# Maximum Principle for Near-Optimality of Mean-Field FBSDEs

Authors: Ruijing Li; Chaozhu Hu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The present paper concerns with a near-optimal control problem for systems governed by mean-field forward-backward stochastic differential equations (FBSDEs) with mixed initial-terminal conditions. Utilizing Ekeland’s variational principle as well as the reduction method, the necessary and sufficient near-optimality conditions are established in the form of Pontryagin’s type. The results are obtained under restriction on the convexity of the control domain. As an application, a linear-quadratic stochastic control problem is solved explicitly.


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## Body

## 1. Introduction

Near-optimal control problems have attracted more attentions in recent years due to its distinct advantages, such as existence under minimal assumptions, availability in most practical cases, and convenience for implementation both analytically and numerically. The study of this theory can be traced back to Ekeland [1] and later greatly developed by Zhou [2–4] for deterministic and stochastic cases. Since then, many works have been devoted to the near-optimality of various stochastic control systems. Without being exhaustive, let us refer to [5–13] and the references therein.In 2015, Zhang et al. [14] investigated the near-optimality necessary conditions for classical linear FBSDEs, where the control domain was with nonconvexity. Via convergence technique as well as reduction method, they established the near-optimal maximum principle. Soon afterwards, under the same assumptions, Zhang [15] presented the near-optimal sufficient conditions for such classical linear FBSDEs. Especially, in 2018, by defining viscosity solution with perturbation factor to dispense the illusory differentiability condition of value function, Zhang and Zhou [16] established the necessary near-optimality conditions for stochastic recursive systems by virtue of dynamic programming principle. Another noteworthy thing is that, for recent years, some authors started research studies on near-optimal control problems for delay systems. For example, Zhang [17] first studied near-optimal control problems for linear stochastic delay systems. By anticipated backward stochastic differential equations method as well as maximum principle, necessary condition and sufficient verification theorem were provided. Then, also under restriction on convexity control domain, Wang and Wu [18] investigated near-optimal control problem for nonlinear stochastic delay systems. By Ekeland’s variational principle and corresponding moment estimations, they presented the sufficient as well as necessary near-optimality conditions. For more details, refer to [19, 20] and the references therein.However, to the best of our knowledge, few papers can be found in the literature on the near-optimality of mean-field backward stochastic differential equations (BSDEs). This new kind of mean-field BSDEs was first introduced by Buckdahn et al. [21], which were derived as a limit of some highly dimensional system of FBSDEs, corresponding to a large number of particles. It has been shown in Buckdahn et al. [22] that, such a mean-field BSDE described the viscosity solution of the associated nonlocal partial differential equations. Henceforth, many authors take into account of this system of McKean-Vlasov type (Lasry and Lions [23]) adapted for different frameworks, for example, Xu and Wu [24] presented a maximum principle for optimal control problems governed by backward stochastic partial differential equations of mean-field type, and for other related works, refer to [25–28].As we can see that all the above literature studies are about mean-field problems involving expectations as mean-field terms. In fact, there is another line dealing with mean-field problems, which involve large-population as mean-field terms to describe the impact of the population’s collective behaviors on all agents (Huang et al. [29]) such as the work of Huang [30] and Xu and Shi [31] as well as the work of Xu and Zhang [32] all concerned with general mean-field linear-quadratic-Gaussian (LQG) games of stochastic large-population systems; through the consistency condition, they derived the decentralized strategies and further verified the asymptotic near-optimality property (namely, ε-Nash equilibrium) of decentralized strategies for the LQ games. On the contrary, a relevant work of Hafayed and Abbas [8] dealing with near-optimal control problems has established necessary and sufficient conditions for mean-field singular stochastic systems in the case of controlled diffusion coefficient. Particularly, in the concluding section, it is pointed out that the establishment of necessary and sufficient near-optimal conditions for mean-field FBSDEs also remains an open problem. Motivated by this fact with the addition of above described mean-field theory application background in economics and finance, this paper is to discuss near-optimal control problems for mean-field FBSDEs, where the controlled state systems are with mixed initial-terminal conditions.The main contribution of this paper lies in the initial introduction of three first-order adjoint equations to eliminate the corresponding variational processes during dual analysis; another is rooted in the usage of reduction method to guarantee the well-posedness of the first-order adjoint equations with mixed initial-terminal conditions. Via classical convex variational technique and Ekeland’s variational principle, a necessary condition of Pontryagin’s type is derived. Then, under some additional assumptions, we prove that the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. It is remarkable that our results extend those of [5] essentially to the framework of mean-field theory.The rest of this paper is organized as follows. In Section2, we state some preliminaries and basic definitions. In Sections 3 and 4, we establish the main theorems and provide its detailed proof. In Section 5, an example of a linear-quadratic control problem is worked out to illustrate the theoretical applications. Finally, some concluding remarks are given in Section 6.


## 2. Preliminaries

LetΩ,ℱ,ℱtt≥0,P be a filtered probability space satisfying the usual condition, on which a one-dimensional standard Brownian motion Wtt≥0 is defined, ℱ=ℱs,0≤s≤T be the natural filtration generated by Wtt≥0 and augmented by all P-null sets, i.e.,(1)ℱs=σWr,r≤s∨Np,s∈0,T,where Np is the set of all P-null subsets. We now introduce some spaces of random variables and stochastic processes.Lℱ2Ω;R=ℱ−measurable random variableX:EX2<∞Sℱ20,T;R=ℱ−adapted and continuous processψ:Esupt∈0,Tψt2<∞ℋℱ20,T;R=ℱ−adapted processψ:E∫0Tψt2dt<∞M20,T≔Sℱ20,T;R×Sℱ20,T;R×ℋℱ20,T;RClearly,M20,T is a Banach space. Any process in M20,T is defined by Θ≔x,y,z with the norm(2)ΘM20,T≔Esupt∈0,Txt2+supt∈0,Tyt2+∫0Tzt2dt1/2.We study the near-optimal control problem of the following controlled mean-field FBSDEs having mixed initial-terminal conditions:(3)dxt=bt,xt,Ext,utdt+σt,xt,Ext,utdWt,−dyt=ft,xt,yt,zt,Ext,Eyt,Ezt,utdt−ztdWt,x0=γxT,y0,yT=hxT,y0,where b,σ:0,T×R2×U⟶R; f:0,T×R6×U⟶R; h,γ:R×R⟶R; and U is a given convex closed set of R. The cost functional to be minimized over the space U=Lℱ20,T;U of admissible controls takes the form(4)Ju=E∫0Tlt,xt,yt,zt,Ext,Eyt,Ezt,utdt+φxT,y0,with l:0,T×R6×U⟶R; φ:R×R⟶R.Definition 1 (see [4]).
Both a family of admissible pairsxε,yε,zε,uε parameterized by ε>0 and any element xε,yε,zε,uε in the family are called near-optimal if(5)Juε−infu∈UJu≤rε,holds for sufficiently small ε, where r is a function of ε satisfying rε⟶0 as ε⟶0. The estimate rε is called an error bound. If rε=Cεδ for some δ>0 independent of the constant C, then uε is called near-optimal with order εδ. Particularly, when rε=ε, uε is called ε-optimal. The near-optimal control problem under consideration in this paper is as follows.ProblemA.
Finduε∈U such that(6)Juε=infv∈UJv+ε.
Some notations and assumptions are presented before giving the well-posedness of system (3). We denote the norm by ⋅ of an Euclidean space.A1 The functions b, σ, f, and l are ℱ-progressively measurable in u, continuously differentiable in x,y,z,x˜,y˜,andz˜, and the derivatives of b, σ, f, and l with respect to x,y,z,x˜,y˜,andz˜ are bounded. Moreover, for some constant C>0,(7)1+x+x˜−1bt,x,x˜,u+1+x+x˜−1σt,x,x˜,u+1+x+y+z+x˜+y˜+z˜−1ft,x,y,z,x˜,y˜,z˜,u+1+x+y+z+x˜+y˜+z˜−1lt,x,y,z,x˜,y˜,z˜,u≤C.A2h, γ, and φ are continuously differentiable in xandy, and the derivatives of h, γ, and φ with respect to xandy are bounded. Moreover, for some constant C>0, ρ=h,γ,φ(8)1+x+y−1ρx,y≤C.A3 There is a constant C>0 and β∈0,1 such that(9)bxt,x,x˜,u−bxt,x′,x˜′,u+bx˜t,x,x˜,u−bx˜t,x′,x˜′,u+σxt,x,x˜,u−σxt,x′,x˜′,u+σx˜t,x,x˜,u−σx˜t,x′,x˜′,u≤Cx−x′β+x˜−x˜′β,fit,x,y,z,x˜,y˜,z˜,u−fit,x′,y′,z′,x˜′,y˜′,z˜′,u+lit,x,y,z,x˜,y˜,z˜,u−lit,x′,y′,z′,x˜′,y˜′,z˜′,u≤Cx−x′β+y−y′β+z−z′β+x˜−x˜′β+y˜−y˜′β+z˜−z˜′β.
Furthermore,(10)ρxx,y−ρxx′,y′+ρyx,y−ρyx′,y′≤Cx−x′β+y−y′β,where i=x,y,z,x˜,y˜,z˜ and ρ=h,γ,φ.Remark 1.
Under assumptionsA1−A3 via Theorem 2 in [33], the mean-field stochastic system (3) admits a unique adapted solution x,y,z∈M20,T.
In fact, due to the mixed initial-terminal conditions in the state equation, even if we have the well-posedness of the state equation via the Lyapunov operator introduced in [34], the well-posedness of the first-order adjoint equation seems to be not guaranteed. To overcome this difficulty, we introduce a reduction method inspired by the study of optimality variational principle for controlled FBSDEs with mixed initial-terminal conditions [35]. First, we pose the following problem.Problemℬ.
Findx0ε,y0ε,uε∈ℛ≔R×R×U such that(11)Jx0ε,y0ε,uε=infx0,y0,u∈UJx0,y0,u+ε,where x0,y0,u is subject to the forward control system:(12)dxt=bt,xt,Ext,utdt+σt,xt,Ext,utdWt,−dyt=ft,xt,yt,zt,Ext,Eyt,Ezt,utdt−ztdWt,x0=x0,y0=y0,with the mixed initial-terminal state constraints:(13)x0=γxT,y0,yT=hxT,y0.
It is remarkable that, for ProblemA, the mean-field system (3) has a unique solution x,y,z under A1−A3, which implies that y0 is unique and completely determinate. While, for Problem ℬ, y0 is arbitrary and viewed as a control variable. It just needs to satisfy the near-optimal state constraints at time T. So, Problem A is embedded into Problem ℬ. Hence, if the triple x0ε,y0ε,uε is the near-optimal control of ℬ, then uε is near-optimal for Problem A. In the following section, we will adopt the classical convex variational technique to solve Problem ℬ.


## 3. Necessary Condition of Near-Optimality

This section is devoted to the study of the main theorem. For simplicity, we denote(14)bj≔bjt,xt,Ext,ut,σj≔σjt,xt,Ext,ut,fj≔fjt,Γt,ut,lj≔ljt,Γt,ut,ρxT=ρxTxT,y0,ρy0=ρy0xT,y0,j∈x,y,z,x˜,y˜,z˜,Γt=xt,yt,zt,Ext,Eyt,Ezt,ρ=h,γ,φ.For anyu∈U and the corresponding state processes x,y,z, we define the first-order adjoint equation as(15)dξt=fyξt+ly+Efy˜ξt+ly˜dt+fzξt+lz+Efz˜ξt+lz˜dWt,−dηt=−fxξt+bxηt+σxζt−lxdt+E−fx˜ξt+bx˜ηt+σx˜ζt−lx˜dt−ζtdWt,ξ0=−θ0φy0−η0γy0+ξThy0,ηT=−ξThxT+η0γxT+θ0φxT.Remark 2.
Under assumptionsA1−A3, the adjoint equation (15) admits a unique adapted solution ξ,η,ζ∈M20,T. The well-posedness of the corresponding adjoint system will be provided in the derivation process of Theorem 1.
Define a metric onU by(16)du,u′=E∫0Tu−u′2dt1/2,∀u,u′∈U.
SinceU is closed, it can be shown that U,d is a complete metric space. Next, we will present some continuity of the state processes and adjoint processes with respect to the metric d.Lemma 1.
For any0<α<1 and 0<p≤2, there is a constant C=Cα,p>0 such that, for any u,u¯∈U along with the corresponding trajectories x,y,z and x¯,y¯,z¯, it follows that(17)Esupt∈0,Txt−x¯tp≤Cdu,u¯αp/2,supt∈0,TEyt−y¯tp+E∫0Tzt−z¯tpdt≤Cdu,u¯αp/2.Proof.
Applying the classical methods as Lemma 4 in [5] for dealing with mean-field FBSDEs, together with Burkholder–Davis–Gundy inequality and Gronwall’s inequality, we can logically obtain the estimates.Lemma 2.
LetA1−A3 hold, for any 0<α<1 and 1<p<2 satisfying 1+αβp<2, and there is a constant C=Cα,β,p>0 such that, for any u,u¯∈U, along with the corresponding trajectories x,y,z and x¯,y¯,z¯ and the solutions ξ,η,ζ and ξ¯,η¯,ζ¯ of the corresponding adjoint equation (15), it holds that(18)E∫0Tξt−ξ¯tpdt≤Cdu,u¯αβp/2,E∫0Tηt−η¯tp+ζt−ζ¯tpdt≤Cdu,u¯αβp/2.Proof.
Applying the classical methods as Lemma 5 in [5] for dealing with mean-field FBSDEs, we can naturally obtain the estimates.Theorem 1.
LetA1−A3 hold, for any ε>0, uε is an ε-optimal control of problem ℬ. Then, for any κ∈0,1/3, there exist three parameters θ0ε, θ¯0ε, and θ¯Tε with θ0ε2+Eθ¯0ε2+Eθ¯Tε2=1, and θ0ε≥0 holds that(19)−Cεκθ0ε≤E∫0Tηtεbt,xtε,Extε,ut−bt,xtε,Extε,utε+ζtεσt,xtε,Extε,ut−σt,xtε,Extε,utε−ξtεft,Γtε,ut−ft,Γtε,utε−lt,Γtε,ut−lt,Γtε,utεdt,∀u∈U,where ξε,ηε,ζε∈M20,T is the solution of the first-order adjoint equation (15) corresponding to uε.Proof.
Under the assumptionA2, it is easy to check that Jx0,y0,u is lower semicontinuous on ℛ≔R×R×U, which is a complete metric space under the following metric:(20)dℛΘ,Θ˜=x0−x˜02+y0−y˜02+du,u˜21/2,∀Θ=x0,y0,u,Θ˜=x˜0,y˜0,u˜∈ℛ.
By Ekeland’s variational principle [1], there exists an admissible control x˜0ε,y˜0ε,u˜ε∈ℛ such that(21)dℛx0ε,y0ε,uε,x˜0ε,y˜0ε,u˜ε≤ε2/3,J˜εx˜0ε,y˜0ε,u˜ε≤J˜εx0,y0,u,∀x0,y0,u∈ℛ,where(22)J˜εx0,y0,u=Jx0,y0,u+ε1/3dℛx0,y0,u,x˜0ε,y˜0ε,u˜ε.
It means thatx˜0ε,y˜0ε,u˜ε is optimal for system (12) with the new cost functional J˜εx0,y0,u. On the contrary, due to the mixed initial-terminal endpoint constraints in problem ℬ, we need to introduce the penalty functional to transform the original problem with endpoint constraints to the penalized optimal control problem with no endpoint constraints.
Letx˜0ε,y˜0ε,u˜ε be an optimal control of problem ℬ, with the corresponding optimal state process x˜ε,y˜ε,z˜ε. Without loss of generality, we assume that J˜εx˜0ε,y˜0ε,u˜ε=0. For any δ>0 and x0,y0,u∈ℛ, we define the penalty functional:(23)Jδ,εx0,y0,u=J˜εx0,y0,u+δ2+Ex0−γxT,y02+yT−hxT,y021/2.
Obviously,(24)Jδ,εx0,y0,u>0,Jδ,εx˜0ε,y˜0ε,u˜ε=δ≤infx0,y0,u∈ℛJδ,εx0,y0,u+δ.
By Ekeland’s variational principle, there exists a 3-tuplex0δ,ε,y0δ,ε,uδ,ε∈ℛ such that(25)Jδ,εx0δ,ε,y0δ,ε,uδ,ε≤Jδ,εx˜0ε,y˜0ε,u˜ε=δ,x0δ,ε−x˜0ε2+y0δ,ε−y˜0ε2+duδ,ε,u˜ε2≤δ,−δx0δ,ε−x02+y0δ,ε−y02+duδ,ε,u21/2≤Jδ,εx0,y0,u−Jδ,εx0δ,ε,y0δ,ε,uδ,ε,∀x0,y0,u∈ℛ.
Therefore,x0δ,ε,y0δ,ε,uδ,ε is optimal for system (13) with the new cost functional:(26)Jδ,εx0,y0,u+δx0δ,ε−x02+y0δ,ε−y02+duδ,ε,u21/2.
So far, we have transformed the original problem with endpoint constraints to the penalized optimal control problem with no endpoint constraints, and the optimal 3-tuplex0δ,ε,y0δ,ε,uδ,ε approaches x˜0ε,y˜0ε,u˜ε as δ⟶0. In the following, a convex perturbation is employed to obtain a maximum principle for x0δ,ε,y0δ,ε,uδ,ε. To this end, let Θ=x0,y0,u∈ℛ such that Θδ,ε+Θ=x0δ,ε+x0,y0δ,ε+y0,uδ,ε+u∈ℛ; then, for any ε>0,(27)Θtε,δ,ε=Θtδ,ε+εΘt∈ℛ,t∈0,T.
Letxε,δ,ε,yε,δ,ε,zε,δ,ε be the state processes corresponding to x0ε,δ,ε,y0ε,δ,ε,uε,δ,ε, and the processes x1ε,δ,ε,y1ε,δ,ε,z1ε,δ,ε be the solution of the following variational equations:(28)dx1,tε,δ,ε=bxδ,εx1,tε,δ,ε+bx˜δ,εEx1,tε,δ,ε+Δbδ,εISε,tdt+σxδ,εx1,tε,δ,ε+σx˜δ,εEx1,tε,δ,ε+Δσδ,εISε,tdWt,−dy1,tε,δ,ε=fxδ,εx1,tε,δ,ε+fyδ,εy1,tε,δ,ε+fzδ,εz1,tε,δ,ε+fx˜δ,εEx1,tε,δ,ε+fy˜δ,εEy1,tε,δ,ε+fz˜δ,εEz1,tε,δ,ε+Δfδ,εISε,tdt−z1,tε,δ,εdWt,x1,0ε,δ,ε=εx0,y1,0ε,δ,ε=εy0,where for simplicity of notations, we still use ρjδ,ε corresponding to xδ,ε,yδ,ε,zδ,ε,uδ,ε, ρ=b,σ,f. Then, we have the following estimates, whose proofs are similar to those given in [27](29)Esupt∈0,Tx1,tε,δ,ε2+Esupt∈0,Txtε,δ,ε−xtδ,ε2≤Cε,Esupt∈0,Txtε,δ,ε−xtδ,ε−x1,tε,δ,ε2≤Cε2,Esupt∈0,Ty1,tε,δ,ε2+Esupt∈0,Tytε,δ,ε−ytδ,ε2≤Cε,Esupt∈0,Tytε,δ,ε−ytδ,ε−y1,tε,δ,ε2≤Cε2,E∫0Tz1,tε,δ,ε2dt+E∫0Tztε,δ,ε−ztδ,ε2dt≤Cε,E∫0Tztε,δ,ε−ztδ,ε−z1,tε,δ,ε2dt≤Cε2.
Noting thatdutε,δ,ε,utδ,ε≤Cε, from the last relation in (25), we derive(30)−εδx02+y02+C≤Jδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−Jδ,εx0δ,ε,y0δ,ε,uδ,ε=Jδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε2−Jδ,εx0δ,ε,y0δ,ε,uδ,ε2Jδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+Jδ,εx0δ,ε,y0δ,ε,uδ,ε=J˜εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+δ2−J˜εx0δ,ε,y0δ,ε,uδ,ε+δ2Jδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+Jδ,εx0δ,ε,y0δ,ε,uδ,ε+Ex0ε,δ,ε−γxTε,δ,ε,y0ε,δ,ε2−x0δ,ε−γxTδ,ε,y0δ,ε2Jδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+Jδ,εx0δ,ε,y0δ,ε,uδ,ε+EyTε,δ,ε−hxTε,δ,ε,y0ε,δ,ε2−yTδ,ε−hxTδ,ε,y0δ,ε2Jδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+Jδ,εx0δ,ε,y0δ,ε,uδ,ε=θ0ε,δ,εJ˜εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−J˜εx0δ,ε,y0δ,ε,uδ,ε+Eθ¯0ε,δ,εx0ε,δ,ε−γxTε,δ,ε,y0ε,δ,ε−x0δ,ε−γxTδ,ε,y0δ,ε+Eθ¯Tε,δ,εyTε,δ,ε−hxTε,δ,ε,y0ε,δ,ε−yTδ,ε−hxTδ,ε,y0δ,ε=θ0δ,ε+o1J˜εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−J˜εx0δ,ε,y0δ,ε,uδ,ε+Eθ¯0δ,ε+o1x0ε,δ,ε−γxTε,δ,ε,y0ε,δ,ε−x0δ,ε−γxTδ,ε,y0δ,ε+Eθ¯Tδ,ε+o1yTε,δ,ε−hxTε,δ,ε,y0ε,δ,ε−yTδ,ε−hxTδ,ε,y0δ,ε,with(31)θ0ε,δ,ε=2∫01λJ˜εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+1−λJ˜εx0δ,ε,y0δ,ε,uδ,ε+δdλJδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+Jδ,εx0δ,ε,y0δ,ε,uδ,ε,θ¯0ε,δ,ε=x0ε,δ,ε−γxTε,δ,ε,y0ε,δ,ε+x0δ,ε−γxTδ,ε,y0δ,εJδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+Jδ,εx0δ,ε,y0δ,ε,uδ,ε,θTε,δ,ε=yTε,δ,ε−hxTε,δ,ε,y0ε,δ,ε+yTδ,ε−hxTδ,ε,y0δ,εJδ,εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+Jδ,εx0δ,ε,y0δ,ε,uδ,ε,and(32)θ0δ,ε=J˜εx0δ,ε,y0δ,ε,uδ,ε+δJδ,εx0δ,ε,y0δ,ε,uδ,ε,θ¯0δ,ε=x0δ,ε−γxTδ,ε,y0δ,εJδ,εx0δ,ε,y0δ,ε,uδ,ε,θTδ,ε=yTδ,ε−hxTδ,ε,y0δ,εJδ,εx0δ,ε,y0δ,ε,uδ,ε.
It is necessary to point out thatθ0δ,ε≥0, and θ0δ,ε2+Eθ¯0δ,ε2+EθTδ,ε2=1. Thus, there exists a subsequence still denoted by θ0δ,ε,θ¯0δ,ε,θTδ,ε convergent, i.e.,(33)limδ⟶0θ0δ,ε,θ¯0δ,ε,θTδ,ε=θ0ε,θ¯0ε,θTε.
We claim thatθ0ε≠0. The detailed illustration of this point refers to [35]. Here, θ0ε,θ¯0ε,θTε is called the Lagrange multiplier of the corresponding optimal 3-tuple x˜0ε,y˜0ε,u˜ε. On the contrary,(34)θ0ε,δ,εJ˜εx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−J˜εx0δ,ε,y0δ,ε,uδ,ε=θ0ε,δ,εJx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε+ε1/3dℛx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε,x˜0ε,y˜0ε,u˜ε−Jx0δ,ε,y0δ,ε,uδ,ε−ε1/3dℛx0δ,ε,y0δ,ε,uδ,ε,x˜0ε,y˜0ε,u˜ε=θ0ε,δ,εJx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−Jx0δ,ε,y0δ,ε,uδ,ε+ε1/3θ0ε,δ,εdℛx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε,x˜0ε,y˜0ε,u˜ε−dℛx0δ,ε,y0δ,ε,uδ,ε,x˜0ε,y˜0ε,u˜ε≤θ0ε,δ,εJx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−Jx0δ,ε,y0δ,ε,uδ,ε+ε1/3θ0ε,δ,εdℛx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε,x0δ,ε,y0δ,ε,uδ,ε≤θ0ε,δ,εJx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−Jx0δ,ε,y0δ,ε,uδ,ε+εε1/3θ0ε,δ,εx02+y02+C.
FromA3 and (29), we can deduce(35)Jx0ε,δ,ε,y0ε,δ,ε,uε,δ,ε−Jx0δ,ε,y0δ,ε,uδ,ε=E∫0Tlt,Γtε,δ,ε,utε,δ,ε−lt,Γtδ,ε,utδ,εdt+EφxTε,δ,ε,y0ε,δ,ε−φxTδ,ε,y0δ,ε=E∫0Tlt,Γtδ,ε,utε,δ,ε−lt,Γtδ,ε,utδ,εdt+E∫0T∫01lxε,δ,εxtε,δ,ε−xtδ,ε+lyε,δ,εytε,δ,ε−ytδ,ε+lzε,δ,εztε,δ,ε−ztδ,εdλdt+E∫0T∫01lx˜ε,δ,εExtε,δ,ε−Extδ,ε+ly˜ε,δ,εEytε,δ,ε−Eytδ,ε+lz˜ε,δ,εEztε,δ,ε−Eztδ,εdλdt+EφxTε,δ,ε,y0ε,δ,ε−φxTδ,ε,y0ε,δ,ε+φxTδ,ε,y0ε,δ,ε−φxTδ,ε,y0δ,ε=E∫0Tlt,Γtδ,ε,utε,δ,ε−lt,Γtδ,ε,utδ,ε+lxδ,εx1,tε,δ,ε+lx˜δ,εEx1,tε,δ,ε+lyδ,εy1,tε,δ,ε+ly˜δ,εEy1,tε,δ,ε+lzδ,εz1,tε,δ,ε+lz˜δ,εEz1,tε,δ,εdt+E∫0T∫01lxε,δ,ε−lxδ,εx1,tε,δ,ε+lyε,δ,ε−lyδ,εy1,tε,δ,ε+lzε,δ,ε−lzδ,εz1,tε,δ,ε+lx˜ε,δ,ε−lx˜δ,εEx1,tε,δ,ε+ly˜ε,δ,ε−ly˜δ,εEy1,tε,δ,ε+lz˜ε,δ,ε−lz˜δ,εEz1,tε,δ,εdλdt+E∫01φxTxTδ,ε+λxTε,δ,ε−xTδ,ε,y0ε,δ,εxTε,δ,ε−xTδ,εdλ+E∫01φy0xTδ,ε,y0δ,ε+λy0ε,δ,ε−y0δ,εy0ε,δ,ε−y0δ,εdλ=E∫0Tlt,Γtδ,ε,utε,δ,ε−lt,Γtδ,ε,utδ,ε+lxδ,εx1,tε,δ,ε+lx˜δ,εEx1,tε,δ,ε+lyδ,εy1,tε,δ,ε+ly˜δ,εEy1,tε,δ,ε+lzδ,εz1,tε,δ,ε+lz˜δ,εEz1,tε,δ,εdt+EφxTxTδ,ε,y0δ,εx1,Tε,δ,ε+φy0xTδ,ε,y0δ,εy1,0ε,δ,ε+E∫01φxTxTδ,ε+λxTε,δ,ε−xTδ,ε,y0ε,δ,ε−φxTxTδ,ε,y0δ,εx1,Tε,δ,εdλ+E∫01φy0xTδ,ε,y0δ,ε+λy0ε,δ,ε−y0δ,ε−φy0xTδ,ε,y0δ,εy1,0ε,δ,εdλ+oε=E∫0Tlt,Γtδ,ε,utε,δ,ε−lt,Γtδ,ε,utδ,ε+lxδ,εx1,tε,δ,ε+lx˜δ,εEx1,tε,δ,ε+lyδ,εy1,tε,δ,ε+ly˜δ,εEy1,tε,δ,ε+lzδ,εz1,tε,δ,ε+lz˜δ,εEz1,tε,δ,εdt+EφxTxTδ,ε,y0δ,εx1,Tε,δ,ε+φy0xTδ,ε,y0δ,εy1,0ε,δ,ε+oε,where ljε,δ,ε=ljt,Γtδ,ε+εΓtε,δ,ε−Γtδ,ε,utε,δ,ε, j=x,y,z,x˜,y˜,z˜. Similarly,(36)Eθ¯0ε,δ,εx0ε,δ,ε−γxTε,δ,ε,y0ε,δ,ε−x0δ,ε−γxTδ,ε,y0δ,ε=Eθ¯0ε,δ,εx1,0ε,δ,ε−γxTxTδ,ε,y0δ,εx1,Tε,δ,ε−γy0xTδ,ε,y0δ,εy1,0ε,δ,ε+oε,Eθ¯Tε,δ,εyTε,δ,ε−hxTε,δ,ε,y0ε,δ,ε−yTδ,ε−hxTδ,ε,y0δ,ε=Eθ¯Tε,δ,εy1,Tε,δ,ε−hxTxTδ,ε,y0δ,εx1,Tε,δ,ε−hy0xTδ,ε,y0δ,εy1,0ε,δ,ε+oε.
Then, taking notice of (30), we can further obtain(37)−εx02+y02+Cδ+ε1/3θ0ε,δ,ε≤θ0ε,δ,εE∫0Tlt,Γtδ,ε,utε,δ,ε−lt,Γtδ,ε,utδ,ε+lxδ,εx1,tε,δ,ε+lx˜δ,εEx1,tε,δ,ε+lyδ,εy1,tε,δ,ε+ly˜δ,εEy1,tε,δ,ε+lzδ,εz1,tε,δ,ε+lz˜δ,εEz1,tε,δ,εdt+θ0ε,δ,εEφxTxTδ,ε,y0δ,εx1,Tε,δ,ε+φy0xTδ,ε,y0δ,εy1,0ε,δ,ε+Eθ¯0ε,δ,εx1,0ε,δ,ε−γxTxTδ,ε,y0δ,εx1,Tε,δ,ε−γy0xTδ,ε,y0δ,εy1,0ε,δ,ε+Eθ¯Tε,δ,εy1,Tε,δ,ε−hxTxTδ,ε,y0δ,εx1,Tε,δ,ε−hy0xTδ,ε,y0δ,εy1,0ε,δ,ε+oε.
Let us introduce the following first-order BSDEs:(38)dξ˜tε,δ,ε=fyδ,εξ˜tε,δ,ε+lyδ,ε+Efy˜δ,εξ˜tε,δ,ε+ly˜δ,εdt+fzδ,εξ˜tε,δ,ε+lzδ,ε+Efz˜δ,εξ˜tε,δ,ε+lz˜δ,εdWt,−dη˜tε,δ,ε=−fxδ,εξ˜tε,δ,ε+bxδ,εη˜tε,δ,ε+σxδ,εζ˜tε,δ,ε−lxδ,ε+E−fx˜δ,εξ˜tε,δ,ε+bx˜δ,εη˜tε,δ,ε+σx˜δ,εζ˜tε,δ,ε−lx˜δ,εdt−ζ˜tε,δ,εdWt,ξ˜Tε,δ,ε=θ¯Tε,δ,ε,η˜Tε,δ,ε=−θ¯Tε,δ,εhxTδ,ε−θ¯0ε,δ,εγxTδ,ε+θ0ε,δ,εφxTδ,ε,where ρxTδ,ε=ρxTxTδ,ε,y0δ,ε, ρ=h,γ,φ. Applying Itô’s formula to ξ˜tε,δ,εy1,tε,δ,ε+η˜tε,δ,εx1,tε,δ,ε fulfills(39)Eθ¯Tε,δ,εy1,Tε,δ,ε−θ¯Tε,δ,εhxTδ,ε+θ¯0ε,δ,εγxTδ,ε−θ0ε,δ,εφxTδ,εx1,Tε,δ,ε−εEξ˜0ε,δ,εy0+η˜0ε,δ,εx0=E∫0Tlxδ,ε+Elx˜δ,εx1,tε,δ,ε+lyδ,ε+Ely˜δ,εy1,tε,δ,ε+lzδ,ε+Elz˜δ,εz1,tε,δ,ε+η˜tε,δ,εΔbδ,ε+ζ˜tε,δ,εΔσδ,ε−ξ˜tε,δ,εΔfδ,εISε,tdt.
Combining (37) and (39), yields(40)−εx02+y02+Cδ+ε1/3θ0ε,δ,ε≤εEθ¯0ε,δ,ε+η˜0ε,δ,εx0+θ0ε,δ,εφy0δ,ε−θ¯0ε,δ,εγy0δ,ε−θ¯Tε,δ,εhy0δ,ε+ξ˜0ε,δ,εy0+E∫0Tη˜tε,δ,εΔbδ,ε+ζ˜tε,δ,εΔσδ,ε−ξ˜tε,δ,εΔfδ,ε−Δlδ,εISε,tdt+oε.
To derive the first-order adjoint equation with mixed initial-terminal conditions, divideε in (40) and then send ε⟶0,δ⟶0, and we see that(41)Eθ¯0ε+η˜0εx0+θ0εφy0ε−θ¯0εγy0ε−θ¯Tεhy0ε+ξ˜0εy0≥0,∀x0,y0∈R×R,which implies(42)η˜0ε=−θ¯0ε,ξ˜0ε=−θ0εφy0ε−η˜0εγy0ε+ξ˜Tεhy0ε.
Meanwhile, by takingx0,y0=0,0 in (40), dividing this inequality by ε, and then sending ε⟶0, δ⟶0, the variational inequality follows:(43)−Cε1/3θ0ε≤E∫0Tη˜tεbt,x˜tε,Ex˜tε,ut−bt,x˜tε,Ex˜tε,u˜tε+ζ˜tεσt,x˜tε,Ex˜tε,ut−σt,x˜tε,Ex˜tε,u˜tε−ξ˜tεft,Γ˜tε,ut−ft,Γ˜tε,u˜tε−lt,Γ˜tε,ut−lt,Γ˜tε,u˜tεdt,where Γ˜tε=x˜tε,y˜tε,z˜tε,Ex˜tε,Ey˜tε,Ez˜tε. On the contrary, from (38) and (42), we can present the adjoint equation with mixed initial-terminal conditions as follows:(44)dξ˜tε=fyεξ˜tε+lyε+Efy˜εξ˜tε+ly˜εdt+fzεξ˜tε+lzε+Efz˜εξ˜tε+lz˜εdWt,−dη˜tε=−fxεξ˜tε+bxεη˜tε+σxεζ˜tε−lxε+E−fx˜εξ˜tε+bx˜εη˜tε+σx˜εζ˜tε−lx˜εdt−ζ˜tεdWt,ξ˜0ε=−θ0εφy0ε−η˜0εγy0ε+ξ˜Tεhy0ε,η˜Tε=−ξ˜TεhxTε+η˜0εγxTε+θ0εφxTε,whose well-posedness can be deduced directly virus the above derivative process (Remark 2). Now, consider (43) again but with x˜ε,y˜ε,z˜ε,u˜ε, etc., replaced by xε,yε,zε,uε, etc. We are about to derive an estimate for the term similar to the right side of (43) with respect to xε,yε,zε,uε, etc. To this end, we first estimate the following difference:(45)E∫0Tη˜tεbt,x˜tε,Ex˜tε,ut−bt,x˜tε,Ex˜tε,u˜tε−ηtεbt,xtε,Extε,ut−bt,xtε,Extε,utεdt=E∫0Tη˜tε−ηtεbt,x˜tε,Ex˜tε,ut−bt,x˜tε,Ex˜tε,u˜tεdt+E∫0Tηtεbt,x˜tε,Ex˜tε,ut−bt,xtε,Extε,utdt−E∫0Tηtεbt,x˜tε,Ex˜tε,u˜tε−bt,xtε,Extε,utεdt=Ξ1+Ξ2+Ξ3.
Due to Lemma2, for any κ∈0,1/3, by using the similar arguments as developed in [7] the proof of Theorem 1, we can also prove that(46)Ξ1=E∫0Tη˜tε−ηtεbt,x˜tε,Ex˜tε,ut−bt,x˜tε,Ex˜tε,u˜tεdt≤Cεκ,Ξ2=E∫0Tηtεbt,x˜tε,Ex˜tε,ut−bt,xtε,Extε,utdt≤Cεκ,Ξ3=E∫0Tηtεbt,x˜tε,Ex˜tε,u˜tε−bt,xtε,Extε,utεdt≤Cεκ,i.e.,(47)E∫0Tη˜tεbt,x˜tε,Ex˜tε,ut−bt,x˜tε,Ex˜tε,u˜tε−ηtεbt,xtε,Extε,ut−bt,xtε,Extε,utεdt≤Cεκ.
Similarly, via Lemma2, we also have(48)E∫0Tζ˜tεσt,x˜tε,Ex˜tε,ut−σt,x˜tε,Ex˜tε,u˜tε−ζtεσt,xtε,Extε,ut−σt,xtε,Extε,utε−ξ˜tεft,Γ˜tε,ut−ft,Γ˜tε,u˜tε+ξtεft,Γtε,ut−ft,Γtε,utε−lt,Γ˜tε,ut−lt,Γ˜tε,u˜tε+lt,Γtε,ut−lt,Γtε,utεdt≤Cεκ.
Therefore, the desired result (20) follows immediately by combining (43)–(48).
Sincex0,y0,u is arbitrary, we draw the desired conclusion and summarize it as follows.Theorem 2.
SupposeA1−A3 hold. For any ε>0, uε is an ε-optimal control of problem A. Then, for any κ∈0,1/3, there exist three nonnegative parameters θ0ε, θ¯0ε, and θ¯Tε with θ0ε2+Eθ¯0ε2+Eθ¯Tε2=1 and θ0ε≥0 such that, for any x0∈R, y0∈R, and u∈U, the necessary condition (20) holds a.e. a.s., where ξε,ηε,ζε is the solution of (15) corresponding to uε.
Define the HamiltonianHt,x,y,z,u,ξ,η,ζ by(49)Ht,x,y,z,u,ξ,η,ζ=−ξft,Γ,u+ηbt,x,Ex,u+ζσt,x,Ex,u−lt,Γ,u,then we have the following form of necessity conditions.Corollary 1.
Under the assumptions of Theorem2, it holds that(50)E∫0THt,xtε,ytε,ztε,utε,ξtε,ηtε,ζtεdt≥supu∈UE∫0THt,xtε,ytε,ztε,ut,ξtε,ηtε,ζtεdt−Cθ0εεκ.Proof.
According to the definition of the controluε,δ,ε, the point u∈U can be replaced by any admissible control u∈U, and the subsequent arguments still go through. Therefore, the conclusion in Theorem 2 holds for any u∈U, which is an easy variant of our corollary.Remark 3.
If the coefficients of system (3) do not depend on the expected values of the states, Theorem 2 reduces to the near-maximum condition for the classical system under convex control domain.Remark 4.
For exact optimality, the integral form and the pointwise form of the maximum condition is equivalent; however, it is not the case for near-optimality. We can only deduce the near maximum condition in an integral form.Remark 5.
Ifε=0, we can obtain a stochastic maximum principle for controlled mean-field FBSDEs with the control domain of convexity assumption.


## 4. Sufficient Condition of Near-Optimality

In this section, we will prove that the near-maximum condition of the HamiltonianH in the integral form is sufficient for near-optimality under some additional assumptions.A4 Let b, σ, f, and l be differentiable in u, and there exists a constant C>0 such that(51)bt,x,x˜,u1−bt,x,x˜,u2+but,x,x˜,u1−but,x,x˜,u2+σt,x,x˜,u1−σt,x,x˜,u2+σut,x,x˜,u1−σut,x,x˜,u2≤Cu1−u2,ft,Γ,u1−ft,Γ,u2+fut,Γ,u1−fut,Γ,u2+lt,Γ,u1−lt,Γ,u2+lut,Γ,u1−lut,Γ,u2≤Cu1−u2.Theorem 3.
LetA1−A4 hold, and assume that the Hamiltonian Ht,⋅,⋅,⋅,⋅,ξtε,ηtε,ζtε is concave for a.e. t∈0,T, P-a.s., h is concave, and φ and γ are convex. Let Λtε=ξε,ηε,ζε be the solution of the adjoint equation (15) associated with xε,yε,zε,uε. If, for any u∈U and some ε>0,(52)supu∈UE∫0THt,xtε,ytε,ztε,ut,Λtεdt≤E∫0THt,xtε,ytε,ztε,utε,Λtεdt+ε,holds, then we have(53)Juε≤infu∈UJu+Cε1/2,where C>0 is a constant independent of ε.Proof.
Fixε>0, for any u, v∈U, and define a new metric d¯ on U as follows:(54)d¯u,v=E∫0Tνtεut−vtdt,with νtε=1+ξtε+ηtε+ζtε. Obviously, d¯ is a complete metric on U as a weighted L1 norm. Define a functional J¯ on U by(55)J¯u=E∫0THt,xtε,ytε,ztε,ut,Λtεdt.
A simple calculation shows that(56)J¯u−J¯v≤CE∫0Tνtεut−vtdt.
Therefore,J¯ is continuous on U with respect to d¯. Then, by using (52) and Ekeland’s variational principle, there exists a u^ε∈U such that(57)d¯u^ε,uε≤ε1/2,and(58)E∫0TH˜t,xtε,ytε,ztε,u^tεdt=maxu∈UE∫0TH˜t,xtε,ytε,ztε,utdt,where(59)H˜t,x,y,z,u=Ht,x,y,z,u,Λ−ε1/2νtεu−u^ε.
The integral-form maximum condition (58) implies a pointwise maximum condition, that is, for a.e. t∈0,T and P-a.s.,(60)H˜t,xtε,ytε,ztε,u^tε=maxu∈UH˜t,xtε,ytε,ztε,ut.
Then, by Lemma 2.3 of Yong and Zhou [36], we have 0∈∂uH˜t,xtε,ytε,ztε,u^tε. By using (59) and the fact that the generalized gradient of the sum of two functions is contained in the sum of the generalized gradients of the two functions, we deduce(61)∂uH˜t,xtε,ytε,ztε,u^tε=∂uHt,xtε,ytε,ztε,u^tε,Λtε+−ε1/2νtε,ε1/2νtε.
Furthermore, sinceH is differentiable in u, there exists a ϑtε∈−ε1/2νtε,ε1/2νtε, such that(62)Hut,xtε,ytε,ztε,u^tε,Λtε=−ϑtε.
Consequently, byA4, we can prove that(63)Hut,xtε,ytε,ztε,utε,Λtε≤Hut,xtε,ytε,ztε,utε,Λtε−Hut,xtε,ytε,ztε,u^tε,Λtε+Hut,xtε,ytε,ztε,u^tε,Λtε≤Cνtεutε−u^tε+ε1/2νtε.
By the concavity ofHt,⋅,⋅,⋅,⋅,Λtε, we have(64)Ht,xt,yt,zt,ut,Λtε−Ht,xtε,ytε,ztε,utε,Λtε≤Hx˜εExt−Extε+Hy˜εEyt−Eytε+Hz˜εEzt−Eztε+Hxεxt−xtε+Hyεyt−ytε+Hzεzt−ztε+Huεut−utε,for any x,y,z,u, where Hjε=Hjt,xtε,ytε,ztε,utε,Λtε, j=x,y,z,x˜,y˜,z˜.
Taking integrations, from (57) and (63), follows(65)E∫0THt,xt,yt,zt,ut,Λtε−Ht,xtε,ytε,ztε,utε,Λtεdt≤E∫0THx˜εExt−Extεdt+E∫0THy˜εEyt−Eytεdt+E∫0THz˜εEzt−Eztεdt+E∫0THxεxt−xtεdt+E∫0THyεyt−ytεdt+E∫0THzεzt−ztεdt+Cε1/2.
Applying Itô’s formula toξtεyt−ytε+ηtεxt−xtε, we obtain(66)EξTεhxT,y0−hxTε,y0ε+E−ξTεhxTε+η0εγxTε+θ0εφxTεxT−xTε−E−θ0εφy0ε−η0εγy0ε+ξTεhy0εy0−y0ε−Eη0εγxT,y0−γxTε,y0ε=E∫0THt,xt,yt,zt,ut,Λtε−Ht,xtε,ytε,ztε,utε,Λtε+lt,Γt,ut−lt,Γtε,utεdt−E∫0TEHx˜εxt−xtεdt−E∫0TEHy˜εyt−ytεdt−E∫0TEHz˜εzt−ztεdt−E∫0THxεxt−xtεdt−E∫0THyεyt−ytεdt−E∫0THzεzt−ztεdt.
Sinceh is concave, we have(67)EξTεhxT,y0−hxTε,y0ε−ξTεhxTεxT−xTε−ξTεhy0εy0−y0ε≥0.
Similarly, by the convexity ofφ and γ yielding(68)EφxTεxT−xTε+φy0εy0−y0ε≤EφxT,y0−φxTε,y0ε,(69)EγxTεxT−xTε+γy0εy0−y0ε≤EγxT,y0−γxTε,y0ε.
Combining (65) and (67)–(69), we can finally get the following result:(70)Juε≤Ju+Cε1/2.
Sinceu is arbitrary, we can draw the conclusion.Corollary 2.
Under the assumptions of Theorem3, a sufficient condition for an admissible control uε to be ε-optimal is(71)supu∈UE∫0THt,xtε,ytε,ztε,u,Λtεdt≤E∫0THt,xtε,ytε,ztε,utε,Λtεdt+εC2.


## 5. A Linear-Quadratic Problem

Consider the near-optimal control problem.ProblemPε.
MinimizeJεu=E∫012/2ut2+2ε/2ut2dt+2/2x12+2/2y02, subject to(72)dxt=xt+Ext+utdt+xt+Ext+utdWt,−dyt=xt+yt+zt+Ext+Eyt+Ezt+utdt−ztdWt,x0=x1−12y0,y1=12x1+12y0,where ε>0 is a small parameter, and the control domain is limited as U=−1,1.
In view of the practical point, it is difficult to find a near-optimal control forPε directly. So, we adopt the approach, in which Pε is firstly approximated by a simpler one (Problem P), and then an optimal control u∗ for P is obtained. Finally, it is proved that u∗ is near-optimal for Pε. This method has been applied to the hierarchical controls of stochastic manufacturing systems [37].ProblemP.
MinimizeJu=E∫012/2ut2dt+2/2x12+2/2y02, subject to(73)dxt=xt+Ext+utdt+xt+Ext+utdWt,−dyt=xt+yt+zt+Ext+Eyt+Ezt+utdt−ztdWt,x0=x1−12y0,y1=12x1+12y0.
Letu be an optimal admissible control of P, and the corresponding optimal trajectory is denoted by x,y,z. Set θ0=2/2, for a given admissible triple x,y,z,u, the corresponding first-order adjoint equation is presented as(74)dξt=ξt+Eξtdt+ξt+EξtdWt,−dηt=−ξt+ηt+ζt+E−ξt+ηt+ζtdt−ζtdWt,ξ0=−y0+12η0+12ξ1,η1=−12ξ1+η0+x1.
From Remark5 and Theorem 2, the candidate optimal control u should satisfy(75)22vt2−ηt+ζt−ξtvt≤22ut2−ηt+ζt−ξtut,∀v∈U.
Then, we have(76)ut=μt,μt∈−1,1,1,μt∈−∞,−1,−1,μt∈1,+∞,with μt=2/2ηt+ζt−ξt.
We claim that the controlu defined by (76) is optimal for Problem P, which will be illustrated in the following proposition. Now, we are about to show that the same optimal control is near-optimal for Pε when ε is sufficiently small. Denote by x∗,y∗,z∗,u∗ the optimal state and optimal control under (76) and ξ∗,η∗,ζ∗ the corresponding solution of (74). Then, the Hamiltonian function for P is(77)Ht,xt,yt,zt,ut,Λt=−22ut2−ξt−ηt−ζtut+ηt+ζtxt+Ext−ξtxt+yt+zt+Ext+Eyt+Ezt.
Sinceu∗ is optimal, it necessarily maximizes the Hamiltonian function a.s., namely,(78)ut∗+22ξt∗−ηt∗−ζt∗=0,P−a.s.a.e.
However, the Hamiltonian function forPε is(79)Hεt,xt,yt,zt,ut,Λt=−221+εut2−ξt−ηt−ζtut+ηt+ζtxt+Ext−ξtxt+yt+zt+Ext+Eyt+Ezt.
Obviously, it is concave. Moreover, it is maximized atuε, which satisfies(80)utε=ut∗1+ε.
Hence,(81)supu∈UE∫01Hεt,xt∗,yt∗,zt∗,ut,Λt∗dt−E∫01Hεt,xt∗,yt∗,zt∗,ut∗,Λt∗dt≤E∫01supu∈UHεt,xt∗,yt∗,zt∗,ut,Λt∗dt−E∫01Hεt,xt∗,yt∗,zt∗,ut∗,Λt∗dt=E∫01Hεt,xt∗,yt∗,zt∗,utε,Λt∗−Hεt,xt∗,yt∗,zt∗,ut∗,Λt∗dt=E∫01ε221+εut∗2dt≤Cε.
According to Theorem3, u∗ is near-optimal for Pε with an error order of ε when ε is sufficiently small.Proposition 1.
The controlu∗ defined by (76) together with the corresponding trajectory x∗,y∗,z∗ is an optimal solution for Problem P.Proof.
Supposex,y,z is the trajectory of system (73) controlled by u∈U. By the convexity of a function, we have(82)22x12−22x1∗2≥2x1∗x1−x1∗,22y02−22y0∗2≥2y0∗y0−y0∗.
Applying Itô’s formula toξt∗yt−yt∗+ηt∗xt−xt∗, we obtain(83)Ex1∗x1−x1∗+y0∗y0−y0∗=E∫01ξt∗−ηt∗−ζt∗ut−ut∗dt.
Then,(84)Ju−Ju∗≥E∫0122ut2−22ut∗2dt+2x1∗x1−x1∗+2y0∗y0−y0∗=E∫0122ut2−22ut∗2+2ξt∗−ηt∗−ζt∗ut−ut∗dt.
From (76), we have(85)22ut2+2ξt∗−ηt∗−ζt∗ut≥22ut∗2+2ξt∗−ηt∗−ζt∗ut∗,∀u∈U.
The inequality above implies that(86)Ju−Ju∗≥0,∀u∈U.
Therefore,u∗ is the optimal control for P, and x∗,y∗,z∗ is the optimal trajectory.


## 6. Conclusion

This paper discussed near-optimal control problems for mean-field FBSDEs with mixed initial-terminal conditions. Firstly, we initially introduce three first-order adjoint equations to operate dual analysis with corresponding variational processes. Secondly, the reduction method is adopted to guarantee the well-posedness of the first-order adjoint equations also with mixed initial-terminal conditions. Furthermore, by introducing the penalty functional, the original problem with endpoint constraints is transformed as penalized optimal control problem with no endpoint constraints. Via convex variational technique as well as Ekeland’s variational principle, the necessary condition of Pontryagin’s type is established. Finally, to illustrate the application of our theoretical results, a linear-quadratic problem is worked out. In our future work, we will develop the research on near-optimality to solve both theoretical and practical problems.



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