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

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.

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): −𝑑𝑥_𝑡=𝐹𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝑡+𝐺𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝑊_𝑡−𝑧_𝑡𝑑𝐵_𝑡𝑥,0≤𝑡≤𝑇,₀=𝜉,−𝑑𝑦_𝑡=𝑓𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝑡+𝑔𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝐵_𝑡−𝑞_𝑡𝑑𝑊_𝑡𝑦,0≤𝑡≤𝑇,_𝑇=𝜂, (1.1) which generalized the general FBSDEs. Here the forward equation is “forward” with respect to a standard stochastic integral 𝑑𝑊_𝑡, as well as “backward” with respect to a backward stochastic integral 𝑑𝐵_𝑡; the coupled “backward equation” is “forward” under the backward stochastic integral 𝑑𝐵_𝑡 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 state 𝜉 and the terminal state 𝜂 fall in two convex sets, respectively, and the corresponding states 𝑥_𝑇^{(𝜉,𝜂,𝑢(⋅))} and 𝑦₀^{(𝜉,𝜂,𝑢(⋅))} satisfy the constraints 𝐸(𝜓(𝑥_𝑇^{(𝜉,𝜂,𝑢(⋅))}))=𝑎 and 𝐸(ℎ(𝑦₀^{(𝜉,𝜂,𝑢(⋅))}))=𝑏, respectively. Then we minimize the following cost function: 𝐽𝜉,𝜂,𝑢(⋅)≜𝐸^𝑇₀𝑙𝑥(𝑡),𝑧(𝑡),𝑦(𝑡),𝑞(𝑡),𝑢(𝑡),𝑡𝑑𝑡+𝜒(𝜉)+𝜆𝜂+𝜙(𝑥(𝑇))+𝛾𝑦(0). (1.2) 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 Section 2.1, 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 {𝐖_𝑡,0≤𝑡≤𝑇} and {𝐁_𝑡,0≤𝑡≤𝑇} be two mutually independent standard Brownian motion processes, with values in ℝ^𝑑,ℝ^𝑙, respectively, defined on (Ω,ℱ,𝑃). Let ℕ denote the class of 𝑃-null set of ℱ. For each 𝑡∈0,𝑇], we define ℱ_𝑡≜ℱ^𝑊_𝑡∨ℱ^𝐵_𝑡,𝑇, where ℱ^𝑊_𝑡=ℱ^𝑊_0,𝑡=𝜎𝑊_𝑟−𝑊₀;0≤𝑟≤𝑡∨ℕ,ℱ^𝐵_𝑡=ℱ^𝐵_0,𝑡=𝜎𝐵_𝑟−𝐵_𝑡;𝑡≤𝑟≤𝑇∨ℕ. (2.1) Note that the collection {ℱ_𝑡,𝑡∈[0,𝑇]} is neither increasing nor decreasing, and it does not constitute a filtration.

For any Euclidean space 𝐻, we denote by ⟨⋅,⋅⟩ the scale product of 𝐻. The Euclidean norm of a vector 𝑦∈ℝ^𝑘 will be denoted by |𝑦|, and for a 𝑑×𝑛 matrix A, we define ||𝐴||=𝑇𝑟(𝐴𝐴^∗).

For any 𝑛∈𝑁, let 𝑀²(0,𝑇;ℝ^𝑛) denote the set of (classes of 𝑑𝑃⊗𝑑𝑡 a.e. equal) 𝑛-dimensional jointly measurable stochastic processes {𝜑_𝑡;𝑡∈[0,𝑇]} which satisfy

(i) 𝐸∫^𝑇₀|𝜑_𝑡|²𝑑𝑡<∞; (ii) 𝜑_𝑡 is ℱ_𝑡-measurable, for a.e. 𝑡∈[0,𝑇].

We denote by 𝑆²(0,𝑇;ℝ^𝑛) the set of continuous 𝑛-dimensional stochastic processes which satisfy:

(i) 𝐸(sup_{0≤𝑡≤𝑇}|𝜑_𝑡|²)<∞; (ii) 𝜑_𝑡 is ℱ_𝑡-measurable, for any 𝑡∈[0,𝑇].

Let 𝑓∶Ω×[0,𝑇]×ℝ^𝑘×ℝ^𝑘×𝑑→ℝ^𝑘,𝑔∶Ω×[0,𝑇]×ℝ^𝑘×ℝ^𝑘×𝑑→ℝ^𝑘×𝑙 (2.2) be jointly measurable such that for any (𝑦,𝑞)∈ℝ^𝑘×ℝ^𝑘×𝑑,𝑓(⋅,𝑦,𝑞)∈𝑀²(0,𝑇;ℝ^𝑘),𝑔(⋅,𝑦,𝑞)∈𝑀²(0,𝑇;ℝ^𝑘×𝑙).

Moreover, we assume that there exist constants 𝐶>0 and 0<𝛼<1 such that for any (𝜔,𝑡)∈Ω×[0,𝑇], (𝑦₁,𝑞₁),(𝑦₂,𝑞₂)∈ℝ^𝑘×ℝ^𝑘×𝑙, 𝑓𝑡,𝑦₁,𝑞₁−𝑓𝑡,𝑦₂,𝑞₂²≤𝐶𝑦₁−𝑦₂²+𝑞₁−𝑞₂²,𝑔𝑡,𝑦₁,𝑞₁−𝑔𝑡,𝑦₂,𝑞₂²≤𝐶𝑦₁−𝑦₂²+𝛼𝑞₁−𝑞₂². (H) Given 𝜂∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑘), we consider the following BDSDE: 𝑦_𝑡=𝜂+^𝑇_𝑡𝑓𝑠,𝑦_𝑠,𝑞_𝑠𝑑𝑠+^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑞_𝑠𝑑𝐵_𝑠−^𝑇_𝑡𝑞_𝑠𝑑𝑊_𝑠,0≤𝑡≤𝑇. (2.3)

We note that the integral with respect to {𝐵_𝑡} is a “backward Itô integral” and the integral with respect to {𝑊_𝑡} 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;ℝ^𝑘)×𝑀²(0,𝑇;ℝ^𝑘×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. Let 𝛼∈𝑆²(0,𝑇;ℝ^𝑘), 𝛽∈𝑀²(0,𝑇;ℝ^𝑘), 𝛾∈𝑀²(0,𝑇;ℝ^𝑘×𝑙),𝛿∈𝑀²(0,𝑇;ℝ^𝑘×𝑑) be such that 𝛼_𝑡=𝛼₀+^𝑡₀𝛽_𝑠𝑑𝑠+^𝑡₀𝛾_𝑠𝑑𝐵_𝑠+^𝑡₀𝛿_𝑠𝑑𝑊_𝑠,0≤𝑡≤𝑇. (2.4) Then, 𝛼_𝑡²=𝛼₀²+2^𝑡₀𝛼_𝑠,𝛽_𝑠𝑑𝑠+2^𝑡₀𝛼_𝑠,𝛾_𝑠𝑑𝐵_𝑠−^𝑡₀𝛾_𝑠²𝑑𝑠+^𝑡₀𝛿_𝑠²𝑑𝑠+2^𝑡₀𝛼_𝑠,𝛿_𝑠𝑑𝑊_𝑠,𝐸𝛼_𝑡²=𝐸𝛼₀²+2𝐸^𝑡₀𝛼_𝑠,𝛽_𝑠𝑑𝑠−𝐸^𝑡₀𝛾_𝑠²𝑑𝑠+𝐸^𝑡₀𝛿_𝑠²𝑑𝑠. (2.5) Generally, for 𝜙∈𝐶²(ℝ^𝑘), 𝜙𝛼_𝑡=𝜙𝛼₀+^𝑡₀𝜙^′𝛼_𝑠,𝛽_𝑠𝑑𝑠+^𝑡₀𝜙^′𝛼_𝑠,𝛾_𝑠𝑑𝐵_𝑠+^𝑡₀𝜙^′𝛼_𝑠,𝛿_𝑠𝑑𝑊_𝑠−12^𝑡₀𝑇𝑟𝜙^′′𝛼_𝑠𝛾_𝑠𝛾^∗_𝑠1𝑑𝑠+2^𝑡₀𝑇𝑟𝜙^′′𝛼_𝑠𝛿_𝑠𝛿^∗_𝑠𝑑𝑠. (2.6)

2.2. Problem Formulation

Let 𝐾 be a nonempty convex subset of ℝ^𝑛×𝑑. We set 𝑈[0,𝑇]=𝑢(⋅)∣𝑢(𝑡)∈𝐾,a.e.,a.s.,0≤𝑡≤𝑇;𝑢(⋅)∈𝑀²0,𝑇;ℝ^𝑛×𝑑. (2.7) An element of 𝑈[0,𝑇] is called an admissible control. Now let 𝐹∶Ω×[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑×ℝ^𝑛×𝑑→ℝ^𝑛,𝐺∶Ω×[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑×ℝ^𝑛×𝑑→ℝ^𝑛×𝑑,𝑓∶Ω×[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑×ℝ^𝑛×𝑑→ℝ^𝑘,𝑔∶Ω×[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑×ℝ^𝑛×𝑑→ℝ^𝑘×𝑙 (2.8) be jointly measurable such that for any (𝑥,𝑧,𝑦,𝑞)∈ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑 and any 𝑢(⋅) ∈𝑈[0,𝑇]𝐹⋅,𝑥,𝑧,𝑦,𝑞,𝑢(⋅)∈𝑀²0,𝑇;ℝ^𝑛,𝐺⋅,𝑥,𝑧,𝑦,𝑞,𝑢(⋅)∈𝑀²0,𝑇;ℝ^𝑛×𝑑,𝑓⋅,𝑥,𝑧,𝑦,𝑞,𝑢(⋅)∈𝑀²0,𝑇;ℝ^𝑘,𝑔⋅,𝑥,𝑧,𝑦,𝑞,𝑢(⋅)∈𝑀²0,𝑇;ℝ^𝑘×𝑙. (2.9) Let 𝜁(𝑡)=𝑥(𝑡),𝑧(𝑡),𝑦(𝑡),𝑞(𝑡)^𝑇,𝐴(𝑡,𝜁)=−𝐹,−𝐺,−𝑓,−𝑔^𝑇(𝑡,𝜁). (2.10) We assume the following.(H1)∀𝜁₁=(𝑥₁,𝑧₁,𝑦₁,𝑞₁), 𝜁₂=(𝑥₂,𝑧₂,𝑦₂,𝑞₂)∈ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑 and 𝑡∈[0,𝑇], there exists a constant 𝜇>0 such that the following monotonicity condition holds for any 𝑢(⋅) ∈𝑈[0,𝑇]: ⟨𝐴𝑡,𝜁₁−𝐴𝑡,𝜁₂,𝜁₁−𝜁₂⟩≤−𝜇𝜁₁−𝜁₂². (2.11)(H2)There exist constants 𝐶>0 and 0<𝛼<1/2 such that for any (𝜔,𝑡)∈Ω×[0,𝑇], 𝑢∈ℝ^𝑛×𝑑, (𝑥₁,𝑧₁,𝑦₁,𝑞₁),(𝑥₂,𝑧₂,𝑦₂,𝑞₂)∈ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑 the following conditions hold: 𝜗𝑡,𝑥₁,𝑧₁,𝑦₁,𝑞₁,𝑢−𝜗𝑡,𝑥₂,𝑧₂,𝑦₁,𝑞₁,𝑢²≤𝐶𝑥₁−𝑥₂²+𝑧₁−𝑧₂²+𝑦₁−𝑦₂²+𝑞₁−𝑞₂²,𝜘𝑡,𝑥₁,𝑧₁,𝑦₁,𝑞₁,𝑢−𝜘𝑡,𝑥₂,𝑧₂,𝑦₁,𝑞₁,𝑢²≤𝐶𝑥₁−𝑥₂²+𝑦₁−𝑦₂²+𝛼𝑧₁−𝑧₂²+𝑞₁−𝑞₂², (2.12)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<‖𝑔_𝑧(⋅)‖<1/2, 0<‖𝐺_𝑧(⋅)‖<1/2, 0<‖𝑔_𝑞(⋅)‖<1/2, 0<‖𝐺_𝑞(⋅)‖<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|).

Given 𝜉∈𝐿²(Ω,ℱ₀,𝑃;ℝ^𝑛), 𝜂∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑘), and ∀𝑢(⋅)∈𝑈[0,𝑇], let us consider the following time-symmetric FBDSDE: −𝑑𝑥_𝑡=𝐹𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝑡+𝐺𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝑊_𝑡−𝑧_𝑡𝑑𝐵_𝑡,𝑥₀=𝜉,−𝑑𝑦_𝑡=𝑓𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝑡+𝑔𝑡,𝑥_𝑡,𝑧_𝑡,𝑦_𝑡,𝑞_𝑡,𝑢_𝑡𝑑𝐵_𝑡−𝑞_𝑡𝑑𝑊_𝑡,𝑦_𝑇=𝜂. (2.13) Recall Theorem 2.2 in [16]. We have the following.

Theorem 2.2. For given 𝜉∈𝐿²(Ω,ℱ₀,𝑃;ℝ^𝑛), 𝜂∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑘) and ∀𝑢(⋅)∈𝑈[0,𝑇], assume (H1)~(H3); then (2.13) exists as a unique ℱ_𝑡-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: 𝑈=𝜉,𝜂,𝑢(⋅)∣𝜉∈𝐾₁⊂ℝ^𝑛a.s.,𝜂∈𝐾₂⊂ℝ^𝑘a.s.,𝐸𝜉²<∞,𝐸𝜂²<∞,𝑢(⋅)∈𝑈[0,𝑇], (2.14) 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 𝐸𝜓𝑥_𝑇^{(𝜉,𝜂,𝑢(⋅))}=𝑎,𝐸ℎ𝑦₀^{(𝜉,𝜂,𝑢(⋅))}=𝑏. (2.15) For each (𝜉,𝜂,𝑢(⋅))∈𝑈, consider the following cost function: 𝐽𝜉,𝜂,𝑢(⋅)≜𝐸^𝑇₀𝑙𝑥(𝑡),𝑧(𝑡),𝑦(𝑡),𝑞(𝑡),𝑢(𝑡),𝑡𝑑𝑡+𝜒(𝜉)+𝜆𝜂+𝜙(𝑥(𝑇))+𝛾𝑦(0). (2.16) Our optimization problem is inf_{(𝜉,𝜂,𝑢(⋅))∈𝑈}𝐽(⋅)𝜓𝑥𝜉,𝜂,𝑢subjectto𝐸_𝑇^{(𝜉,𝜂,𝑢(⋅))}=𝑎;𝐸ℎ𝑦₀^{(𝜉,𝜂,𝑢(⋅))}=𝑏. (2.17)

Definition 2.4. A triple of random variable (𝜉,𝜂,𝑢(⋅))∈𝑈 is called feasible for given 𝑎∈ℝ^𝑛, 𝑏∈ℝ^𝑘 if the solution (2.13) satisfies 𝐸(𝜓(𝑥_𝑇^{(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 𝑑𝜉¹,𝜂¹,𝑢¹(⋅),𝜉²,𝜂²,𝑢²(⋅)≜𝐸𝜉¹−𝜉^221/2+𝐸𝜂¹−𝜂^221/2+𝑢¹(⋅)−𝑢²(⋅)^21/2. (2.18) 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, 𝜉^𝜌,𝜂^𝜌,𝑢^𝜌(⋅)≜𝜉^∗+𝜌𝜉−𝜉^∗,𝜂^∗+𝜌𝜂−𝜂^∗,𝑢^∗(⋅)+𝜌𝑢(⋅)−𝑢^∗(⋅)∈𝑈. (2.19) Let (𝑥^𝜌(⋅),𝑧^𝜌(⋅),𝑦^𝜌(⋅),𝑞^𝜌(⋅)) be the state processes of (2.13) associated with (𝜉^𝜌,𝜂^𝜌,𝑢^𝜌(⋅)).

To derive the first-order necessary condition, we let (𝑥(⋅),𝑧(⋅),𝑦(⋅),𝑞(⋅)) be the solution of the following time-symmetric FBDSDE: −𝑑𝑥(𝑡)=𝐹^∗_𝑥(𝑡)𝑥(𝑡)+𝐹^∗_𝑧(𝑡)𝑧(𝑡)+𝐹^∗_𝑦(𝑡)𝑦(𝑡)+𝐹^∗_𝑞(𝑡)𝑞(𝑡)+𝐹^∗_𝑢(𝑡)𝑢(⋅)−𝑢^∗(⋅+𝐺)𝑑𝑡_∗𝑥(𝑡)𝑥(𝑡)+𝐺^∗_𝑧(𝑡)𝑧(𝑡)+𝐺^∗_𝑦(𝑡)𝑦(𝑡)+𝐺^∗_𝑞(𝑡)𝑞(𝑡)+𝐺^∗_𝑢(𝑡)𝑢(⋅)−𝑢^∗(⋅)𝑑𝑊_𝑡−𝑧(𝑡)(𝑡),𝑥(0)𝑑𝐵=𝜉−𝜉^∗,−𝑑𝑦(𝑡)=𝑓^∗_𝑥(𝑡)𝑥(𝑡)+𝑓^∗_𝑧(𝑡)𝑧(𝑡)+𝑓^∗_𝑦(𝑡)𝑦(𝑡)+𝑓^∗_𝑞(𝑡)𝑞(𝑡)+𝑓^∗_𝑢(𝑡)𝑢(⋅)−𝑢^∗(⋅)+𝑔𝑑𝑡_∗𝑥(𝑡)𝑥(𝑡)+𝑔^∗_𝑧(𝑡)𝑧(𝑡)+𝑔^∗_𝑦(𝑡)𝑦(𝑡)+𝑔^∗_𝑞(𝑡)𝑞(𝑡)+𝑔^∗_𝑢(𝑡)𝑢(⋅)−𝑢^∗(⋅)𝑑𝐵_𝑡−𝑞(𝑡)(𝑡),𝑦(𝑇)𝑑𝑊=𝜂−𝜂^∗, (2.20) where𝐻^∗_𝑘(𝑡)=𝐻_𝑘(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 𝑥_𝜌(𝑡)=𝜌⁻¹𝑥_𝜌(𝑡)−𝑥^∗(𝑡)−𝑥(𝑡),𝑧_𝜌(𝑡)=𝜌⁻¹𝑧_𝜌(𝑡)−𝑧^∗(𝑡−𝑧(𝑡),𝑦_𝜌)(𝑡)=𝜌⁻¹𝑦_𝜌(𝑡)−𝑦^∗(𝑡)−𝑦(𝑡),𝑞_𝜌(𝑡)=𝜌⁻¹𝑞_𝜌(𝑡)−𝑞^∗(𝑡)−𝑞(𝑡). (2.21) We have the following convergence.

Lemma 2.5. Assuming (H1)~(H3) one has lim_𝜌→0sup_{0≤𝑡≤𝑇}𝐸𝑥_𝜌(𝑡)²=0,lim_𝜌→0𝐸^𝑇₀𝑧_𝜌(𝑡)²𝑑𝑡=0,lim_𝜌→0sup_{0≤𝑡≤𝑇}𝐸𝑦_𝜌(𝑡)²=0,lim_𝜌→0𝐸^𝑇₀𝑞_𝜌(𝑡)²𝑑𝑡=0. (2.22)

Proof. From (2.13) and (2.20), we have −𝑑𝑦_𝜌(𝑡)=𝜌⁻¹𝑓𝑥_𝜌(𝑡),𝑧_𝜌(𝑡),𝑦_𝜌(𝑡),𝑞_𝜌(𝑡),𝑢_𝜌(𝑡),𝑡−𝑓𝑥^∗(𝑡),𝑧^∗(𝑡),𝑦^∗(𝑡),𝑞^∗(𝑡),𝑢^∗(𝑡),𝑡−𝜌𝑓^∗_𝑥(𝑡)𝑥(𝑡)−𝜌𝑓^∗_𝑧(𝑡)𝑧(𝑡)−𝜌𝑓^∗_𝑦(𝑡)𝑦(𝑡)−𝜌𝑓^∗_𝑞(𝑡)𝑞(𝑡)−𝜌𝑓^∗_𝑢(𝑡)𝑢(⋅)−𝑢^∗(⋅)𝑑𝑡+𝜌⁻¹𝑔𝑥_𝜌(𝑡),𝑧_𝜌(𝑡),𝑦_𝜌(𝑡),𝑞_𝜌(𝑡),𝑢_𝜌(𝑡),𝑡−𝑔𝑥^∗(𝑡),𝑧^∗(𝑡),𝑦^∗(𝑡),𝑞^∗(𝑡),𝑢^∗(𝑡),𝑡−𝜌𝑔^∗_𝑥(𝑡)𝑥(𝑡)−𝜌𝑔^∗_𝑧(𝑡)𝑧(𝑡)−𝜌𝑔^∗_𝑦(𝑡)𝑦(𝑡)−𝜌𝑔^∗_𝑞(𝑡)𝑞(𝑡)−𝜌𝑔^∗_𝑢(𝑡)𝑢(⋅)−𝑢^∗(⋅)𝑑𝐵_𝑡−𝑞_𝜌(𝑡)𝑑𝑊_𝑡,𝑦_𝜌(𝑇)=0. (2.23) Let 𝐴^𝜌(𝑡)=¹₀𝑓_𝑥𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡)𝐵,𝑡𝑑𝜆,_𝜌(𝑡)=¹₀𝑓_𝑧𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡)𝐶,𝑡𝑑𝜆,_𝜌(𝑡)=¹₀𝑓_𝑦𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡)𝐷,𝑡𝑑𝜆,_𝜌(𝑡)=¹₀𝑓_𝑞𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡)𝐸,𝑡𝑑𝜆,_𝜌(𝑡)=𝐴^𝜌(𝑡)−𝑓^∗_𝑥(𝑡)𝑥(𝑡)+𝐵^𝜌(𝑡)−𝑓^∗_𝑧(𝑡)𝑧(𝑡)+𝐶^𝜌(𝑡)−𝑓^∗_𝑦(𝑡)𝑦(𝑡)+𝐷^𝜌(𝑡)−𝑓^∗_𝑞(𝑡𝑞(𝑡)+₁₀)𝑓_𝑢𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡),𝑡−𝑓^∗_𝑢(𝑡)𝑢(𝑡)−𝑢^∗(𝑡)𝐴𝑑𝜆,_𝜌1(𝑡)=¹₀𝑔_𝑥𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡)𝐵,𝑡𝑑𝜆,_𝜌1(𝑡)=¹₀𝑔_𝑧𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡,𝑞_∗)(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡)𝐶,𝑡𝑑𝜆,_𝜌1(𝑡)=¹₀𝑔_𝑦𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡,𝑞_∗)(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡)𝐷,𝑡𝑑𝜆,_𝜌1(𝑡)=¹₀𝑔_𝑞𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡𝐸),𝑡𝑑𝜆,_𝜌1(𝑡)=𝐴^𝜌₁(𝑡)−𝑔^∗_𝑥(𝑡)𝑥(𝑡)+𝐵^𝜌₁(𝑡)−𝑔^∗_𝑧(𝑡)𝑧(𝑡)+𝐶^𝜌₁(𝑡)−𝑔^∗_𝑦(𝑡)𝑦(𝑡)+𝐷^𝜌₁(𝑡)−𝑔^∗_𝑞(𝑡𝑞(𝑡)+₁₀)𝑔_𝑢𝑥^∗(𝑡)+𝜆𝜌𝑥(𝑡)+𝑥_𝜌(𝑡),𝑧^∗(𝑡)+𝜆𝜌𝑧(𝑡)+𝑧_𝜌(𝑡),𝑦^∗(𝑡)+𝜆𝜌𝑦(𝑡)+𝑦_𝜌(𝑡),𝑞^∗(𝑡)+𝜆𝜌𝑞(𝑡)+𝑞_𝜌(𝑡),𝑢^∗(𝑡)+𝜆𝜌𝑢(𝑡)−𝑢^∗(𝑡),𝑡−𝑔^∗_𝑢(𝑡)𝑢(𝑡)−𝑢^∗(𝑡)𝑑𝜆. (2.24) Thus −𝑑𝑦_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+𝐴)𝑑𝑡_𝜌1(𝑡)𝑥_𝜌(𝑡)+𝐵^𝜌₁(𝑡)𝑧_𝜌(𝑡)+𝐶^𝜌₁(𝑡)𝑦_𝜌(𝑡)+𝐷^𝜌₁(𝑡)𝑞_𝜌(𝑡)+𝐸^𝜌₁(𝑡)𝑑𝐵_𝑡−𝑞_𝜌(𝑡)(𝑡),𝑦𝑑𝑊_𝜌(𝑇)=0. (2.25) Using Lemma 2.1 to |𝑦_𝜌(𝑡)|², we get 𝐸𝑦_𝜌(𝑡)²+𝐸^𝑇_𝑡𝑞_𝜌(𝑠)²𝑑𝑠=2𝐸^𝑇_𝑡𝑦_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(𝑠)+𝐸^𝜌(𝑠)𝑑𝑠+𝐸^𝑇_𝑡𝐴^𝜌₁(𝑠)𝑥_𝜌(𝑠)+𝐵^𝜌₁(𝑠)𝑧_𝜌(𝑠)+𝐶^𝜌₁(𝑠)𝑦_𝜌(𝑠)+𝐷^𝜌₁(𝑠)𝑞_𝜌(𝑠)+𝐸^𝜌₁(𝑠)²𝐸𝑑𝑠≤𝐾^𝑇_𝑡𝑦_𝜌(𝑠)²𝑑𝑠+𝐸^𝑇_𝑡𝑥_𝜌(𝑠)²𝑑𝑠+𝛽₁𝐸^𝑇_𝑡𝑧_𝜌(𝑠)²𝑑𝑠+𝐸^𝑇_𝑡𝑞_𝜌(𝑠)²𝑑𝑠+𝐽_𝜌, (2.26) where 𝐾>0, 0<𝛽₁<1/2 are constants and 𝐽_𝜌=𝐸∫^𝑇_𝑡|𝐸^𝜌(𝑠)|²𝑑𝑠+𝐸∫^𝑇_𝑡|𝐸^𝜌₁(𝑠)|²𝑑𝑠.
Similar analysis shows that −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)+𝐴𝑑𝑡_′𝜌1(𝑡)𝑥_𝜌(𝑡)+𝐵^′𝜌₁(𝑡)𝑧_𝜌(𝑡)+𝐶^′𝜌₁(𝑡)𝑦_𝜌(𝑡)+𝐷^′𝜌₁(𝑡)𝑞_𝜌(𝑡)+𝐸^′𝜌₁(𝑡)𝑑𝑊_𝑡−𝑧_𝜌(𝑡)(𝑡),𝑥𝑑𝐵_𝜌(0)=0, (2.27) where 𝐻^′𝜌(𝑡) and 𝐻^′𝜌₁(𝑡) (𝐻=𝐴,𝐵,𝐶,𝐷and𝐸) are similarly defined as before.
It yields that 𝐸𝑥_𝜌(𝑡)²+𝐸^𝑇_𝑡𝑧_𝜌(𝑠)²𝑑𝑠=2𝐸^𝑇_𝑡𝑥_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^𝑇_𝑡𝐴^′𝜌₁(𝑠)𝑥_𝜌(𝑠)+𝐵^′𝜌₁(𝑠)𝑧_𝜌(𝑠)+𝐶^′𝜌₁(𝑠)𝑦_𝜌(𝑠)+𝐷^′𝜌₁(𝑠)𝑞_𝜌(𝑠)+𝐸^′𝜌₁(𝑠)²𝑑𝑠≤𝐾𝐸^𝑇_𝑡𝑦_𝜌(𝑠)²𝑑𝑠+^𝑇_𝑡𝑥_𝜌(𝑠)²𝑑𝑠+𝛽₂𝐸^𝑇_𝑡𝑧_𝜌(𝑠)²𝑑𝑠+^𝑇_𝑡𝑞_𝜌(𝑠)²𝑑𝑠+𝐽^′_𝜌, (2.28) where 𝐾>0, 0<𝛽₂<1/2 are constants and 𝐽^′_𝜌=𝐸[∫^𝑇_𝑡|𝐸^′𝜌(𝑠)|²𝑑𝑠+∫^𝑇_𝑡|𝐸^′𝜌₁(𝑠)|²𝑑𝑠]. Since 0<𝛽₁+𝛽₂<1 and 𝐾>0, there exists 𝐾₁>0 such that 𝐸𝑥_𝜌(𝑡)²+𝐸𝑦_𝜌(𝑡)²+1−𝛽₁−𝛽₂𝐸^𝑇_𝑡𝑧_𝜌(𝑠)²𝑑𝑠+𝐸^𝑇_𝑡𝑞_𝜌(𝑠)²𝑑𝑠≤𝐾₁𝐸^𝑇_𝑡𝑥_𝜌(𝑠)²𝑑𝑠+𝐸^𝑇_𝑡𝑦_𝜌(𝑠)²𝑑𝑠+𝐽_𝜌+𝐽^′_𝜌. (2.29) Since the Lebesgue dominated convergence theorem implies lim_𝜌→0𝐽_𝜌=0,lim_𝜌→0𝐽^′_𝜌=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: 𝐸𝜓𝑥_𝑇^{(𝜉,𝜂,𝑢(⋅))}=𝑎,𝐸ℎ𝑦₀^{(𝜉,𝜂,𝑢(⋅))}=𝑏. (2.30) Define 𝐹_𝜀𝜉,𝜂,𝑢(⋅)≜𝐸(𝜓(𝑥_𝑇^{(𝜉,𝜂,𝑢(⋅))}))−𝑎²+𝐸(ℎ(𝑦₀^{(𝜉,𝜂,𝑢(⋅))}))−𝑏²+max0,𝐸𝜙𝑥^∗(𝑇)−𝜙𝑥_𝑇^{(𝜉,𝜂,𝑢(⋅))}+𝜀²+max(0,𝐸[𝛾(𝑦^∗(0))−𝛾(𝑦₀^{(𝜉,𝜂,𝑢(⋅))})]+𝜀)^21/2, (2.31) 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(𝑥_𝑇^{(𝜉,𝜂,𝑢(⋅))}))−𝑎|², |𝐸(ℎ(𝑦₀^{(𝜉,𝜂,𝑢(⋅))}))−𝑏|², 𝜙(𝑥_𝑇^{(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: ℎ₃𝐸𝜓_𝑥𝑥^∗(𝑇),𝑥(𝑇)+ℎ₂𝐸ℎ_𝑦𝑦^∗(0),𝑦(0)+ℎ₁𝐸⟨𝜙_𝑥𝑥^∗(𝑇),𝑥(𝑇)⟩+ℎ₀𝐸⟨𝛾_𝑦𝑦^∗(0),𝑦(0)⟩≥0, (2.32) 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 that 𝐹_𝜀(⋅) is continuous on 𝑈 such that 𝐹_𝜀𝜉^∗,𝜂^∗,𝑢^∗(⋅)=√𝐹2𝜀,_𝜀(⋅)(⋅)𝐹𝜉,𝜂,𝑢>0,∀𝜉,𝜂,𝑢∈𝑈,_𝜀𝜉^∗,𝜂^∗,𝑢^∗(⋅)≤inf_{(𝜉,𝜂,𝑢(⋅))∈𝑈}𝐹_𝜀𝜉,𝜂,𝑢(⋅)+√2𝜀. (2.33) Thus, from Ekeland's variational principle [28], ∃(𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅))∈𝑈 such that(i)𝐹_𝜀(𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅))≤𝐹_𝜀(𝜉^∗,𝜂^∗,𝑢^∗(⋅)), (ii)𝑑((𝜉^∗,𝜂^∗,𝑢^∗(⋅)),(𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)))≤√2𝜀, (iii)𝐹_𝜀(𝜉,𝜂,𝑢(⋅))+√2𝜀𝑑((𝜉,𝜂,𝑢(⋅)),(𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)))≥𝐹_𝜀(𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)),forall(𝜉,𝜂,𝑢(⋅))∈𝑈. For any (𝜉,𝜂,𝑢(⋅))∈𝑈 and 0≤𝜌≤1, set (𝜉^𝜀_𝜌,𝜂^𝜀_𝜌,𝑢^𝜀_𝜌(⋅))=(𝜉^𝜀+𝜌(𝜉−𝜉^𝜀),𝜂^𝜀+𝜌(𝜂−𝜂^𝜀),𝑢^𝜀(⋅)+𝜌(𝑢(⋅)−𝑢^𝜀(⋅))). Let (𝑥^𝜀_𝜌(⋅),𝑦^𝜀_𝜌(⋅),𝑧^𝜀_𝜌(⋅),𝑞^𝜀_𝜌(⋅)) (resp., (𝑥^𝜀(⋅),𝑦^𝜀(⋅),𝑧^𝜀(⋅),𝑞^𝜀(⋅))) be the solution of (2.13) under (𝜉^𝜀_𝜌,𝜂^𝜀_𝜌,𝑢^𝜀_𝜌(⋅)) (resp., (𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)), and let (𝑥^𝜀(⋅),𝑦^𝜀(⋅),𝑧^𝜀(⋅),𝑞^𝜀(⋅)) be the solution of (2.20) in which (𝜉^∗,𝜂^∗,𝑢^∗(⋅)) is substituted by (𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)).
From (iii), we know that 𝐹_𝜀𝜉^𝜀_𝜌,𝜂^𝜀_𝜌,𝑢^𝜀_𝜌(⋅)−𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)+√2𝜀𝑑𝜉^𝜀_𝜌,𝜂^𝜀_𝜌,𝑢^𝜀_𝜌(⋅),𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)≥0. (2.34) On the other hand, similarly to Lemma 2.5 we have lim_𝜌→0sup_{0≤𝑡≤𝑇}𝐸𝜌⁻¹𝑥^𝜀_𝜌(𝑡)−𝑥^𝜀(𝑡)−𝑥^𝜀(𝑡)²=0,lim_𝜌→0sup_{0≤𝑡≤𝑇}𝐸𝜌⁻¹𝑦^𝜀_𝜌(𝑡)−𝑦^𝜀(𝑡)−𝑦^𝜀(𝑡)²=0. (2.35) This leads to the following expansions: 𝐸(𝜙(𝑥^𝜀_𝜌(𝑇))−𝑎²−𝐸(𝜙(𝑥^𝜀(𝑇))−𝑎²=2𝜌𝐸𝜙𝑥^𝜀(𝑇)−𝑎,𝐸𝜙_𝑥𝑥^𝜀(𝑇)𝑥^𝜀(𝑇)+𝑜𝜌,𝐸(ℎ(𝑦^𝜀_𝜌(0))−𝑏²−𝐸(ℎ(𝑦𝜀(0))−𝑏²=2𝜌𝐸ℎ𝑦^𝜀(0)−𝑏,𝐸ℎ_𝑦𝑦^𝜀(0)𝑦^𝜀(0)+𝑜𝜌. (2.36) Applying the linearization technique, then 𝐸𝜙𝑥^𝜀_𝜌(𝑇)−𝜙𝑥^𝜀(𝑇)=𝐸^𝑇₀𝜙_𝑥𝑥^{(𝜉𝜀̂𝜉+𝜆𝜌𝜀,𝜂𝜀𝜂+𝜆𝜌𝜀,𝑢𝜀𝑢(⋅)+𝜆𝜌𝜀)}(𝑇)𝑑𝜆⋅𝜌𝑥^𝜀(𝑇,𝐸𝛾𝑦_𝜀𝜌)(0)−𝛾𝑦^𝜀(0)=𝐸^𝑇₀𝛾_𝑦𝑦^{(𝜉𝜀̂𝜉+𝜆𝜌𝜀,𝜂𝜀𝜂+𝜆𝜌𝜀,𝑢𝜀𝑢(⋅)+𝜆𝜌𝜀(⋅))}(0)𝑑𝜆⋅𝜌𝑦^𝜀(0). (2.37) So we have the following expansions: 𝐸[𝜙(𝑥^∗(𝑇))−𝜙(𝑥^𝜀_𝜌(𝑇)]+𝜀²−𝐸[𝜙(𝑥^∗(𝑇))−𝜙(𝑥^𝜀(𝑇)]+𝜀²=−2𝜌𝐸𝜙_𝑥𝑥^𝜀(𝑇)⋅𝜌𝑥^𝜀_𝑇𝐸𝜙𝑥^∗(𝑇)−𝜙(𝑥^𝜀(𝑇)+𝜀,𝐸[𝛾(𝑦^∗(0)−𝛾(𝑦^𝜀_𝜌(0)]+𝜀²−𝐸[𝛾(𝑦^∗(0))−𝛾(𝑦^𝜀(0))]+𝜀²=−2𝜌𝐸𝛾_𝑦𝑦^𝜀(0)⋅𝜌𝑦^𝜀₀𝐸𝛾𝑦^∗(0)−𝛾𝑦^𝜀(0)+𝜀. (2.38) For the given 𝜀, we consider the following four cases.
Case 1. There exists 𝜌₀>0 such that 𝐸𝜙𝑥^𝜀_𝜌(𝑇)−𝜙𝑥^𝜀(𝑇)+𝜀>0,𝐸𝛾𝑦^𝜀_𝜌(0)−𝛾𝑦^𝜀(0)+𝜀>0 (2.39) for all 𝜌∈(0,𝜌₀).
In this case, lim_𝜌→0𝐹_𝜀𝜉^𝜀_𝜌,𝜂^𝜀_𝜌,𝑢^𝜀_𝜌(⋅)−𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝜌=lim_𝜌→01𝐹_𝜀𝜉^𝜀_𝜌,𝜂^𝜀_𝜌,𝑢^𝜀_𝜌(⋅)+𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐹²_𝜀𝜉^𝜀_𝜌,𝜂^𝜀_𝜌,𝑢^𝜀_𝜌(⋅)−𝐹²_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝜌=1𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸𝜓𝑥^𝜀(𝑇)−𝑎,𝐸𝜓_𝑥𝑥^𝜀(𝑇)𝑥^𝜀(𝑇+𝑦)𝐸ℎ^𝜀(0)−𝑏,𝐸ℎ_𝑦𝑦^𝜀(0)𝑦^𝜀(0−𝐸𝜙𝑥_∗)(𝑇)−𝜙𝑥^𝜀(𝑇)+𝜀,𝐸𝜙_𝑥𝑥^𝜀(𝑇)𝑥^𝜀(𝑇𝛾𝑦)−⟨𝐸^∗(0)−𝛾𝑦^𝜀(0)+𝜀,𝐸𝛾_𝑦𝑦^𝜀(0)𝑦^𝜀(0)⟩. (2.40) Dividing (2.34) by 𝜌 and sending 𝜌 to 0, we obtain ℎ^𝜀₃𝐸⟨𝜓_𝑥𝑥^𝜀(𝑇),𝑥^𝜀(𝑇)⟩+ℎ^𝜀₂𝐸⟨ℎ_𝑦𝑦^𝜀(0),𝑦^𝜀(0)⟩+ℎ^𝜀₁𝐸⟨𝜙_𝑥𝑥^𝜀(𝑇),𝑥^𝜀(𝑇)⟩+ℎ^𝜀₀𝐸⟨𝛾_𝑦(𝑦^𝜀(0),𝑦^𝜀(0)⟩≥−√2𝜀𝑑𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅),𝜉,𝜂,𝑢(⋅), (2.41) where ℎ^𝜀₀1=−𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸𝛾𝑦^∗(0)−𝛾𝑦^𝜀(0ℎ)+𝜀≤0,_𝜀11=−𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸𝜙𝑥^∗(𝑇)−𝜙𝑥^𝜀(𝑇)ℎ+𝜀≤0,_𝜀2=1𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸ℎ𝑦^𝜀(0),ℎ−𝑏_𝜀3=1𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸𝜓𝑥^𝜀(𝑇)−𝑎. (2.42)Case 2. There exists a position sequence {𝜌_𝑛} satisfying 𝜌_𝑛→0 such that 𝐸𝜙𝑥^𝜀_𝜌𝑛(𝑇)−𝜙𝑥^∗(𝑇)+𝜀≤0,𝐸𝛾𝑦^𝜀_𝜌𝑛(0)−𝛾𝑦^∗(0)+𝜀>0. (2.43) Then 𝐹_𝜀𝜉^𝜀_𝜌𝑛,𝜂^𝜀_𝜌𝑛,𝑢^𝜀_𝜌𝑛(⋅)=𝐸𝜓(𝑥^𝜀_𝜌𝑛(𝑇))−𝑎²+𝐸ℎ(𝑦^𝜀_𝜌𝑛(0))−𝑏²+max(0,𝐸[𝛾(𝑦^∗(0))−𝛾(𝑦^𝜀_𝜌𝑛(0))]+𝜀)^21/2. (2.44) For sufficiently large 𝑛, since 𝐹_𝜀(⋅) is continuous, we conclude 𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)=𝐸𝜓(𝑥^𝜀(𝑇))−𝑎^21/2+𝐸ℎ(𝑦^𝜀(0))−𝑏²+max0,𝐸𝛾𝑦^∗(0)−𝛾𝑦^𝜀(0)+𝜀^21/2. (2.45) Now lim_𝑛→∞𝐹_𝜀𝜉^𝜀_𝜌𝑛,𝜂^𝜀_𝜌𝑛,𝑢^𝜀_𝜌𝑛(⋅)−𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝜌_𝑛=lim_𝑛→∞1𝐹_𝜀𝜉^𝜀_𝜌𝑛,𝜂^𝜀_𝜌𝑛,𝑢^𝜀_𝜌𝑛(⋅)+𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐹²_𝜀𝜉^𝜀_𝜌𝑛,𝜂^𝜀_𝜌𝑛,𝑢^𝜀_𝜌𝑛(⋅)−𝐹²_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝜌_𝑛=1𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)⟨𝐸𝜓𝑥^𝜀(𝑇)−𝑎,𝐸𝜓_𝑥𝑥^𝜀(𝑇)𝑥^𝜀(𝑇⟩𝑦)+⟨𝐸ℎ^𝜀(0)−𝑏,𝐸ℎ_𝑦𝑦^𝜀(0)𝑦^𝜀(0⟩−𝐸𝛾𝑦_∗)(0)−𝛾𝑦^𝜀(0)+𝜀,𝐸𝛾_𝑦𝑦^𝜀(0)𝑦^𝜀(0). (2.46) Similar to Case 1 we get ℎ^𝜀₃𝐸⟨𝜓_𝑥𝑥^𝜀(𝑇),𝑥^𝜀(𝑇)⟩+ℎ^𝜀₂𝐸⟨ℎ_𝑦𝑦^𝜀(0),𝑦^𝜀(0)⟩+ℎ^𝜀₀𝐸⟨𝛾_𝑦𝑦^𝜀(0),𝑦^𝜀(0)⟩≥−√2𝜀𝑑𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅),𝜉,𝜂,𝑢(⋅), (2.47) where ℎ^𝜀₀1=−𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸𝛾𝑦^∗(0)−𝛾𝑦^𝜀(0ℎ)+𝜀≤0,_𝜀1ℎ=0,_𝜀2=1𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸ℎ𝑦^𝜀(0),ℎ−𝑏_𝜀3=1𝐹_𝜀𝜉^𝜀,𝜂^𝜀,𝑢^𝜀(⋅)𝐸𝜓𝑥^𝜀(𝑇)−𝑎. (2.48)Case 3. There exists a positive sequence {𝜌_𝑛} satisfying 𝜌_𝑛→0 such that 𝐸𝜙𝑥^𝜀_𝜌𝑛(𝑇)−𝜙𝑥^∗(𝑇)+𝜀>0,𝐸𝛾𝑦^𝜀_𝜌𝑛(0)−𝛾𝑦^∗(0)+𝜀≤0. (2.49)Case 4. There exists a positive sequence {𝜌_𝑛} satisfying 𝜌_𝑛→0 such that 𝐸𝜙𝑥^𝜀_𝜌𝑛(𝑇)−𝜙𝑥^∗(𝑇)+𝜀≤0,𝐸𝛾𝑦^𝜀_𝜌𝑛(0)−𝛾𝑦^∗(0)+𝜀≤0. (2.50) Similar techniques can be used for both Case 3 and Case 4.
In summary, for all those cases, we have ℎ^𝜀₀≤0,ℎ^𝜀₁≤0 and |ℎ^𝜀₀|²+|ℎ^𝜀₁|²+|ℎ^𝜀₂|²+|ℎ^𝜀₃|²=1 by the definition of 𝐹_𝜀(⋅). Then there exists a convergent subsequence of (ℎ^𝜀₃,ℎ^𝜀₂,ℎ^𝜀₁,ℎ^𝜀₀) 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 where 𝑙(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: −𝑑𝑚(𝑡)=−𝐹^∗_𝑥(𝑡)𝑚(𝑡)+𝐺^∗_𝑥(𝑡)𝑝(𝑡)+𝑓^∗_𝑥(𝑡)𝑛(𝑡)+𝑔^∗_𝑥(𝑡)𝛿(𝑡)+𝐹𝑑𝑡_∗𝑧(𝑡)𝑚(𝑡)+𝐺^∗_𝑧(𝑡)𝑝(𝑡)+𝑓^∗_𝑧(𝑡)𝑛(𝑡)+𝑔^∗_𝑧(𝑡)𝛿(𝑡)𝑑𝐵_𝑡−𝑝_𝑡𝑑𝑊_𝑡,𝑚(𝑇)=−ℎ₃𝜓_𝑥𝑥^∗_𝑇+ℎ₁𝜙_𝑥𝑥^∗_𝑇,𝑑𝑛(𝑡)=𝐹^∗_𝑦(𝑡)𝑚(𝑡)+𝐺^∗_𝑦(𝑡)𝑝(𝑡)+𝑓^∗_𝑦(𝑡)𝑛(𝑡)+𝑔^∗_𝑦(𝑡)𝛿(𝑡)+𝐹𝑑𝑡_∗𝑞(𝑡)𝑚(𝑡)+𝐺^∗_𝑞(𝑡)𝑝(𝑡)+𝑓^∗_𝑞(𝑡)𝑛(𝑡)+𝑔^∗_𝑞(𝑡)𝛿(𝑡)𝑑𝑊_𝑡−𝛿_𝑡𝑑𝐵_𝑡,𝑛(0)=ℎ₂ℎ_𝑦𝑦^∗₀+ℎ₀𝛾_𝑦𝑦^∗₀, (2.51) where 𝐹^∗_𝑘(𝑡), 𝑓^∗_𝑘(𝑡), 𝐺^∗_𝑘(𝑡), 𝑔^∗_𝑘(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], ⟨𝑚(0),𝜉−𝜉^∗(𝑇)⟩≤0,⟨𝑛,𝜂−𝜂^∗(𝑡)𝐹⟩≥0,⟨𝑚^∗_𝑢(𝑡)+𝑝(𝑡)𝐺^∗_𝑢(𝑡)+𝑛(𝑡)𝑓^∗_𝑢(𝑡)+𝑔^∗_𝑢(𝑡)𝛿(𝑡),𝑢(𝑡)−𝑢^∗(𝑡)⟩≥0. (2.52)

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 𝐸⟨𝑚(𝑇),𝑥(𝑇)⟩+𝐸𝑛(𝑇),𝑦(𝑇)−𝐸⟨𝑚(0),𝑥(0)⟩−𝐸𝑛(0),𝑦(0)=𝐸^𝑇₀𝑚_𝑡−𝐹^∗_𝑥(𝑡)𝑥(𝑡)−𝐹^∗_𝑧(𝑡)𝑧(𝑡)−𝐹^∗_𝑦(𝑡)𝑦(𝑡)−𝐹^∗_𝑞(𝑡)𝑞(𝑡)−𝐹^∗_𝑢(𝑡)𝑢(𝑡)𝑑𝑡+𝐸^𝑇₀𝑥(𝑡)𝐹^∗_𝑥(𝑡)𝑚(𝑡)+𝐺^∗_𝑥(𝑡)𝑝(𝑡)+𝑓^∗_𝑥(𝑡)𝑛(𝑡)+𝑔^∗_𝑥(𝑡)𝛿(𝑡)𝑑𝑡+𝐸^𝑇₀𝑧(𝑡)𝐹^∗_𝑧(𝑡)𝑚(𝑡)+𝐺^∗_𝑧(𝑡)𝑝(𝑡)+𝑓^∗_𝑧(𝑡)𝑛(𝑡)+𝑔^∗_𝑧(𝑡)𝛿(𝑡)𝑑𝑡−𝐸^𝑇₀𝑝(𝑡)𝐺^∗_𝑥(𝑡)𝑥(𝑡)+𝐺^∗_𝑧(𝑡)𝑧(𝑡)+𝐺^∗_𝑦(𝑡)𝑦(𝑡)+𝐺^∗_𝑞(𝑡)𝑞(𝑡)+𝐺^∗_𝑢(𝑡)𝑢(𝑡)𝑑𝑡−𝐸^𝑇₀𝑛(𝑡)𝑓^∗_𝑥(𝑡)𝑥(𝑡)+𝑓^∗_𝑧(𝑡)𝑧(𝑡)+𝑓^∗_𝑦(𝑡)𝑦(𝑡)+𝑓^∗_𝑞(𝑡)𝑞(𝑡)+𝑓^∗_𝑢(𝑡)𝑢(𝑡)𝑑𝑡+𝐸^𝑇₀𝑦(𝑡)𝐹^∗_𝑦(𝑡)𝑚(𝑡)+𝐺^∗_𝑦(𝑡)𝑝(𝑡)+𝑓^∗_𝑦(𝑡)𝑛(𝑡)+𝑔^∗_𝑦(𝑡)𝛿(𝑡)𝑑𝑡+𝐸^𝑇₀𝑞(𝑡)𝐹^∗_𝑞(𝑡)𝑚(𝑡)+𝐺^∗_𝑞(𝑡)𝑝(𝑡)+𝑓^∗_𝑞(𝑡)𝑛(𝑡)+𝑞^∗_𝑞(𝑡)𝛿(𝑡)𝑑𝑡−𝐸^𝑇₀𝛿(𝑡)𝑔^∗_𝑥(𝑡)𝑥(𝑡)+𝑔^∗_𝑧(𝑡)𝑧(𝑡)+𝑔^∗_𝑦(𝑡)𝑦(𝑡)+𝑔^∗_𝑞(𝑡)𝑞(𝑡)+𝑔^∗_𝑢(𝑡)𝑢(𝑡)𝑑𝑡=−𝐸^𝑇₀⟨𝑚(𝑡)𝐹^∗_𝑢(𝑡)+𝑝(𝑡)𝐺^∗_𝑢(𝑡)+𝑛(𝑡)𝑓^∗_𝑢(𝑡)+𝑔^∗_𝑢(𝑡)𝛿(𝑡),𝑢(𝑡)⟩𝑑𝑡. (2.53) This yields 𝐸⟨𝑛(𝑇),𝑦(𝑇)⟩−𝐸⟨𝑚(0),𝑥(0)⟩+𝐸^𝑇₀⟨𝑚(𝑡)𝐹^∗_𝑢(𝑡)+𝑝(𝑡)𝐺^∗_𝑢(𝑡)+𝑛(𝑡)𝑓^∗_𝑢(𝑡)+𝛿(𝑡)𝑔^∗_𝑢(𝑡),𝑢(𝑡)−𝑢^∗(𝑡)⟨𝑚(𝑇),𝑥(𝑇𝑛(0),𝑦(0⟩𝑑𝑡=−𝐸)⟩+𝐸)=ℎ₃𝐸𝜓_𝑥𝑥^∗(𝑇),𝑥(𝑇)+ℎ₂𝐸ℎ_𝑦𝑦^∗(0),𝑦(0)+ℎ₁𝐸𝜙_𝑥𝑥^∗(𝑇),𝑥(𝑇)+ℎ₀𝐸𝛾_𝑦𝑦^∗(0),𝑦(0)≥0. (2.54) For any (𝜉,𝜂,𝑢(⋅))∈𝑈, we get 𝐸⟨𝑛(𝑇),𝜂−𝜂^∗⟩−𝐸⟨𝑚(0),𝜉−𝜉^∗⟩+𝐸^𝑇₀⟨𝑚(𝑡)𝐹^∗_𝑢(𝑡)+𝑝(𝑡)𝐺^∗_𝑢(𝑡)+𝑛(𝑡)𝑓^∗_𝑢(𝑡)+𝛿(𝑡)𝑔^∗_𝑢(𝑡),𝑢(𝑡)−𝑢^∗(𝑡)⟩𝑑𝑡≥0. (2.55) Thus, it is easy to see that (2.52) holds.

2.3.4. The General Case

Define the Hamiltonian 𝐻∶ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑×ℝ^𝑛×𝑑×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑘×𝑑×[0,𝑇]→ℝ (2.56) by 𝐻=𝐹(𝑡+𝐺(𝑡(𝑡)(𝑡)⟩.𝑥,𝑧,𝑦,𝑞,𝑢,𝑚,𝑝,𝑛,𝛿,𝑡𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝑚)𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝑝)+⟨𝑓𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝑛⟩+⟨𝑔𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝛿+𝑙𝑥,𝑧,𝑦,𝑞,𝑢,𝑡 (2.57) 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: −𝑑𝑚(𝑡)=−𝐹^∗_𝑥(𝑡)𝑚(𝑡)+𝐺^∗_𝑥(𝑡)𝑝(𝑡)+𝑓^∗_𝑥(𝑡)𝑛(𝑡)+𝑔^∗_𝑥(𝑡)𝛿(𝑡)+𝑙^∗_𝑥(𝑡)+𝐹𝑑𝑡_∗𝑧(𝑡)𝑚(𝑡)+𝐺^∗_𝑧(𝑡)𝑝(𝑡)+𝑓^∗_𝑧(𝑡)𝑛(𝑡)+𝑔^∗_𝑧(𝑡)𝛿(𝑡)+𝑙^∗_𝑧(𝑡)𝑑𝐵_𝑡−𝑝_𝑡𝑑𝑊_𝑡,𝑚(𝑇)=−ℎ₃𝜓_𝑥𝑥^∗_𝑇+ℎ₁𝜙_𝑥𝑥^∗_𝑇,𝑑𝑛(𝑡)=𝐹^∗_𝑦(𝑡)𝑚(𝑡)+𝐺^∗_𝑦(𝑡)𝑝(𝑡)+𝑓^∗_𝑦(𝑡)𝑛(𝑡)+𝑔^∗_𝑦(𝑡)𝛿(𝑡)+𝑙^∗_𝑦(𝑡)+𝐹𝑑𝑡_∗𝑞(𝑡)𝑚(𝑡)+𝐺^∗_𝑞(𝑡)𝑝(𝑡)+𝑓^∗_𝑞(𝑡)𝑛(𝑡)+𝑔^∗_𝑞(𝑡)𝛿(𝑡)+𝑙^∗_𝑞(𝑡)𝑑𝑊_𝑡−𝛿_𝑡𝑑𝐵_𝑡,𝑛(0)=ℎ₂ℎ_𝑦𝑦^∗₀+ℎ₀𝛾_𝑦𝑦^∗₀, (2.58) where 𝑙^∗_𝑎(𝑡)=𝑙_𝑎(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]: 𝑚(0)−𝜒_𝑥𝜉^∗,𝜉−𝜉^∗(𝑇)≤0,⟨𝑛+𝜆_𝑦𝜂^∗,𝜂−𝜂^∗⟩≥0,⟨𝐻_𝑢𝑡,𝑥^∗(𝑡),𝑧^∗(𝑡),𝑦^∗(𝑡),𝑞^∗(𝑡),𝑢^∗(𝑡),𝑚(𝑡),𝑝(𝑡),𝑛(𝑡),𝛿(𝑡),𝑢(𝑡)−𝑢^∗(𝑡)⟩≥0. (2.59)

Remark 2.9. Let us denote the boundary of 𝐾₁ by 𝜕𝐾₁. Set 𝑀≜𝑤∈Ω∣𝜉^∗(𝜔)∈𝜕𝐾₁. (2.60) Then 𝑚(0)−𝜒_𝑥𝜉^∗𝑚(0)≤0a.s.on𝑀,−𝜒_𝑥𝜉^∗=0a.s.on𝑀^𝑐. (2.61) 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 given 𝜉∈𝐿²(Ω,ℱ₀,𝑃;ℝ^𝑛) and 𝑢(⋅)∈𝑈[0,𝑇], we consider the following controlled system composed of a FSDE and a BDSDE: 𝑑𝑦(𝑡)=𝑏𝑡,𝑦(𝑡),𝑢(𝑡)𝑑𝑡+𝜎𝑡,𝑦(𝑡),𝑢(𝑡)𝑑𝑊_𝑡,𝑦(0)(𝑡)==𝑏,−𝑑𝑥𝑓𝑡,𝑥(𝑡),𝑧(𝑡),𝑦(𝑡),𝑢(𝑡)𝑑𝑡+𝑔𝑡,𝑥(𝑡),𝑧(𝑡),𝑦(𝑡),𝑢(𝑡)𝑑𝑊_𝑡−𝑧(𝑡)𝑑𝐵_𝑡,𝑥(0)=𝜉, (3.1) where 𝑏∈𝑅^𝑘 is given, 𝑥(0)=𝜉∈𝐾₁, a.s., where 𝐾₁ is a given nonempty convex subset in 𝑅^𝑛.
Set the mappings 𝑏∶Ω×[0,𝑇]×ℝ^𝑘×ℝ^𝑛×𝑑→ℝ^𝑘,𝜎∶Ω×[0,𝑇]×ℝ^𝑘×ℝ^𝑛×𝑑→ℝ^𝑘×𝑑,𝑓∶Ω×[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑛×𝑑→ℝ^𝑛,𝑔∶Ω×[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑛×𝑑→ℝ^𝑛×𝑑. (3.2) In this case, we regard 𝑢(⋅) and 𝜉 as the control variables. Define the following cost function: 𝐽(𝜉,𝑢(⋅))=𝐸^𝑇₀𝑙𝑡,𝑥(𝑡),𝑧(𝑡),𝑦(𝑡),𝑢(𝑡)𝑑𝑡+𝜒(𝜉)+𝜆𝑦(𝑇)+𝜙(𝑥(𝑇)), (3.3) where 𝑙∶Ω×[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘×ℝ^𝑛×𝑑→ℝ,𝜒∶ℝ^𝑛→ℝ,𝜆∶ℝ^𝑘→ℝ,𝜙∶ℝ^𝑛→ℝ. (3.4) 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 constants 𝐶>0 and 0<𝛼<1/2 such that for any (𝜔,𝑡)∈Ω×[0,𝑇],  𝑢∈ℝ^𝑛×𝑑, (𝑥₁,𝑧₁,𝑦₁),(𝑥₂,𝑧₂,𝑦₂)∈ℝ^𝑛×ℝ^𝑛×𝑙×ℝ^𝑘 the following conditions hold: 𝑓(𝑡,𝑥₁,𝑧₁,𝑦₁,𝑢)−𝑓(𝑡,𝑥₂,𝑧₂,𝑦₂,𝑢)²≤𝐶𝑥₁−𝑥₂²+𝑧₁−𝑧₂²+𝑦₁−𝑦₂²,𝑔(𝑡,𝑥₁,𝑧₁,𝑦₁,𝑢)−𝑔(𝑡,𝑥₂,𝑧₂,𝑦₂,𝑢)²≤𝐶𝑥₁−𝑥₂²+𝑦₁−𝑦₂²+𝛼𝑧₁−𝑧₂². (3.5)(H3) The derivatives of 𝑏,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 given 𝜉∈𝐿²(Ω,ℱ₀,𝑃;ℝ^𝑛) and 𝑢(⋅)∈𝑈[0,𝑇], there exists a unique triple 𝑥(⋅),𝑦(⋅),𝑧(⋅)∈𝑀²0,𝑇;ℝ^𝑛×𝑀²0,𝑇;ℝ^𝑘×𝑀²0,𝑇;ℝ^𝑛×𝑙. (3.6) which solves (3.1).

We assume an additional terminal state constraint 𝑦(𝑇)=𝜂∈𝐾₂, 𝑎.𝑠., where 𝐾₂ is a given nonempty convex subset in ℝ^𝑘. Our stochastic control problem is ((⋅(⋅)[]inf𝐽𝜉,𝑢))subjectto𝑢∈𝑈0,𝑇;𝜉∈𝐾₁,a.s,𝜂∈𝐾₂,a.s. (3.7)

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 exists 𝛼>0 such that |𝜎(𝑦,𝑢₁,𝑡)−𝜎(𝑦,𝑢₂,𝑡)|≥𝛼|𝑢₁−𝑢₂| for all 𝑦∈ℝ^𝑘,𝑡∈[0,𝑇] and 𝑢₁, 𝑢₂∈ℝ^𝑛×𝑑.
Note (H1) and (H4) imply the mapping 𝑢→𝜎(𝑦,𝑢,𝑡) is a bijection from ℝ^𝑛×𝑑 onto itself for any (𝑦,𝑡).
Let 𝑞≡𝜎(𝑦,𝑢,𝑡) and denote the inverse function by 𝑢=𝜎(𝑦,𝑞,𝑡). Then system (3.1) can be rewritten as −𝑑𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑞(𝑡)𝑑𝑡−𝑞(𝑡)𝑑𝑊_𝑡,𝑦(0)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)𝑞(𝑡)=𝑏,−𝑑𝑥=𝐹𝑡,𝑥,𝑧,𝑦,𝑞𝑑𝑡+𝐺𝑡,𝑥,𝑧,𝑦𝑑𝑊_𝑡−𝑧(𝑡)𝑑𝐵_𝑡,𝑥(0)=𝜉, (3.8) 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: −𝑑𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑞(𝑡)𝑑𝑡−𝑞(𝑡)𝑑𝑊_𝑡,𝑦(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)=𝜂,−𝑑𝑥=𝐹𝑡,𝑥,𝑧,𝑦,𝑞𝑑𝑡+𝐺𝑡,𝑥,𝑧,𝑦,𝑞𝑑𝑊_𝑡−𝑧(𝑡)𝑑𝐵_𝑡,𝑥(0)=𝜉, (3.9) where the control variables are the random variables 𝜉 and 𝜂 to be chosen from the following set: 𝑈=𝜉,𝜂∣𝜉∈𝐾₁,a.s.𝐸𝜉²<∞,𝜂∈𝐾₂,a.s.𝐸𝜂²<∞. (3.10) For each (𝜉,𝜂)∈𝑈, consider the following cost: 𝐽𝜉,𝜂=𝐸^𝑇₀𝑙𝑡,𝑥(𝑡),𝑧(𝑡),𝑦(𝑡),𝑞(𝑡)𝑑𝑡+𝜒(𝜉)+𝜆𝜂+𝜙(𝑥(𝑇)), (3.11) 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: (0)inf𝐽𝜉,𝜂subjectto𝜉,𝜂∈𝑈;𝑦=𝑏, (3.12) 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 𝑑𝑛(𝑡)=𝐹^∗_𝑦(𝑡)𝑚(𝑡)+𝐺^∗_𝑦(𝑡)𝑝(𝑡)+𝑓^∗_𝑦(𝑡)𝑛(𝑡)+𝑙^∗_𝑦(𝑡+𝐹)𝑑𝑡_∗𝑞(𝑡)𝑚(𝑡)+𝐺^∗_𝑞(𝑡)𝑝(𝑡)+𝑓^∗_𝑞(𝑡)𝑛(𝑡)+𝑙^∗_𝑞(𝑡)𝑑𝑊_𝑡,𝑛(0)=ℎ₂,−𝑑𝑚(𝑡)=−𝐹^∗_𝑥(𝑡)𝑚(𝑡)+𝐺^∗_𝑥(𝑡)𝑝(𝑡)+𝑙^∗_𝑥(𝑡)+𝐹𝑑𝑡_∗𝑧(𝑡)𝑚(𝑡)+𝐺^∗_𝑧(𝑡)𝑝(𝑡)+𝑙^∗_𝑧(𝑡)𝑑𝐵_𝑡−𝑝_𝑡𝑑𝑊_𝑡,𝑚(𝑇)=−ℎ₁𝜙_𝑥𝑥^∗(𝑇), (3.13) where 𝐻^∗_𝑘(𝑡)=𝐻_𝑘(𝑡,𝑥^∗(𝑡),𝑧^∗(𝑡),𝑦^∗(𝑡),𝑞^∗(𝑡)) (𝐻=𝐹,𝐺,𝑙) and 𝑓^∗_𝑥(𝑡)=𝑓_𝑥(𝑡,𝑥^∗(𝑡),𝑧^∗(𝑡)), 𝑓^∗_𝑞(𝑡)=𝑓_𝑞(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: ⟨𝑚(0)−𝜒_𝑥𝜉^∗,𝜉−𝜉^∗(𝑇)⟩≤0,⟨𝑛+𝜆_𝑦𝜂^∗,𝜂−𝜂^∗⟩≥0. (3.14)

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. Given 𝜂∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑘) and 𝑢(⋅)∈𝑈[0,𝑇], consider the following BDSDE: −𝑑𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑞(𝑡),𝑢(𝑡)𝑑𝑡+𝑔𝑡,𝑦(𝑡),𝑞(𝑡),𝑢(𝑡)𝑑𝐵_𝑡−𝑞(𝑡)𝑑𝑊_𝑡𝑦(𝑇),0≤𝑡≤𝑇,=𝜂. (3.15) 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 that 𝑦(0)^{(𝜂,𝑢(⋅))} is a ℱ^𝐵_0,𝑇-measurable variable, and 𝐸ℎ𝑦^{(𝜂,𝑢(⋅))}(0)=𝑏. (3.16) Now we regard 𝜂 and 𝑢(⋅) as the control variables to be chosen from the following set: 𝑈=𝜂,𝑢(⋅)∣𝜂∈𝐾₁,a.s.,𝐸𝜂²<∞,𝑢(⋅)∈𝑈[0,𝑇]. (3.17) For each (𝜂,𝑢(⋅))∈𝑈, consider the following cost function: 𝐽𝜂,𝑢(⋅)=𝐸^𝑇₀𝑙𝑡,𝑦(𝑡),𝑞(𝑡),𝑢(𝑡)𝑑𝑡+𝜆𝜂+𝛾𝑦(0), (3.18) which gives rise to the following optimization problem: inf_{(⋅𝜂,𝑢)∈𝑈}𝐽(⋅)ℎ𝑦𝜂,𝑢subjectto𝐸^{(𝜂,𝑢(⋅))}(0)=𝑏. (3.19)

Maximum Principle
Let 𝐻𝑦,𝑞,𝑢,𝑛,𝛿≜⟨𝑓𝑡,𝑦,𝑞,𝑢,𝑛(𝑡)⟩+⟨𝑔𝑡,𝑦,𝑞,𝑢,𝛿(𝑡)⟩+𝑙𝑡,𝑦,𝑞,𝑢. (3.20) Then the adjoint equation is 𝑑𝑛(𝑡)=𝑓^∗_𝑦(𝑡)𝑛(𝑡)+𝑔^∗_𝑦(𝑡)𝛿(𝑡)+𝑙^∗_𝑦(𝑡)𝑑𝑡−𝛿(𝑡)𝑑𝐵_𝑡+𝑓^∗_𝑞(𝑡)𝑛(𝑡)+𝑔^∗_𝑞(𝑡)𝛿(𝑡)+𝑙^∗_𝑞(𝑡)𝑑𝑊_𝑡,𝑛(0)=ℎ₂ℎ_𝑦𝑦^∗(0)+ℎ₀𝛾_𝑦𝑦^∗(0), (3.21) where 𝐻^∗_𝑘(𝑡)=𝐻_𝑘(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 ⟨𝑛(𝑇)+𝜆_𝑦𝜂^∗,𝜂−𝜂^∗⟩≥0,⟨𝐻_𝑢𝑡,𝑦^∗,𝑞^∗,𝑢^∗,𝑛,𝛿,𝑢(𝑡)−𝑢^∗(𝑡)⟩≥0. (3.22)

3.3. Backward Doubly Stochastic LQ Problem without State Constraints

Consider the following linear system: (𝑡)=𝐴(𝑡)𝑥(𝑡)(𝑡)𝑧(𝑡)(𝑡)𝑦(𝑡)(𝑡)𝑞(𝑡)(𝑡)𝑢(𝑡+𝐴−𝑑𝑥+𝐵+𝐶+𝐷+𝐸)𝑑𝑡_′(𝑡)𝑥(𝑡)+𝐵^′(𝑡)𝑧(𝑡)+𝐶^′(𝑡)𝑦(𝑡)+𝐷^′(𝑡)𝑞(𝑡)+𝐸^′(𝑡)𝑢(𝑡)𝑑𝑊_𝑡−𝑧(𝑡)𝑑𝐵_𝑡,𝑥(0)(𝑡)=𝐴=𝑥,−𝑑𝑦^′′(𝑡)𝑥(𝑡)+𝐵^′′(𝑡)𝑧(𝑡)+𝐶^′′(𝑡)𝑦(𝑡)+𝐷^′′(𝑡)𝑞(𝑡)+𝐸^′′(𝑡)𝑢(𝑡+𝐴)𝑑𝑡_′′′(𝑡)𝑥(𝑡)+𝐵^′′′(𝑡)𝑧(𝑡)+𝐶^′′′(𝑡)𝑦(𝑡)+𝐷^′′′(𝑡)𝑞(𝑡)+𝐸^′′′(𝑡)𝑢(𝑡)𝑑𝐵_𝑡−𝑞(𝑡)𝑑𝑊_𝑡,𝑦(𝑇)=𝑦, (3.23) 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 𝑙(𝑡)(𝑡)(𝑡)(𝑡)(𝑡=1𝑡,𝑥,𝑧,𝑦,𝑞,𝑢)2𝐹(𝑡)𝑥(𝑡)(𝑡)+1⋅𝑥2𝐺(𝑡)𝑧(𝑡)(𝑡)+1⋅𝑧2𝐻(𝑡)𝑦(𝑡)(𝑡)+1⋅𝑦2𝐼(𝑡)𝑞(𝑡)(𝑡)+1⋅𝑞2𝐽(𝑡)𝑢(𝑡)(𝑡),𝜒(𝑥)𝑦(𝑥)=1⋅𝑢=0,𝜆=0,𝜙2𝑈(𝑡)𝑥²𝑦=1,𝛾2𝑄(𝑡)𝑦², (3.24) 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 −𝑑𝑚(𝑡)=−𝐴(𝑡)𝑚(𝑡)+𝐴^′(𝑡)𝑝(𝑡)+𝐴^′′(𝑡)𝑛(𝑡)+𝐴^′′′(𝑡)𝛿(𝑡)+𝐹(𝑡)𝑥^∗(𝑡+𝐵(𝑡)𝑚(𝑡))𝑑𝑡+𝐵^′(𝑡)𝑝(𝑡)+𝐵^′′(𝑡)𝑛(𝑡)+𝐵^′′′(𝑡)𝛿(𝑡)+𝐺(𝑡)𝑧^∗(𝑡)𝑑𝐵_𝑡−𝑝(𝑡)𝑑𝑊_𝑡,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+𝐷(𝑡)𝑚(𝑡))𝑑𝑡+𝐷^′(𝑡)𝑝(𝑡)+𝐷^′′(𝑡)𝑛(𝑡)+𝐷^′′′(𝑡)𝛿(𝑡)+𝐼(𝑡)𝑞^∗(𝑡)𝑑𝑊_𝑡−𝛿(𝑡)𝑑𝐵_𝑡,𝑛(0)=𝑄(0)𝑦^∗(0). (3.25) Define 𝐻(𝑡)(𝑡)⟩(𝑡)(𝑡)⟩.𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝑚,𝑝,𝑛,𝛿≜⟨𝐹𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝑚⟩+⟨𝐺𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝑝+⟨𝑓𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝑛⟩+⟨𝑔𝑡,𝑥,𝑧,𝑦,𝑞,𝑢,𝛿+𝑙𝑡,𝑥,𝑧,𝑦,𝑞,𝑢 (3.26) Suppose that 𝐾 is an open set. Then we have the following result: 𝐻_𝑢𝑡,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. (3.27) Thus, 𝑢^∗(𝑡)=𝐽⁻¹(𝑡)𝐸(𝑡)𝑚(𝑡)+𝐸^′(𝑡)𝑝(𝑡)+𝐸^′′(𝑡)𝑛(𝑡)+𝐸^′′′(𝑡)𝛿(𝑡). (3.28) However, the maximum principle gives only the necessary condition for optimal control. We also have the following theorem.

Theorem 3.3. The function 𝑢^∗(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 the 𝑢^∗(⋅) is the optimal control. For all 𝑣(⋅)∈𝐾, let (𝑥^𝑣(⋅),𝑧^𝑣(⋅),𝑦^𝑣(⋅),𝑞^𝑣(⋅)) be the corresponding trajectory of (3.23), we get 𝐽(𝑣(⋅))−𝐽𝑢^∗(⋅)=12^𝑇₀⟨𝐹(𝑡)𝑥^𝑣(𝑡),𝑥^𝑣(𝑡)⟩−⟨𝐹(𝑡)𝑥^∗(𝑡),𝑥^∗(𝑡)⟩+⟨𝐺(𝑡)𝑧^𝑣(𝑡),𝑧^𝑣(𝑡)⟩−𝐺(𝑡)𝑧^∗(𝑡),𝑧^∗(𝑡)+𝐻(𝑡)𝑦^𝑣(𝑡),𝑦^𝑣(𝑡)−𝐻(𝑡)𝑦^∗(𝑡),𝑦^∗(𝑡)+𝐼(𝑡)𝑞^𝑣(𝑡),𝑞^𝑣(𝑡−𝐼(𝑡)𝑞_∗)(𝑡),𝑞^∗(𝑡)+⟨𝐽(𝑡)𝑣(𝑡),𝑣(𝑡)⟩−𝐽(𝑡)𝑢^∗(𝑡),𝑢^∗(𝑡(𝑇)𝑥)𝑑𝑡+⟨𝑈^𝑣(𝑇),𝑥^𝑣(𝑇)⟩−𝑈(𝑇)𝑥^∗(𝑇),𝑥^∗(𝑇)+𝑄(0)𝑦^𝑣(0),𝑦^𝑣(0)−𝑄(0)𝑦^∗(0),𝑦^∗(0)≥𝐸^𝑇₀𝐹(𝑡)𝑥^∗(𝑡),𝑥^𝑣(𝑡)−𝑥^∗(𝑡)+𝐺(𝑡)𝑧^∗(𝑡),𝑧^𝑣(𝑡)−𝑧^∗(𝑡+𝐻(𝑡)𝑦_∗)(𝑡),𝑦^𝑣(𝑡)−𝑦^∗(𝑡)+𝐼(𝑡)𝑞^∗(𝑡),𝑞^𝑣(𝑡)−𝑞^∗(𝑡)+𝐽(𝑡)𝑢^∗(𝑡),𝑣(𝑡)−𝑢^∗(𝑡(𝑇)𝑥)𝑑𝑡+⟨𝑈^∗(𝑇),𝑥^𝑣(𝑇)−𝑥^∗(𝑇)⟩+𝑄(0)𝑦^∗(0),𝑦^𝑣(0)−𝑦^∗(0). (3.29) Using Lemma 2.1 to ⟨𝑥^𝑣(𝑡)−𝑥^∗(𝑡),𝑚(𝑡)⟩+⟨𝑦^𝑣(𝑡)−𝑦^∗(𝑡),𝑛(𝑡)⟩, we obtain ⟨𝑈(𝑇)𝑥^∗(𝑇),𝑥^𝑣(𝑇)−𝑥^∗(𝑇)⟩+⟨𝑄(0)𝑦^∗(0),𝑦^𝑣(0)−𝑦^∗(0)⟩=−𝐸^𝑇₀⟨𝐹(𝑡)𝑥^∗(𝑡),𝑥^𝑣(𝑡)−𝑥^∗(𝑡)⟩+⟨𝐺(𝑡)𝑧^∗(𝑡),𝑧^𝑣(𝑡)−𝑧^∗(𝑡)⟩+⟨𝐻(𝑡)𝑦^∗(𝑡),𝑦^𝑣(𝑡)−𝑦^∗(𝑡)⟩+⟨𝐼(𝑡)𝑞^∗(𝑡),𝑞^𝑣(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. (3.30) So by the definition of 𝑢^∗(𝑡), 𝐽(𝑣(⋅))−𝐽𝑢^∗(⋅)≥𝐸^𝑇₀𝐸(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. (3.31) 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 that 𝑢¹(⋅) 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 𝐽𝑢¹(⋅)=𝐽𝑢²(⋅𝑢)=𝛼≥0,2𝛼=𝐽¹(⋅)+𝐽𝑢²(⋅)𝑢≥2𝐽¹(⋅)+𝑢²(⋅)2+2𝐸^𝑇₀𝐽(𝑡)𝑢¹(⋅)−𝑢²(⋅)2,𝑢¹(⋅)−𝑢²(⋅)2𝛽𝑑𝑡≥2𝛼+2𝐸^𝑇₀𝑢¹(⋅)−𝑢²(⋅)²𝑑𝑡, (3.32) here 𝛽>0. So 𝐸^𝑇₀𝑢¹(⋅)−𝑢²(⋅)²𝑑𝑡≤0, (3.33) which shows that 𝑢¹(⋅)=𝑢²(⋅).

Example 3.4. Consider the following backward doubly stochastic LQ problem, where 𝑡∈[0,1],𝑢(⋅)∈𝑈[0,1]=[−1,1] and 𝑛=𝑘=𝑑=𝑙=1: (𝑡)=1−𝑑𝑥2𝑦(𝑡)+2𝑢(𝑡)𝑑𝑊_𝑡−𝑧(𝑡)𝑑𝐵_𝑡,𝑥(0)(𝑡)=1=0,−𝑑𝑦2(𝑥(𝑡)+𝑢(𝑡))𝑑𝐵_𝑡−𝑞(𝑡)𝑑𝑊_𝑡,𝑦(1)=0. (3.34) We want to minimize the following cost function: 𝐽(𝑢(⋅))=𝐸¹₀𝑥(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. (3.35) From (3.34), we get, for 𝑡∈[0,1], 𝑥(𝑡)=−^𝑡₀𝑦(𝑠)+2𝑢(𝑠)𝑑𝑊_𝑠+^𝑡₀𝑧(𝑠)𝑑𝐵_𝑠,𝑦(𝑡)=¹_𝑡(𝑥(𝑠)+𝑢(𝑠))𝑑𝐵_𝑠−¹_𝑡𝑞(𝑠)𝑑𝑊_𝑠. (3.36) By substituting 𝑥(⋅) and 𝑦(⋅) into the cost function, we obtain 𝐽(𝑢(⋅))=𝐸¹₀3𝑢(𝑠)²𝑑𝑠. (3.37) Thus, the optimal control is 𝑢^∗(𝑡)≡0,𝑡∈[0,1] with the optimal state trajectory 𝑥^∗(𝑡),𝑧^∗(𝑡),𝑦^∗(𝑡),𝑞^∗(𝑡)≡0,𝑡∈[0,1]. (3.38) The adjoint equations are 𝑑𝑚(𝑡)=𝑛(𝑡)𝑑𝑡+𝑝(𝑡)𝑑𝑊_𝑡,𝑚(1)(𝑡)(𝑡)(𝑡)=0,𝑑𝑛=𝑝𝑑𝑡−𝛿𝑑𝐵_𝑡,𝑛(0)=0. (3.39) It is obvious that (𝑚(⋅),𝑝(⋅),𝑛(⋅),𝛿(⋅))=(0,0,0,0) is the unique solution of the above equation.

Acknowledgments

The authors would like to thank Professor Shige Peng for some useful conversations. This work was supported by National Natural Science Foundation of China (no. 11171187, no. 10871118, and no. 10921101); supported by the Programme of Introducing Talents of Discipline to Universities of China (no. B12023); supported by Program for New Century Excellent Talents in University of China.