# Maximum Principle for Mean-Field FBSDEs with Mixed Initial-Terminal Conditions

Authors: Ruijing Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The present paper concerns with optimal control problems allowing for time inconsistent utility functions of mean-field FBSDEs with mixed initial-terminal conditions. Moreover, the control variable enters the diffusion coefficient, and the control domain is with nonconvexity. Via extended Ekeland’s variational principle as well as the reduction method, a general stochastic maximum principle is established in the framework of mean-field theory. Finally, a linear-quadratic example is worked out to illustrate the application of the results.


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## Body

## 1. Introduction

It is well known that the key tools in solving optimal control problems are based upon Pontryagin’s maximum principle and the Bellman dynamic programming principle. Recently, the mean-field approaches have been widely applied in various fields, such as in Finance, Statistical Mechanics, and Game theory. The systems these fields involve are mainly of mean-field type, where the performance functionals, drifts, and diffusion coefficients depend not only on the state but also on the probability distribution of the state. In this case, the mean-variance utility functions are of time inconsistence, for which the Bellman dynamic programming principle does not hold. Therefore, problems involving such utilities cannot be approached by the classical Hamiltonian–Jacobi–Bellman equation. In this paper, we study a mean-variance setting of the Principle-Agent problem by means of Pontryagin’s maximum principle.The pioneering paper in which nonlinear mean-field backward stochastic differential equations (BSDEs) first appeared is in [1]. Taking advantage of a BSDE approach, the paper [2] considered the mean-field BSDE and gave a probabilistic interpretation to related nonlocal partial differential equations. Since then, the theory of mean-field forward-backward stochastic differential equations (FBSDEs) has been intensively investigated, and rich results have been obtained for the system of this kind of McKean–Vlasov type (see [3]) in the literature. For instance, by using the penalization method to construct approximation solution, the authors in [4] proved the existence of solutions to first-order mean-field games arising in optimal switching. In [5], by translating the splitting method to BSDEs, the author proved the existence and the uniqueness of the solution of the split equations; moreover, under appropriate regularity assumptions on the coefficients, the value function turned out to be the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation of mean-field type. On the other hand, with the development of mean-field FBSDEs, research on stochastic control problems in mean-field context got rapid growth. For instance, by utilizing a spike variation of the optimal control, the authors in [6] established Peng’s-type stochastic maximum principle. When the controlled system contains the Markov jump parameters, the authors in [7] established the corresponding maximum principle for mean-field stochastic linear-quadratic optimal control problems. Furthermore, with the help of maximum principle, the authors in [8] presented necessary as well as sufficient solvability condition of the finite horizon mean-field optimal control problem in an explicit expression form, and the author in [9] established necessary and sufficient optimality conditions for the mean-field jump-diffusion system with moving-average as well as pointwise delays. Further, by using a backward separation method with a dominated growth rate as well as an approximate technique, the authors in [10] presented a maximum principle for optimality of mean-field stochastic differential equations, where the state is partially observed via a noisy process. For some related topics, we refer the readers to [4, 11–13], [14], [7, 15, 16], [17], and the references therein.However, to the best of our knowledge, when the control domain is not assumed convexity, for the stochastic optimal control problems with mixed initial-terminal constraints, their corresponding stochastic maximum principle is all established on the basis of requirements of lower semicontinuity of the penalty functional. So, how to reduce this requirement is a new problem. This paper is trying to remedy this issue and establish stochastic maximum principle for mean-field FBSDEs under general control domains. With the help of extended Ekeland’s variational principle and the reduction method, a general stochastic maximum principle will be acquired. It is necessary to point out that, due to the application of extended Ekeland’s variational principle, the reduction method we adopt is different from those given in [18]; meantime, our results extend those of [19] essentially to the framework of mean-field theory.The rest of this paper is organized as follows. Preliminaries are in Section2. In Section 3, we give some prior estimates and establish the general stochastic maximum principle in the framework of mean-field theory. In Section 4, a linear-quadratic stochastic control problem is worked out to illustrate the theoretical results. Finally, some concluding remarks are given in Section 5.


## 2. Preliminaries

LetΩ,ℱ,ℱtt≥0,P be a filtered probability space satisfying the usual condition, on which a one-dimensional Brownian motion Btt≥0 is defined, and ℱ=ℱs,0≤s≤T be the natural filtration generated by Btt≥0 and augmented by all P-null sets, i.e.,(1)ℱs=σBr,r≤s∨Np,s∈0,T,where Np is the set of all P-null subsets. We denote by Sℱ20,T;R the set of ℱ-adapted and continuous stochastic processes ψ such that Esupt∈0,Tψt2<∞, by ℋℱ20,T;R the set of ℱ-adapted stochastic processes ψ such that E∫0Tψt2dt<∞, and M20,T:=Sℱ20,T;R×Sℱ20,T;R×ℋℱ20,T;R.LetΩ¯,ℱ¯,P¯=Ω×Ω,ℱ⊗ℱ,P⊗P be the (noncompleted) product of Ω,ℱ,P with itself, and this product space is endowed with the filtration F¯=ℱ¯t=ℱ⊗ℱt,0≤t≤T. Then, any random variable χ∈L0Ω,ℱ,P originally defined on Ω can be extended canonically to Ω¯ : χ′w′,w=χw′, w′,w∈Ω¯=Ω×Ω. For any ρ∈L1Ω¯,ℱ¯,P¯, the variable ρ⋅,w : Ω⟶R belongs to L1Ω,ℱ,P, Pdw-a.s., and its expectation is denoted by(2)E′ρ⋅,w=∫Ωρw′,wPdw′.Notice thatE′ρ=E′ρ⋅,w, and E¯ρ=∫Ω¯ρdP¯=∫ΩE′ρ⋅,wPdw=EE′ρ.Consider a fully coupled mean-field stochastic differential equation with mixed initial-terminal conditions as follows:(3)dXt=E′bt,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdt+E′σt,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdBt,−dYt=E′ft,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdt−ZtdBt,X0=γXT,Y0,YT=hXT,Y0,where b,σ,f:0,T×R6×U⟶R; γ,h:R2⟶R; U is a nonempty closed and nonconvex subset of R.Remark 1.
The drift termsb and f and the diffusion term σ of (3) have to be interpreted as follows:(4)E′bs,Xs′,Ys′,Zs′,Xs,Ys,Zs,usw=∫Ωbw′,w,s,Xsw′,Ysw′,Zsw′,Xsw,Ysw,Zsw,usw,w′Pdw′,E′fs,Xs′,Ys′,Zs′,Xs,Ys,Zs,usw=∫Ωfw′,w,s,Xsw′,Ysw′,Zsw′,Xsw,Ysw,Zsw,usw,w′Pdw′,E′σs,Xs′,Ys′,Zs′,Xs,Ys,Zs,usw=∫Ωσw′,w,s,Xsw′,Ysw′,Zsw′,Xsw,Ysw,Zsw,usw,w′Pdw′.
The cost functional associated with (3) is of the form as follows:(5)Ju=E∫0TE′lt,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdt+EφXT,Y0,where l:0,T×R6×U⟶R; φ:R2⟶R. The admissible control set is defined as(6)U=ut∈LF¯20,T,U|utw′,w:0,T×Ω×Ω⟶U.
Then, the optimal control problem we are concerned with is as follows.Problem 1.
A Find a u¯∈U such that(7)Ju¯=infu∈UJu.
Ifu¯ attains the infimum, then it is called optimal; the solution X¯,Y¯,Z¯ corresponding to u¯ is called an optimal trajectory. In what follows, some notations and assumptions shall be assumed to ensure the well-posedness of stochastic system (3). We denote the scalar product by ⋅,⋅ and the norm by ⋅ of an Euclidean space. For Λ:=x,y,z,x˜,y˜,z˜, Ξ:=XT,Y0 and define Ft,Λ,u:=−ft,Λ,u,bt,Λ,u,σt,Λ,u, ΥΞ:=γΞ,hΞ.H1 The functions bs, σ, f, l, φ, h, and γ are twice continuously differentiable with respect to x,y,x˜,y˜. Moreover, all their derivatives with respect to x,y,x˜,y˜ up to second-order are continuous in x,y,x˜,y˜,u and bounded.H2 The functions b, σ, f, l, φ, h, and γ grow linearly with respect to x˜,y˜,z˜,x,y,z,u and are continuous in t,u, respectively.H3 For any Λ,u∈U, F⋅,Λ,u∈ℋℱ20,T;R3, there exists a constant C>0 such that(8)Ft,Λ1,u−Ft,Λ2,u≤CΛ1−Λ2,∀Λj=xj,yj,zj,x˜j,y˜j,z˜j,j=1,2.H4 (monotonic conditions). E¯Ft,Λ,u−Ft,Λ1,u,Θ−Θ1≤−β1E¯Θ−Θ12,P−a.s., ΥΞ−ΥΞ˜,Ξ−Ξ˜≥μ1Ξ−Ξ˜2, H4′E¯Ft,Λ,u−Ft,Λ1,u,Θ−Θ1≥β2E¯Θ−Θ12,P−a.s., ΥΞ−ΥΞ˜,Ξ−Ξ˜≤−μ2Ξ−Ξ˜2, ∀Θ=x,y,z,Θ1=x1,y1,z1 where β1, μ1 and β2, μ2 are given nonnegative constants.Remark 2.
Under the assumptionsH1−H4 or H1−H4′, via Theorem 1 in [20], the mean-field system (3) admits a unique adapted solution Xt,Yt,Ztt∈0,T∈Sℱ20,T;R×Sℱ20,T;R×ℋℱ20,T;R.
In fact, due to the mixed initial-terminal conditions in the state equation, even if the well-posedness of the state equation is ensured via the Lyapunov operator introduced in [21], the well-posedness of the first-order adjoint equation seems to be not guaranteed. On the other hand, since both Z and u appear in the coefficients of the backward equation (3), the regularity of process Z (as a part of the state process) is not enough to obtain a second-order adjoint equation. Thus, the classical method cannot be directly used. To overcome this difficulty, we introduce a reduction method inspired by the study of optimality variational principle for controlled FBSDEs [18]. First, a variation of problem A is formulated as follows.Problem 2.
ℬ Minimize(9)JX0,Y0,Z,u=E∫0TE′lt,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdt+EφXT,Y0,subject to(10)dXt=E′bt,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdt+E′σt,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdBt,−dYt=E′ft,Xt′,Yt′,Zt′,Xt,Yt,Zt,utdt−ZtdBt,X0=X0,Y0=Y0.
OverX0,Y0,Z,u∈ℛ:=R×R×ℋℱ20,T;R×U with the mixed initial-terminal state constraints,(11)X0=γXT,Y0,YT=hXT,Y0.Remark 3.
Via Theorem 2 in [22], under the assumptions H1−H2, the mean-field forward stochastic system (10) admits a unique solution Xt,Yt,Ztt∈0,T∈Sℱ20,T;R×Sℱ20,T;R×ℋℱ20,T;R.
It is remarkable that problemA is embedded into problem ℬ. Hence, if X0¯,Y0¯,Z¯,u¯ is the optimal control of ℬ, then u¯ is optimal for A. In the following section, the classical second-order variational technique will be adopted to solve problem ℬ.
Before presenting the main results, we introduce the extended Ekeland’s variational principle, which plays an important role in obtaining the stochastic maximum principle.Theorem 1.
(extended Ekeland’s variational principle [23]). Let K be a nonempty closed subset of a complete metric space S,d and G:K×K⟶R be a bifunction. Assume that ε>0 and the following assumptions are satisfied:(1)
i For all x∈K, L:=y∈K:Gx,y+εdx,y≤0 is closed(2)
iiGx,x=0, for all x∈K(3)
iiiGx,y≤Gx,z+Gz,y, for all x,y,z∈K
Ifinfy∈KGx0,y>−∞ for some x0∈K, then there exists x¯∈K such that(12)Gx0,x¯+εdx0,x¯≤0,Gx¯,x+εdx¯,x>0,for allx∈K,x≠x¯.


## 3. Stochastic Maximum Principle

In this section, we will try to solve problemℬ. Firstly, suppose X¯0,Y¯0,Z¯,u¯ is an optimal control of ℬ, with the corresponding optimal state process X¯,Y¯. Without loss of generality, we assume that JX¯0,Y¯0,Z¯,u¯=0. For any δ>0 and X0,Y0,Z,u∈ℛ, the penalty functional is defined as(13)JδX0,Y0,Z,u=JX0,Y0,Z,u+δ2+EX0−γXT,Y02+YT−hXT,X021/2.Endow the setU and ℋℱ20,T;R with the metric d,(14)dv,v˜=E¯t∈0,T|vt≠v˜t,∀v,v˜∈Uorℋℱ20,T;R.where ⋅ denotes the Lebesgue measure. Then, it is easy to verify as in [24] that ℛ is a complete metric space with the metric d˜:(15)d˜=X0−X˜02+Y0−Y˜02+dZ,Z˜2+du,u˜21/2,∀X0,Y0,Z,u,X˜0,Y˜0,Z˜,u˜∈ℛ.Clearly, (13) subject to (10) is a forward stochastic control problem without the state constraints. However, we have to notice that the unboundedness of ℛ cannot assure lower semicontinuity of JδX0,Y0,Z,u in ℛ,d˜. So, Ekeland’s variational principle cannot be directly used. Fortunately, extended Ekeland’s variational principle can reduce the requirement of lower semicontinuity of JδX0,Y0,Z,u. Hence, Theorem 1 can be adopted to discuss this issue. First, we consider the case that X0, Y0, and Z take values in ℳ⊂R, ℳ⊂R, and N⊂R, and ℳ is convex. Moreover, ℳ and N are all closed.SetFς,τ=Jδτ−Jδς with ς=X˜0,Y˜0,Z˜,u˜∈ℛ1=ℳ×ℳ×ℋℱ20,T;N×U and τ=X0,Y0,Z,u∈ℛ1. Then, Fς,τ satisfies the conditions of Theorem 1, the demonstration process is similar as the appendix in [25], and thus the detail is omitted. Hence, there exists X0δ,Y0δ,Zδ,uδ∈ℛ1 such that(16)JδX0δ,Y0δ,Zδ,uδ−JδX¯0,Y¯0,Z¯,u¯+δd˜X0δ,Y0δ,Zδ,uδ,X¯0,Y¯0,Z¯,u¯≤0,JδX0,Y0,Z,u−JδX0δ,Y0δ,Zδ,uδ+δd˜X0δ,Y0δ,Zδ,uδ,X0,Y0,Z,u≥0.Note that,X0¯,Y0¯,Z¯,u¯ is the optimal control variables for problem ℬ; they satisfy the optimal state constraints at time T. Thus, from the definition of penalty functional (13), we can derive JδX¯0,Y¯0,Z¯,u¯=δ. Further, we can get(17)JδX0δ,Y0δ,Zδ,uδ≤JδX¯0,Y¯0,Z¯,u¯=δ,d˜X0δ,Y0δ,Zδ,uδ,X¯0,Y¯0,Z¯,u¯≤δ,JδX0δ,Y0δ,Zδ,uδ≤JδX0,Y0,Z,u+δd˜X0δ,Y0δ,Zδ,uδ,X0,Y0,Z,u.Then, (17) means that the control process X0δ,Y0δ,Zδ,uδ is a global minimum point of the following penalized cost functional:(18)JδX0,Y0,Z,u+δd˜X0δ,Y0δ,Zδ,uδ,X0,Y0,Z,u.SinceZ and u are dynamic, spike technique for Zδ and uδ is employed. For sufficiently small ε>0, define(19)μtδ,ε=μtδ,t∈0,T\Sε,μt,t∈Sε.For someSε⊆0,T with Sε=εT, where ⋅ also denotes the Lebesgue measure, μ=Z,u. For X0 and Y0, since they are independent of the time variable and range in a convex control domain, we adopt the convex perturbation for X0δ and Y0δ. Let X0,Y0∈ℳ be control variables such that X0δ+X0∈ℳ and Y0δ+Y0∈ℳ. Then, for any ε∈0,1, X0δ,ε=X0δ+εX0 and Y0δ,ε=Y0δ+εY0 are also control variables with values in ℳ.For the sake of convenience, we denote(20)X=XY,v=Z′Zu,X0=X0Y0,XT=XTYT,Bt,X′,X,v=bt,Xt′,Yt′,Zt′,Xt,Yt,Zt,ut−ft,Xt′,Yt′,Zt′,Xt,Yt,Zt,ut,Σt,X′,X,v=σt,Xt′,Yt′,Zt′,Xt,Yt,Zt,utz,ΓX0,XT=X0−γXT,Y0YT−hXT,Y0,FX0,XT=φXT,Y0.These notations also work for their corresponding derivatives.SupposeXδ,ε and Xδ are solutions of (10) corresponding to X0δ,ε,vδ,ε and X0δ,vδ, respectively. We introduce the first-order and second-order variational equations.(21)dX1,tδ,ε=E′BX˜X1,tδ,ε+BXX1,tδ,ε+ΔBISε,tdt+E′ΣX˜X1,tδ,ε+ΣXX1,tδ,ε+ΔΣISε,tdBt,X1,0δ,ε=εX0.(22)dX2,tδ,ε=E′BX˜X2,tδ,ε′+BXX2,tδ,ε+ΔBX˜X1,tδ,ε′+ΔBXX1,tδ,εISε,t+12BX˜X˜X1,tδ,ε′2+12BXXX1,tδ,ε2dt+E′∑X˜X2,tδ,ε′+∑XX2,tδ,ε+Δ∑X˜X1,tδ,ε′+Δ∑XX1,tδ,εISε,t+12∑X˜X˜X1,tδ,ε′2+12∑XXX1,tδ,ε2dBt,X2,0δ,ε=0,where ISε denotes the indicator function of a set Sε. It is easy to verify that (21) and (22) admit unique solutions, and these solutions satisfy the following prior estimates.Lemma 1.
LetH1−H4 or H1−H4′ hold, then(23)Esupt∈0,TX1,tδ,ε2k+Esupt∈0,TXtδ,ε−Xtδ2k≤Cεk,Esupt∈0,TXtδ,ε−Xtδ−X1,tδ,ε2k+Esupt∈0,TX2,tδ,ε2k≤Cε2k,Esupt∈0,TXtδ,ε−Xtδ−X1,tδ,ε−X2,tδ,ε2k≤ε2kρkε,where ρk:0,∞⟶0,∞ such that limε⟶0ρkε=0.Proof 1.
The detail refers to Lemma1 in [6].Theorem 2.
SupposeH1−H4 or H1−H4′ hold. X¯0,Y¯0,Z¯,u¯ is an optimal control for ℬ with the corresponding optimal trajectory X¯,Y¯. Then, there exist three nonnegative parameters θ0, θ¯0, and θ¯T with θ02+Eθ¯02+Eθ¯T2=1 and an adapted solution ξ,η,ζ∈M20,T of the following FBSDE:(24)dξt=−E′bxξt+σxζt−fxηt+bx˜ξt′+σx˜ξt′−fx˜ηt′dt+ζtdBt,−dηt=E′byξt+σyζt−fyηt+by˜ξt′+σy˜ξt′−fy˜ηt′dt−ζ˜tdBt,ξT=θ0φXT−θ¯0γXT−θ¯ThXT,ηT=θ¯T,such that(25)P0≥0,0,0,θ¯0γY0Y0+θ¯ThY0Y0−θ0φY0Y0.,and(26)E¯Ht,X¯t,X¯′t,vt,Φt,Ψt,θt−Ht,X¯t,X¯′t,v¯t,Φt,Ψt,θt,+12△ΣtTP△Σ≥0.∀X0∈ℳ,Y0∈ℳ,Z∈N,u∈U,a.e.,a.s..where H is the Hamiltonian function defined by(27)Ht,X,X˜,v,Φ,Ψ,θ=Φ,Bt,X,X˜,v+Ψ,Σt,X,X˜,v+θ,lt,X,X˜,v.
AndP,Q∈Sℱ20,T;R2×2×ℋℱ20,T;R2×2 is the unique adapted solution of the following BSDE:(28)dPt=−E′BXTPt+PtBX+BX˜TPt′+Pt′BX˜+ΣXTPtΣXδ+ΣX˜TPt′ΣX˜δ+ΣXTQt+QtΣX+ΣX˜TQt′+Qt′ΣX˜+HXX+HX˜X˜dt+QtdBt,PT=θ0FXTXT+ΓXTXTθ.Proof 2.
This state constrained stochastic maximum principle will be proved into two steps.Step 1.
LetX0,Y0, and Z take values in ℳ⊂R, ℳ⊂R, and N⊂R, respectively, and ℳ be convex. Moreover, ℳ and N are all closed.
From (17) and the definition of d˜, we have(29)JδX0δ,ε,Y0δ,ε,Zδ,ε,uδ,ε−JδX0δ,Y0δ,Zδ,uδ+δd˜X0δ,Y0δ,Zδ,uδ,X0δ,ε,Y0δ,ε,Zδ,ε,uδ,ε=JδX0δ,ε,vδ,ε−JδX0δ,vδ+δd˜X0δ,vδ,X0δ,ε,vδ,ε=JδX0δ,ε,vδ,ε2−JδX0δ,vδ2JδX0δ,ε,vδ,ε+JδX0δ,vδ+εδ2T2+X02+Y02≥0.
By using Taylor expansion with addition of Lemma1, we deduce(30)JδX0δ,ε,vδ,ε2−JδX0δ,vδ2JδX0δ,ε,vδ,ε+JδX0δ,vδ=θ0δ,εJX0δ,ε,vδ,ε−JX0δ,vδ+Eθ¯δ,ε,ΓX0δ,ε,XTδ,ε−ΓX0δ,XTδ·θ0δ+o1JX0δ,ε,vδ,ε−JX0δ,vδ+Eθ¯δ+o1,ΓX0δ,ε,XTδ,ε−ΓX0δ,XTδ,where o1 stands for certain vectors that tend to 0 as ε⟶0 and , stands for inner product of vectors. Here, θ¯δ,ε=θ¯0δ,ε,θ¯Tδ,ε, θ¯δ=θ¯0δ,θ¯Tδ, and(31)θ0δ,ε=2∫01λJX0δ,ε,vδ,ε+1−λJX0δ,vδ+δdλJδX0δ,ε,vδ,ε+JδX0δ,vδ,θ¯0δ,ε=X0δ,ε−γXTδ,ε,Y0δ,ε+X0δ−γXTδ,Y0δJδX0δ,ε,vδ,ε+JδX0δ,vδ,θ¯Tδ,ε=YTδ,ε−hXTδ,ε,Y0δ,ε+YTδ−hXTδ,Y0δJδX0δ,ε,vδ,ε+JδX0δ,vδ,θ0δ=JX0δ,vδ+δJδX0δ,vδ,θ¯0δ=X0δ−γXTδ,Y0δJδX0δ,vδ,θ¯Tδ=YTδ−hXTδ,Y0δJδX0δ,vδ.
Then,θ0δ≥0 and θ0δ2+Eθ¯0δ2+Eθ¯Tδ2=1. Thus, there exists a subsequence still denoted by θ0δ,θ¯0δ,θ¯Tδ convergent, i.e.,(32)limδ⟶0θ0δ,θ¯0δ,θ¯Tδ=θ0,θ¯0,θ¯T.
We claim thatθ0≠0. The detailed illustration of this point refers to [19]. Here, θ0,θ¯0,θ¯T is called the Lagrange multiplier of the corresponding optimal 4-tuple X¯0,Y¯0,Z¯,u¯. While, by Taylor expansion,(33)JX0δ,ε,vδ,ε−JX0δ,vδ=E∫0TE′lt,Xtδ,ε,Xtδ,ε′,vtδ,ε−lt,Xtδ,Xtδ′,vtδdt+FX0δ,ε,XTδ,ε−FX0δ,XTδ=E∫0TE′lX˜δ,X1,tδ,ε+X2,tδ,ε′+lXδ,X1,tδ,ε+X2,tδ,ε+12lX˜X˜δX1,tδ,ε′,X1,tδ,ε′+lXXδX1,tδ,ε,X1,tδ,ε+ΔlδISε,tdt+EFX0δ,X0δ,ε−X0δ+FXTδ,XTδ,ε−XTδ+12FX0X0δX0δ,ε−X0δ,X0δ,ε−X0δ+FXTXTδXTδ,ε−XTδ,XTδ,ε−XTδ+2FXTX0δX0δ,ε−X0δ,XTδ,ε−XTδ+oε=E∫0TE′lX˜δ,X1,tδ,ε+X2,tδ,ε′+lXδ,X1,tδ,ε+X2,tδ,ε+12lX˜X˜δX1,tδ,ε′,X1,tδ,ε′+lXXδX1,tδ,ε,X1,tδ,ε+ΔlδISε,tdt+EFX0δ,εX0+FXTδ,X1,Tδ+X2,Tδ+12FX0X0δεX0,εX0+FXTXTδX1,Tδ,X1,Tδ+2FXTX0δεX0,X1,Tδ+oε,where(34)liδ=lit,Xtδ,Xtδ′,vtδ,lijδ=lijt,Xtδ,Xtδ′,vtδ,Fjδ=FjX0δ,XTδ,Fijδ=FijX0δ,XTδ,i,j=X,X˜.
Similarly,(35)Eθ¯δ,ε,ΓX0δ,ε,XTδ,ε−ΓX0δ,XTδ=EΓX0δθ¯δ,ε,X0δ,ε−X0δ+ΓXTδθ¯δ,ε,XTδ,ε−XTδ+12ΓX0X0δθ¯δ,εX0δ,ε−X0δ,X0δ,ε−X0δ+ΓXTXTδθ¯δ,εXTδ,ε−XTδ,XTδ,ε−XTδ+2ΓXTX0δθ¯δ,εX0δ,ε−X0δ,XTδ,ε−XTδ+oε=EΓX0δθ¯δ,ε,εX0+ΓXTδθ¯δ,ε,X1,Tδ+X2,Tδ+12ΓX0X0δθ¯δ,εεX0,εX0+ΓXTXTδθ¯δ,εX1,Tδ,X1,Tδ+2ΓXTX0δθ¯δ,εεX0,X1,Tδ+oε,where Γjδ=ΓjX0δ,XTδandΓijδ=ΓijX0δ,XTδ,i,j=X0,XT.
Consequently, from29−35, the variational inequality is derived as(36)−εδ2T2+|X0|2+|Y0|2≤θ0δ,εJX0δ,ε,vδ,ε−JX0δ,vδ+Eθ¯δ,ε,ΓX0δ,ε,XTδ,ε−ΓX0δ,XTδ=θ0δ,εE∫0TE′lX˜δ,X1,tδ,ε+X2,tδ,ε′+lXδ,X1,tδ,ε+X2,tδ,ε+12lX˜X˜δX1,tδ,ε′,X1,tδ,ε′+lXXδX1,tδ,ε,X1,tδ,ε+ΔlδISε,tdt+Eθ0δ,εFX0δ+ΓX0δθ¯δ,ε,εX0+θ0δ,εFXTδ+ΓXTδθ¯δ,ε,X1,Tδ+X2,Tδ+12θ0δ,εFX0X0δ+ΓX0X0δθ¯δ,εεX0,εX0+θ0δ,εFXTXTδ+ΓXTXTδθ¯δ,εX1,Tδ,X1,Tδ+2θ0δ,εFXTX0δ+ΓXTX0δθ¯δ,εεX0,X1,Tδ+oε.
To get rid ofX2δ in the above inequality, we introduce the first-order adjoint equations as(37)dΦtδ,ε=−E′Φtδ,εBXδ+Φtδ,ε′BX˜δ+Ψtδ,εΣXδ+Ψtδ,ε′ΣX˜δdt+Ψtδ,εdBt,ΦTδ,ε=θ0δ,εFXTδ+ΓXTδθ¯δ,ε.
Applying Itô’s formula toΦtδ,ε,X1,tδ,ε+X2,tδ,ε fulfills(38)Eθ0δ,εFXTδ+ΓXTδθ¯δ,ε,X1,Tδ,ε+X2,Tδ,ε=EΦ0δ,ε,εX0+∫0TE′Φtδ,ε,ΔBX˜X1,tδ,ε′+ΔBXX1,tδ,ε+ΔBISε,t+12BX˜X˜X1,tδ,ε′2+12BXXX1,tδ,ε2+Ψtδ,ε,ΔΣX˜X1,tδ,ε′+ΔΣXX1,tδ,ε+ΔΣISε,t+12ΣX˜X˜X1,tδ,ε′2+12ΣXXX1,tδ,ε2dt.
Combining (36) and (38), again with the help of Lemma 1, the variational inequality (36) can be changed as(39)−εδ2T2+X02+Y02≤E∫0TE′Φtδ,ε,△Bδ+Ψtδ,ε,△Σδ+θ0δ,ε,△lδISε,t+12HXXδ,εX1,tδ,ε,X1,tδ,ε+12HX˜X˜δ,εX1,tδ,ε′,X1,tδ,ε′dt+Eθ0δ,εFX0δ+ΓX0δθ¯δ,ε+Φ0δ,ε,εX0+12θ0δ,εFX0X0δ+ΓX0X0δθ¯δ,εεX0,εX0+θ0δ,εFXTXTδ+ΓXTXTδθ¯δ,εX1,Tδ,X1,Tδ+2θ0δ,εFXTX0δ+ΓXTX0δθ¯δ,εεX0,X1,Tδ+oε,with Hiiδ,ε=Hiit,Xtδ,X˜tδ,vtδ,Φtδ,ε,Ψtδ,ε,θtδ,ε being the Hessian of(40)Ht,X,X˜,v,Φ,Ψ,θ=Φ,Bt,X,X˜,v+Ψ,Σt,X,X˜,v+θ,lt,X,X˜,v.
We see that in (39), there left some second-order terms in X1δ,ε. Hence, we want to further get rid of them. Firstly, the equations satisfied by Θδ,ε=X1δ,εX1δ,εT are presented by appealing to Itô’s formula.(41)dΘtδ,ε=E′BXδΘtδ,ε+Θtδ,εBXδT+BX˜δΘtδ,ε′+Θtδ,ε′BX˜δT+ΣXδΘtδ,εΣXδT+ΣX˜˜δΘtδ,ε′TΣX˜δT+Λ1,tδ,ε+△Σδ△ΣδT+Λ2,tδ,εISε,tdt+E′ΣXδΘtδ,ε+Θtδ,εΣXδT+ΣX˜δΘtδ,ε′+Θtδ,ε′ΣX˜δT+Λ3,tδ,εdBt,Θ0δ,ε=ε2X0X0T,where(42)Λ1,tδ,ε=ΣXδX1,tδ,εX1,tδ,ε′TΣX˜δT+ΣX˜δX1,tδ,ε′X1,tδ,εTΣXδT,Λ2,tδ,ε=ΣXδX1,tδ,ε△ΣδT+△ΣδX1,tδ,εT△ΣδT+X1,tδ,εΔBδT+ΣX˜δX1,tδ,ε′△ΣδT+△ΣδX1,tδ,ε′TΣX˜δT,Λ3,tδ,ε=X1,tδ,εX1,tδ,ε′TΣX˜δT+ΣX˜δX1,tδ,ε′X1,tδ,εT+X1,tδ,ε△ΣδT+△ΣδX1,tδ,εTISε,t.
Secondly, we introduce the second-order adjoint equations corresponding touδ,ε, respectively, to remove X1δX1δT in (39). Now, let Pδ,ε,Qδ,ε be the adapted solution of the following BSDE:(43)dPtδ,ε=−E′BXδTPtδ,ε+Ptδ,εBXδ+BXδTPtδ,ε′+Ptδ,ε′BX˜δ+ΣXδTPtδ,εΣXδ+ΣX˜δTPtδ,ε′ΣX˜δ+ΣXδTQtδ,ε+Qtδ,εΣXδ+ΣX˜δTQtδ,ε′+Qtδ,ε′ΣX˜δ+HXXδ,ε+HX˜X˜δ,εdt+Qtδ,εdBt,PTδ,ε=θ0δ,εFXTXTδ+ΓXTXTδθδ,ε.
Then, by Lemma1 and applying Itô’s formula to Ptδ,ε,Θtδ,ε yield(44)Eθ0δ,εFXTXTδ+ΓXTXTδθδ,εX1,Tδ,ε,X1,Tδ,ε−ε2P0δ,εX0,X0=E∫0TE′△ΣδTPtδ,ε△ΣδISε,t−HXXδ,εX1,tδ,ε,X1,tδ,ε−HX˜X˜δ,εX1,tδ,ε′,X1,tδ,ε′dt+oε.
Inserting (44) into (39), we can derive(45)−εδ2T2+X02+Y02≤E∫0TE′Ht,Xtδ,Xδt′,vtδ,ε,Φtδ,ε,Ψtδ,ε,θtδ,ε−Ht,Xtδ,Xδt′,vtδ,Φtδ,ε,Ψtδ,ε,θtδ,ε+12△ΣδTPtδ,ε△ΣδISε,tdt+Eθ0δ,εFX0δ+ΓX0δθ¯δ,ε+Φ0δ,ε,εX0+12θ0δ,εFX0X0δ+ΓX0X0δθ¯δ,ε+P0δ,εεX0,εX0+2θ0δ,εFXTX0δ+ΓXTX0δθ¯δ,εεX0,X1,Tδ+oε,where(46)FX0=0,φY0,FXT=φXT,0,FX0X0=000φY0Y0,FXTX0=0φXTY000,FX0XT=00φY0XT0,FXTXT=φXTXT000.
By the Frechet differentiability, forΓ, we have(47)ΓX0,XT,θ¯=X0−γXT,Y0θ¯0+YT−hXT,Y0θ¯T.
Hence,(48)ΓX0θ¯=ΓX0,XT,θ¯X0=θ¯0,−θ¯0γY0−θ¯ThY0,ΓXTθ¯=ΓX0,XT,θ¯XT=−θ¯0γXT−θ¯ThXT,θ¯T,ΓX0X0θ¯=ΓX0,XT,θ¯X0X0=000−θ¯0γY0Y0−θ¯ThY0Y0,ΓX0XTθ¯=ΓX0,XT,θ¯X0XT=00−θ¯0γY0XT−θ¯ThY0XT0,ΓXTX0θ¯=ΓX0,XT,θ¯XTX0=0−θ¯0γXTY0−θ¯ThXTY000,ΓXTXTθ¯=ΓX0,XT,θ¯XTXT=−θ¯0γXTXT−θ¯ThXTXT000.
Note thatlimε⟶0θ¯δ,ε2+Eθ¯δ,ε2=1; there exists a subsequence of θ0δ,ε,θ¯δ,ε still denoted by θ0δ,ε,θ¯δ,ε such that limε⟶0θ0δ,ε,θ¯δ,ε=θ0δ,θ¯δ,withθ0δ2+Eθ¯δ2=1. Consequently, by the continuous dependence of the solution of mean-field BSDEs on parameters, we have the following conclusions, as ε⟶0, Φδ,ε,Ψδ,ε⟶Φδ,Ψδ,Pδ,ε,Qδ,ε⟶Pδ,Qδinℋℱ20,T;R×R where Φδ,Ψδ and Pδ,Qδ are solutions of first-order and second-order adjoint equations corresponding to uδ, respectively. Hence, let 3.17×1/ε and send ε⟶0, δ⟶0, we see that(49)Eθ0FX0+ΓX0θ¯+Φ0,X0≥0,∀X0∈R2,which implies(50)Φ0=−θ0FX0−ΓX0θ¯.
Also, dividingε2X02 in (45) and sending X0⟶∞, ε⟶0,δ⟶0, we get(51)Eθ0FX0X0+ΓX0X0θ¯+P0≥0,twhich implies that the necessary condition (25) holds. Finally, by taking X0=0 in (45) and sending ε⟶0,δ⟶0, then using a standard argument, the variational inequality (26) follows.
Finally, let(52)Φ=ξ,η,Ψ=ζ,ζ˜.
Then, it follows from (37) and (50) that(53)ξ0=−θ¯0,η0=−θ0φY0+θ¯0γY0+θ¯ThY0,ξT=θ0φXT−θ¯0γXT−θ¯ThXT,ηT=θ¯T.
By use of (52), we may rewrite the first-order adjoint (37), further giving the one-dimensional first-order adjoint (24).Step 2.
The general case of control domains.
First, we set(54)ℳn=X0∈R2;X0≤X¯0+n,Nn=Zt∈R;Zt≤Z¯t+n,ℋℱ20,T,Nn=Zt∈ℋℱ20,T,R;Zt∈Nn,Un=u∈U|supt0,TE¯ut≤supt∈0,TE¯u¯t+n,n=1,2,3,…
It is easy to see thatℳn is convex and(55)X¯0∈ℳn⊆ℳn+1,R2=Un=1∞ℳn,u¯∈Un⊆Un+1,U=Un=1∞Un,Z¯t∈ℋℱ20,T,Nn⊆ℋℱ20,T,Nn+1,ℋℱ20,T,R=Un=1∞ℋℱ20,T,Nn.
Then, by using the similar methods in [25], the desired conclusion (26) can be drawn.
SinceX0,Y0,Z,u is arbitrary, we draw the desired conclusion and summarize it as follows.Theorem 3 (general maximum principle).
SupposeH1−H4 or H1−H4 hold. X¯0,Y¯0,Z¯,u¯ is an optimal tuple for problem A with the corresponding optimal trajectory X¯,Y¯. Then, there exist three nonnegative parameters θ0, θ¯0, and θ¯T with θ02+Eθ¯02+Eθ¯T2=1, such that for any X0∈R, Y0∈R, Z∈ℋℱ20,T,R, and u∈U, the necessary conditions (25) and (26) hold a.e. a.s., where H is the Hamiltonian function and Φt,Ψt,Pt,Qt,θt are solutions of (24) and (28) corresponding to u¯, respectively.Remark 4.
When the coefficientsb, σ, and f do not depend on w′, system (3) reduces to the standard one, then Theorem 3 reduces to the necessary conditions for classical fully coupled FBSDEs, and the corresponding results were obtained by [19]. So our result is nontrivially more general of [19].Remark 5.
Compared with reference [19], the extended Ekeland’s variational principle is firstly introduced to reduce the requirement of lower semicontinuity of Jδ. Secondly, a convergence technique is used to establish the stochastic maximum principle in general control domains. These methods are all different from the ones in [19].Remark 6.
For tackling the mixed initial-terminal conditions, a reduction method is adopted to transform the original mixed initial-terminal conditions as mixed initial-terminal state constraints. This treatment not only ensures the well-posedness of the first-order adjoint equations but also simplifies the derivation of the second-order adjoint equations.


## 4. Application

Consider the optimal control problem.Problem 3.
D Minimize Ju=1/2E∫0Tl1Zt2+l2ut2dt+φ1XT2+φ2Y02 over U, subject to(56)dXt=b1EXt+b2EYt+b3EZt+b4utdt+σ1EXt+σ2EYt+σ3EZt+σ4utdBt,−dYt=f1EXt+f2EYt+f3EZt+f4utdt−ZtdBt,X0=γ1XT+γ2Y0,YT=h1XT+h2Y0,where bi, σi, fi, i∈1,…,4 and φj, γj, hj, j∈1,2 are real constants of R. Moreover, l1>0, l2>0, φ1≥0, and φ2>0. In practice, the range of control variable is limited for some reasons such as U=−∞,−1∪1,+∞ (see [18]). Assume that(57)−2f1b1−f2σ1−f3b1−f22b2σ2+b3σ1−f3σ2+b32σ3=−f1−f2−f3b1b2b3σ1σ2σ3+−f1b1σ1−f2b2σ2−f3b3σ3,is negative definite matrix, and(58)2h1h2−γ1h2−γ1−2γ2,is positive definite matrix. Then, for any u∈U0,T, there exists a unique state process X,Y,Z. Now, suppose u¯ is optimal for problem D, then the adjoint equations are read as follows:(59)dξt=−b1Eξt+σ1Eζt−f1Eηtdt+ζtdBt,−dηt=b2Eξt+σ2Eζt−f2Eηtdt−ζ˜tdBt,ξT=θ0φ1XT−θ¯0γ1−θ¯Th1,ηT=θ¯T.
Clearly, (59) admits unique solution ξ,η,ζ by assumptions H1−H4. And the BSDE for P,Q becomes(60)dPt=−EBTPt+PtB+ΣTPtΣ+ΣTQt+QtΣdt+QtdBt,PT=2θ0φ1000,where B=b1b2−f1−f2 and Σ=σ1σ200. According to the existence and uniqueness of solutions,(61)P˙t=−EBTPt+PtB+ΣTPtΣ+ΣTQt+QtΣ,Q=0.
Hence, inequality (25) holds naturally. Then, let Z=Z¯ in the maximum condition (26); a candidate optimal control u¯ should satisfy(62)12θl2ut2+b4ξt−f4ηt+σ4ζtut+12σ42P1,tut2−σ42P1,tu¯tut≥12θl2u¯t2+b4ξt−f4ηt+σ4ζtu¯t−12σ42P1,tu¯t2,∀u∈U.
Then, we have(63)u¯t=μt,μt∈−∞,−1∪1,+∞,1,μt∈0,1,−1,μt∈−1,0,with μt=−b4ξt−f4ηt+σ4ζt/θl2.Remark 7.
From the above proof process, we note that when the coefficients of system (56) only depend on the expected values of the solution states, the second-order adjoint equation is mean-field independent and solvable.Proposition 1.
The controlu¯ defined by (63) is optimal, together with the corresponding trajectory X¯,Y¯,Z¯ which is an optimal solution for problem D.Proof 3.
SupposeX,Y,Z is the trajectory of system (56) controlled by u∈U. By the convexity of a function, we have(64)12φ1XT2−12φ1X¯T2≥φ1X¯TXT−X¯T,and(65)12φ2Y02−12φ2Y¯02≥φ2Y¯0Y0−Y¯0.
Applying Itô’s formula toξtXt−X¯t+ηtYt−Y¯t, one has(66)Eθ0φ1X¯TXT−X¯T+θ0φ2Y¯0Y0−Y¯0=E∫0Tb4ξt−f4ηt+σ4ζtut−u¯t+ζ˜tZt−Z¯tdt.
Taking notice of (64)–(66), (67)Ju−Ju¯≥12E∫0Tl1Zt2−Z¯t2+l2ut2−u¯t2dt+Eφ1X¯TXT−X¯T+φ2Y¯0Y0−Y¯0=12E∫0Tl1Zt2−Z¯t2+l2ut2−u¯t2dt+E∫0Tb4ξt−f4ηt+σ4ζtθ0ut−u¯t+ζ˜tθ0Zt−Z¯tdt=12E∫0Tl1Zt−Z¯t2+l2ut2−u¯t2dt+E∫0Tb4ξt−f4ηt+σ4ζtθ0ut−u¯tdt.
From (63), we know that(68)12l2ut2+b4ξt−f4ηt+σ4ζtθ0ut≥12l2u¯t2+b4ξt−f4ηt+σ4ζtθ0u¯t,holds for all u∈U. This means that Ju−Ju¯≥0, ∀u∈U, which yields the optimality of u¯.


## 5. Conclusion

This paper discussed optimal control problems of mean-field FBSDEs with mixed initial-terminal conditions. Firstly, we initially introduce extended Ekeland’s variational principle to reduce requirements of lower semicontinuity of the penalty functional. Secondly, the reduction method is adopted to guarantee the well-posedness of the first-order and second-order adjoint equations also with mixed initial-terminal conditions. Via spike variational technique, a general stochastic maximum principle is established in the framework of mean-field theory. Finally, a linear-quadratic example is worked out to illustrate the application of the results.



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