# A Maximum Principle for Controlled Time-Symmetric Forward-Backward Doubly Stochastic Differential Equation with Initial-Terminal Sate Constraints

Authors: Shaolin Ji; Qingmeng Wei; Xiumin Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the optimal control problem of a controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal state constraints. Applying the terminal perturbation method and Ekeland’s variation principle, a necessary condition of the stochastic optimal control, that is, stochastic maximum principle, is derived. Applications to backward doubly stochastic linear-quadratic control models are investigated.


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## Body

## 1. Introduction

It is well known that general coupled forward-backward stochastic differential equations (FBSDEs) consist of a forward SDE of Itô's type and a backward SDE of Pardoux-Peng's (see [1, 2]). Since Antonelli [3] first studied FBSDEs in early 1990s, FBSDEs have been studied widely in many papers (see [4–7]). FBSDEs are often encountered in the optimization problem when applying stochastic maximum principle (see [8, 9]). In finance, FBSDEs are used when considering problems with the large investors; see [6, 10, 11]. Such equations are also used in the potential theory (see [12]). Moreover, one can apply FBSDEs to study Homogenization and singular perturbation of certain quasilinear parabolic PDEs with periodic structures (see [13, 14]).In order to produce a probabilistic representation of certain quasilinear stochastic partial differential equations (SPDEs), Pardoux and Peng [15] first introduced backward doubly stochastic differential equations (BDSDEs) and proved the existence and uniqueness theorem of BDSDEs. Using such BDSDEs they proved the existence and uniqueness theorem of those quasilinear SPDEs and thus significantly extended the famous Feynman-Kac formula for such SPDEs.Peng and Shi [16] studied the following time-symmetric forward-backward doubly stochastic differential equations (FBDSDEs):
(1.1)-dxt=F(t,xt,zt,yt,qt,ut)dt+G(t,xt,zt,yt,qt,ut)dWt-ztdBt,0≤t≤T,x0=ξ,-dyt=f(t,xt,zt,yt,qt,ut)dt+g(t,xt,zt,yt,qt,ut)dBt-qtdWt,0≤t≤T,yT=η,
which generalized the general FBSDEs. Here the forward equation is “forward” with respect to a standard stochastic integral dWt, as well as “backward” with respect to a backward stochastic integral dBt; the coupled “backward equation” is “forward” under the backward stochastic integral dBt and “backward” under the forward one. In other words, both the forward equation and the backward one are BDSDEs with different directions of stochastic integral. Under certain monotonicity conditions, they proved the uniqueness and existence theorem for these equations. In [17], when deriving the stochastic maximum principle of backward doubly stochastic optimal control problems, Han et al. showed that this kind of equations are just the state equation and adjoint equation of their optimal control problem.In this paper, we study a stochastic optimal control problem with initial-terminal state constraints where the controlled system is described by the above time-symmetric FBDSDEs. We suppose that the initial stateξ and the terminal state η fall in two convex sets, respectively, and the corresponding states xT(ξ,η,u(·)) and y0(ξ,η,u(·)) satisfy the constraints E(ψ(xT(ξ,η,u(·))))=a and E(h(y0(ξ,η,u(·))))=b, respectively. Then we minimize the following cost function:
(1.2)J(ξ,η,u(·))≜E[∫0T‍l(x(t),z(t),y(t),q(t),u(t),t)dt+χ(ξ)+λ(η)+ϕ(x(T))+γ(y(0))].
It is well known that the maximum principle is an important approach to study optimal control problems. The systematic account of this theory can be found in [9, 18]. When the controlled system under consideration is assumed to be with state constraints, especially with sample-wise constraints, the corresponding stochastic optimal control problems are difficult to solve. A sample-wise constraint requires that the state be in a given set with probability 1, for example, a nonnegativity constraint on the wealth process, that is, bankruptcy prohibition in financial markets. In order to deal with such optimal control problems, an approach named “terminal perturbation method” was introduced and applied in financial optimization problems recently (see [19–22]). This method is based on the dual method or martingale method introduced by Bielecki et al. in [23] and El Karoui et al. in [24]. It mainly applies Ekeland's variational principle to tackle the state constraints and derive a stochastic maximum principle which characterizes the optimal solution. For other works about the optimization problem with state constraints, the readers may refer to [25, 26]. In this paper, a stochastic maximum principle is obtained for the controlled time-symmetric FBDSDEs with initial-terminal state constraints by using Ekeland's variational principle.We give three specific applications to illustrate our theoretical results. In the first application, the controlled state equations are composed of a normal FSDE and a BDSDE. By introducing a backward formulation of the controlled system (inspired by [21]), we present the stochastic maximum principle for the optimal control. As a special case, we only consider one BDSDE as our state equation in the second application. As stated in the last application, our results can be applied in forward-backward doubly stochastic linear-quadratic (LQ) optimal control problems. The explicit expression of the optimal control is derived. Since the control system of SPDEs can be transformed to the relevant control system of FBDSDEs, our results can be used to solve the optimal control problem of one kind of SPDEs.This paper is organized as follows. In Section2.1, we recall some preliminaries. And we formulate our control problem in Section 2.2. In Section 2.3, by applying Ekeland's variation principle we obtain a stochastic maximum principle of this controlled time-symmetric FBDSDEs with initial-terminal state constraints. Some applications are given in the last section.


## 2. The Main Problem

### 2.1. Preliminaries

Let us first recall the existence and uniqueness results of the BDSDE which was introduced by Pardoux and Peng [15] and an extension of the well-known Itô's formula which would be often used in this paper.Let(Ω,ℱ,P) be a probability space, and let T>0 be fixed throughout this paper. Let {Wt,0≤t≤T} and {Bt,0≤t≤T} be two mutually independent standard Brownian motion processes, with values in ℝd,ℝl, respectively, defined on (Ω,ℱ,P). Let ℕ denote the class of P-null set of ℱ. For each t∈0,T], we define  ℱt≜ℱtW∨ℱt,TB, where
(2.1)ℱtW=ℱ0,tW=σ{Wr-W0;0≤r≤t}∨ℕ,ℱtB=ℱ0,tB=σ{Br-Bt;t≤r≤T}∨ℕ.
Note that the collection {ℱt,t∈[0,T]} is neither increasing nor decreasing, and it does not constitute a filtration.For any Euclidean spaceH, we denote by 〈·,·〉 the scale product of H. The Euclidean norm of a vector y∈ℝk will be denoted by |y|, and for a d×n matrix A, we define ||A||=Tr(AA*).For anyn∈N, let M2(0,T;ℝn) denote the set of (classes of dP⊗dt a.e. equal) n-dimensional jointly measurable stochastic processes {φt;t∈[0,T]} which satisfy(i)E∫0T‍|φt|2dt<∞; (ii) φt is ℱt-measurable, for a.e. t∈[0,T].We denote byS2(0,T;ℝn) the set of continuous n-dimensional stochastic processes which satisfy:(i)E(sup0≤t≤T|φt|2)<∞; (ii) φt is ℱt-measurable, for any t∈[0,T].Let(2.2)f:Ω×[0,T]×ℝk×ℝk×d→ℝk,g:Ω×[0,T]×ℝk×ℝk×d→ℝk×l
be jointly measurable such that for any (y,q)∈ℝk×ℝk×d,f(·,y,q)∈M2(0,T;ℝk),g(·,y,q)∈M2(0,T;ℝk×l).Moreover, we assume that there exist constantsC>0 and 0<α<1 such that for any (ω,t)∈Ω×[0,T], (y1,q1),(y2,q2)∈ℝk×ℝk×l,
(H)|f(t,y1,q1)-f(t,y2,q2)|2≤C(|y1-y2|2+∥q1-q2∥2),∥g(t,y1,q1)-g(t,y2,q2)∥2≤C|y1-y2|2+α∥q1-q2∥2.
Given η∈L2(Ω,ℱT,P;ℝk), we consider the following BDSDE:
(2.3)yt=η+∫tT‍f(s,ys,qs)ds+∫tT‍g(s,ys,qs)dBs-∫tT‍qsdWs,0≤t≤T.We note that the integral with respect to{Bt} is a “backward Itô integral” and the integral with respect to {Wt} is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Skorohod integral; see Nualart and Pardoux [27].By Theorem 1.1 in [15], (2.3) has a unique solution(y,q)∈S2(0,T;ℝk)×M2(0,T;ℝk×d).Next let us recall an extension of the well-known Itô's formula in [21] which would be often used in this paper.Lemma 2.1.
Letα∈S2(0,T;ℝk), β∈M2(0,T;ℝk), γ∈M2(0,T;ℝk×l),δ∈M2(0,T;ℝk×d) be such that
(2.4)αt=α0+∫0t‍βsds+∫0t‍γsdBs+∫0t‍δsdWs,0≤t≤T.
Then,
(2.5)|αt|2=|α0|2+2∫0t‍(αs,βs)ds+2∫0t‍(αs,γsdBs)-∫0t‍∥γs∥2ds+∫0t‍∥δs∥2ds+2∫0t‍(αs,δsdWs),E|αt|2=E|α0|2+2E∫0t‍(αs,βs)ds-E∫0t‍∥γs∥2ds+E∫0t‍∥δs∥2ds.
Generally, for ϕ∈C2(ℝk),
(2.6)ϕ(αt)=ϕ(α0)+∫0t‍(ϕ′(αs),βs)ds+∫0t‍(ϕ′(αs),γsdBs)+∫0t‍(ϕ′(αs),δsdWs)-12∫0t‍Tr[ϕ′′(αs)γsγs*]ds+12∫0t‍Tr[ϕ′′(αs)δsδs*]ds.


### 2.2. Problem Formulation

LetK be a nonempty convex subset of ℝn×d. We set
(2.7)U[0,T]={u(·)∣u(t)∈K,a.e.,a.s.,0≤t≤T;u(·)∈M2(0,T;ℝn×d)}.
An element of U[0,T] is called an admissible control. Now let
(2.8)F:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝn,G:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝn×d,f:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝk,g:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝk×l
be jointly measurable such that for any (x,z,y,q)∈ℝn×ℝn×l×ℝk×ℝk×d and any u(·)∈U[0,T](2.9)F(·,x,z,y,q,u(·))∈M2(0,T;ℝn),G(·,x,z,y,q,u(·))∈M2(0,T;ℝn×d),f(·,x,z,y,q,u(·))∈M2(0,T;ℝk),g(·,x,z,y,q,u(·))∈M2(0,T;ℝk×l).
Let
(2.10)ζ(t)=(x(t),z(t),y(t),q(t))T,A(t,ζ)=(-F,-G,-f,-g)T(t,ζ).
We assume the following.(H1)
∀ζ1=(x1,z1,y1,q1), ζ2=(x2,z2,y2,q2)∈ℝn×ℝn×l×ℝk×ℝk×d and t∈[0,T], there exists a constant μ>0 such that the following monotonicity condition holds for any u(·)∈U[0,T]:
(2.11)〈A(t,ζ1)-A(t,ζ2),ζ1-ζ2〉≤-μ|ζ1-ζ2|2.(H2)
There exist constantsC>0 and 0<α<1/2 such that for any (ω,t)∈Ω×[0,T], u∈ℝn×d, (x1,z1,y1,q1),(x2,z2,y2,q2)∈ℝn×ℝn×l×ℝk×ℝk×d the following conditions hold:
(2.12)|ϑ(t,x1,z1,y1,q1,u)-ϑ(t,x2,z2,y1,q1,u)|2≤C(|x1-x2|2+∥z1-z2∥2+|y1-y2|2+∥q1-q2∥2),∥ϰ(t,x1,z1,y1,q1,u)-ϰ(t,x2,z2,y1,q1,u)∥2≤C(|x1-x2|2+|y1-y2|2)+α(∥z1-z2∥2+∥q1-q2∥2),where ϑ=(F,f), ϰ=(G,g).(H3)
F,G,f,g,ψ,h,l,χ,λ,ϕ and γ are continuous in their arguments and continuously differentiable in (x,z,y,q,u), and the derivatives of F,G,f,g in (x,z,y,q,u) are bounded and 0<∥gz(·)∥<1/2,0<∥Gz(·)∥<1/2,0<∥gq(·)∥<1/2,0<∥Gq(·)∥<1/2; the derivatives of l in (x,y,z,q,u) are bounded by C(1+|x|+|z|+|y|+|q|+∥u∥), and the derivatives of ϕ, χ, and ψ in x are bounded by C(1+|x|); γ, λ, and h in y are bounded by C(1+|y|).Givenξ∈L2(Ω,ℱ0,P;ℝn), η∈L2(Ω,ℱT,P;ℝk), and ∀u(·)∈U[0,T], let us consider the following time-symmetric FBDSDE:
(2.13)-dxt=F(t,xt,zt,yt,qt,ut)dt+G(t,xt,zt,yt,qt,ut)dWt-ztdBt,x0=ξ,-dyt=f(t,xt,zt,yt,qt,ut)dt+g(t,xt,zt,yt,qt,ut)dBt-qtdWt,yT=η.
Recall Theorem 2.2 in [16]. We have the following.Theorem 2.2.
For givenξ∈L2(Ω,ℱ0,P;ℝn), η∈L2(Ω,ℱT,P;ℝk) and ∀u(·)∈U[0,T], assume (H1)~(H3); then (2.13) exists as a unique ℱt-adapted solution (x(t),z(t),y(t),q(t)).In (2.13), we regard ξ,η,u(·) as controls. ξ,η,u(·) can be chosen from the following admissible set:
(2.14)U={(ξ,η,u(·))∣ξ∈K1⊂ℝna.s.,η∈K2⊂ℝka.s.,E[|ξ|2]<∞,E[|η|2]<∞,u(·)∈U[0,T][|ξ|2]},
where K1 and K2 are convex.Remark 2.3.
A main assumption in this paper is that the control domains are convex. For the terminal perturbation method, it is difficult to weaken or completely remove these assumptions. Until now, it remains an interesting and challenging open problem.We also assume the state constraints(2.15)E(ψ(xT(ξ,η,u(·))))=a,E(h(y0(ξ,η,u(·))))=b.
For each (ξ,η,u(·))∈U, consider the following cost function:
(2.16)J(ξ,η,u(·))≜E[∫0T‍l(x(t),z(t),y(t),q(t),u(t),t)dt+χ(ξ)+λ(η)+ϕ(x(T))+γ(y(0))].
Our optimization problem is
(2.17)inf(ξ,η,u(·))∈UJ(ξ,η,u(·))subjecttoE(ψ(xT(ξ,η,u(·))))=a;E(h(y0(ξ,η,u(·))))=b.Definition 2.4.
A triple of random variable(ξ,η,u(·))∈U is called feasible for given a∈ℝn, b∈ℝk if the solution (2.13) satisfies E(ψ(xT(ξ,η,u(·))))=a and E(h(y0(ξ,η,u(·))))=b. We will denote by ℕ(a,b) the set of all feasible (ξ,η,u(·)) for any given a and b.A feasible(ξ*,η*,u*(·)) is called optimal if it attains the minimum of J(ξ,η,u(·)) over ℕ(a,b).The aim of this paper is to obtain a characterization of(ξ*,η*,u*(·)), that is, the stochastic maximum principle.


### 2.3. Stochastic Maximum Principle

Using Ekeland's variational principle, we derive maximum principle for the optimization problem (2.17) in this section. For simplicity, we first study the case where l(y(t),z(t),y(t),q(t),u(t),t)=0, χ(x)=0 and λ(y)=0 in Sections 2.3.1–2.3.3 and then present the results for the general case in Section 2.3.4.


#### 2.3.1. Variational Equations

For(ξ1,η1,u1(·)),(ξ2,η2,u2(·))∈U, we define a metric in U by
(2.18)d(ξ1,η1,u1(·)),(ξ2,η2,u2(·))≜((E|ξ1-ξ2|2)1/2+E|η1-η2|2)1/2+(∥u1(·)-u2(·)∥2)1/2.
It is obvious that (U,d(·,·)) is a complete metric space.Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding state processes of (2.13). for all(ξ,η,u(·))∈U and for all 0≤ρ≤1,
(2.19)(ξρ,ηρ,uρ(·))≜(ξ*+ρ(ξ-ξ*),η*+ρ(η-η*),u*(·)+ρ(u(·)-u*(·)))∈U.
Let (xρ(·),zρ(·),yρ(·),qρ(·)) be the state processes of (2.13) associated with (ξρ,ηρ,uρ(·)).To derive the first-order necessary condition, we let(x^(·),z^(·),y^(·),q^(·)) be the solution of the following time-symmetric FBDSDE:
(2.20)-dx^(t)=(Fx*(t)x^(t)+Fz*(t)z^(t)+Fy*(t)y^(t)+Fq*(t)q^(t)+Fu*(t)(u(·)-u*(·)))dt+(Gx*(t)x^(t)+Gz*(t)z^(t)+Gy*(t)y^(t)+Gq*(t)q^(t)+Gu*(t)(u(·)-u*(·)))dWt-z^(t)dB(t),x^(0)=ξ-ξ*,-dy^(t)=(fx*(t)x^(t)+fz*(t)z^(t)+fy*(t)y^(t)+fq*(t)q^(t)+fu*(t)(u(·)-u*(·)))dt+(gx*(t)x^(t)+gz*(t)z^(t)+gy*(t)y^(t)+gq*(t)q^(t)+gu*(t)(u(·)-u*(·)))dBt-q^(t)dW(t),y^(T)=η-η*,
whereHk*(t)=Hk(t,x*(·),z*(·),y*(·),q*(·),u*(·)) for H=F,G,f,g, k=x,z,y,q,u, respectively. Equation (2.20) is called the variation equation.Set(2.21)x~ρ(t)=ρ-1[xρ(t)-x*(t)]-x^(t),z~ρ(t)=ρ-1[zρ(t)-z*(t)]-z^(t),y~ρ(t)=ρ-1[yρ(t)-y*(t)]-y^(t),q~ρ(t)=ρ-1[qρ(t)-q*(t)]-q^(t).
We have the following convergence.Lemma 2.5.
Assuming (H1)~(H3) one has(2.22)limρ→0sup0≤t≤TE[|x~ρ(t)|2]=0,limρ→0E[∫0T‍∥z~ρ(t)∥2dt]=0,limρ→0sup0≤t≤TE[|y~ρ(t)|2]=0,limρ→0E[∫0T‍∥q~ρ(t)∥2dt]=0.Proof.
From (2.13) and  (2.20), we have
(2.23)-dy~ρ(t)=ρ-1[ρfy*f(xρ(t),zρ(t),yρ(t),qρ(t),uρ(t),t)-f(x*(t),z*(t),y*(t),q*(t),u*(t),t)-ρfx*(t)x^(t)-ρfz*(t)z^(t)-ρfy*(t)y^(t)-ρfq*(t)q^(t)-ρfu*(t)(u(·)-u*(·))]dt+ρ-1[ρgz*(t)z^(t)gx*(t)x^(t)g(xρ(t),zρ(t),yρ(t),qρ(t),uρ(t),t)-g(x*(t),z*(t),y*(t),q*(t),u*(t),t)-ρgx*(t)x^(t)-ρgz*(t)z^(t)-ρgy*(t)y^(t)-ρgq*(t)q^(t)-ρgu*(t)(u(·)-u*(·))]dBt-q~ρ(t)dWt,y~ρ(T)=0.
Let
(2.24)Aρ(t)=∫01‍fx(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Bρ(t)=∫01‍fz(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Cρ(t)=∫01‍fy(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Dρ(t)=∫01‍fq(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Eρ(t)=[Aρ(t)-fx*(t)]x^(t)+[Bρ(t)-fz*(t)]z^(t)+[Cρ(t)-fy*(t)]y^(t)+[Dρ(t)-fq*(t)]q^(t)+∫01‍[fu(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)-fu*(t)](u(t)-u*(t))dλ,A1ρ(t)=∫01‍gx(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,B1ρ(t)=∫01‍gz(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,C1ρ(t)=∫01‍gy(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,D1ρ(t)=∫01‍gq(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,E1ρ(t)=[A1ρ(t)-gx*(t)]x^(t)+[B1ρ(t)-gz*(t)]z^(t)+[C1ρ(t)-gy*(t)]y^(t)+[D1ρ(t)-gq*(t)]q^(t)+∫01‍[gu(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)-gu*(t)](u(t)-u*(t))dλ.
Thus
(2.25)-dy~ρ(t)=(Aρ(t)x~ρ(t)+Bρ(t)z~ρ(t)+Cρ(t)y~ρ(t)+Dρ(t)q~ρ(t)+Eρ(t))dt+(A1ρ(t)x~ρ(t)+B1ρ(t)z~ρ(t)+C1ρ(t)y~ρ(t)+D1ρ(t)q~ρ(t)+E1ρ(t))dBt-q~ρ(t)dW(t),y~ρ(T)=0.
Using Lemma 2.1 to |y~ρ(t)|2, we get
(2.26)E|y~ρ(t)|2+E∫tT‍∥q~ρ(s)∥2ds=2E∫tT‍(y~ρ(t),Aρ(s)x~ρ(s)+Bρ(s)z~ρ(s)+Cρ(s)y~ρ(s)+Dρ(s)q~ρ(s)+Eρ(s))ds+E∫tT‍∥A1ρ(s)x~ρ(s)+B1ρ(s)z~ρ(s)+C1ρ(s)y~ρ(s)+D1ρ(s)q~ρ(s)+E1ρ(s)∥2ds≤K(E∫tT‍|y~ρ(s)|2ds+E∫tT‍|x~ρ(s)|2ds)+β1(E∫tT‍∥z~ρ(s)∥2ds+E∫tT‍∥q~ρ(s)∥2ds+Jρ),
where K>0, 0<β1<1/2 are constants and Jρ=E∫tT‍|Eρ(s)|2ds+E∫tT‍|E1ρ(s)|2ds.
Similar analysis shows that(2.27)-dx~ρ(t)=(A'ρ(t)x~ρ(t)+B'ρ(t)z~ρ(t)+C'ρ(t)y~ρ(t)+D'ρ(t)q~ρ(t)+E'ρ(t))dt+(A1'ρ(t)x~ρ(t)+B1'ρ(t)z~ρ(t)+C1'ρ(t)y~ρ(t)+D1'ρ(t)q~ρ(t)+E1'ρ(t))dWt-z~ρ(t)dB(t),x~ρ(0)=0,
where H'ρ(t) and H1'ρ(t) (H=A,B,C,DandE) are similarly defined as before.
It yields that(2.28)E|x~ρ(t)|2+E∫tT‍∥z~ρ(s)∥2ds=2E∫tT‍(x~ρ(t),A'ρ(s)x~ρ(s)+B'ρ(s)z~ρ(s)+C'ρ(s)y~ρ(s)+D'ρ(s)q~ρ(ts)+E'ρ(s))ds+E∫tT‍∥A1'ρ(s)x~ρ(s)+B1'ρ(s)z~ρ(s)+C1'ρ(s)y~ρ(s)+D1'ρ(s)q~ρ(s)+E1'ρ(s)∥2ds≤KE[∫tT‍|y~ρ(s)|2ds+∫tT‍|x~ρ(s)|2ds]+β2E[∫tT‍∥z~ρ(s)∥2ds+∫tT‍∥q~ρ(s)∥2ds+Jρ′],
where K>0, 0<β2<1/2 are constants and Jρ′=E[∫tT‍|E'ρ(s)|2ds+∫tT‍|E1'ρ(s)|2ds]. Since 0<β1+β2<1 and K>0, there exists K1>0 such that
(2.29)E|x~ρ(t)|2+E|y~ρ(t)|2+(1-β1-β2)(E∫tT‍∥z~ρ(s)∥2ds+E∫tT‍∥q~ρ(s)∥2ds)≤K1(E∫tT‍|x~ρ(s)|2ds+E∫tT‍|y~ρ(s)|2ds+Jρ+Jρ′).
Since the Lebesgue dominated convergence theorem implies limρ→0Jρ=0,limρ→0Jρ′=0, we obtain the result by Gronwall's inequality.


#### 2.3.2. Variational Inequality

In this subsection, we apply Ekeland's variational principle [28] to deal with initial-terminal state constraints:
(2.30)E(ψ(xT(ξ,η,u(·))))=a,E(h(y0(ξ,η,u(·))))=b.
Define
(2.31)Fε((ξ,η,u(·)))≜{|E(ψ(xT(ξ,η,u(·))))-a|2+|E(h(y0(ξ,η,u(·))))-b|2+(max(0,E[ϕ(x*(T))-ϕ(xT(ξ,η,u(·)))]+ε))2+(max(0,E[γ(y*(0))-γ(y0(ξ,η,u(·)))]+ε))2}1/2,
where a and b are the given initial and terminal state constraints and ε is an arbitrary positive constant.It is easy to check that the mappings|E(ψ(xT(ξ,η,u(·))))-a|2, |E(h(y0(ξ,η,u(·))))-b|2, ϕ(xT(ξ,η,u(·))) and γ(y0(ξ,η,u(·))) are all continuous functionals from U to ℝ.Theorem 2.6.
Suppose (H1)~(H3). Let(ξ*,η*,u*(·)) be an optimal solution to (2.17). Then there exist h0,h1,h2,h3∈ℝ with h0,h1≤0 and |h0|+|h1|+|h2|+|h3|≠0 such that the following variational inequality holds:
(2.32)h3E〈ψx(x*(T)),x^(T)〉+h2E〈hy(y*(0)),y^(0)〉+h1E〈ϕx(x*(T)),x^(T)〉+h0E〈γy(y*(0)),y^(0)〉≥0,
where x^(T) is the solution x^(·) of (2.20) at time T and y^(0) is the solution y^(·) of (2.20) at time 0.Proof.
It is easy to check thatFε(·) is continuous on U such that
(2.33)Fε(ξ*,η*,u*(·))=2ε,Fε(ξ,η,u(·))>0,∀(ξ,η,u(·))∈U,Fε(ξ*,η*,u*(·))≤inf(ξ,η,u(·))∈UFε(ξ,η,u(·))+2ε.
Thus, from Ekeland's variational principle [28], ∃(ξε,ηε,uε(·))∈U such that(i)
Fε(ξε,ηε,uε(·))≤Fε(ξ*,η*,u*(·)),(ii)
d((ξ*,η*,u*(·)),(ξε,ηε,uε(·)))≤2ε,(iii)
Fε(ξ,η,u(·))+2εd((ξ,η,u(·)),(ξε,ηε,uε(·)))≥Fε(ξε,ηε,uε(·)),forall(ξ,η,u(·))∈U.For any (ξ,η,u(·))∈U and 0≤ρ≤1, set (ξρε,ηρε,uρε(·))=(ξε+ρ(ξ-ξε),ηε+ρ(η-ηε),uε(·)+ρ(u(·)-uε(·))). Let (xρε(·),yρε(·),zρε(·),qρε(·)) (resp., (xε(·),yε(·),zε(·),qε(·))) be the solution of (2.13) under (ξρε,ηρε,uρε(·)) (resp., (ξε,ηε,uε(·)), and let (x^ε(·),y^ε(·),z^ε(·),q^ε(·)) be the solution of (2.20) in which (ξ*,η*,u*(·)) is substituted by (ξε,ηε,uε(·)).
From (iii), we know that(2.34)Fε(ξρε,ηρε,uρε(·))-Fε(ξε,ηε,uε(·))+2εd((ξρε,ηρε,uρε(·)),(ξε,ηε,uε(·)))≥0.
On the other hand, similarly to Lemma 2.5 we have
(2.35)limρ→0sup0≤t≤TE[ρ-1[xρε(t)-xε(t)]-x^ε(t)]2=0,limρ→0sup0≤t≤TE[ρ-1[yρε(t)-yε(t)]-y^ε(t)]2=0.
This leads to the following expansions:
(2.36)|E(ϕ(xρε(T))-a|2-|E(ϕ(xε(T))-a|2=2ρ〈Eϕ(xε(T))-a,E[ϕx(xε(T))x^ε(T)]〉+o(ρ),|E(h(yρε(0))-b|2-|E(h(yε(0))-b|2=2ρ〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉+o(ρ).
Applying the linearization technique, then
(2.37)E[ϕ(xρε(T))-ϕ(xε(T))]=E[∫0T‍ϕx(x(ξε+λρξ^ε,ηε+λρη^ε,uε(·)+λρu^ε)(T))dλ·ρx^ε(T)],E[γ(yρε(0))-γ(yε(0))]=E[∫0T‍γy(y(ξε+λρξ^ε,ηε+λρη^ε,uε(·)+λρu^ε(·))(0))dλ·ρy^ε(0)].
So we have the following expansions:
(2.38)(E[ϕ(x*(T))-ϕ(xρε(T)]+ε)2-(E[ϕ(x*(T))-ϕ(xε(T)]+ε)2=-2ρE[ϕx(xε(T))·ρx^Tε][E[ϕ(x*(T))-ϕ(xε(T)]+ε],(E[γ(y*(0)-γ(yρε(0)]+ε)2-(E[γ(y*(0))-γ(yε(0))]+ε)2=-2ρE[γy(yε(0))·ρy^0ε][E[γ(y*(0))-γ(yε(0))]+ε].
For the given ε, we consider the following four cases.
Case
1.There exists ρ0>0 such that
(2.39)E[ϕ(xρε(T))-ϕ(xε(T))]+ε>0,E[γ(yρε(0))-γ(yε(0))]+ε>0
for all ρ∈(0,ρ0).
In this case,(2.40)limρ→0Fε(ξρε,ηρε,uρε(·))-Fε(ξε,ηε,uε(·))ρ=limρ→01Fε(ξρε,ηρε,uρε(·))+Fε(ξε,ηε,uε(·))Fε2(ξρε,ηρε,uρε(·))-Fε2(ξε,ηε,uε(·))ρ=1Fε(ξε,ηε,uε(·)){〈Eψ(xε(T))-a,E[ψx(xε(T))x^ε(T)]〉+〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉-〈E[ϕ(x*(T))-ϕ(xε(T))]+ε,E[ϕx(xε(T))x^ε(T)]〉-〈E[γ(y*(0))-γ(yε(0))]+ε,E[γy(yε(0))y^ε(0)]〉}.
Dividing (2.34) by ρ and sending ρ to 0, we obtain
(2.41)h3εE〈ψx(xε(T)),x^ε(T)〉+h2εE〈hy(yε(0)),y^ε(0)〉+h1εE〈ϕx(xε(T)),x^ε(T)〉+h0εE〈γy(yε(0),y^ε(0)〉≥-2εd((ξε,ηε,uε(·)),(ξ,η,u(·))),
where
(2.42)h0ε=-1Fε(ξε,ηε,uε(·))[E[γ(y*(0))-γ(yε(0))]+ε]≤0,h1ε=-1Fε(ξε,ηε,uε(·))[E[ϕ(x*(T))-ϕ(xε(T))]+ε]≤0,h2ε=1Fε(ξε,ηε,uε(·))[Eh(yε(0))-b],h3ε=1Fε(ξε,ηε,uε(·))[Eψ(xε(T))-a].Case2.There exists a position sequence {ρn} satisfying ρn→0 such that
(2.43)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε≤0,E[γ(yρnε(0))-γ(y*(0))]+ε>0.
Then
(2.44)Fε(ξρnε,ηρnε,uρnε(·))={|Eψ(xρnε(T))-a|2+|Eh(yρnε(0))-b|2+(max(0,E[γ(y*(0))-γ(yρnε(0))]+ε))2}1/2.
For sufficiently large n, since Fε(·) is continuous, we conclude
(2.45)Fε(ξε,ηε,uε(·))={|Eψ(xε(T))-a|2}1/2+|Eh(yε(0))-b|2+(max(0,E[γ(y*(0))-γ(yε(0))]+ε))2}1/2.
Now
(2.46)limn→∞Fε(ξρnε,ηρnε,uρnε(·))-Fε(ξε,ηε,uε(·))ρn=limn→∞1Fε(ξρnε,ηρnε,uρnε(·))+Fε(ξε,ηε,uε(·))Fε2(ξρnε,ηρnε,uρnε(·))-Fε2(ξε,ηε,uε(·))ρn=1Fε(ξε,ηε,uε(·)){[γy(yε(0))y^ε(0)]〈Eψ(xε(T))-a,E[ψx(xε(T))x^ε(T)]〉+〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉-〈E[γ(y*(0))-γ(yε(0))]+ε,E[γy(yε(0))y^ε(0)]〉(}.
Similar to Case 1 we get
(2.47)h3εE〈ψx(xε(T)),x^ε(T)〉+h2εE〈hy(yε(0)),y^ε(0)〉+h0εE〈γy(yε(0)),y^ε(0)〉≥-2εd((ξε,ηε,uε(·)),(ξ,η,u(·))),
where
(2.48)h0ε=-1Fε(ξε,ηε,uε(·))[E[γ(y*(0))-γ(yε(0))]+ε]≤0,h1ε=0,h2ε=1Fε(ξε,ηε,uε(·))[Eh(yε(0))-b],h3ε=1Fε(ξε,ηε,uε(·))[Eψ(xε(T))-a].Case3.There exists a positive sequence {ρn} satisfying ρn→0 such that
(2.49)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε>0,E[γ(yρnε(0))-γ(y*(0))]+ε≤0.Case4.There exists a positive sequence {ρn} satisfying ρn→0 such that
(2.50)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε≤0,E[γ(yρnε(0))-γ(y*(0))]+ε≤0.
Similar techniques can be used for both Case 3 and Case 4.
In summary, for all those cases, we haveh0ε≤0,h1ε≤0 and |h0ε|2+|h1ε|2+|h2ε|2+|h3ε|2=1 by the definition of Fε(·). Then there exists a convergent subsequence of (h3ε,h2ε,h1ε,h0ε)  whose limit is denoted by (h3,h2,h1,h0). On the other hand, it is easy to check that x^ε(T)→0, y^ε(0)→0 as ε→0. Thus (2.32) holds.


#### 2.3.3.   Maximum Principle

In this subsection we derive the maximum principle for the case wherel(x,z,y,q,u,t)=0, χ(x)=0, λ(y)=0 and then present the results for the general case in Section 2.3.4. To this end, we introduce the adjoint process (m(·),p(·)), and (n(·),δ(·)) associated with the optimal solution (ξ*,η*,u*(·)) to (2.13), which is the solution of the following time-symmetric FBDSDE:
(2.51)-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t))dBt-ptdWt,m(T)=-(h3ψx(xT*)+h1ϕx(xT*)),dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+gq*(t)δ(t))dWt-δtdBt,n(0)=h2hy(y0*)+h0γy(y0*),
where Fk*(t), fk*(t), Gk*(t), gk*(t) for k=x,y,z,q are defined as in (2.20). It is easy to check that there exist unique processes (m(·),p(·)), (n(·),δ(·)) which solve the above equations.Theorem 2.7.
We assume (H1)~(H4). Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η,u(·))∈U one has for any t∈[0,T],
(2.52)〈m(0),ξ-ξ*〉≤0,〈n(T),η-η*〉≥0,〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+gu*(t)δ(t),u(t)-u*(t)〉≥0.Proof.
For any(ξ,η,u(·))∈U, let (x^(·),z^(·),y^(·),q^(·)) be the solution to (2.20). Applying Lemma 2.1 to 〈m(t),x^(t)〉+〈n(t),y^(t)〉, we have
(2.53)E〈m(T),x^(T)〉+E〈n(T),y^(T)〉-E〈m(0),x^(0)〉-E〈n(0),y^(0)〉=E∫0T‍mt(-Fx*(t)x^(t)-Fz*(t)z^(t)-Fy*(t)y^(t)-Fq*(t)q^(t)-Fu*(t)u^(t))dt+E∫0T‍x^(t)(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t))dt+E∫0T‍z^(t)(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t))dt-E∫0T‍p(t)(Gx*(t)x^(t)+Gz*(t)z^(t)+Gy*(t)y^(t)+Gq*(t)q^(t)+Gu*(t)u^(t))dt-E∫0T‍n(t)(fx*(t)x^(t)+fz*(t)z^(t)+fy*(t)y^(t)+fq*(t)q^(t)+fu*(t)u^(t))dt+E∫0T‍y^(t)(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t))dt+E∫0T‍q^(t)(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+qq*(t)δ(t))dt-E∫0T‍δ(t)(gx*(t)x^(t)+gz*(t)z^(t)+gy*(t)y^(t)+gq*(t)q^(t)+gu*(t)u^(t))dt=-E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+gu*(t)δ(t),u^(t)〉dt.
This yields
(2.54)E〈n(T),y^(T)〉-E〈m(0),x^(0)〉+E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+δ(t)gu*(t),u(t)-u*(t)〉dt=-E〈m(T),x^(T)〉+E〈n(0),y^(0)〉=h3E〈ψx(x*(T)),x^(T)〉+h2E〈hy(y*(0)),y^(0)〉+h1E〈ϕx(x*(T)),x^(T)〉+h0E〈γy(y*(0)),y^(0)〉≥0.
For any (ξ,η,u(·))∈U, we get
(2.55)E〈n(T),η-η*〉-E〈m(0),ξ-ξ*〉+E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+δ(t)gu*(t),u(t)-u*(t)〉dt≥0.
Thus, it is easy to see that (2.52) holds.


#### 2.3.4. The General Case

Define the Hamiltonian(2.56)H:ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d×ℝn×ℝn×l×ℝk×ℝk×d×[0,T]→ℝ
by
(2.57)H(x,z,y,q,u,m,p,n,δ,t)=〈F(t,x,z,y,q,u),m(t)〉+〈G(t,x,z,y,q,u),p(t)〉+〈f(t,x,z,y,q,u),n(t)〉+〈g(t,x,z,y,q,u),δ(t)〉+l(x,z,y,q,u,t).
Now we consider the general case where l(x,z,y,q,u,t)≠0, χ(x)≠0, λ(y)≠0.Since the proof of the maximum principle is essentially similar as in the preceding subsection, we only present the result without proof.Let(ξ*,η*,u*(·)) be optimal to (2.17) with (x*(·),z*(·),y*(·),q*(·)) being the corresponding optimal trajectory of (2.13). We define the following adjoint equations:
(2.58)-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t)+lx*(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t)+lz*(t))dBt-ptdWt,m(T)=-(h3ψx(xT*)+h1ϕx(xT*)),dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t)+ly*(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+gq*(t)δ(t)+lq*(t))dWt-δtdBt,n(0)=h2hy(y0*)+h0γy(y0*),
where la*(t)=la(x*(t),z*(t),y*(t),q*(t),u*(t),t), a=x,z,y,q, respectively.Theorem 2.8.
We assume (H1)~(H3). Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η,u(·))∈U, one has that the following inequalities hold, for any t∈[0,T]:
(2.59)〈m(0)-χx(ξ*),ξ-ξ*〉≤0,〈n(T)+λy(η*),η-η*〉≥0,〈Hu(t,x*(t),z*(t),y*(t),q*(t),u*(t),m(t),p(t),n(t),δ(t)),u(t)-u*(t)〉≥0.Remark 2.9.
Let us denote the boundary ofK1 by ∂K1. Set
(2.60)M≜{w∈Ω∣ξ*(ω)∈∂K1}.
Then
(2.61)m(0)-χx(ξ*)≤0a.s.onM,m(0)-χx(ξ*)=0a.s.onMc.
Similar analysis can be used to the boundaries of K2 and K.


## 2.1. Preliminaries

Let us first recall the existence and uniqueness results of the BDSDE which was introduced by Pardoux and Peng [15] and an extension of the well-known Itô's formula which would be often used in this paper.Let(Ω,ℱ,P) be a probability space, and let T>0 be fixed throughout this paper. Let {Wt,0≤t≤T} and {Bt,0≤t≤T} be two mutually independent standard Brownian motion processes, with values in ℝd,ℝl, respectively, defined on (Ω,ℱ,P). Let ℕ denote the class of P-null set of ℱ. For each t∈0,T], we define  ℱt≜ℱtW∨ℱt,TB, where
(2.1)ℱtW=ℱ0,tW=σ{Wr-W0;0≤r≤t}∨ℕ,ℱtB=ℱ0,tB=σ{Br-Bt;t≤r≤T}∨ℕ.
Note that the collection {ℱt,t∈[0,T]} is neither increasing nor decreasing, and it does not constitute a filtration.For any Euclidean spaceH, we denote by 〈·,·〉 the scale product of H. The Euclidean norm of a vector y∈ℝk will be denoted by |y|, and for a d×n matrix A, we define ||A||=Tr(AA*).For anyn∈N, let M2(0,T;ℝn) denote the set of (classes of dP⊗dt a.e. equal) n-dimensional jointly measurable stochastic processes {φt;t∈[0,T]} which satisfy(i)E∫0T‍|φt|2dt<∞; (ii) φt is ℱt-measurable, for a.e. t∈[0,T].We denote byS2(0,T;ℝn) the set of continuous n-dimensional stochastic processes which satisfy:(i)E(sup0≤t≤T|φt|2)<∞; (ii) φt is ℱt-measurable, for any t∈[0,T].Let(2.2)f:Ω×[0,T]×ℝk×ℝk×d→ℝk,g:Ω×[0,T]×ℝk×ℝk×d→ℝk×l
be jointly measurable such that for any (y,q)∈ℝk×ℝk×d,f(·,y,q)∈M2(0,T;ℝk),g(·,y,q)∈M2(0,T;ℝk×l).Moreover, we assume that there exist constantsC>0 and 0<α<1 such that for any (ω,t)∈Ω×[0,T], (y1,q1),(y2,q2)∈ℝk×ℝk×l,
(H)|f(t,y1,q1)-f(t,y2,q2)|2≤C(|y1-y2|2+∥q1-q2∥2),∥g(t,y1,q1)-g(t,y2,q2)∥2≤C|y1-y2|2+α∥q1-q2∥2.
Given η∈L2(Ω,ℱT,P;ℝk), we consider the following BDSDE:
(2.3)yt=η+∫tT‍f(s,ys,qs)ds+∫tT‍g(s,ys,qs)dBs-∫tT‍qsdWs,0≤t≤T.We note that the integral with respect to{Bt} is a “backward Itô integral” and the integral with respect to {Wt} is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Skorohod integral; see Nualart and Pardoux [27].By Theorem 1.1 in [15], (2.3) has a unique solution(y,q)∈S2(0,T;ℝk)×M2(0,T;ℝk×d).Next let us recall an extension of the well-known Itô's formula in [21] which would be often used in this paper.Lemma 2.1.
Letα∈S2(0,T;ℝk), β∈M2(0,T;ℝk), γ∈M2(0,T;ℝk×l),δ∈M2(0,T;ℝk×d) be such that
(2.4)αt=α0+∫0t‍βsds+∫0t‍γsdBs+∫0t‍δsdWs,0≤t≤T.
Then,
(2.5)|αt|2=|α0|2+2∫0t‍(αs,βs)ds+2∫0t‍(αs,γsdBs)-∫0t‍∥γs∥2ds+∫0t‍∥δs∥2ds+2∫0t‍(αs,δsdWs),E|αt|2=E|α0|2+2E∫0t‍(αs,βs)ds-E∫0t‍∥γs∥2ds+E∫0t‍∥δs∥2ds.
Generally, for ϕ∈C2(ℝk),
(2.6)ϕ(αt)=ϕ(α0)+∫0t‍(ϕ′(αs),βs)ds+∫0t‍(ϕ′(αs),γsdBs)+∫0t‍(ϕ′(αs),δsdWs)-12∫0t‍Tr[ϕ′′(αs)γsγs*]ds+12∫0t‍Tr[ϕ′′(αs)δsδs*]ds.


## 2.2. Problem Formulation

LetK be a nonempty convex subset of ℝn×d. We set
(2.7)U[0,T]={u(·)∣u(t)∈K,a.e.,a.s.,0≤t≤T;u(·)∈M2(0,T;ℝn×d)}.
An element of U[0,T] is called an admissible control. Now let
(2.8)F:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝn,G:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝn×d,f:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝk,g:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d→ℝk×l
be jointly measurable such that for any (x,z,y,q)∈ℝn×ℝn×l×ℝk×ℝk×d and any u(·)∈U[0,T](2.9)F(·,x,z,y,q,u(·))∈M2(0,T;ℝn),G(·,x,z,y,q,u(·))∈M2(0,T;ℝn×d),f(·,x,z,y,q,u(·))∈M2(0,T;ℝk),g(·,x,z,y,q,u(·))∈M2(0,T;ℝk×l).
Let
(2.10)ζ(t)=(x(t),z(t),y(t),q(t))T,A(t,ζ)=(-F,-G,-f,-g)T(t,ζ).
We assume the following.(H1)
∀ζ1=(x1,z1,y1,q1), ζ2=(x2,z2,y2,q2)∈ℝn×ℝn×l×ℝk×ℝk×d and t∈[0,T], there exists a constant μ>0 such that the following monotonicity condition holds for any u(·)∈U[0,T]:
(2.11)〈A(t,ζ1)-A(t,ζ2),ζ1-ζ2〉≤-μ|ζ1-ζ2|2.(H2)
There exist constantsC>0 and 0<α<1/2 such that for any (ω,t)∈Ω×[0,T], u∈ℝn×d, (x1,z1,y1,q1),(x2,z2,y2,q2)∈ℝn×ℝn×l×ℝk×ℝk×d the following conditions hold:
(2.12)|ϑ(t,x1,z1,y1,q1,u)-ϑ(t,x2,z2,y1,q1,u)|2≤C(|x1-x2|2+∥z1-z2∥2+|y1-y2|2+∥q1-q2∥2),∥ϰ(t,x1,z1,y1,q1,u)-ϰ(t,x2,z2,y1,q1,u)∥2≤C(|x1-x2|2+|y1-y2|2)+α(∥z1-z2∥2+∥q1-q2∥2),where ϑ=(F,f), ϰ=(G,g).(H3)
F,G,f,g,ψ,h,l,χ,λ,ϕ and γ are continuous in their arguments and continuously differentiable in (x,z,y,q,u), and the derivatives of F,G,f,g in (x,z,y,q,u) are bounded and 0<∥gz(·)∥<1/2,0<∥Gz(·)∥<1/2,0<∥gq(·)∥<1/2,0<∥Gq(·)∥<1/2; the derivatives of l in (x,y,z,q,u) are bounded by C(1+|x|+|z|+|y|+|q|+∥u∥), and the derivatives of ϕ, χ, and ψ in x are bounded by C(1+|x|); γ, λ, and h in y are bounded by C(1+|y|).Givenξ∈L2(Ω,ℱ0,P;ℝn), η∈L2(Ω,ℱT,P;ℝk), and ∀u(·)∈U[0,T], let us consider the following time-symmetric FBDSDE:
(2.13)-dxt=F(t,xt,zt,yt,qt,ut)dt+G(t,xt,zt,yt,qt,ut)dWt-ztdBt,x0=ξ,-dyt=f(t,xt,zt,yt,qt,ut)dt+g(t,xt,zt,yt,qt,ut)dBt-qtdWt,yT=η.
Recall Theorem 2.2 in [16]. We have the following.Theorem 2.2.
For givenξ∈L2(Ω,ℱ0,P;ℝn), η∈L2(Ω,ℱT,P;ℝk) and ∀u(·)∈U[0,T], assume (H1)~(H3); then (2.13) exists as a unique ℱt-adapted solution (x(t),z(t),y(t),q(t)).In (2.13), we regard ξ,η,u(·) as controls. ξ,η,u(·) can be chosen from the following admissible set:
(2.14)U={(ξ,η,u(·))∣ξ∈K1⊂ℝna.s.,η∈K2⊂ℝka.s.,E[|ξ|2]<∞,E[|η|2]<∞,u(·)∈U[0,T][|ξ|2]},
where K1 and K2 are convex.Remark 2.3.
A main assumption in this paper is that the control domains are convex. For the terminal perturbation method, it is difficult to weaken or completely remove these assumptions. Until now, it remains an interesting and challenging open problem.We also assume the state constraints(2.15)E(ψ(xT(ξ,η,u(·))))=a,E(h(y0(ξ,η,u(·))))=b.
For each (ξ,η,u(·))∈U, consider the following cost function:
(2.16)J(ξ,η,u(·))≜E[∫0T‍l(x(t),z(t),y(t),q(t),u(t),t)dt+χ(ξ)+λ(η)+ϕ(x(T))+γ(y(0))].
Our optimization problem is
(2.17)inf(ξ,η,u(·))∈UJ(ξ,η,u(·))subjecttoE(ψ(xT(ξ,η,u(·))))=a;E(h(y0(ξ,η,u(·))))=b.Definition 2.4.
A triple of random variable(ξ,η,u(·))∈U is called feasible for given a∈ℝn, b∈ℝk if the solution (2.13) satisfies E(ψ(xT(ξ,η,u(·))))=a and E(h(y0(ξ,η,u(·))))=b. We will denote by ℕ(a,b) the set of all feasible (ξ,η,u(·)) for any given a and b.A feasible(ξ*,η*,u*(·)) is called optimal if it attains the minimum of J(ξ,η,u(·)) over ℕ(a,b).The aim of this paper is to obtain a characterization of(ξ*,η*,u*(·)), that is, the stochastic maximum principle.


## 2.3. Stochastic Maximum Principle

Using Ekeland's variational principle, we derive maximum principle for the optimization problem (2.17) in this section. For simplicity, we first study the case where l(y(t),z(t),y(t),q(t),u(t),t)=0, χ(x)=0 and λ(y)=0 in Sections 2.3.1–2.3.3 and then present the results for the general case in Section 2.3.4.


### 2.3.1. Variational Equations

For(ξ1,η1,u1(·)),(ξ2,η2,u2(·))∈U, we define a metric in U by
(2.18)d(ξ1,η1,u1(·)),(ξ2,η2,u2(·))≜((E|ξ1-ξ2|2)1/2+E|η1-η2|2)1/2+(∥u1(·)-u2(·)∥2)1/2.
It is obvious that (U,d(·,·)) is a complete metric space.Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding state processes of (2.13). for all(ξ,η,u(·))∈U and for all 0≤ρ≤1,
(2.19)(ξρ,ηρ,uρ(·))≜(ξ*+ρ(ξ-ξ*),η*+ρ(η-η*),u*(·)+ρ(u(·)-u*(·)))∈U.
Let (xρ(·),zρ(·),yρ(·),qρ(·)) be the state processes of (2.13) associated with (ξρ,ηρ,uρ(·)).To derive the first-order necessary condition, we let(x^(·),z^(·),y^(·),q^(·)) be the solution of the following time-symmetric FBDSDE:
(2.20)-dx^(t)=(Fx*(t)x^(t)+Fz*(t)z^(t)+Fy*(t)y^(t)+Fq*(t)q^(t)+Fu*(t)(u(·)-u*(·)))dt+(Gx*(t)x^(t)+Gz*(t)z^(t)+Gy*(t)y^(t)+Gq*(t)q^(t)+Gu*(t)(u(·)-u*(·)))dWt-z^(t)dB(t),x^(0)=ξ-ξ*,-dy^(t)=(fx*(t)x^(t)+fz*(t)z^(t)+fy*(t)y^(t)+fq*(t)q^(t)+fu*(t)(u(·)-u*(·)))dt+(gx*(t)x^(t)+gz*(t)z^(t)+gy*(t)y^(t)+gq*(t)q^(t)+gu*(t)(u(·)-u*(·)))dBt-q^(t)dW(t),y^(T)=η-η*,
whereHk*(t)=Hk(t,x*(·),z*(·),y*(·),q*(·),u*(·)) for H=F,G,f,g, k=x,z,y,q,u, respectively. Equation (2.20) is called the variation equation.Set(2.21)x~ρ(t)=ρ-1[xρ(t)-x*(t)]-x^(t),z~ρ(t)=ρ-1[zρ(t)-z*(t)]-z^(t),y~ρ(t)=ρ-1[yρ(t)-y*(t)]-y^(t),q~ρ(t)=ρ-1[qρ(t)-q*(t)]-q^(t).
We have the following convergence.Lemma 2.5.
Assuming (H1)~(H3) one has(2.22)limρ→0sup0≤t≤TE[|x~ρ(t)|2]=0,limρ→0E[∫0T‍∥z~ρ(t)∥2dt]=0,limρ→0sup0≤t≤TE[|y~ρ(t)|2]=0,limρ→0E[∫0T‍∥q~ρ(t)∥2dt]=0.Proof.
From (2.13) and  (2.20), we have
(2.23)-dy~ρ(t)=ρ-1[ρfy*f(xρ(t),zρ(t),yρ(t),qρ(t),uρ(t),t)-f(x*(t),z*(t),y*(t),q*(t),u*(t),t)-ρfx*(t)x^(t)-ρfz*(t)z^(t)-ρfy*(t)y^(t)-ρfq*(t)q^(t)-ρfu*(t)(u(·)-u*(·))]dt+ρ-1[ρgz*(t)z^(t)gx*(t)x^(t)g(xρ(t),zρ(t),yρ(t),qρ(t),uρ(t),t)-g(x*(t),z*(t),y*(t),q*(t),u*(t),t)-ρgx*(t)x^(t)-ρgz*(t)z^(t)-ρgy*(t)y^(t)-ρgq*(t)q^(t)-ρgu*(t)(u(·)-u*(·))]dBt-q~ρ(t)dWt,y~ρ(T)=0.
Let
(2.24)Aρ(t)=∫01‍fx(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Bρ(t)=∫01‍fz(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Cρ(t)=∫01‍fy(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Dρ(t)=∫01‍fq(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Eρ(t)=[Aρ(t)-fx*(t)]x^(t)+[Bρ(t)-fz*(t)]z^(t)+[Cρ(t)-fy*(t)]y^(t)+[Dρ(t)-fq*(t)]q^(t)+∫01‍[fu(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)-fu*(t)](u(t)-u*(t))dλ,A1ρ(t)=∫01‍gx(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,B1ρ(t)=∫01‍gz(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,C1ρ(t)=∫01‍gy(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,D1ρ(t)=∫01‍gq(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,E1ρ(t)=[A1ρ(t)-gx*(t)]x^(t)+[B1ρ(t)-gz*(t)]z^(t)+[C1ρ(t)-gy*(t)]y^(t)+[D1ρ(t)-gq*(t)]q^(t)+∫01‍[gu(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)-gu*(t)](u(t)-u*(t))dλ.
Thus
(2.25)-dy~ρ(t)=(Aρ(t)x~ρ(t)+Bρ(t)z~ρ(t)+Cρ(t)y~ρ(t)+Dρ(t)q~ρ(t)+Eρ(t))dt+(A1ρ(t)x~ρ(t)+B1ρ(t)z~ρ(t)+C1ρ(t)y~ρ(t)+D1ρ(t)q~ρ(t)+E1ρ(t))dBt-q~ρ(t)dW(t),y~ρ(T)=0.
Using Lemma 2.1 to |y~ρ(t)|2, we get
(2.26)E|y~ρ(t)|2+E∫tT‍∥q~ρ(s)∥2ds=2E∫tT‍(y~ρ(t),Aρ(s)x~ρ(s)+Bρ(s)z~ρ(s)+Cρ(s)y~ρ(s)+Dρ(s)q~ρ(s)+Eρ(s))ds+E∫tT‍∥A1ρ(s)x~ρ(s)+B1ρ(s)z~ρ(s)+C1ρ(s)y~ρ(s)+D1ρ(s)q~ρ(s)+E1ρ(s)∥2ds≤K(E∫tT‍|y~ρ(s)|2ds+E∫tT‍|x~ρ(s)|2ds)+β1(E∫tT‍∥z~ρ(s)∥2ds+E∫tT‍∥q~ρ(s)∥2ds+Jρ),
where K>0, 0<β1<1/2 are constants and Jρ=E∫tT‍|Eρ(s)|2ds+E∫tT‍|E1ρ(s)|2ds.
Similar analysis shows that(2.27)-dx~ρ(t)=(A'ρ(t)x~ρ(t)+B'ρ(t)z~ρ(t)+C'ρ(t)y~ρ(t)+D'ρ(t)q~ρ(t)+E'ρ(t))dt+(A1'ρ(t)x~ρ(t)+B1'ρ(t)z~ρ(t)+C1'ρ(t)y~ρ(t)+D1'ρ(t)q~ρ(t)+E1'ρ(t))dWt-z~ρ(t)dB(t),x~ρ(0)=0,
where H'ρ(t) and H1'ρ(t) (H=A,B,C,DandE) are similarly defined as before.
It yields that(2.28)E|x~ρ(t)|2+E∫tT‍∥z~ρ(s)∥2ds=2E∫tT‍(x~ρ(t),A'ρ(s)x~ρ(s)+B'ρ(s)z~ρ(s)+C'ρ(s)y~ρ(s)+D'ρ(s)q~ρ(ts)+E'ρ(s))ds+E∫tT‍∥A1'ρ(s)x~ρ(s)+B1'ρ(s)z~ρ(s)+C1'ρ(s)y~ρ(s)+D1'ρ(s)q~ρ(s)+E1'ρ(s)∥2ds≤KE[∫tT‍|y~ρ(s)|2ds+∫tT‍|x~ρ(s)|2ds]+β2E[∫tT‍∥z~ρ(s)∥2ds+∫tT‍∥q~ρ(s)∥2ds+Jρ′],
where K>0, 0<β2<1/2 are constants and Jρ′=E[∫tT‍|E'ρ(s)|2ds+∫tT‍|E1'ρ(s)|2ds]. Since 0<β1+β2<1 and K>0, there exists K1>0 such that
(2.29)E|x~ρ(t)|2+E|y~ρ(t)|2+(1-β1-β2)(E∫tT‍∥z~ρ(s)∥2ds+E∫tT‍∥q~ρ(s)∥2ds)≤K1(E∫tT‍|x~ρ(s)|2ds+E∫tT‍|y~ρ(s)|2ds+Jρ+Jρ′).
Since the Lebesgue dominated convergence theorem implies limρ→0Jρ=0,limρ→0Jρ′=0, we obtain the result by Gronwall's inequality.


### 2.3.2. Variational Inequality

In this subsection, we apply Ekeland's variational principle [28] to deal with initial-terminal state constraints:
(2.30)E(ψ(xT(ξ,η,u(·))))=a,E(h(y0(ξ,η,u(·))))=b.
Define
(2.31)Fε((ξ,η,u(·)))≜{|E(ψ(xT(ξ,η,u(·))))-a|2+|E(h(y0(ξ,η,u(·))))-b|2+(max(0,E[ϕ(x*(T))-ϕ(xT(ξ,η,u(·)))]+ε))2+(max(0,E[γ(y*(0))-γ(y0(ξ,η,u(·)))]+ε))2}1/2,
where a and b are the given initial and terminal state constraints and ε is an arbitrary positive constant.It is easy to check that the mappings|E(ψ(xT(ξ,η,u(·))))-a|2, |E(h(y0(ξ,η,u(·))))-b|2, ϕ(xT(ξ,η,u(·))) and γ(y0(ξ,η,u(·))) are all continuous functionals from U to ℝ.Theorem 2.6.
Suppose (H1)~(H3). Let(ξ*,η*,u*(·)) be an optimal solution to (2.17). Then there exist h0,h1,h2,h3∈ℝ with h0,h1≤0 and |h0|+|h1|+|h2|+|h3|≠0 such that the following variational inequality holds:
(2.32)h3E〈ψx(x*(T)),x^(T)〉+h2E〈hy(y*(0)),y^(0)〉+h1E〈ϕx(x*(T)),x^(T)〉+h0E〈γy(y*(0)),y^(0)〉≥0,
where x^(T) is the solution x^(·) of (2.20) at time T and y^(0) is the solution y^(·) of (2.20) at time 0.Proof.
It is easy to check thatFε(·) is continuous on U such that
(2.33)Fε(ξ*,η*,u*(·))=2ε,Fε(ξ,η,u(·))>0,∀(ξ,η,u(·))∈U,Fε(ξ*,η*,u*(·))≤inf(ξ,η,u(·))∈UFε(ξ,η,u(·))+2ε.
Thus, from Ekeland's variational principle [28], ∃(ξε,ηε,uε(·))∈U such that(i)
Fε(ξε,ηε,uε(·))≤Fε(ξ*,η*,u*(·)),(ii)
d((ξ*,η*,u*(·)),(ξε,ηε,uε(·)))≤2ε,(iii)
Fε(ξ,η,u(·))+2εd((ξ,η,u(·)),(ξε,ηε,uε(·)))≥Fε(ξε,ηε,uε(·)),forall(ξ,η,u(·))∈U.For any (ξ,η,u(·))∈U and 0≤ρ≤1, set (ξρε,ηρε,uρε(·))=(ξε+ρ(ξ-ξε),ηε+ρ(η-ηε),uε(·)+ρ(u(·)-uε(·))). Let (xρε(·),yρε(·),zρε(·),qρε(·)) (resp., (xε(·),yε(·),zε(·),qε(·))) be the solution of (2.13) under (ξρε,ηρε,uρε(·)) (resp., (ξε,ηε,uε(·)), and let (x^ε(·),y^ε(·),z^ε(·),q^ε(·)) be the solution of (2.20) in which (ξ*,η*,u*(·)) is substituted by (ξε,ηε,uε(·)).
From (iii), we know that(2.34)Fε(ξρε,ηρε,uρε(·))-Fε(ξε,ηε,uε(·))+2εd((ξρε,ηρε,uρε(·)),(ξε,ηε,uε(·)))≥0.
On the other hand, similarly to Lemma 2.5 we have
(2.35)limρ→0sup0≤t≤TE[ρ-1[xρε(t)-xε(t)]-x^ε(t)]2=0,limρ→0sup0≤t≤TE[ρ-1[yρε(t)-yε(t)]-y^ε(t)]2=0.
This leads to the following expansions:
(2.36)|E(ϕ(xρε(T))-a|2-|E(ϕ(xε(T))-a|2=2ρ〈Eϕ(xε(T))-a,E[ϕx(xε(T))x^ε(T)]〉+o(ρ),|E(h(yρε(0))-b|2-|E(h(yε(0))-b|2=2ρ〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉+o(ρ).
Applying the linearization technique, then
(2.37)E[ϕ(xρε(T))-ϕ(xε(T))]=E[∫0T‍ϕx(x(ξε+λρξ^ε,ηε+λρη^ε,uε(·)+λρu^ε)(T))dλ·ρx^ε(T)],E[γ(yρε(0))-γ(yε(0))]=E[∫0T‍γy(y(ξε+λρξ^ε,ηε+λρη^ε,uε(·)+λρu^ε(·))(0))dλ·ρy^ε(0)].
So we have the following expansions:
(2.38)(E[ϕ(x*(T))-ϕ(xρε(T)]+ε)2-(E[ϕ(x*(T))-ϕ(xε(T)]+ε)2=-2ρE[ϕx(xε(T))·ρx^Tε][E[ϕ(x*(T))-ϕ(xε(T)]+ε],(E[γ(y*(0)-γ(yρε(0)]+ε)2-(E[γ(y*(0))-γ(yε(0))]+ε)2=-2ρE[γy(yε(0))·ρy^0ε][E[γ(y*(0))-γ(yε(0))]+ε].
For the given ε, we consider the following four cases.
Case
1.There exists ρ0>0 such that
(2.39)E[ϕ(xρε(T))-ϕ(xε(T))]+ε>0,E[γ(yρε(0))-γ(yε(0))]+ε>0
for all ρ∈(0,ρ0).
In this case,(2.40)limρ→0Fε(ξρε,ηρε,uρε(·))-Fε(ξε,ηε,uε(·))ρ=limρ→01Fε(ξρε,ηρε,uρε(·))+Fε(ξε,ηε,uε(·))Fε2(ξρε,ηρε,uρε(·))-Fε2(ξε,ηε,uε(·))ρ=1Fε(ξε,ηε,uε(·)){〈Eψ(xε(T))-a,E[ψx(xε(T))x^ε(T)]〉+〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉-〈E[ϕ(x*(T))-ϕ(xε(T))]+ε,E[ϕx(xε(T))x^ε(T)]〉-〈E[γ(y*(0))-γ(yε(0))]+ε,E[γy(yε(0))y^ε(0)]〉}.
Dividing (2.34) by ρ and sending ρ to 0, we obtain
(2.41)h3εE〈ψx(xε(T)),x^ε(T)〉+h2εE〈hy(yε(0)),y^ε(0)〉+h1εE〈ϕx(xε(T)),x^ε(T)〉+h0εE〈γy(yε(0),y^ε(0)〉≥-2εd((ξε,ηε,uε(·)),(ξ,η,u(·))),
where
(2.42)h0ε=-1Fε(ξε,ηε,uε(·))[E[γ(y*(0))-γ(yε(0))]+ε]≤0,h1ε=-1Fε(ξε,ηε,uε(·))[E[ϕ(x*(T))-ϕ(xε(T))]+ε]≤0,h2ε=1Fε(ξε,ηε,uε(·))[Eh(yε(0))-b],h3ε=1Fε(ξε,ηε,uε(·))[Eψ(xε(T))-a].Case2.There exists a position sequence {ρn} satisfying ρn→0 such that
(2.43)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε≤0,E[γ(yρnε(0))-γ(y*(0))]+ε>0.
Then
(2.44)Fε(ξρnε,ηρnε,uρnε(·))={|Eψ(xρnε(T))-a|2+|Eh(yρnε(0))-b|2+(max(0,E[γ(y*(0))-γ(yρnε(0))]+ε))2}1/2.
For sufficiently large n, since Fε(·) is continuous, we conclude
(2.45)Fε(ξε,ηε,uε(·))={|Eψ(xε(T))-a|2}1/2+|Eh(yε(0))-b|2+(max(0,E[γ(y*(0))-γ(yε(0))]+ε))2}1/2.
Now
(2.46)limn→∞Fε(ξρnε,ηρnε,uρnε(·))-Fε(ξε,ηε,uε(·))ρn=limn→∞1Fε(ξρnε,ηρnε,uρnε(·))+Fε(ξε,ηε,uε(·))Fε2(ξρnε,ηρnε,uρnε(·))-Fε2(ξε,ηε,uε(·))ρn=1Fε(ξε,ηε,uε(·)){[γy(yε(0))y^ε(0)]〈Eψ(xε(T))-a,E[ψx(xε(T))x^ε(T)]〉+〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉-〈E[γ(y*(0))-γ(yε(0))]+ε,E[γy(yε(0))y^ε(0)]〉(}.
Similar to Case 1 we get
(2.47)h3εE〈ψx(xε(T)),x^ε(T)〉+h2εE〈hy(yε(0)),y^ε(0)〉+h0εE〈γy(yε(0)),y^ε(0)〉≥-2εd((ξε,ηε,uε(·)),(ξ,η,u(·))),
where
(2.48)h0ε=-1Fε(ξε,ηε,uε(·))[E[γ(y*(0))-γ(yε(0))]+ε]≤0,h1ε=0,h2ε=1Fε(ξε,ηε,uε(·))[Eh(yε(0))-b],h3ε=1Fε(ξε,ηε,uε(·))[Eψ(xε(T))-a].Case3.There exists a positive sequence {ρn} satisfying ρn→0 such that
(2.49)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε>0,E[γ(yρnε(0))-γ(y*(0))]+ε≤0.Case4.There exists a positive sequence {ρn} satisfying ρn→0 such that
(2.50)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε≤0,E[γ(yρnε(0))-γ(y*(0))]+ε≤0.
Similar techniques can be used for both Case 3 and Case 4.
In summary, for all those cases, we haveh0ε≤0,h1ε≤0 and |h0ε|2+|h1ε|2+|h2ε|2+|h3ε|2=1 by the definition of Fε(·). Then there exists a convergent subsequence of (h3ε,h2ε,h1ε,h0ε)  whose limit is denoted by (h3,h2,h1,h0). On the other hand, it is easy to check that x^ε(T)→0, y^ε(0)→0 as ε→0. Thus (2.32) holds.


### 2.3.3.   Maximum Principle

In this subsection we derive the maximum principle for the case wherel(x,z,y,q,u,t)=0, χ(x)=0, λ(y)=0 and then present the results for the general case in Section 2.3.4. To this end, we introduce the adjoint process (m(·),p(·)), and (n(·),δ(·)) associated with the optimal solution (ξ*,η*,u*(·)) to (2.13), which is the solution of the following time-symmetric FBDSDE:
(2.51)-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t))dBt-ptdWt,m(T)=-(h3ψx(xT*)+h1ϕx(xT*)),dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+gq*(t)δ(t))dWt-δtdBt,n(0)=h2hy(y0*)+h0γy(y0*),
where Fk*(t), fk*(t), Gk*(t), gk*(t) for k=x,y,z,q are defined as in (2.20). It is easy to check that there exist unique processes (m(·),p(·)), (n(·),δ(·)) which solve the above equations.Theorem 2.7.
We assume (H1)~(H4). Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η,u(·))∈U one has for any t∈[0,T],
(2.52)〈m(0),ξ-ξ*〉≤0,〈n(T),η-η*〉≥0,〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+gu*(t)δ(t),u(t)-u*(t)〉≥0.Proof.
For any(ξ,η,u(·))∈U, let (x^(·),z^(·),y^(·),q^(·)) be the solution to (2.20). Applying Lemma 2.1 to 〈m(t),x^(t)〉+〈n(t),y^(t)〉, we have
(2.53)E〈m(T),x^(T)〉+E〈n(T),y^(T)〉-E〈m(0),x^(0)〉-E〈n(0),y^(0)〉=E∫0T‍mt(-Fx*(t)x^(t)-Fz*(t)z^(t)-Fy*(t)y^(t)-Fq*(t)q^(t)-Fu*(t)u^(t))dt+E∫0T‍x^(t)(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t))dt+E∫0T‍z^(t)(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t))dt-E∫0T‍p(t)(Gx*(t)x^(t)+Gz*(t)z^(t)+Gy*(t)y^(t)+Gq*(t)q^(t)+Gu*(t)u^(t))dt-E∫0T‍n(t)(fx*(t)x^(t)+fz*(t)z^(t)+fy*(t)y^(t)+fq*(t)q^(t)+fu*(t)u^(t))dt+E∫0T‍y^(t)(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t))dt+E∫0T‍q^(t)(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+qq*(t)δ(t))dt-E∫0T‍δ(t)(gx*(t)x^(t)+gz*(t)z^(t)+gy*(t)y^(t)+gq*(t)q^(t)+gu*(t)u^(t))dt=-E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+gu*(t)δ(t),u^(t)〉dt.
This yields
(2.54)E〈n(T),y^(T)〉-E〈m(0),x^(0)〉+E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+δ(t)gu*(t),u(t)-u*(t)〉dt=-E〈m(T),x^(T)〉+E〈n(0),y^(0)〉=h3E〈ψx(x*(T)),x^(T)〉+h2E〈hy(y*(0)),y^(0)〉+h1E〈ϕx(x*(T)),x^(T)〉+h0E〈γy(y*(0)),y^(0)〉≥0.
For any (ξ,η,u(·))∈U, we get
(2.55)E〈n(T),η-η*〉-E〈m(0),ξ-ξ*〉+E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+δ(t)gu*(t),u(t)-u*(t)〉dt≥0.
Thus, it is easy to see that (2.52) holds.


### 2.3.4. The General Case

Define the Hamiltonian(2.56)H:ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d×ℝn×ℝn×l×ℝk×ℝk×d×[0,T]→ℝ
by
(2.57)H(x,z,y,q,u,m,p,n,δ,t)=〈F(t,x,z,y,q,u),m(t)〉+〈G(t,x,z,y,q,u),p(t)〉+〈f(t,x,z,y,q,u),n(t)〉+〈g(t,x,z,y,q,u),δ(t)〉+l(x,z,y,q,u,t).
Now we consider the general case where l(x,z,y,q,u,t)≠0, χ(x)≠0, λ(y)≠0.Since the proof of the maximum principle is essentially similar as in the preceding subsection, we only present the result without proof.Let(ξ*,η*,u*(·)) be optimal to (2.17) with (x*(·),z*(·),y*(·),q*(·)) being the corresponding optimal trajectory of (2.13). We define the following adjoint equations:
(2.58)-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t)+lx*(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t)+lz*(t))dBt-ptdWt,m(T)=-(h3ψx(xT*)+h1ϕx(xT*)),dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t)+ly*(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+gq*(t)δ(t)+lq*(t))dWt-δtdBt,n(0)=h2hy(y0*)+h0γy(y0*),
where la*(t)=la(x*(t),z*(t),y*(t),q*(t),u*(t),t), a=x,z,y,q, respectively.Theorem 2.8.
We assume (H1)~(H3). Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η,u(·))∈U, one has that the following inequalities hold, for any t∈[0,T]:
(2.59)〈m(0)-χx(ξ*),ξ-ξ*〉≤0,〈n(T)+λy(η*),η-η*〉≥0,〈Hu(t,x*(t),z*(t),y*(t),q*(t),u*(t),m(t),p(t),n(t),δ(t)),u(t)-u*(t)〉≥0.Remark 2.9.
Let us denote the boundary ofK1 by ∂K1. Set
(2.60)M≜{w∈Ω∣ξ*(ω)∈∂K1}.
Then
(2.61)m(0)-χx(ξ*)≤0a.s.onM,m(0)-χx(ξ*)=0a.s.onMc.
Similar analysis can be used to the boundaries of K2 and K.


## 2.3.1. Variational Equations

For(ξ1,η1,u1(·)),(ξ2,η2,u2(·))∈U, we define a metric in U by
(2.18)d(ξ1,η1,u1(·)),(ξ2,η2,u2(·))≜((E|ξ1-ξ2|2)1/2+E|η1-η2|2)1/2+(∥u1(·)-u2(·)∥2)1/2.
It is obvious that (U,d(·,·)) is a complete metric space.Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding state processes of (2.13). for all(ξ,η,u(·))∈U and for all 0≤ρ≤1,
(2.19)(ξρ,ηρ,uρ(·))≜(ξ*+ρ(ξ-ξ*),η*+ρ(η-η*),u*(·)+ρ(u(·)-u*(·)))∈U.
Let (xρ(·),zρ(·),yρ(·),qρ(·)) be the state processes of (2.13) associated with (ξρ,ηρ,uρ(·)).To derive the first-order necessary condition, we let(x^(·),z^(·),y^(·),q^(·)) be the solution of the following time-symmetric FBDSDE:
(2.20)-dx^(t)=(Fx*(t)x^(t)+Fz*(t)z^(t)+Fy*(t)y^(t)+Fq*(t)q^(t)+Fu*(t)(u(·)-u*(·)))dt+(Gx*(t)x^(t)+Gz*(t)z^(t)+Gy*(t)y^(t)+Gq*(t)q^(t)+Gu*(t)(u(·)-u*(·)))dWt-z^(t)dB(t),x^(0)=ξ-ξ*,-dy^(t)=(fx*(t)x^(t)+fz*(t)z^(t)+fy*(t)y^(t)+fq*(t)q^(t)+fu*(t)(u(·)-u*(·)))dt+(gx*(t)x^(t)+gz*(t)z^(t)+gy*(t)y^(t)+gq*(t)q^(t)+gu*(t)(u(·)-u*(·)))dBt-q^(t)dW(t),y^(T)=η-η*,
whereHk*(t)=Hk(t,x*(·),z*(·),y*(·),q*(·),u*(·)) for H=F,G,f,g, k=x,z,y,q,u, respectively. Equation (2.20) is called the variation equation.Set(2.21)x~ρ(t)=ρ-1[xρ(t)-x*(t)]-x^(t),z~ρ(t)=ρ-1[zρ(t)-z*(t)]-z^(t),y~ρ(t)=ρ-1[yρ(t)-y*(t)]-y^(t),q~ρ(t)=ρ-1[qρ(t)-q*(t)]-q^(t).
We have the following convergence.Lemma 2.5.
Assuming (H1)~(H3) one has(2.22)limρ→0sup0≤t≤TE[|x~ρ(t)|2]=0,limρ→0E[∫0T‍∥z~ρ(t)∥2dt]=0,limρ→0sup0≤t≤TE[|y~ρ(t)|2]=0,limρ→0E[∫0T‍∥q~ρ(t)∥2dt]=0.Proof.
From (2.13) and  (2.20), we have
(2.23)-dy~ρ(t)=ρ-1[ρfy*f(xρ(t),zρ(t),yρ(t),qρ(t),uρ(t),t)-f(x*(t),z*(t),y*(t),q*(t),u*(t),t)-ρfx*(t)x^(t)-ρfz*(t)z^(t)-ρfy*(t)y^(t)-ρfq*(t)q^(t)-ρfu*(t)(u(·)-u*(·))]dt+ρ-1[ρgz*(t)z^(t)gx*(t)x^(t)g(xρ(t),zρ(t),yρ(t),qρ(t),uρ(t),t)-g(x*(t),z*(t),y*(t),q*(t),u*(t),t)-ρgx*(t)x^(t)-ρgz*(t)z^(t)-ρgy*(t)y^(t)-ρgq*(t)q^(t)-ρgu*(t)(u(·)-u*(·))]dBt-q~ρ(t)dWt,y~ρ(T)=0.
Let
(2.24)Aρ(t)=∫01‍fx(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Bρ(t)=∫01‍fz(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Cρ(t)=∫01‍fy(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Dρ(t)=∫01‍fq(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,Eρ(t)=[Aρ(t)-fx*(t)]x^(t)+[Bρ(t)-fz*(t)]z^(t)+[Cρ(t)-fy*(t)]y^(t)+[Dρ(t)-fq*(t)]q^(t)+∫01‍[fu(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)-fu*(t)](u(t)-u*(t))dλ,A1ρ(t)=∫01‍gx(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,B1ρ(t)=∫01‍gz(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,C1ρ(t)=∫01‍gy(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,D1ρ(t)=∫01‍gq(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)dλ,E1ρ(t)=[A1ρ(t)-gx*(t)]x^(t)+[B1ρ(t)-gz*(t)]z^(t)+[C1ρ(t)-gy*(t)]y^(t)+[D1ρ(t)-gq*(t)]q^(t)+∫01‍[gu(x*(t)+λρ(x^(t)+x~ρ(t)),z*(t)+λρ(z^(t)+z~ρ(t)),y*(t)+λρ(y^(t)+y~ρ(t)),q*(t)+λρ(q^(t)+q~ρ(t)),u*(t)+λρ(u(t)-u*(t)),t)-gu*(t)](u(t)-u*(t))dλ.
Thus
(2.25)-dy~ρ(t)=(Aρ(t)x~ρ(t)+Bρ(t)z~ρ(t)+Cρ(t)y~ρ(t)+Dρ(t)q~ρ(t)+Eρ(t))dt+(A1ρ(t)x~ρ(t)+B1ρ(t)z~ρ(t)+C1ρ(t)y~ρ(t)+D1ρ(t)q~ρ(t)+E1ρ(t))dBt-q~ρ(t)dW(t),y~ρ(T)=0.
Using Lemma 2.1 to |y~ρ(t)|2, we get
(2.26)E|y~ρ(t)|2+E∫tT‍∥q~ρ(s)∥2ds=2E∫tT‍(y~ρ(t),Aρ(s)x~ρ(s)+Bρ(s)z~ρ(s)+Cρ(s)y~ρ(s)+Dρ(s)q~ρ(s)+Eρ(s))ds+E∫tT‍∥A1ρ(s)x~ρ(s)+B1ρ(s)z~ρ(s)+C1ρ(s)y~ρ(s)+D1ρ(s)q~ρ(s)+E1ρ(s)∥2ds≤K(E∫tT‍|y~ρ(s)|2ds+E∫tT‍|x~ρ(s)|2ds)+β1(E∫tT‍∥z~ρ(s)∥2ds+E∫tT‍∥q~ρ(s)∥2ds+Jρ),
where K>0, 0<β1<1/2 are constants and Jρ=E∫tT‍|Eρ(s)|2ds+E∫tT‍|E1ρ(s)|2ds.
Similar analysis shows that(2.27)-dx~ρ(t)=(A'ρ(t)x~ρ(t)+B'ρ(t)z~ρ(t)+C'ρ(t)y~ρ(t)+D'ρ(t)q~ρ(t)+E'ρ(t))dt+(A1'ρ(t)x~ρ(t)+B1'ρ(t)z~ρ(t)+C1'ρ(t)y~ρ(t)+D1'ρ(t)q~ρ(t)+E1'ρ(t))dWt-z~ρ(t)dB(t),x~ρ(0)=0,
where H'ρ(t) and H1'ρ(t) (H=A,B,C,DandE) are similarly defined as before.
It yields that(2.28)E|x~ρ(t)|2+E∫tT‍∥z~ρ(s)∥2ds=2E∫tT‍(x~ρ(t),A'ρ(s)x~ρ(s)+B'ρ(s)z~ρ(s)+C'ρ(s)y~ρ(s)+D'ρ(s)q~ρ(ts)+E'ρ(s))ds+E∫tT‍∥A1'ρ(s)x~ρ(s)+B1'ρ(s)z~ρ(s)+C1'ρ(s)y~ρ(s)+D1'ρ(s)q~ρ(s)+E1'ρ(s)∥2ds≤KE[∫tT‍|y~ρ(s)|2ds+∫tT‍|x~ρ(s)|2ds]+β2E[∫tT‍∥z~ρ(s)∥2ds+∫tT‍∥q~ρ(s)∥2ds+Jρ′],
where K>0, 0<β2<1/2 are constants and Jρ′=E[∫tT‍|E'ρ(s)|2ds+∫tT‍|E1'ρ(s)|2ds]. Since 0<β1+β2<1 and K>0, there exists K1>0 such that
(2.29)E|x~ρ(t)|2+E|y~ρ(t)|2+(1-β1-β2)(E∫tT‍∥z~ρ(s)∥2ds+E∫tT‍∥q~ρ(s)∥2ds)≤K1(E∫tT‍|x~ρ(s)|2ds+E∫tT‍|y~ρ(s)|2ds+Jρ+Jρ′).
Since the Lebesgue dominated convergence theorem implies limρ→0Jρ=0,limρ→0Jρ′=0, we obtain the result by Gronwall's inequality.


## 2.3.2. Variational Inequality

In this subsection, we apply Ekeland's variational principle [28] to deal with initial-terminal state constraints:
(2.30)E(ψ(xT(ξ,η,u(·))))=a,E(h(y0(ξ,η,u(·))))=b.
Define
(2.31)Fε((ξ,η,u(·)))≜{|E(ψ(xT(ξ,η,u(·))))-a|2+|E(h(y0(ξ,η,u(·))))-b|2+(max(0,E[ϕ(x*(T))-ϕ(xT(ξ,η,u(·)))]+ε))2+(max(0,E[γ(y*(0))-γ(y0(ξ,η,u(·)))]+ε))2}1/2,
where a and b are the given initial and terminal state constraints and ε is an arbitrary positive constant.It is easy to check that the mappings|E(ψ(xT(ξ,η,u(·))))-a|2, |E(h(y0(ξ,η,u(·))))-b|2, ϕ(xT(ξ,η,u(·))) and γ(y0(ξ,η,u(·))) are all continuous functionals from U to ℝ.Theorem 2.6.
Suppose (H1)~(H3). Let(ξ*,η*,u*(·)) be an optimal solution to (2.17). Then there exist h0,h1,h2,h3∈ℝ with h0,h1≤0 and |h0|+|h1|+|h2|+|h3|≠0 such that the following variational inequality holds:
(2.32)h3E〈ψx(x*(T)),x^(T)〉+h2E〈hy(y*(0)),y^(0)〉+h1E〈ϕx(x*(T)),x^(T)〉+h0E〈γy(y*(0)),y^(0)〉≥0,
where x^(T) is the solution x^(·) of (2.20) at time T and y^(0) is the solution y^(·) of (2.20) at time 0.Proof.
It is easy to check thatFε(·) is continuous on U such that
(2.33)Fε(ξ*,η*,u*(·))=2ε,Fε(ξ,η,u(·))>0,∀(ξ,η,u(·))∈U,Fε(ξ*,η*,u*(·))≤inf(ξ,η,u(·))∈UFε(ξ,η,u(·))+2ε.
Thus, from Ekeland's variational principle [28], ∃(ξε,ηε,uε(·))∈U such that(i)
Fε(ξε,ηε,uε(·))≤Fε(ξ*,η*,u*(·)),(ii)
d((ξ*,η*,u*(·)),(ξε,ηε,uε(·)))≤2ε,(iii)
Fε(ξ,η,u(·))+2εd((ξ,η,u(·)),(ξε,ηε,uε(·)))≥Fε(ξε,ηε,uε(·)),forall(ξ,η,u(·))∈U.For any (ξ,η,u(·))∈U and 0≤ρ≤1, set (ξρε,ηρε,uρε(·))=(ξε+ρ(ξ-ξε),ηε+ρ(η-ηε),uε(·)+ρ(u(·)-uε(·))). Let (xρε(·),yρε(·),zρε(·),qρε(·)) (resp., (xε(·),yε(·),zε(·),qε(·))) be the solution of (2.13) under (ξρε,ηρε,uρε(·)) (resp., (ξε,ηε,uε(·)), and let (x^ε(·),y^ε(·),z^ε(·),q^ε(·)) be the solution of (2.20) in which (ξ*,η*,u*(·)) is substituted by (ξε,ηε,uε(·)).
From (iii), we know that(2.34)Fε(ξρε,ηρε,uρε(·))-Fε(ξε,ηε,uε(·))+2εd((ξρε,ηρε,uρε(·)),(ξε,ηε,uε(·)))≥0.
On the other hand, similarly to Lemma 2.5 we have
(2.35)limρ→0sup0≤t≤TE[ρ-1[xρε(t)-xε(t)]-x^ε(t)]2=0,limρ→0sup0≤t≤TE[ρ-1[yρε(t)-yε(t)]-y^ε(t)]2=0.
This leads to the following expansions:
(2.36)|E(ϕ(xρε(T))-a|2-|E(ϕ(xε(T))-a|2=2ρ〈Eϕ(xε(T))-a,E[ϕx(xε(T))x^ε(T)]〉+o(ρ),|E(h(yρε(0))-b|2-|E(h(yε(0))-b|2=2ρ〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉+o(ρ).
Applying the linearization technique, then
(2.37)E[ϕ(xρε(T))-ϕ(xε(T))]=E[∫0T‍ϕx(x(ξε+λρξ^ε,ηε+λρη^ε,uε(·)+λρu^ε)(T))dλ·ρx^ε(T)],E[γ(yρε(0))-γ(yε(0))]=E[∫0T‍γy(y(ξε+λρξ^ε,ηε+λρη^ε,uε(·)+λρu^ε(·))(0))dλ·ρy^ε(0)].
So we have the following expansions:
(2.38)(E[ϕ(x*(T))-ϕ(xρε(T)]+ε)2-(E[ϕ(x*(T))-ϕ(xε(T)]+ε)2=-2ρE[ϕx(xε(T))·ρx^Tε][E[ϕ(x*(T))-ϕ(xε(T)]+ε],(E[γ(y*(0)-γ(yρε(0)]+ε)2-(E[γ(y*(0))-γ(yε(0))]+ε)2=-2ρE[γy(yε(0))·ρy^0ε][E[γ(y*(0))-γ(yε(0))]+ε].
For the given ε, we consider the following four cases.
Case
1.There exists ρ0>0 such that
(2.39)E[ϕ(xρε(T))-ϕ(xε(T))]+ε>0,E[γ(yρε(0))-γ(yε(0))]+ε>0
for all ρ∈(0,ρ0).
In this case,(2.40)limρ→0Fε(ξρε,ηρε,uρε(·))-Fε(ξε,ηε,uε(·))ρ=limρ→01Fε(ξρε,ηρε,uρε(·))+Fε(ξε,ηε,uε(·))Fε2(ξρε,ηρε,uρε(·))-Fε2(ξε,ηε,uε(·))ρ=1Fε(ξε,ηε,uε(·)){〈Eψ(xε(T))-a,E[ψx(xε(T))x^ε(T)]〉+〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉-〈E[ϕ(x*(T))-ϕ(xε(T))]+ε,E[ϕx(xε(T))x^ε(T)]〉-〈E[γ(y*(0))-γ(yε(0))]+ε,E[γy(yε(0))y^ε(0)]〉}.
Dividing (2.34) by ρ and sending ρ to 0, we obtain
(2.41)h3εE〈ψx(xε(T)),x^ε(T)〉+h2εE〈hy(yε(0)),y^ε(0)〉+h1εE〈ϕx(xε(T)),x^ε(T)〉+h0εE〈γy(yε(0),y^ε(0)〉≥-2εd((ξε,ηε,uε(·)),(ξ,η,u(·))),
where
(2.42)h0ε=-1Fε(ξε,ηε,uε(·))[E[γ(y*(0))-γ(yε(0))]+ε]≤0,h1ε=-1Fε(ξε,ηε,uε(·))[E[ϕ(x*(T))-ϕ(xε(T))]+ε]≤0,h2ε=1Fε(ξε,ηε,uε(·))[Eh(yε(0))-b],h3ε=1Fε(ξε,ηε,uε(·))[Eψ(xε(T))-a].Case2.There exists a position sequence {ρn} satisfying ρn→0 such that
(2.43)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε≤0,E[γ(yρnε(0))-γ(y*(0))]+ε>0.
Then
(2.44)Fε(ξρnε,ηρnε,uρnε(·))={|Eψ(xρnε(T))-a|2+|Eh(yρnε(0))-b|2+(max(0,E[γ(y*(0))-γ(yρnε(0))]+ε))2}1/2.
For sufficiently large n, since Fε(·) is continuous, we conclude
(2.45)Fε(ξε,ηε,uε(·))={|Eψ(xε(T))-a|2}1/2+|Eh(yε(0))-b|2+(max(0,E[γ(y*(0))-γ(yε(0))]+ε))2}1/2.
Now
(2.46)limn→∞Fε(ξρnε,ηρnε,uρnε(·))-Fε(ξε,ηε,uε(·))ρn=limn→∞1Fε(ξρnε,ηρnε,uρnε(·))+Fε(ξε,ηε,uε(·))Fε2(ξρnε,ηρnε,uρnε(·))-Fε2(ξε,ηε,uε(·))ρn=1Fε(ξε,ηε,uε(·)){[γy(yε(0))y^ε(0)]〈Eψ(xε(T))-a,E[ψx(xε(T))x^ε(T)]〉+〈Eh(yε(0))-b,E[hy(yε(0))y^ε(0)]〉-〈E[γ(y*(0))-γ(yε(0))]+ε,E[γy(yε(0))y^ε(0)]〉(}.
Similar to Case 1 we get
(2.47)h3εE〈ψx(xε(T)),x^ε(T)〉+h2εE〈hy(yε(0)),y^ε(0)〉+h0εE〈γy(yε(0)),y^ε(0)〉≥-2εd((ξε,ηε,uε(·)),(ξ,η,u(·))),
where
(2.48)h0ε=-1Fε(ξε,ηε,uε(·))[E[γ(y*(0))-γ(yε(0))]+ε]≤0,h1ε=0,h2ε=1Fε(ξε,ηε,uε(·))[Eh(yε(0))-b],h3ε=1Fε(ξε,ηε,uε(·))[Eψ(xε(T))-a].Case3.There exists a positive sequence {ρn} satisfying ρn→0 such that
(2.49)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε>0,E[γ(yρnε(0))-γ(y*(0))]+ε≤0.Case4.There exists a positive sequence {ρn} satisfying ρn→0 such that
(2.50)E[ϕ(xρnε(T))-ϕ(x*(T))]+ε≤0,E[γ(yρnε(0))-γ(y*(0))]+ε≤0.
Similar techniques can be used for both Case 3 and Case 4.
In summary, for all those cases, we haveh0ε≤0,h1ε≤0 and |h0ε|2+|h1ε|2+|h2ε|2+|h3ε|2=1 by the definition of Fε(·). Then there exists a convergent subsequence of (h3ε,h2ε,h1ε,h0ε)  whose limit is denoted by (h3,h2,h1,h0). On the other hand, it is easy to check that x^ε(T)→0, y^ε(0)→0 as ε→0. Thus (2.32) holds.


## 2.3.3.   Maximum Principle

In this subsection we derive the maximum principle for the case wherel(x,z,y,q,u,t)=0, χ(x)=0, λ(y)=0 and then present the results for the general case in Section 2.3.4. To this end, we introduce the adjoint process (m(·),p(·)), and (n(·),δ(·)) associated with the optimal solution (ξ*,η*,u*(·)) to (2.13), which is the solution of the following time-symmetric FBDSDE:
(2.51)-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t))dBt-ptdWt,m(T)=-(h3ψx(xT*)+h1ϕx(xT*)),dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+gq*(t)δ(t))dWt-δtdBt,n(0)=h2hy(y0*)+h0γy(y0*),
where Fk*(t), fk*(t), Gk*(t), gk*(t) for k=x,y,z,q are defined as in (2.20). It is easy to check that there exist unique processes (m(·),p(·)), (n(·),δ(·)) which solve the above equations.Theorem 2.7.
We assume (H1)~(H4). Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η,u(·))∈U one has for any t∈[0,T],
(2.52)〈m(0),ξ-ξ*〉≤0,〈n(T),η-η*〉≥0,〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+gu*(t)δ(t),u(t)-u*(t)〉≥0.Proof.
For any(ξ,η,u(·))∈U, let (x^(·),z^(·),y^(·),q^(·)) be the solution to (2.20). Applying Lemma 2.1 to 〈m(t),x^(t)〉+〈n(t),y^(t)〉, we have
(2.53)E〈m(T),x^(T)〉+E〈n(T),y^(T)〉-E〈m(0),x^(0)〉-E〈n(0),y^(0)〉=E∫0T‍mt(-Fx*(t)x^(t)-Fz*(t)z^(t)-Fy*(t)y^(t)-Fq*(t)q^(t)-Fu*(t)u^(t))dt+E∫0T‍x^(t)(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t))dt+E∫0T‍z^(t)(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t))dt-E∫0T‍p(t)(Gx*(t)x^(t)+Gz*(t)z^(t)+Gy*(t)y^(t)+Gq*(t)q^(t)+Gu*(t)u^(t))dt-E∫0T‍n(t)(fx*(t)x^(t)+fz*(t)z^(t)+fy*(t)y^(t)+fq*(t)q^(t)+fu*(t)u^(t))dt+E∫0T‍y^(t)(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t))dt+E∫0T‍q^(t)(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+qq*(t)δ(t))dt-E∫0T‍δ(t)(gx*(t)x^(t)+gz*(t)z^(t)+gy*(t)y^(t)+gq*(t)q^(t)+gu*(t)u^(t))dt=-E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+gu*(t)δ(t),u^(t)〉dt.
This yields
(2.54)E〈n(T),y^(T)〉-E〈m(0),x^(0)〉+E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+δ(t)gu*(t),u(t)-u*(t)〉dt=-E〈m(T),x^(T)〉+E〈n(0),y^(0)〉=h3E〈ψx(x*(T)),x^(T)〉+h2E〈hy(y*(0)),y^(0)〉+h1E〈ϕx(x*(T)),x^(T)〉+h0E〈γy(y*(0)),y^(0)〉≥0.
For any (ξ,η,u(·))∈U, we get
(2.55)E〈n(T),η-η*〉-E〈m(0),ξ-ξ*〉+E∫0T‍〈m(t)Fu*(t)+p(t)Gu*(t)+n(t)fu*(t)+δ(t)gu*(t),u(t)-u*(t)〉dt≥0.
Thus, it is easy to see that (2.52) holds.


## 2.3.4. The General Case

Define the Hamiltonian(2.56)H:ℝn×ℝn×l×ℝk×ℝk×d×ℝn×d×ℝn×ℝn×l×ℝk×ℝk×d×[0,T]→ℝ
by
(2.57)H(x,z,y,q,u,m,p,n,δ,t)=〈F(t,x,z,y,q,u),m(t)〉+〈G(t,x,z,y,q,u),p(t)〉+〈f(t,x,z,y,q,u),n(t)〉+〈g(t,x,z,y,q,u),δ(t)〉+l(x,z,y,q,u,t).
Now we consider the general case where l(x,z,y,q,u,t)≠0, χ(x)≠0, λ(y)≠0.Since the proof of the maximum principle is essentially similar as in the preceding subsection, we only present the result without proof.Let(ξ*,η*,u*(·)) be optimal to (2.17) with (x*(·),z*(·),y*(·),q*(·)) being the corresponding optimal trajectory of (2.13). We define the following adjoint equations:
(2.58)-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+fx*(t)n(t)+gx*(t)δ(t)+lx*(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+fz*(t)n(t)+gz*(t)δ(t)+lz*(t))dBt-ptdWt,m(T)=-(h3ψx(xT*)+h1ϕx(xT*)),dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+gy*(t)δ(t)+ly*(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+gq*(t)δ(t)+lq*(t))dWt-δtdBt,n(0)=h2hy(y0*)+h0γy(y0*),
where la*(t)=la(x*(t),z*(t),y*(t),q*(t),u*(t),t), a=x,z,y,q, respectively.Theorem 2.8.
We assume (H1)~(H3). Let(ξ*,η*,u*(·)) be optimal and let (x*(·),z*(·),y*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η,u(·))∈U, one has that the following inequalities hold, for any t∈[0,T]:
(2.59)〈m(0)-χx(ξ*),ξ-ξ*〉≤0,〈n(T)+λy(η*),η-η*〉≥0,〈Hu(t,x*(t),z*(t),y*(t),q*(t),u*(t),m(t),p(t),n(t),δ(t)),u(t)-u*(t)〉≥0.Remark 2.9.
Let us denote the boundary ofK1 by ∂K1. Set
(2.60)M≜{w∈Ω∣ξ*(ω)∈∂K1}.
Then
(2.61)m(0)-χx(ξ*)≤0a.s.onM,m(0)-χx(ξ*)=0a.s.onMc.
Similar analysis can be used to the boundaries of K2 and K.


## 3. Applications

In this section, we give three specific cases to illustrate the applications of our obtained results.


### 3.1. System Composed of a Forward SDE and a BDSDE

Classical Formulation
For givenξ∈L2(Ω,ℱ0,P;ℝn) and u(·)∈U[0,T], we consider the following controlled system composed of a FSDE and a BDSDE:
(3.1)dy(t)=b-(t,y(t),u(t))dt+σ(t,y(t),u(t))dWt,y(0)=b,-dx(t)=f-(t,x(t),z(t),y(t),u(t))dt+g-(t,x(t),z(t),y(t),u(t))dWt-z(t)dBt,x(0)=ξ,
where b∈Rk is given, x(0)=ξ∈K1, a.s., where K1 is a given nonempty convex subset in Rn.
Set the mappings(3.2)b-:Ω×[0,T]×ℝk×ℝn×d→ℝk,σ:Ω×[0,T]×ℝk×ℝn×d→ℝk×d,f-:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝn×d→ℝn,g-:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝn×d→ℝn×d.
In this case, we regard u(·) and ξ as the control variables. Define the following cost function:
(3.3)J(ξ,u(·))=E[∫0T‍l-(t,x(t),z(t),y(t),u(t))dt+χ(ξ)+λ(y(T))+ϕ(x(T))],
where
(3.4)l-:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝn×d→ℝ,χ:ℝn→ℝ,λ:ℝk→ℝ,ϕ:ℝn→ℝ.
We assume(H1)
b-,σ,f-,g-,l-,χ,λ, and ϕ are continuous in their arguments and continuously differentiable in (x,z,y,u).(H2)
There exist constantsC>0 and 0<α<1/2 such that for any (ω,t)∈Ω×[0,T],  u∈ℝn×d, (x1,z1,y1),(x2,z2,y2)∈ℝn×ℝn×l×ℝk the following conditions hold:
(3.5)|f-(t,x1,z1,y1,u)-f-(t,x2,z2,y2,u)|2≤C(|x1-x2|2+∥z1-z2∥2+|y1-y2|2),∥g-(t,x1,z1,y1,u)-g-(t,x2,z2,y2,u)∥2≤C(|x1-x2|2+|y1-y2|2)+α∥z1-z2∥2.(H3)
The derivatives ofb-,σ,f-,g- in (x,y,z,u) are bounded; the derivatives of l- in (x,y,z,u) are bounded by C(1+|x|+|y|+|z|+∥u∥); the derivatives of χ and ϕ in x are bounded by C(1+|x|); the derivatives of λ in y are bounded by C(1+|y|).Then, for givenξ∈L2(Ω,ℱ0,P;ℝn) and u(·)∈U[0,T], there exists a unique triple
(3.6)(x(·),y(·),z(·))∈M2(0,T;ℝn)×M2(0,T;ℝk)×M2(0,T;ℝn×l).
which solves (3.1).We assume an additional terminal state constrainty(T)=η∈K2, a.s., where K2 is a given nonempty convex subset in ℝk. Our stochastic control problem is
(3.7)infJ(ξ,u(·))subjecttou(·)∈U[0,T];ξ∈K1,a.s,η∈K2,a.s.Backward Formulation
From now on, we give an equivalent backward formulation of the previously mentioned stochastic optimal problem (3.7). To do so we need an additional assumption:(H4)
there existsα>0 such that |σ(y,u1,t)-σ(y,u2,t)|≥α|u1-u2| for all y∈ℝk,t∈[0,T] and u1, u2∈ℝn×d.
Note (H1) and (H4)  imply the mappingu→σ(y,u,t) is a bijection from ℝn×d onto itself for any (y,t).
Letq≡σ(y,u,t)  and denote the inverse function by u=σ~(y,q,t). Then system (3.1) can be rewritten as
(3.8)-dy(t)=f(t,y(t),q(t))dt-q(t)dWt,y(0)=b,-dx(t)=F(t,x(t),z(t),y(t),q(t))dt+G(t,x(t),z(t),y(t)q(t))dWt-z(t)dBt,x(0)=ξ,
where f(t,y,q)=-b(t,y,σ~(y,q,t)) and F(t,x,z,y,q)=f-(t,x,z,y,σ~(y,q,t)), G(t,x,z,y,q)=g-(t,x,z,y,σ~(y,q,t)).
A key observation that inspires our approach of solving problem (3.7) is that, since u→σ(x,u,t) is a bijection, q(·) can be regarded as the control; moreover, by the BSDE theory selecting q(·) is equivalent to selecting the terminal value y(T). Hence we introduce the following “controlled” system:
(3.9)-dy(t)=f(t,y(t),q(t))dt-q(t)dWt,y(t)=η,-dx(t)=F(t,x(t),z(t),y(t),q(t))dt+G(t,x(t),z(t),y(t),q(t))dWt-z(t)dBt,x(0)=ξ,
where the control variables are the random variables ξ and η to be chosen from the following set:
(3.10)U={(ξ,η)∣ξ∈K1,a.s.E|ξ|2<∞,η∈K2,a.s.E|η|2<∞}.
For each (ξ,η)∈U, consider the following cost:
(3.11)J(ξ,η)=E[∫0T‍l(t,x(t),z(t),y(t),q(t))dt+χ(ξ)+λ(η)+ϕ(x(T))],
where l(t,x,z,y,q)=l-(t,x,z,y,σ~(y,q,t)).
This gives rise to the following auxiliary optimization problem:(3.12)infJ(ξ,η)subjectto(ξ,η)∈U;y(0)=b,
where y(0)(ξ,η) is the solution of (3.9) at time 0 under ξ and η.
It is clear that the original problem (3.7) is equivalent to the auxiliary one (3.12).
Hence, hereafter we focus ourselves on solving (3.12). The advantage of doing this is that, since ξ and η now are the control variables, the state constraint in (3.7) becomes a control constraint in (3.12), whereas it is well known in control theory that a control constraint is much easier to deal with than a state constraint. There is, nonetheless, a cost of doing; that is, the original initial condition y(0)(ξ,η)=b now becomes a constraint, as shown in (3.12).
From now on, we denote the solution of (3.9) by (x(ξ,η)(·),y(ξ,η)(·),q(ξ,η)(·),z(ξ,η)(·)), whenever necessary, to show the dependence on (ξ,η). We also denote x(ξ,η)(0) and y(ξ,η)(0) by x(0)(ξ,η) and y(0)(ξ,η), respectively. Finally, it is easy to check that f,F,G and l satisfy similar conditions in Assumptions (H1)−(H3).
We note that this is a special case of (2.17), so by the same method we have the following result.Stochastic Maximum Principle
We define(3.13)dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+ly*(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+lq*(t))dWt,n(0)=h2,-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+lx*(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+lz*(t))dBt-ptdWt,m(T)=-h1ϕx(x*(T)),
where Hk*(t)=Hk(t,x*(t),z*(t),y*(t),q*(t)) (H=F,G,l) and fx*(t)=fx(t,x*(t),z*(t)), fq*(t)=fq(t,x*(t),z*(t)), h1 and h2 are defined as in (2.32). It is easy to check that there exist unique processes n(·) and (m(·),p(·)) which solve the above equations.Theorem 3.1.
One assumes (H1)~(H4). Let (ξ*,η*) be optimal to (3.12), and let (x*(·),y*(·),z*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η)∈U, one has that the following inequalities hold:
(3.14)〈m(0)-χx(ξ*),ξ-ξ*〉≤0,〈n(T)+λy(η*),η-η*〉≥0.


### 3.2.   System Composed of a BDSDE with State Constraints

Although this case describes a controlled BDSDE system with state constraints, it seems trivial. Thus, we only give a brief illustration. Givenη∈L2(Ω,ℱT,P;ℝk) and u(·)∈U[0,T], consider the following BDSDE:
(3.15)-dy(t)=f(t,y(t),q(t),u(t))dt+g(t,y(t),q(t),u(t))dBt-q(t)dWt,0≤t≤T,y(T)=η.
For given u(·)∈U[0,T] and f,g satisfying (H2) and (H3), from the Theorem 1.1 in [15], it is easy to check that there exists a unique solution (y(·),q(·)) of (3.15).Note thaty(0)(η,u(·)) is a ℱ0,TB-measurable variable, and
(3.16)E(h(y(η,u(·))(0)))=b.
Now we regard η and u(·) as the control variables to be chosen from the following set:
(3.17)U={(η,u(·))∣η∈K1,a.s.,E[|η|2]<∞,u(·)∈U[0,T]}.
For each (η,u(·))∈U, consider the following cost function:
(3.18)J(η,u(·))=E[∫0T‍l(t,y(t),q(t),u(t))dt+λ(η)+γ(y(0))],
which gives rise to the following optimization problem:
(3.19)inf(η,u(·))∈UJ(η,u(·))subjecttoE(h(y(η,u(·))(0)))=b.Maximum Principle
Let(3.20)H(y,q,u,n,δ)≜〈f(t,y,q,u),n(t)〉+〈g(t,y,q,u),δ(t)〉+l(t,y,q,u).
Then the adjoint equation is
(3.21)dn(t)=(fy*(t)n(t)+gy*(t)δ(t)+ly*(t))dt-δ(t)dBt+(fq*(t)n(t)+gq*(t)δ(t)+lq*(t))dWt,n(0)=h2hy(y*(0))+h0γy(y*(0)),
where Hk*(t)=Hk(t,y*(t),q*(t)) for H=f,g; k=y,q, respectively.Theorem 3.2.
One assumes (H1)~(H3). Let η* and u*(·) be the optimal controls and let (y*(·),z*(·)) be the corresponding optimal trajectory. Then one has
(3.22)〈n(T)+λy(η*),η-η*〉≥0,〈Hu(t,y*,q*,u*,n,δ),u(t)-u*(t)〉≥0.


### 3.3. Backward Doubly Stochastic LQ Problem without State Constraints

Consider the following linear system:(3.23)-dx(t)=[A(t)x(t)+B(t)z(t)+C(t)y(t)+D(t)q(t)+E(t)u(t)]dt+[A′(t)x(t)+B′(t)z(t)+C′(t)y(t)+D′(t)q(t)+E′(t)u(t)]dWt-z(t)dBt,x(0)=x,-dy(t)=[A′′(t)x(t)+B′′(t)z(t)+C′′(t)y(t)+D′′(t)q(t)+E′′(t)u(t)]dt+[A′′′(t)x(t)+B′′′(t)z(t)+C′′′(t)y(t)+D′′′(t)q(t)+E′′′(t)u(t)]dBt-q(t)dWt,y(T)=y,
where x and y are given constants and H,H′,H′′,H′′′(H=A,B,C,DandE) are corresponding matrices.The cost function (2.16) becomes
(3.24)l(t,x(t),z(t),y(t),q(t),u(t))=12F(t)x(t)·x(t)+12G(t)z(t)·z(t)+12H(t)y(t)·y(t)+12I(t)q(t)·q(t)+12J(t)u(t)·u(t),χ(x)=0,λ(y)=0,ϕ(x)=12U(t)x2,γ(y)=12Q(t)y2,
where all functions of t are bounded and F(t),G(t),H(t),I(t) are symmetric non-negative definite, J(t),U(t),Q(t) are symmetric uniformly positive definite.Then from (2.51), the adjoint equations become
(3.25)-dm(t)=-[A(t)m(t)+A′(t)p(t)+A′′(t)n(t)+A′′′(t)δ(t)+F(t)x*(t)]dt+[B(t)m(t)+B′(t)p(t)+B′′(t)n(t)+B′′′(t)δ(t)+G(t)z*(t)]dBt-p(t)dWt,m(T)=-U(T)x*(T),dn(t)=[C(t)m(t)+C′(t)p(t)+C′′(t)n(t)+C′′′(t)δ(t)+H(t)y*(t)]dt+[D(t)m(t)+D′(t)p(t)+D′′(t)n(t)+D′′′(t)δ(t)+I(t)q*(t)]dWt-δ(t)dBt,n(0)=Q(0)y*(0).
Define
(3.26)H(t,x,z,y,q,u,m,p,n,δ)≜〈F(t,x,z,y,q,u),m(t)〉+〈G(t,x,z,y,q,u),p(t)〉+〈f(t,x,z,y,q,u),n(t)〉+〈g(t,x,z,y,q,u),δ(t)〉+l(t,x,z,y,q,u).
Suppose that K is an open set. Then we have the following result:
(3.27)Hu(t,x*,z*,y*,q*,u*,m,p,n,δ)=E(t)m(t)+E′(t)p(t)+E′′(t)n(t)+E′′′(t)δ(t)+J(t)u(t)=0.
Thus,
(3.28)u*(t)=J-1(t)[E(t)m(t)+E′(t)p(t)+E′′(t)n(t)+E′′′(t)δ(t)].
However, the maximum principle gives only the necessary condition for optimal control. We also have the following theorem.Theorem 3.3.
The functionu*(t)=J-1(t)[E(t)m(t)+E′(t)p(t)+E′′(t)n(t)+E′′′(t)δ(t)] is the unique optimal control for backward doubly stochastic LQ problems, where (x*(·),z*(·),y*(·),q*(·)) and (m(·),p(·),n(·),δ(·)) are solutions of above equations.Proof.
First let us prove that theu*(·) is the optimal control. For all v(·)∈K, let (xv(·),zv(·),yv(·),qv(·)) be the corresponding trajectory of (3.23), we get
(3.29)J(v(·))-J(u*(·))=12{∫0T‍[q*(t)〈F(t)xv(t),xv(t)〉-〈F(t)x*(t),x*(t)〉+〈G(t)zv(t),zv(t)〉-〈G(t)z*(t),z*(t)〉+〈H(t)yv(t),yv(t)〉-〈H(t)y*(t),y*(t)〉+〈I(t)qv(t),qv(t)〉-〈I(t)q*(t),q*(t)〉+〈J(t)v(t),v(t)〉-〈J(t)u*(t),u*(t)〉]dt+〈U(T)xv(T),xv(T)〉-〈U(T)x*(T),x*(T)〉+〈Q(0)yv(0),yv(0)〉-〈Q(0)y*(0),y*(0)〉∫0T‍}≥E{∫0T‍[q*〈F(t)x*(t),xv(t)-x*(t)〉+〈G(t)z*(t),zv(t)-z*(t)〉+〈H(t)y*(t),yv(t)-y*(t)〉+〈I(t)q*(t),qv(t)-q*(t)〉+〈J(t)u*(t),v(t)-u*(t)〉]dt+〈U(T)x*(T),xv(T)-x*(T)〉+〈Q(0)y*(0),yv(0)-y*(0)〉∫0T‍}.
Using Lemma 2.1 to 〈xv(t)-x*(t),m(t)〉+〈yv(t)-y*(t),n(t)〉, we obtain
(3.30)〈U(T)x*(T),xv(T)-x*(T)〉+〈Q(0)y*(0),yv(0)-y*(0)〉=-E∫0T‍[E′′′〈F(t)x*(t),xv(t)-x*(t)〉+〈G(t)z*(t),zv(t)-z*(t)〉+〈H(t)y*(t),yv(t)-y*(t)〉+〈I(t)q*(t),qv(t)-q*(t)〉〈J(t)u*(t),v(t)-u*(t)〉+〈E(t)m(t),v(t)-u*(t)〉+〈E′(t)p(t),v(t)-u*(t)〉+〈E′′(t)n(t),v(t)-u*(t)〉+〈E′′′(t)δ(t),v(t)-u*(t)〉]dt.
So by the definition of u*(t),
(3.31)J(v(·))-J(u*(·))≥E∫0T[〈E(t)m(t),v(t)-u*(t)〉+〈E′(t)p(t),v(t)-u*(t)〉+〈E′′(t)n(t),v(t)-u*(t)〉+〈E′′′(t)δ(t),v(t)-u*(t)〉+〈J(t)u*(t),v(t)-u*(t)〉]dt=0.
From the arbitrariness of v(·)∈K, we deduce u*(t) is the optimal control.
The proof of the uniqueness of the optimal control is classical. Assume thatu1(·) and u2(·) are both optimal controls, and the corresponding trajectories are (x1(·),z1(·),y1(·),q1(·)) and (x2(·),z2(·),y2(·),q2(·)). By the uniqueness of solutions of (3.23), we know that the trajectory corresponding to (u1(·)+u2(·))/2 is ((x1(·)+x2(·))/2,(z1(·)+z2(·))/2,(y1(·)+y2(·))/2,(q1(·)+q2(·))/2), and notice that J(t),U(t),Q(t) are positive, F(t),G(t),H(t),I(t) are nonnegative, we have
(3.32)J(u1(·))=J(u2(·))=α≥0,2α=J(u1(·))+J(u2(·))≥2J(u1(·)+u2(·)2)+2E∫0T‍〈J(t)u1(·)-u2(·)2,u1(·)-u2(·)2〉dt≥2α+β2E∫0T‍|u1(·)-u2(·)|2dt,
here β>0. So
(3.33)E∫0T‍|u1(·)-u2(·)|2dt≤0,
which shows that u1(·)=u2(·).Example 3.4.
Consider the following backward doubly stochastic LQ problem, wheret∈[0,1],u(·)∈U[0,1]=[-1,1] and n=k=d=l=1:
(3.34)-dx(t)=12(y(t)+2u(t))dWt-z(t)dBt,x(0)=0,-dy(t)=12(x(t)+u(t))dBt-q(t)dWt,y(1)=0.
We want to minimize the following cost function:
(3.35)J(u(·))=E∫01‍(x(s)2-y(s)2+z(s)2-q(s)2+2x(s)u(s)-4y(s)u(s))ds+Ex(1)2+Ey(0)2.
From (3.34), we get, for t∈[0,1],
(3.36)x(t)=-∫0t‍(y(s)+2u(s))dWs+∫0t‍z(s)dBs,y(t)=∫t1‍(x(s)+u(s))dBs-∫t1‍q(s)dWs.
By substituting x(·) and y(·) into the cost function, we obtain
(3.37)J(u(·))=E∫01‍3u(s)2ds.
Thus, the optimal control is u*(t)≡0,t∈[0,1] with the optimal state trajectory
(3.38)(x*(t),z*(t),y*(t),q*(t))≡0,t∈[0,1].
The adjoint equations are
(3.39)dm(t)=n(t)dt+p(t)dWt,m(1)=0,dn(t)=p(t)dt-δ(t)dBt,n(0)=0.
It is obvious that (m(·),p(·),n(·),δ(·))=(0,0,0,0) is the unique solution of the above equation.


## 3.1. System Composed of a Forward SDE and a BDSDE

Classical Formulation
For givenξ∈L2(Ω,ℱ0,P;ℝn) and u(·)∈U[0,T], we consider the following controlled system composed of a FSDE and a BDSDE:
(3.1)dy(t)=b-(t,y(t),u(t))dt+σ(t,y(t),u(t))dWt,y(0)=b,-dx(t)=f-(t,x(t),z(t),y(t),u(t))dt+g-(t,x(t),z(t),y(t),u(t))dWt-z(t)dBt,x(0)=ξ,
where b∈Rk is given, x(0)=ξ∈K1, a.s., where K1 is a given nonempty convex subset in Rn.
Set the mappings(3.2)b-:Ω×[0,T]×ℝk×ℝn×d→ℝk,σ:Ω×[0,T]×ℝk×ℝn×d→ℝk×d,f-:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝn×d→ℝn,g-:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝn×d→ℝn×d.
In this case, we regard u(·) and ξ as the control variables. Define the following cost function:
(3.3)J(ξ,u(·))=E[∫0T‍l-(t,x(t),z(t),y(t),u(t))dt+χ(ξ)+λ(y(T))+ϕ(x(T))],
where
(3.4)l-:Ω×[0,T]×ℝn×ℝn×l×ℝk×ℝn×d→ℝ,χ:ℝn→ℝ,λ:ℝk→ℝ,ϕ:ℝn→ℝ.
We assume(H1)
b-,σ,f-,g-,l-,χ,λ, and ϕ are continuous in their arguments and continuously differentiable in (x,z,y,u).(H2)
There exist constantsC>0 and 0<α<1/2 such that for any (ω,t)∈Ω×[0,T],  u∈ℝn×d, (x1,z1,y1),(x2,z2,y2)∈ℝn×ℝn×l×ℝk the following conditions hold:
(3.5)|f-(t,x1,z1,y1,u)-f-(t,x2,z2,y2,u)|2≤C(|x1-x2|2+∥z1-z2∥2+|y1-y2|2),∥g-(t,x1,z1,y1,u)-g-(t,x2,z2,y2,u)∥2≤C(|x1-x2|2+|y1-y2|2)+α∥z1-z2∥2.(H3)
The derivatives ofb-,σ,f-,g- in (x,y,z,u) are bounded; the derivatives of l- in (x,y,z,u) are bounded by C(1+|x|+|y|+|z|+∥u∥); the derivatives of χ and ϕ in x are bounded by C(1+|x|); the derivatives of λ in y are bounded by C(1+|y|).Then, for givenξ∈L2(Ω,ℱ0,P;ℝn) and u(·)∈U[0,T], there exists a unique triple
(3.6)(x(·),y(·),z(·))∈M2(0,T;ℝn)×M2(0,T;ℝk)×M2(0,T;ℝn×l).
which solves (3.1).We assume an additional terminal state constrainty(T)=η∈K2, a.s., where K2 is a given nonempty convex subset in ℝk. Our stochastic control problem is
(3.7)infJ(ξ,u(·))subjecttou(·)∈U[0,T];ξ∈K1,a.s,η∈K2,a.s.Backward Formulation
From now on, we give an equivalent backward formulation of the previously mentioned stochastic optimal problem (3.7). To do so we need an additional assumption:(H4)
there existsα>0 such that |σ(y,u1,t)-σ(y,u2,t)|≥α|u1-u2| for all y∈ℝk,t∈[0,T] and u1, u2∈ℝn×d.
Note (H1) and (H4)  imply the mappingu→σ(y,u,t) is a bijection from ℝn×d onto itself for any (y,t).
Letq≡σ(y,u,t)  and denote the inverse function by u=σ~(y,q,t). Then system (3.1) can be rewritten as
(3.8)-dy(t)=f(t,y(t),q(t))dt-q(t)dWt,y(0)=b,-dx(t)=F(t,x(t),z(t),y(t),q(t))dt+G(t,x(t),z(t),y(t)q(t))dWt-z(t)dBt,x(0)=ξ,
where f(t,y,q)=-b(t,y,σ~(y,q,t)) and F(t,x,z,y,q)=f-(t,x,z,y,σ~(y,q,t)), G(t,x,z,y,q)=g-(t,x,z,y,σ~(y,q,t)).
A key observation that inspires our approach of solving problem (3.7) is that, since u→σ(x,u,t) is a bijection, q(·) can be regarded as the control; moreover, by the BSDE theory selecting q(·) is equivalent to selecting the terminal value y(T). Hence we introduce the following “controlled” system:
(3.9)-dy(t)=f(t,y(t),q(t))dt-q(t)dWt,y(t)=η,-dx(t)=F(t,x(t),z(t),y(t),q(t))dt+G(t,x(t),z(t),y(t),q(t))dWt-z(t)dBt,x(0)=ξ,
where the control variables are the random variables ξ and η to be chosen from the following set:
(3.10)U={(ξ,η)∣ξ∈K1,a.s.E|ξ|2<∞,η∈K2,a.s.E|η|2<∞}.
For each (ξ,η)∈U, consider the following cost:
(3.11)J(ξ,η)=E[∫0T‍l(t,x(t),z(t),y(t),q(t))dt+χ(ξ)+λ(η)+ϕ(x(T))],
where l(t,x,z,y,q)=l-(t,x,z,y,σ~(y,q,t)).
This gives rise to the following auxiliary optimization problem:(3.12)infJ(ξ,η)subjectto(ξ,η)∈U;y(0)=b,
where y(0)(ξ,η) is the solution of (3.9) at time 0 under ξ and η.
It is clear that the original problem (3.7) is equivalent to the auxiliary one (3.12).
Hence, hereafter we focus ourselves on solving (3.12). The advantage of doing this is that, since ξ and η now are the control variables, the state constraint in (3.7) becomes a control constraint in (3.12), whereas it is well known in control theory that a control constraint is much easier to deal with than a state constraint. There is, nonetheless, a cost of doing; that is, the original initial condition y(0)(ξ,η)=b now becomes a constraint, as shown in (3.12).
From now on, we denote the solution of (3.9) by (x(ξ,η)(·),y(ξ,η)(·),q(ξ,η)(·),z(ξ,η)(·)), whenever necessary, to show the dependence on (ξ,η). We also denote x(ξ,η)(0) and y(ξ,η)(0) by x(0)(ξ,η) and y(0)(ξ,η), respectively. Finally, it is easy to check that f,F,G and l satisfy similar conditions in Assumptions (H1)−(H3).
We note that this is a special case of (2.17), so by the same method we have the following result.Stochastic Maximum Principle
We define(3.13)dn(t)=(Fy*(t)m(t)+Gy*(t)p(t)+fy*(t)n(t)+ly*(t))dt+(Fq*(t)m(t)+Gq*(t)p(t)+fq*(t)n(t)+lq*(t))dWt,n(0)=h2,-dm(t)=-(Fx*(t)m(t)+Gx*(t)p(t)+lx*(t))dt+(Fz*(t)m(t)+Gz*(t)p(t)+lz*(t))dBt-ptdWt,m(T)=-h1ϕx(x*(T)),
where Hk*(t)=Hk(t,x*(t),z*(t),y*(t),q*(t)) (H=F,G,l) and fx*(t)=fx(t,x*(t),z*(t)), fq*(t)=fq(t,x*(t),z*(t)), h1 and h2 are defined as in (2.32). It is easy to check that there exist unique processes n(·) and (m(·),p(·)) which solve the above equations.Theorem 3.1.
One assumes (H1)~(H4). Let (ξ*,η*) be optimal to (3.12), and let (x*(·),y*(·),z*(·),q*(·)) be the corresponding optimal trajectory. Then for arbitrary (ξ,η)∈U, one has that the following inequalities hold:
(3.14)〈m(0)-χx(ξ*),ξ-ξ*〉≤0,〈n(T)+λy(η*),η-η*〉≥0.


## 3.2.   System Composed of a BDSDE with State Constraints

Although this case describes a controlled BDSDE system with state constraints, it seems trivial. Thus, we only give a brief illustration. Givenη∈L2(Ω,ℱT,P;ℝk) and u(·)∈U[0,T], consider the following BDSDE:
(3.15)-dy(t)=f(t,y(t),q(t),u(t))dt+g(t,y(t),q(t),u(t))dBt-q(t)dWt,0≤t≤T,y(T)=η.
For given u(·)∈U[0,T] and f,g satisfying (H2) and (H3), from the Theorem 1.1 in [15], it is easy to check that there exists a unique solution (y(·),q(·)) of (3.15).Note thaty(0)(η,u(·)) is a ℱ0,TB-measurable variable, and
(3.16)E(h(y(η,u(·))(0)))=b.
Now we regard η and u(·) as the control variables to be chosen from the following set:
(3.17)U={(η,u(·))∣η∈K1,a.s.,E[|η|2]<∞,u(·)∈U[0,T]}.
For each (η,u(·))∈U, consider the following cost function:
(3.18)J(η,u(·))=E[∫0T‍l(t,y(t),q(t),u(t))dt+λ(η)+γ(y(0))],
which gives rise to the following optimization problem:
(3.19)inf(η,u(·))∈UJ(η,u(·))subjecttoE(h(y(η,u(·))(0)))=b.Maximum Principle
Let(3.20)H(y,q,u,n,δ)≜〈f(t,y,q,u),n(t)〉+〈g(t,y,q,u),δ(t)〉+l(t,y,q,u).
Then the adjoint equation is
(3.21)dn(t)=(fy*(t)n(t)+gy*(t)δ(t)+ly*(t))dt-δ(t)dBt+(fq*(t)n(t)+gq*(t)δ(t)+lq*(t))dWt,n(0)=h2hy(y*(0))+h0γy(y*(0)),
where Hk*(t)=Hk(t,y*(t),q*(t)) for H=f,g; k=y,q, respectively.Theorem 3.2.
One assumes (H1)~(H3). Let η* and u*(·) be the optimal controls and let (y*(·),z*(·)) be the corresponding optimal trajectory. Then one has
(3.22)〈n(T)+λy(η*),η-η*〉≥0,〈Hu(t,y*,q*,u*,n,δ),u(t)-u*(t)〉≥0.


## 3.3. Backward Doubly Stochastic LQ Problem without State Constraints

Consider the following linear system:(3.23)-dx(t)=[A(t)x(t)+B(t)z(t)+C(t)y(t)+D(t)q(t)+E(t)u(t)]dt+[A′(t)x(t)+B′(t)z(t)+C′(t)y(t)+D′(t)q(t)+E′(t)u(t)]dWt-z(t)dBt,x(0)=x,-dy(t)=[A′′(t)x(t)+B′′(t)z(t)+C′′(t)y(t)+D′′(t)q(t)+E′′(t)u(t)]dt+[A′′′(t)x(t)+B′′′(t)z(t)+C′′′(t)y(t)+D′′′(t)q(t)+E′′′(t)u(t)]dBt-q(t)dWt,y(T)=y,
where x and y are given constants and H,H′,H′′,H′′′(H=A,B,C,DandE) are corresponding matrices.The cost function (2.16) becomes
(3.24)l(t,x(t),z(t),y(t),q(t),u(t))=12F(t)x(t)·x(t)+12G(t)z(t)·z(t)+12H(t)y(t)·y(t)+12I(t)q(t)·q(t)+12J(t)u(t)·u(t),χ(x)=0,λ(y)=0,ϕ(x)=12U(t)x2,γ(y)=12Q(t)y2,
where all functions of t are bounded and F(t),G(t),H(t),I(t) are symmetric non-negative definite, J(t),U(t),Q(t) are symmetric uniformly positive definite.Then from (2.51), the adjoint equations become
(3.25)-dm(t)=-[A(t)m(t)+A′(t)p(t)+A′′(t)n(t)+A′′′(t)δ(t)+F(t)x*(t)]dt+[B(t)m(t)+B′(t)p(t)+B′′(t)n(t)+B′′′(t)δ(t)+G(t)z*(t)]dBt-p(t)dWt,m(T)=-U(T)x*(T),dn(t)=[C(t)m(t)+C′(t)p(t)+C′′(t)n(t)+C′′′(t)δ(t)+H(t)y*(t)]dt+[D(t)m(t)+D′(t)p(t)+D′′(t)n(t)+D′′′(t)δ(t)+I(t)q*(t)]dWt-δ(t)dBt,n(0)=Q(0)y*(0).
Define
(3.26)H(t,x,z,y,q,u,m,p,n,δ)≜〈F(t,x,z,y,q,u),m(t)〉+〈G(t,x,z,y,q,u),p(t)〉+〈f(t,x,z,y,q,u),n(t)〉+〈g(t,x,z,y,q,u),δ(t)〉+l(t,x,z,y,q,u).
Suppose that K is an open set. Then we have the following result:
(3.27)Hu(t,x*,z*,y*,q*,u*,m,p,n,δ)=E(t)m(t)+E′(t)p(t)+E′′(t)n(t)+E′′′(t)δ(t)+J(t)u(t)=0.
Thus,
(3.28)u*(t)=J-1(t)[E(t)m(t)+E′(t)p(t)+E′′(t)n(t)+E′′′(t)δ(t)].
However, the maximum principle gives only the necessary condition for optimal control. We also have the following theorem.Theorem 3.3.
The functionu*(t)=J-1(t)[E(t)m(t)+E′(t)p(t)+E′′(t)n(t)+E′′′(t)δ(t)] is the unique optimal control for backward doubly stochastic LQ problems, where (x*(·),z*(·),y*(·),q*(·)) and (m(·),p(·),n(·),δ(·)) are solutions of above equations.Proof.
First let us prove that theu*(·) is the optimal control. For all v(·)∈K, let (xv(·),zv(·),yv(·),qv(·)) be the corresponding trajectory of (3.23), we get
(3.29)J(v(·))-J(u*(·))=12{∫0T‍[q*(t)〈F(t)xv(t),xv(t)〉-〈F(t)x*(t),x*(t)〉+〈G(t)zv(t),zv(t)〉-〈G(t)z*(t),z*(t)〉+〈H(t)yv(t),yv(t)〉-〈H(t)y*(t),y*(t)〉+〈I(t)qv(t),qv(t)〉-〈I(t)q*(t),q*(t)〉+〈J(t)v(t),v(t)〉-〈J(t)u*(t),u*(t)〉]dt+〈U(T)xv(T),xv(T)〉-〈U(T)x*(T),x*(T)〉+〈Q(0)yv(0),yv(0)〉-〈Q(0)y*(0),y*(0)〉∫0T‍}≥E{∫0T‍[q*〈F(t)x*(t),xv(t)-x*(t)〉+〈G(t)z*(t),zv(t)-z*(t)〉+〈H(t)y*(t),yv(t)-y*(t)〉+〈I(t)q*(t),qv(t)-q*(t)〉+〈J(t)u*(t),v(t)-u*(t)〉]dt+〈U(T)x*(T),xv(T)-x*(T)〉+〈Q(0)y*(0),yv(0)-y*(0)〉∫0T‍}.
Using Lemma 2.1 to 〈xv(t)-x*(t),m(t)〉+〈yv(t)-y*(t),n(t)〉, we obtain
(3.30)〈U(T)x*(T),xv(T)-x*(T)〉+〈Q(0)y*(0),yv(0)-y*(0)〉=-E∫0T‍[E′′′〈F(t)x*(t),xv(t)-x*(t)〉+〈G(t)z*(t),zv(t)-z*(t)〉+〈H(t)y*(t),yv(t)-y*(t)〉+〈I(t)q*(t),qv(t)-q*(t)〉〈J(t)u*(t),v(t)-u*(t)〉+〈E(t)m(t),v(t)-u*(t)〉+〈E′(t)p(t),v(t)-u*(t)〉+〈E′′(t)n(t),v(t)-u*(t)〉+〈E′′′(t)δ(t),v(t)-u*(t)〉]dt.
So by the definition of u*(t),
(3.31)J(v(·))-J(u*(·))≥E∫0T[〈E(t)m(t),v(t)-u*(t)〉+〈E′(t)p(t),v(t)-u*(t)〉+〈E′′(t)n(t),v(t)-u*(t)〉+〈E′′′(t)δ(t),v(t)-u*(t)〉+〈J(t)u*(t),v(t)-u*(t)〉]dt=0.
From the arbitrariness of v(·)∈K, we deduce u*(t) is the optimal control.
The proof of the uniqueness of the optimal control is classical. Assume thatu1(·) and u2(·) are both optimal controls, and the corresponding trajectories are (x1(·),z1(·),y1(·),q1(·)) and (x2(·),z2(·),y2(·),q2(·)). By the uniqueness of solutions of (3.23), we know that the trajectory corresponding to (u1(·)+u2(·))/2 is ((x1(·)+x2(·))/2,(z1(·)+z2(·))/2,(y1(·)+y2(·))/2,(q1(·)+q2(·))/2), and notice that J(t),U(t),Q(t) are positive, F(t),G(t),H(t),I(t) are nonnegative, we have
(3.32)J(u1(·))=J(u2(·))=α≥0,2α=J(u1(·))+J(u2(·))≥2J(u1(·)+u2(·)2)+2E∫0T‍〈J(t)u1(·)-u2(·)2,u1(·)-u2(·)2〉dt≥2α+β2E∫0T‍|u1(·)-u2(·)|2dt,
here β>0. So
(3.33)E∫0T‍|u1(·)-u2(·)|2dt≤0,
which shows that u1(·)=u2(·).Example 3.4.
Consider the following backward doubly stochastic LQ problem, wheret∈[0,1],u(·)∈U[0,1]=[-1,1] and n=k=d=l=1:
(3.34)-dx(t)=12(y(t)+2u(t))dWt-z(t)dBt,x(0)=0,-dy(t)=12(x(t)+u(t))dBt-q(t)dWt,y(1)=0.
We want to minimize the following cost function:
(3.35)J(u(·))=E∫01‍(x(s)2-y(s)2+z(s)2-q(s)2+2x(s)u(s)-4y(s)u(s))ds+Ex(1)2+Ey(0)2.
From (3.34), we get, for t∈[0,1],
(3.36)x(t)=-∫0t‍(y(s)+2u(s))dWs+∫0t‍z(s)dBs,y(t)=∫t1‍(x(s)+u(s))dBs-∫t1‍q(s)dWs.
By substituting x(·) and y(·) into the cost function, we obtain
(3.37)J(u(·))=E∫01‍3u(s)2ds.
Thus, the optimal control is u*(t)≡0,t∈[0,1] with the optimal state trajectory
(3.38)(x*(t),z*(t),y*(t),q*(t))≡0,t∈[0,1].
The adjoint equations are
(3.39)dm(t)=n(t)dt+p(t)dWt,m(1)=0,dn(t)=p(t)dt-δ(t)dBt,n(0)=0.
It is obvious that (m(·),p(·),n(·),δ(·))=(0,0,0,0) is the unique solution of the above equation.



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*Source: 537376-2012-12-30.xml*