ℝm×d×ℝn×ℝm×ℝm×d×ℝk⟶ℝn,σ:[0,T]×ℝn×ℝm×ℝm×d×ℝn×ℝm×ℝm×d×ℝk⟶ℝn×d,g:[0,T]×ℝn×ℝm×ℝm×d×ℝn×ℝm×ℝm×d×ℝk⟶ℝm,L:[0,T]×ℝn×ℝm×ℝm×d×ℝn×ℝm×ℝm×d×ℝk⟶ℝ,Φ:ℝn×ℝn⟶ℝm,φ:ℝn×ℝn⟶ℝ,h:ℝm⟶ℝ.
The optimal control problem is to minimize the cost functional J(v(·)) over all admissible controls. An admissible control u(·) is called an optimal control if the cost functional J(v(·)) attains the minimum at u(·). Equation (53) is called the state equation; the solution (x(·),y(·),z(·)) corresponding to u(·) is called the optimal trajectory.We assume the following.(H5.1)
(i)
f
,
g
,
σ
,
L
,
Φ
,
φand h are continuously differentiable to(x,y,z,x~,y~,z~,v);
(ii)
the derivatives off,g,σ,Φ are bounded;
(iii)
the derivatives ofLare bounded byC(1+|x|+|y|+|z|+|x~|+|y~|+|z~|+|v|);
(iv)
the derivatives ofφandhare bounded byC(1+|x|+|x~|)andC(1+|y|),respectively;
(v)
for any given admissible controlv(·),the coefficients satisfy(H3.1)and(H3.2).Letu(·) be an optimal control and let (x(·),y(·),z(·)) be the corresponding optimal trajectory. Let v(·) be such that u(·)+v(·)∈𝒰ad. Since U is convex, then for any 0≤ρ≤1,uρ(·)=u(·)+ρv(·) is also in 𝒰ad.We introduce the following linear mean-field FBSDE:(57)dx1(t)={[fz~(θ(t))(z1(t))′]E′[fx(θ(t))]x1(t)+E′[fy(θ(t))]y1(t)+E′[fz(θ(t))]z1(t)+E′[fv(θ(t))v(t)]+E′[fx~(θ(t))(x1(t))′]+E′[fy~(θ(t))(y1(t))′]+E′[fz~(θ(t))(z1(t))′]}dt+{E′[σy~(θ(t))(y1(t))′]E′[σx(θ(t))]x1(t)+E′[σy(θ(t))]y1(t)+E′[σz(θ(t))]z1(t)+E′[σv(θ(t))v(t)]+E′[σx~(θ(t))(x1(t))′]+E′[σy~(θ(t))(y1(t))′]+E′[σz~(θ(t))(z1(t))′]}dBt,-dy1(t)={[gz~(θ(t))(z1(t))′]E′[gx(θ(t))]x1(t)+E′[gy(θ(t))]y1(t)+E′[gz(θ(t))]z1(t)+E′[gv(θ(t))v(t)]+E′[gx~(θ(t))(x1(t))′]+E′[gy~(θ(t))(y1(t))′]+E′[gz~(θ(t))(z1(t))′]}dt-z1(t)dBt,x1(0)=0,y1(T)=E′[+Φx~(x(T),(x(T))′)(x1(T))′Φx(x(T),(x(T))′)x1(T),+Φx~(x(T),(x(T))′)(x1(T))′],
where θ(t)=(t,x(t),y(t),z(t),(x(t))′,(y(t))′,(z(t))′,u(t)).Remark 14.
Whenl=f,g,σ,L, respectively, lx is the partial derivative of l(t,x,y,z,x~,y~,z~,v) with respect to x; lx~ is the partial derivative of l(t,x,y,z,x~,y~,z~,v) with respect to x~, similar to ly,ly~,lz,lz~,lv.
From (H5.1), it is easy to verify that (57) satisfies (H3.1) and (H3.2); then there exists a unique solution (x1,y1,z1) of mean-field FBSDE (57). Equation (57) is called the variational equation.
We denote by(xρ(·),yρ(·),zρ(·)) the trajectory corresponding to uρ. Then we have the following convergence result.Lemma 15.
One assumes (H5.1) holds. Then,lim⁡ρ→0((xρ(t)-x(t))/ρ)=x1(t), lim⁡ρ→0((yρ(t)-y(t))/ρ)=y1(t), and lim⁡ρ→0((zρ(t)-z(t))/ρ)=z1(t), in M𝔽2(0,T).Proof.
Letx^(t)=xρ(t)-x(t),y^(t)=yρ(t)-y(t),z^(t)=zρ(t)-z(t). Then
(58)dx^(t)=E′[f(χρ(t),u(t)+ρv(t))-f(χ(t),u(t))]dt+E′[σ(χρ(t),u(t)+ρv(t))-σ(χ(t),u(t))σ(χρ(t),u(t)+ρv(t))]dBt,-dy^(t)=E′[g(χρ(t),u(t)+ρv(t))-g(χ(t),u(t))]dt-z^(t)dBt,x^(0)=0,y^(T)=E′[Φ(xρ(T),(xρ(T))′)-Φ(x(T),(x(T))′)Φ(xρ(T),(xρ(T))′)],χρ(t)=(t,xρ(t),yρ(t), zρ(t),(xρ(t))′,(yρ(t))′, (zρ(t))′) and χ(t)=(t,x(t),y(t),z(t),(x(t))′,(y(t))′,(z(t))′). From Theorem 13, it is easy to know that (x^(·),y^(·),z^(·)) converges to 0 in M𝔽2(0,T) as ρ tends to 0. Now, we define
(59)Δx(t)=xρ(t)-x(t)ρ,Δy(t)=yρ(t)-y(t)ρ,Δz(t)=zρ(t)-z(t)ρ,λ(t)=(x(t),y(t),z(t)),λρ(t)=(xρ(t),yρ(t),zρ(t)).
Then,(60)dΔx(t)=1ρE′[f(t,λρ(t),(λρ(t))′,u(t)+ρv(t))-f(t,λ(t),(λ(t))′,u(t))(t,λρ(t),(λρ(t))′,u(t)+ρv(t))]dt+1ρE′[σ(t,λρ(t),(λρ(t))′,u(t)+ρv(t))-σ(t,λ(t),(λ(t))′,u(t))σ(t,λρ(t),(λρ(t))′,u(t)+ρv(t))]dBt,-dΔy(t)=1ρE′[g(t,λρ(t),(λρ(t))′,u(t)+ρv(t))-g(t,λ(t),(λ(t))′,u(t))g(t,λρ(t),(λρ(t))′,u(t)+ρv(t))]dt-Δz(t)dBt,Δx(0)=0,Δy(T)=1ρE′[Φ(xρ(T),(xρ(T))′)-Φ(x(T),(x(T))′)Φ(xρ(T),(xρ(T))′)].
The above Equation(60) can be rewritten as the following:
(61)dΔx(t)=E′[f-(Δχ(t),v(t))]dt+E′[σ-(Δχ(t),v(t))]dBt,-dΔy(t)=E′[g-(Δχ(t),v(t))]dt-Δz(t)dBt,Δx(0)=0,Δy(T)=E′[K-l(T)Δx(T)+M-l(T)(Δx(T))′],
where
(62)Δχ(t)=(t,Δx(t),Δy(t),Δz(t),(Δx(t))′,(Δy(t))′,(Δz(t))′),l-(t,x,y,z,x~,y~,z~,v)=Al(t)x+Bl(t)y+Cl(t)z+Dl(t)x~+El(t)y~+Fl(t)z~+Gl(t)v,
where l=f,g,σ, respectively, and
(63)A(t)=1xρ(t)-x(t)·[l(t,xρ(t),yρ(t),zρ(t),(λρ(t))′,u(t)+ρv(t))-l(t,x(t),yρ(t),zρ(t),(λρ(t))′,u(t)+ρv(t))],B(t)=1yρ(t)-y(t)·[l(t,x(t),yρ(t),zρ(t),(λρ(t))′,u(t)+ρv(t))-l(t,x(t),y(t),zρ(t),(λρ(t))′,u(t)+ρv(t))],C(t)=1zρ(t)-z(t)·[l(t,x(t),y(t),zρ(t),(λρ(t))′,u(t)+ρv(t))-l(t,x(t),y(t),z(t),(λρ(t))′,u(t)+ρv(t))],D(t)=1(xρ(t))′-(x(t))′·[l(t,λ(t),(xρ(t))′,(yρ(t))′,(zρ(t))′,u(t)+ρv(t))-l(t,λ(t),(x(t))′,(yρ(t))′,(zρ(t))′,u(t)+ρv(t))],E(t)=1(yρ(t))′-(y(t))′·[l(t,λ(t),(x(t))′,(yρ(t))′,(zρ(t))′,u(t)+ρv(t))-l(t,λ(t),(x(t))′,(y(t))′,(zρ(t))′,u(t)+ρv(t))],F(t)=1(zρ(t))′-(z(t))′·[l(t,λ(t),(x(t))′,(y(t))′,(zρ(t))′,u(t)+ρv(t))-l(t,λ(t),(x(t))′,(y(t))′,(z(t))′,u(t)+ρv(t))],G(t)=1ρv(t)·[l(t,λ(t),(λ(t))′,u(t)+ρv(t))-l(t,λ(t),(λ(t))′,u(t))],K(T)=1xρ(T)-x(T)×[Φ(xρ(T),(xρ(T))′)-Φ(x(T),(xρ)′(T))],M(T)=1(xρ(T))′-(x(T))′×[Φ(x(T),(xρ(T))′)-Φ(x(T),(x(T))′)],Al(t)={A(t),xρ(t)-x(t)≠0,0,otherwise,Bl(t)={B(t),yρ(t)-y(t)≠0,0,otherwise,Cl(t)={C(t),zρ(t)-z(t)≠0,0,otherwise,Dl(t)={D(t),(xρ(t))′-(x(t))′≠0,0,otherwise,El(t)={E(t),(yρ(t))′-(y(t))′≠0,0,otherwise,Fl(t)={F(t),(zρ(t))′-(z(t))′≠0,0,otherwise,Gl(t)={G(t),ρv(t)≠0,0,otherwise,K-l(T)={K(T),xρ(T)-x(T)≠0,0,otherwise,M-l(T)={M(T),(xρ(T))′-(x(T))′≠0,0,otherwise.
From (H5.1) and the fact that (x^(·),y^(·),z^(·)) converges to 0 in M𝔽2(0,T) as ρ tends to 0, we know that
(64)lim⁡ρ→0(Al(t)-lx(θ(t)))=0,lim⁡ρ→0(Bl(t)-ly(θ(t)))=0,lim⁡ρ→0(Cl(t)-lz(θ(t)))=0,lim⁡ρ→0(Dl(t)-lx~(θ(t)))=0,lim⁡ρ→0(El(t)-ly~(θ(t)))=0,lim⁡ρ→0(Fl(t)-lz~(θ(t)))=0,lim⁡ρ→0(Gl(t)-lv(θ(t)))=0,0=lim⁡ρ→0{l-(t,Δx(t),Δy(t),Δz(t),(Δx(t))′,(Δy(t))′,(Δz(t))′,v(t))-lx(θ(t))Δx(t)-ly(θ(t))Δy(t)-lz(θ(t))Δz(t)-lx~(θ(t))(Δx(t))′-ly~(θ(t))(Δy(t))′-lz~(θ(t))(Δz(t))′-lv(θ(t))v(t)},
where θ(t)=(t,x(t),y(t),z(t), (x(t))′,(y(t))′, (z(t))′,u(t)). Δy(T) has similar results.
As we know, (57) has a unique solution (x1(·),y1(·),z1(·)). Therefore, the solution (Δx(·),Δy(·),Δz(·)) converges to (x1(·),y1(·),z1(·)) in M𝔽2(0,T;ℝn+m+m×d) as ρ tends to 0.Becauseu(·) is an optimal control, then
(65)ρ-1[J(u(·)+ρv(·))-J(u(·))]≥0.
From (65) and Lemma 15, we have the following.Lemma 16.
One supposes that (H5.1) holds. Then, the following variational inequality holds:(66)0≤E∫0TE′[+Lz~(θ(t))(z1(t))′Lx(θ(t))x1(t)+Ly(θ(t))y1(t)+Lz(θ(t))z1(t)+Lx~(θ(t))(x1(t))′+Ly~(θ(t))(y1(t))′+Lz~(θ(t))(z1(t))′+Lv(θ(t))v(t)]dt+E[E′[+φx~(x(T),(x(T))′)(x1(T))′φx(x(T),(x(T))′)x1(T)+φx~(x(T),(x(T))′)(x1(T))′]+hy(y(0))y1(0)+φx~(x(T),(x(T))′)(x1(T))′],
where θ(t)=(t,x(t),y(t),z(t),(x(t))′,(y(t))′,(z(t))′,u(t)).Proof.
Letρ→0 in (65); from Lemma 15 and (H5.1), it is obvious that
(67)ρ-1E(E′[φ(xρ(T)),(xρ(T)′)-φ(x(T),(x(T))′)])⟶EE′[{(x(T),(x(T))′)(x1(T))′}φx(x(T),(x(T))′)x1(T)+φx~(x(T),(x(T))′)(x1(T))′];ρ-1E[h(yρ(0))-h(y(0))]⟶E[hy(y(0))y1(0)];ρ-1E∫0TE′[L(χρ(t),u(t)+ρv(t))-L(θ(t))]dt⟶E∫0TE′[(θ(t))(y1(t))′Lx(θ(t))x1(t)+Ly(θ(t))y1(t)+Lz(θ(t))z1(t)+Lx~(θ(t))(x1(t))′+Ly~(θ(t))(y1(t))′+Lz~(θ(t))(z1(t))′+Lv(θ(t))v(t)(θ(t))(y1(t))′]dt.
The proof is complete.Now we introduce the following adjoint mean-field FBSDE to (57):
(68)dp(t)=E′[(q(t))′gyT(θ(t))p(t)-fyT(θ(t))q(t)-σyT(θ(t))k(t)-Ly(θ(t))+gy~T(ϱ(t))(p(t))′-fy~T(ϱ(t))(q(t))′-σy~T(ϱ(t))(k(t))′-Ly~(ϱ(t))(q(t))′]dt+E′[(q(t))′gzT(θ(t))p(t)-fzT(θ(t))q(t)-σzT(θ(t))k(t)-Lz(θ(t))+gz~T(ϱ(t))(p(t))′-fz~T(ϱ(t))(q(t))′-σz~T(ϱ(t))(k(t))′-Lz~(ϱ(t))(q(t))′]dBt,-dq(t)=E′[(q(t))′-gxT(θ(t))p(t)+fxT(θ(t))q(t)+σxT(θ(t))k(t)+Lx(θ(t))-gx~T(ϱ(t))(p(t))′+fx~T(ϱ(t))(q(t))′+σx~T(ϱ(t))(k(t))′+Lx~(ϱ(t))(q(t))′]dt-k(t)dBtp(0)=-hy(y(0)),q(T)=E′[(q(t))′φx(x(T),(x(T))′)+φx~((x(T))′,x(T))-Φx(x(T),x′(T))p(T)-Φx~(x′(T),x(T))(p(T))′(q(t))′],
where θ(t)=(t,x(t),y(t), z(t),(x(t))′,(y(t))′, (z(t))′,u(t)) and ϱ(t)=(t,(x(t))′,(y(t))′,(z(t))′,x(t),y(t),z(t),(u(t))′). From Theorem 6, we know that there exists a unique triple (p(·),q(·),k(·)) satisfying (68).We define the Hamiltonian functionH as follows:
(69)H(t,x,y,z,x~,y~,z~,v,p,q,k)=〈p,-g(t,x,y,z,x~,y~,z~,v)〉+〈q,f(t,x,y,z,x~,y~,z~,v)〉+〈k,σ(t,x,y,z,x~,y~,z~,v)〉+L(t,x,y,z,x~,y~,z~,v).
Then we have the following maximum principle.Theorem 17.
Letu(·) be an optimal control and let (x(·),y(·),z(·)) be the corresponding trajectory. Then, one has
(70)E′〈Hv(t,χ(t),u(t),p(t),q(t),k(t)),v-u(t)〉≥0,∀v∈U,dtdP-a.e.,
where χ(t)=(x(t),y(t),z(t),(x(t))′,(y(t))′,(z(t))′), (p,q,k) is the solution of the adjoint (68).Proof.
Applying Itô’s formula to〈x1(t),q(t)〉+〈y1(t),p(t)〉, from (57) and (68) and (H3.1), (H3.2), (H3.3), and (H5.1), with the help of (66) and (69), for v(·) such that u(·)+v(·)∈𝒰ad, we get
(71)E∫0T‍E′〈Hv(t,χ(t),u(t),p(t),q(t),k(t)),v〉dt≥0.
Therefore, we have
(72)E′〈Hv(t,χ(t),u(t),p(t),q(t),k(t)),v-u(t)〉≥0,∀v∈U,a.s.a.e.


## 6. Application to the Mean-Field LQ Problems

In this section, we consider a linear-quadratic control problem as an example. For simplicity, we only consider one-dimensional case; that is,m=n=d=1. The state equation can be written as follows:
(73)dx(t)=E′[a1(x(t))′+a2x(t)-a3(a3y(t)+a4z(t))+αv(t)a1(x(t))′]dt+E′[a5(x(t))′+a6x(t)-a4(a3y(t)+a4z(t))+βv(t)(x(t))′]dBt,-dy(t)=E′[a1(y(t))′+a2y(t)+a5(z(t))′+a6z(t)+a7x(t)+γv(t)a1(y(t))′]dt-z(t)dBt,x(0)=a,y(T)=η,
where the constants ai,i=1,…,7, α,β,γ are positive and v∈𝒰. The cost functional is
(74)J(v(·))=12EE′[∫0T[M(t)y2(t)+R(t)z2(t)+S(t)u2(t)]dt+Qy2(0)∫0T[M(t)y2(t)+R(t)z2(t)+S(t)u2(t)]dt],
where M(t),R(t),Q are bounded and nonnegative and S(t) is bounded and positive. Then, the adjoint equation is the following mean-field FBSDE:
(75)dp(t)=E′[(p(t))′a2p(t)+a32q(t)+a3a4k(t)-M(t)y(t)+a1(p(t))′]dt+E′[(p(t))′a6p(t)+a3a4q(t)+a42k(t)-R(t)z+a5(p(t))′]dBt,-dq(t)=E′[(p(t))′-a7p(t)+a2q(t)+a6k(t)+a1q′(t)+a5(k(t))′]dt-k(t)dBt,p(0)=-Qy(0),q(T)=0.
Let
(76)H(t,x,y,z,x~,y~,z~,v,p,q,k)=-p(a1y~+a2y+a5z~+a6z+a7x+γv)+q(a1x~+a2x-a3(a3y+a4z)+αv)+k(a5x~+a6x-a4(a3y+a4z)+βv)+12(M(t)y2(t)+R(t)z2(t)+S(t)u2(t)).Then, from the stochastic maximum principle (Theorem17), we have
(77)-p(t)γ+q(t)α+k(t)β+S(t)u*(t)=0;
that is,
(78)u*(t)=-S-1(t)(-p(t)γ+q(t)α+k(t)β).
However, the maximum principle gives only the necessary condition for optimal control.Now we prove thatu* is the optimal control. For all v∈𝒰, let (xv(·),yv(·),zv(·)) be the corresponding trajectory. Then
(79)J(v(·))-J(u*(·))=12EE′[∫0T[∫0T[M(t)((yv(t))2-(y*(t))2)M(t)((yv(t))2-(y*(t))2)+R(t)((zv(t))2-(z*(t))2)+S(t)((v(t))2-(u*(t))2)∫0T[M(t)((yv(t))2-(y*(t))2)]dt+Q((yv(0))2-(y*(0))2)∫0T[M(t)((yv(t))2-(y*(t))2)]≥EE′∫0T[M(t)y*(t)(yv(t)-y*(t))+R(t)z*(t)(zv(t)-z*(t))+S(t)u*(t)(v(t)-u*(t))]dt+Qy*(0)(yv(0)-y*(0)).
We apply Itô’s formula to p(t)(yv(t)-y*(t))+q(t)(xv(t)-x*(t)), where (p(t),q(t),k(t)) is the solution of adjoint equation with the state process (x*(t),y*(t),z*(t)); notice that when c is a constant, then EE′[cX(t)(X(t))′]=EE′[c(X(t))′X(t)] and we get
(80)EE′[Qy*(0)(yv(0)-y*(0))]=EE′∫0T[(-p(t)γ+q(t)α+k(t)β)(v(t)-u*(t))-M(t)y*(t)(yv(t)-y*(t))-R(t)z*(t)(zv(t)-z*(t))]dt.
Therefore, from the definition of u*(t),
(81)J(v(·))-J(u*(·))≥EE′[∫0T‍(-p(t)γ+q(t)α+k(t)β+S(t)u*(t))×(v(t)-u*(t))∫0T‍(-p(t)γ+q(t)α+k(t)β+S(t)u*(t))]dt=0,
for any v∈𝒰. It means that u*(t) is an optimal control.Remark 18.
Under our assumption, the existence and the uniqueness of the solution of (73) and (75) can be obtained by combining the method of Theorem 3.1 and Theorem 2.1 in [6]. We omit the proof.



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