Relationship between Maximum Principle and Dynamic Programming for Stochastic Recursive Optimal Control Problems and Applications

Abstract

This paper is concerned with the relationship between maximum principle and dynamic programming for stochastic recursive optimal control problems. Under certain differentiability conditions, relations among the adjoint processes, the generalized Hamiltonian function, and the value function are given. A linear quadratic recursive utility portfolio optimization problem in the financial engineering is discussed as an explicitly illustrated example of the main result.

1. Introduction

The nonlinear backward stochastic differential equation (BSDE) was introduced by Pardoux and Peng [1]. Independently, Duffie and Epstein [2] introduced BSDE from economic background. In [2], they presented a stochastic differential formulation of recursive utility. Recursive utility is an extension of the standard additive utility with the instantaneous utility depending not only on the instantaneous consumption rate but also on the future utility. As found by El Karoui et al. [3], the utility process can be regarded as a solution to a special BSDE. The optimal control problem that the cost functional is described by the solution to a BSDE is called a stochastic recursive optimal control problem. In this case, the control systems become forward-backward stochastic differential equations (FBSDEs). This kind of optimal control problems has found important applications in real-world problems such as mathematical economics, mathematical finance, and engineering (see Schroder and Skiadas [4], El Karoui et al. [3, 5], Ji and Zhou [6], Williams [7], and Wang and Wu [8]).

It is well known that Pontryagin’s maximum principle and Bellman’s dynamic programming are two of the most important tools in solving stochastic optimal control problems. See the famous reference book by Yong and Zhou [9] for systematic discussion. For stochastic recursive optimal control problems, Peng [10] first obtained a maximum principle when the control domain is convex. And then Xu [11] studied the nonconvex control domain case, but he needs to assume that the diffusion coefficient does not contain the control variable. Ji and Zhou [6] established a maximum principle when the forward state is constrained in a convex set at the terminal time. Wu [12] established a general maximum principle, where the control domain is nonconvex and the diffusion coefficient depends on the control variable. Maximum principle for stochastic recursive optimal control systems with Poisson jumps, and their applications in finance were studied in Shi and Wu [13], where the control domain is convex.

For another important approach—dynamic programming—to study stochastic recursive optimal control problems, Peng [14] (also see Peng [15]) first obtained the generalized dynamic programming principle and introduced a generalized Hamilton-Jacobi-Bellman (HJB) equation which is a second-order parabolic partial differential equation (PDE). Result that the value function is the viscosity solution to the generalized HJB equation is also proved in [14]. Wu and Yu [16] extended the results of [14, 15] with obstacle constraint for the cost functional described by the solution to a reflected backward stochastic differential equation and proved that the value function is the unique viscosity solution to their generalized HJB equation. Li and Peng [17] generalized the results of [14, 15] by considering the cost functional defined by the controlled BSDE with jumps. They proved that the value function was the viscosity solution to the associated generalized HJB equation with integral-differential operators.

Hence, a natural question arises: are there any relations between these two methods? Such a topic was intuitively discussed by Bismut [18] and Bensoussan [19] and then studied by many researchers. Under certain differentiability conditions, the relationship between the maximum principle and dynamic programming is essentially the relationship between the derivatives of the value function and the solution to the adjoint equation along the optimal state. However, the smoothness conditions do not hold in general and are difficult to verify a priori, see Zhou [20] for the deterministic case and Yong and Zhou [9] for its stochastic counterpart. Zhou [21] first obtained the relationship between general maximum principle and dynamic programming using the viscosity solution theory (see also Zhou [22] or Yong and Zhou [9]), without the assumption that the value function is smooth. For diffusion with jumps, the relationship between maximum principle and dynamic programming was first given by Framstad et al. [23, 24] under certain differentiability conditions, and then Shi and Wu [25] eliminated these restrictions within the framework of viscosity solutions. For singular stochastic optimal control problem, the relationship between maximum principle and dynamic programming was given by Bahlali et al. [26], with the derivatives of the value function. For Markovian regime-switching jump diffusion model, the relationship between maximum principle and dynamic programming was given by Zhang et al. [27], also with the derivatives of the value function.

In this paper, we derive the relationship between maximum principle and dynamic programming for the stochastic recursive optimal control problem. For this problem, we connect the maximum principle of [10] with the dynamic programming of [14, 15], under certain differentiability conditions. Specifically, when the value function is smooth, we give relations among the adjoint processes, the generalized Hamiltonian function, and the value function. For this target, in Section 2, we first adopt some related results of [14, 15], which in this paper are stated as a stochastic verification theorem. Also we prove that under additional convexity conditions, the necessary conditions in the maximum principle of [10] are in fact sufficient. In Section 3, we show the relationship between maximum principle and dynamic programming under certain differentiability conditions for our stochastic recursive optimal control problem, by the martingale representation technique. In Section 4, we discuss a linear quadratic (LQ) recursive utility portfolio optimization problem in the financial engineering. In this problem, the state feedback optimal control is obtained by both the maximum principle and dynamic programming approaches, and the relations we obtained are illustrated explicitly. Finally, we end this paper with some concluding remarks in Section 5.

Notations. Throughout this paper, we denote by 𝐑^𝑛 the space of 𝑛-dimensional Euclidean space, by 𝐑^𝑛×𝑑 the space of 𝑛×𝑑 matrices, and by 𝒮^𝑛 the space of 𝑛×𝑛 symmetric matrices. ⟨⋅,⋅⟩ and |⋅| denote the scalar product and norm in the Euclidean space, respectively. ⊤ appearing in the superscripts denotes the transpose of a matrix.

2. Problem Statement and Preliminaries

Let (Ω,ℱ,𝐏) be a complete probability space equipped with a 𝑑-dimensional standard Brownian motion {𝑊(𝑡)}_𝑡≥0. For fixed 𝑡≥0, the filtration {ℱ^𝑡_𝑠}_𝑠≥𝑡 is generated as ℱ^𝑡_𝑠=𝜎{𝑊(𝑟)−𝑊(𝑡);𝑡≤𝑟≤𝑠}⋁𝒩, where 𝒩 contains all 𝐏-null sets in ℱ and 𝜎₁⋁𝜎₂ denotes the 𝜎-field generated by 𝜎₁∪𝜎₂. In particular, if 𝑡=0, we write ℱ_𝑠≡ℱ^𝑡_𝑠.

Let 𝑇>0 be finite and let 𝐔⊂𝐑^𝑘 be nonempty, convex. For any initial time and state (𝑡,𝑥)∈[0,𝑇)×𝐑^𝑛, consider the state 𝑋^{𝑡,𝑥;𝑢}(⋅)∈𝐑^𝑛 given by the following controlled SDE: 𝑑𝑋^{𝑡,𝑥;𝑢}(𝑠)=𝑏𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)𝑑𝑠+𝜎𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)(𝑠)(𝑠)[],𝑋,𝑢𝑑𝑊,𝑠∈𝑡,𝑇_{𝑡,𝑥;𝑢}(𝑡)=𝑥. (1) Here 𝑏∶[0,𝑇]×𝐑^𝑛×𝐔→𝐑^𝑛, 𝜎∶[0,𝑇]×𝐑^𝑛×𝐔→𝐑^𝑛×𝑑 are given functions.

Given 𝑡∈[0,𝑇), we denote by 𝒰[𝑡,𝑇] the set of {ℱ^𝑡_𝑠}_𝑠≥𝑡-adapted processes. For given 𝑢(⋅)∈𝒰[𝑡,𝑇] and 𝑥∈𝐑^𝑛, an 𝐑^𝑛-valued process 𝑋^{𝑡,𝑥;𝑢}(⋅) is called a solution to (1) if it is an ℱ^𝑡_𝑠-adapted process such that (1) holds. We refer to such 𝑢(⋅)∈𝒰[𝑡,𝑇] as an admissible control and (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑢(⋅)) as an admissible pair. We assume the following.(H1)𝑏, 𝜎 are uniformly continuous in (𝑠,𝑥,𝑢), and there exists a constant 𝐶>0 such that for all 𝑠∈[0,𝑇], 𝑥,𝑥∈𝐑^𝑛, 𝑢,𝑢∈𝐔, |𝑏()(𝑢+|𝜎()(𝑢𝑥|+|𝑢,|𝑏(+|𝜎((|𝑥.𝑠,𝑥,𝑢−𝑏𝑠,𝑥,)|𝑡,𝑥,𝑢−𝜎𝑠,𝑥,)|≤𝐶(|𝑥−𝑢−|)𝑠,𝑥,𝑢)|𝑠,𝑥,𝑢)|≤𝐶1+|) (2)For any 𝑢(⋅)∈𝒰[𝑡,𝑇], under (H1), SDE (1) has a unique solution 𝑋^{𝑡,𝑥;𝑢}(⋅) by the classical SDE theory (see, e.g., Yong and Zhou [9]).

Next, we introduce the following controlled BSDE coupled with controlled SDE (1): −𝑑𝑌^{𝑡,𝑥;𝑢}(𝑠)=𝑓𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)𝑑𝑠−𝑍^{𝑡,𝑥;𝑢}(𝑠)(𝑠)[],𝑌𝑑𝑊,𝑠∈𝑡,𝑇_{𝑡,𝑥;𝑢}(𝑇)=𝜙𝑋^{𝑡,𝑥;𝑢}(𝑇). (3) Here 𝑓∶[0,𝑇]×𝐑^𝑛×𝐑×𝐑^𝑑×𝐔→𝐑, Φ∶𝐑^𝑛→𝐑 are given functions. We assume the following.(H2)𝑓, 𝜙 are uniformly continuous in (𝑠,𝑥,𝑦,𝑧,𝑢) and there exists a constant 𝐶>0 such that for all 𝑠∈[0,𝑇], 𝑥,𝑥∈𝐑^𝑛, 𝑦,𝑦∈𝐑, 𝑧,𝑧∈𝐑^𝑑, 𝑢,𝑢∈𝐔, 𝑓𝑢𝑥|+𝑦+|𝑧|+|𝑢|,𝑓(+𝜙(𝑥(|𝑥,𝜙(𝑥)(𝑥|𝑥|.𝑠,𝑥,𝑦,𝑧,𝑢−𝑓𝑠,𝑥,𝑦,𝑧,≤𝐶|𝑥−𝑦−𝑧−𝑢−𝑠,𝑥,0,0,𝑢))≤𝐶1+|)−𝜙)≤𝐶𝑥− (4)Then for any 𝑢(⋅)∈𝒰[𝑡,𝑇] and the given unique solution 𝑋^{𝑡,𝑥;𝑢}(⋅) to (1), under (H2), BSDE (3) admits a unique solution (𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅)) by the classical BSDE theory (see Pardoux and Peng [1] or Yong and Zhou [9]).

Given 𝑢(⋅)∈𝒰[𝑡,𝑇], we introduce the cost functional 𝐽(𝑡,𝑥;𝑢(⋅))∶=−𝑌^{𝑡,𝑥;𝑢}(𝑠)_𝑠=𝑡,(𝑡,𝑥)∈[0,𝑇]×𝐑^𝑛. (5)

Our recursive stochastic optimal control problem is the following.

Problem 1 (RSOCP). For given (𝑡,𝑥)∈[0,𝑇)×𝐑^𝑛, to minimize (5) subject to (1)–(3) over 𝒰[𝑡,𝑇].
We define the value function 𝑉(𝑡,𝑥)∶=inf_{𝑢(⋅)∈𝒰[𝑡,𝑇]}𝐽(𝑡,𝑥;𝑢(⋅)),(𝑡,𝑥)∈[0,𝑇]×𝐑^𝑛,𝑉(𝑇,𝑥)=−𝜙(𝑥),𝑥∈𝐑^𝑛. (6) Any 𝑢(⋅)∈𝒰[𝑡,𝑇] achieves the above infimum is called an optimal control, and the corresponding solutions 𝑋^{𝑡,𝑥;𝑢}(⋅) to (1) and (𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅)) to (3) are called optimal state. For simplicity, we refer to (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅),𝑢(⋅)) as the optimal quartet.

Remark 1. Because 𝑏, 𝜎, 𝑓, 𝑔 are all deterministic functions, then from [15, Proposition 5.1 of Peng], we know that under (H1) and (H2), the above value function is a deterministic function. So our definition (6) is meaningful.
We introduce the following generalized HJB equation: −𝑣_𝑠(𝑡,𝑥)+sup_𝑢∈𝐔𝐺𝑡,𝑥,−𝑣(𝑡,𝑥),−𝑣_𝑥(𝑡,𝑥),−𝑣_𝑥𝑥()()∈[)𝑡,𝑥,𝑢=0,𝑡,𝑥0,𝑇×𝐑^𝑛,𝑣(𝑇,𝑥)=−𝜙(𝑥),∀𝑥∈𝐑^𝑛, (7) where the generalized Hamiltonian function 𝐺∶[0,𝑇]×𝐑^𝑛×𝐑×𝐑^𝑛×𝒮^𝑛×𝐔→𝐑 is defined as 𝐺1𝑡,𝑥,𝑟,𝑝,𝐴,𝑢∶=2tr𝜎(𝑡,𝑥,𝑢)^⊤(+(()𝐴𝜎𝑡,𝑥,𝑢)𝑝,𝑏𝑡,𝑥,𝑢)+𝑓𝑡,𝑥,𝑟,𝜎𝑡,𝑥,𝑢^⊤𝑝,𝑢. (8)
We have the following result.

Lemma 2 (stochastic verification theorem). Let (H1)-(H2) hold and let (𝑡,𝑥)∈[0,𝑇)×𝐑^𝑛 be fixed. Suppose that 𝑉∈𝐶^1,2([0,𝑇]×𝐑^𝑛) is a solution to (7), then 𝑉()((⋅(⋅)[],()∈[]𝑡,𝑥≤𝐽𝑡,𝑥;𝑢)),∀𝑢∈𝒰𝑡,𝑇𝑡,𝑥0,𝑇×𝐑. (9) Furthermore, an admissible pair (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑢(⋅)) is optimal for Problem (RSOCP) if and only if 𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)=max_𝑢∈𝐔𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢, (10) a.e. 𝑠∈[𝑡,𝑇], 𝐏-a.s.

Proof. For any 𝑢(⋅)∈𝒰[𝑡,𝑇] with the corresponding state 𝑋^{𝑡,𝑥;𝑢}(⋅), applying Itô’s formula to 𝑉(𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)), we obtain the following: 𝑉()𝑋𝑡,𝑥=−𝔼𝜙^{𝑡,𝑥;𝑢}(𝑇)−𝔼^𝑇_𝑡𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑏𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)(𝑠)+1,𝑢2tr𝜎𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)^⊤×𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)×𝜎𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)(𝑠)𝑋,𝑢𝑑𝑠=−𝔼𝜙^{𝑡,𝑥;𝑢}(𝑇)−𝔼^𝑇_𝑡𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)−𝑓𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝜎𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)^⊤×𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)(𝑠)((⋅,𝑢𝑑𝑠=𝐽𝑡,𝑥;𝑢))+𝔼^𝑇_𝑡−𝑉_𝑠𝑠,𝑋_𝑠^{𝑡,𝑥;𝑢}+𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)(𝑠)((⋅,𝑢𝑑𝑠≤𝐽𝑡,𝑥;𝑢))+𝔼^𝑇_𝑡−𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+max_𝑢∈𝐔𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)((⋅.,𝑢𝑑𝑠=𝐽𝑡,𝑥;𝑢)) (11) Thus (9) holds. Next, applying the above inequality to (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑢(⋅)), we have 𝑉(𝑡,𝑥)=𝐽(𝑡,𝑥;𝑢(⋅))+𝔼^𝑇_𝑡−𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)𝑑𝑠. (12) The desired result follows immediately from the fact that −𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)≤0, (13) which is due to the generalized HJB equation (7). The proof is complete.

In convenient to state the maximum principle, we regard the above controlled SDE (1) and BSDE (3) as a controlled FBSDE: 𝑑𝑋^{𝑡,𝑥;𝑢}(𝑠)=𝑏𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)𝑑𝑠+𝜎𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)𝑑𝑊(𝑠),−𝑑𝑌^{𝑡,𝑥;𝑢}(𝑠)=𝑓𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)𝑑𝑠−𝑍^{𝑡,𝑥;𝑢}(𝑠)(𝑠)[],𝑋𝑑𝑊,𝑠∈𝑡,𝑇_{𝑡,𝑥;𝑢}(𝑡)=𝑥,𝑌^{𝑡,𝑥;𝑢}(𝑇)=𝜙𝑋^{𝑡,𝑥;𝑢}(𝑇). (14)

This kind of control system was studied by Peng [10] and a maximum principle was obtained. In order to mention his result, we need the following assumption.(H3)𝑏, 𝜎, 𝜙 are continuously differentiable in (𝑥,𝑢) and 𝑓 is continuously differentiable in (𝑥,𝑦,𝑧,𝑢). Moreover, 𝑏_𝑥, 𝜎_𝑥, 𝑓_𝑥, 𝑓_𝑦, 𝑓_𝑧, 𝑏_𝑢, 𝜎_𝑢, 𝑓_𝑢 are bounded and there exists a constant 𝐶>0 such that 𝜙_𝑥(𝑥)≤𝐶(1+|𝑥|),∀𝑥∈𝐑^𝑛. (15)

Let (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅),𝑢(⋅)) be an optimal quartet. For all 𝑠∈[0,𝑇], we denote 𝑏(𝑠)∶=𝑏𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝜎(𝑠)∶=𝑏𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝑓(𝑠)∶=𝑓𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠), (16) and similar notations are used for all their derivatives.

We introduce the adjoint equation which is an FBSDE: −𝑑𝑝(𝑠)=𝑏_𝑥(𝑠)^⊤𝑝(𝑠)−𝑓_𝑥(𝑠)^⊤𝑞(𝑠)+𝜎_𝑥(𝑠)𝑘(𝑠(𝑠)(𝑠),(𝑠)=)𝑑𝑠−𝑘𝑑𝑊𝑑𝑞𝑓_𝑦(𝑠)^⊤𝑞(𝑠)𝑑𝑠+𝑓_𝑧(𝑠)^⊤𝑞(𝑠)(𝑠)[],𝑝(𝑇)𝑑𝑊,𝑠∈𝑡,𝑇=−𝜙_𝑥𝑋^{𝑡,𝑥;𝑢}(𝑇)^⊤𝑞(𝑇),𝑞(𝑡)=1, (17) and the Hamiltonian function 𝐻∶[0,𝑇]×𝐑^𝑛×𝐑×𝐑^𝑑×𝐔×𝐑^𝑛×𝐑×𝐑^𝑛×𝑑→𝐑 is defined as 𝐻(−𝜎()𝑡,𝑥,𝑦,𝑧,𝑢,𝑝,𝑞,𝑘∶=𝑝,𝑏𝑡,𝑥,𝑢)𝑞,𝑓𝑡,𝑥,𝑦,𝑧,𝑢+tr𝑡,𝑥,𝑢^⊤𝑘. (18)

Under (H1)–(H3), the forward equation in (17) admits an obvious unique solution 𝑝(⋅), and then the backward equation in (17) admits a unique solution (𝑞(⋅),𝑘(⋅)). We call 𝑝, 𝑞, 𝑘 the adjoint processes. Next, the following result holds.

Lemma 3 (necessary maximum principle). Let (H1)–(H3) hold and (𝑡,𝑥)∈[0,𝑇)×𝐑^𝑛 be fixed. Suppose that 𝑢(⋅) is an optimal control for Problem (RSOCP), and (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅)) is the corresponding optimal state. Let (𝑝(⋅),𝑞(⋅),𝑘(⋅)) be the adjoint processes. Then 𝐻_𝑢𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)(𝑠)(𝑠),𝑘(𝑠),𝑝,𝑞,𝑢−𝑢(𝑠)≥0,∀𝑢∈𝐔, (19) a.e. 𝑠∈[𝑡,𝑇], 𝐏-a.s.

Proof. It is an immediate consequence of [10, Theorem 4.4 of Peng].

As we mentioned in the introduction, we can also prove that, under some additional convexity conditions, the above necessary condition in Lemma 3 is also sufficient.

Lemma 4 (sufficient maximum principle). Let (H1)–(H3) hold. Suppose that 𝑢(⋅) is an admissible control, and (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅)) is the corresponding state, with 𝑌(𝑇)=𝑀^⊤_𝑇𝑋(𝑇), 𝑀_𝑇∈𝐑^𝑛. Let (𝑝(⋅),𝑞(⋅),𝑘(⋅)) be the adjoint processes. Suppose that the Hamiltonian function 𝐻 is convex in (𝑥,𝑦,𝑧,𝑢). Then 𝑢(⋅) is an optimal control for Problem (RSOCP) if it satisfies (19).

Proof. For any 𝑢(⋅)∈𝒰[𝑡,𝑇] with the corresponding state (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅)). By Remark 1, we have for fixed 𝑡∈[0,𝑇], 𝐽(𝑡,𝑥;𝑢(𝑡))−𝐽(𝑡,𝑥;𝑢(𝑡))=𝑌^{𝑡,𝑥;𝑢}(𝑡)−𝑌^{𝑡,𝑥;𝑢}(𝑡)=𝔼𝑌^{𝑡,𝑥;𝑢}(𝑡)−𝑌^{𝑡,𝑥;𝑢}(𝑡). (20) Applying Itô’s formula to ⟨𝑌^{𝑡,𝑥;𝑢}(𝑠)−𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑞(𝑠)⟩+⟨𝑋^{𝑡,𝑥;𝑢}(𝑠)−𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑝(𝑠)⟩, noting (14), (17), and 𝑌^{𝑡,𝑥;𝑢}(𝑇)−𝑌^{𝑡,𝑥;𝑢}(𝑇)=𝑀^⊤_𝑇(𝑋^{𝑡,𝑥;𝑢}(𝑇)−𝑋^{𝑡,𝑥;𝑢}(𝑇)), 𝑝(𝑇)=−𝑀_𝑇𝑞(𝑇), we get 𝔼𝑌^{𝑡,𝑥;𝑢}(𝑇)−𝑌^{𝑡,𝑥;𝑢}(𝑇)𝑞(𝑇)−𝔼𝑌^{𝑡,𝑥;𝑢}(𝑡)−𝑌^{𝑡,𝑥;𝑢}(𝑡)𝑞(𝑡)+𝔼𝑋^{𝑡,𝑥;𝑢}(𝑇)−𝑋^{𝑡,𝑥;𝑢}(𝑇),𝑝(𝑇)−𝔼𝑋^{𝑡,𝑥;𝑢}(𝑡)−𝑋^{𝑡,𝑥;𝑢}(𝑡),𝑝(𝑡)=−𝔼𝑌^{𝑡,𝑥;𝑢}(𝑡)−𝑌^{𝑡,𝑥;𝑢}(𝑡)=𝔼^𝑇_𝑡𝑌^{𝑡,𝑥;𝑢}(𝑠)−𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑓_𝑦(𝑠)^⊤𝑞(𝑠+)𝑍^{𝑡,𝑥;𝑢}(𝑠)−𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑓_𝑧(𝑠)^⊤𝑞(𝑠−)𝑓(𝑠)−𝑓𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠)(𝑠),(𝑠+,𝑢,𝑞)𝑋^{𝑡,𝑥;𝑢}(𝑠)−𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑏_𝑥(𝑠)^⊤𝑝(𝑠)+𝑓_𝑥(𝑠)^⊤𝑞(𝑠)−𝜎_𝑥(𝑠)𝑘(𝑠+)𝑏(𝑠)−𝑏𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝑝(𝑠)+tr𝜎(𝑠)−𝜎𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)^⊤𝑘(𝑠)𝑑𝑠=𝔼^𝑇_𝑡𝐻𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝑝(𝑠),𝑞(𝑠),𝑘(𝑠)−𝐻𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)(𝑠)(𝑠)(𝑠)−𝐻,𝑝,𝑞,𝑘_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝑝(𝑠),𝑞(𝑠),𝑘(𝑠),𝑋^{𝑡,𝑥;𝑢}(𝑠)−𝑋^{𝑡,𝑥;𝑢}(𝑠−𝐻_𝑦)𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝑝(𝑠),𝑞(𝑠),𝑘(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠)−𝑌^{𝑡,𝑥;𝑢}(𝑠−𝐻_𝑧)𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝑝(𝑠),𝑞(𝑠),𝑘(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠)−𝑍^{𝑡,𝑥;𝑢}(𝑠)𝑑𝑠. (21) Since 𝐻 is convex in (𝑥,𝑦,𝑧,𝑢), then by the maximum condition (19), 𝐽(𝑡,𝑥;𝑢(𝑡))−𝐽(𝑡,𝑥;𝑢(𝑡))≤𝔼^𝑇_𝑡𝐻_𝑢𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑌^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠),𝑝(𝑠),𝑞(𝑠),𝑘(𝑠),𝑢(𝑠)−𝑢(𝑠)≤0. (22) Thus 𝑢(⋅) is really the optimal control for Problem (RSOCP). The proof is complete.

3. Relationship between Maximum Principle and Dynamic Programming

In this section, we investigate the relationship between the maximum principle and dynamic programming. That is, the connection between the value function 𝑉, the generalized Hamiltonian function 𝐺, and adjoint processes 𝑝, 𝑞, 𝑘. Our main result is the following.

Theorem 5. Let (H1)–(H3) hold and let (𝑡,𝑥)∈[0,𝑇)×𝐑^𝑛 be fixed. Suppose that 𝑢(⋅) is an optimal control for Problem (RSOCP), and (𝑋^{𝑡,𝑥;𝑢}(⋅),𝑌^{𝑡,𝑥;𝑢}(⋅),𝑍^{𝑡,𝑥;𝑢}(⋅)) is the corresponding optimal state. Let (𝑝(⋅),𝑞(⋅),𝑘(⋅)) be the adjoint processes. If 𝑉∈𝐶^1,2([0,𝑇]×𝐑^𝑛), then 𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)=𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)=max_𝑢∈𝐔𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢, (23) a.e. 𝑠∈[𝑡,𝑇], 𝐏-a.s. Furthermore, if 𝑉∈𝐶^1,3([0,𝑇]×𝐑^𝑛) and 𝑉_𝑠𝑥 is also continuous, then 𝑝(𝑠)=𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝑞(𝑠)[]𝑘(𝑠)=𝑉,∀𝑠∈𝑡,𝑇,𝐏-𝑎.𝑠.,_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝜎(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤×𝑓_𝑧𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)𝑞(𝑠),𝑎.𝑒.𝑠∈[𝑡,𝑇],𝐏-𝑎.𝑠., (24) where 𝑞(𝑠)=exp^𝑠_𝑡𝑓_𝑦𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟),−𝑉𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟),−𝑉_𝑥𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟),𝑢(𝑟)𝜎(𝑟),𝑢(𝑟)−12𝑓_𝑧𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟),−𝑉𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟),−𝑉_𝑥𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟)×𝜎(𝑟),𝑢(𝑟)²+𝑑𝑟_𝑠𝑡𝑓_𝑧𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟),−𝑉𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟),−𝑉_𝑥𝑟,𝑋^{𝑡,𝑥;𝑢}(𝑟)×𝜎(𝑟),𝑢(𝑟)𝑑𝑊(𝑟). (25)

Proof. Obviously (25) can be obtained via solving the forward SDE in (17) directly. Now let us prove (24). By [15, Theorem 5.4 of Peng], for fixed 𝑡∈[0,𝑇), it is easy to obtain 𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)=−𝑌^{𝑡,𝑥;𝑢}(𝑠)=−𝔼^𝑇_𝑠𝑓(𝑟)𝑑𝑟+𝜙𝑋^{𝑡,𝑥;𝑢}(𝑇)∣ℱ^𝑡_𝑠,∀𝑠∈[𝑡,𝑇],𝐏-a.s. (26) Define a square-integrable ℱ^𝑡_𝑠-martingale 𝑚(𝑠)∶=−𝔼^𝑇_𝑡𝑓(𝑟)𝑑𝑟+𝜙𝑋^{𝑡,𝑥;𝑢}_𝑇∣ℱ^𝑡_𝑠,𝑠∈[𝑡,𝑇]. (27) Thus, by the martingale representation theorem (see Yong and Zhou [9]), there exists a unique 𝑀(⋅)∈𝐿²_ℱ([𝑡,𝑇];𝐑^𝑑) satisfying 𝑚(𝑠)=𝑚(𝑡)+^𝑠_𝑡𝑀(𝑟)𝑑𝑊(𝑟), (28) where 𝑡∈[0,𝑇) is fixed. Then 𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)=−^𝑇_𝑠𝑓(𝑟)𝑑𝑟−^𝑇_𝑠𝑀(𝑟)𝑑𝑊(𝑟)+𝑉𝑇,𝑋^{𝑡,𝑥;𝑢}(𝑇). (29) On the other hand, applying Itô’s formula to 𝑉(𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)), we obtain 𝑑𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)=𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑏(𝑠+1)2tr𝜎(𝑠)^⊤𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠)𝑑𝑠+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝜎(𝑠)𝑑𝑊(𝑠). (30) Comparing the above two equalities, we conclude that 𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑏(𝑠+1)2tr𝜎(𝑠)^⊤𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠)=𝑓(𝑠),𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝜎(𝑠)=𝑀(𝑠),a.e.𝑠∈[𝑡,𝑇],𝐏-a.s. (31) However, by the uniqueness of solution to BSDE (3), we have 𝑌^{𝑡,𝑥;𝑢}(𝑠)=−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑍^{𝑡,𝑥;𝑢}(𝑠)=−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝜎(𝑠),a.e.𝑠∈[𝑡,𝑇],𝐏-a.s. (32)
Since 𝑉∈𝐶^1,2([0,𝑇]×𝐑^𝑛), it satisfies the generalized HJB equation (7), which implies (23). Also, by (7), we have 0=−𝑉_𝑠𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝐺𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),𝑢(𝑠)≥−𝑉_𝑠(𝑠,𝑥)+𝐺𝑠,𝑥,−𝑉(𝑠,𝑥),−𝑉_𝑥(𝑠,𝑥),−𝑉_𝑥𝑥(𝑠,𝑥),𝑢(𝑠),∀𝑥∈𝐑^𝑛. (33) Consequently, if 𝑉∈𝐶^1,3([0,𝑇]×𝐑^𝑛) and 𝑉_𝑠𝑥 is also continuous, then 𝜕𝜕𝑥−𝑉_𝑠(𝑠,𝑥)+𝐺𝑠,𝑥,−𝑉(𝑠,𝑥),−𝑉_𝑥(𝑠,𝑥),−𝑉_𝑥𝑥(𝑠,𝑥),𝑢(𝑠)_𝑥=𝑋^{𝑡,𝑥;𝑢}_(𝑠)[].=0,∀𝑠∈𝑡,𝑇 (34) This is equivalent to (recall (8)), for all 𝑠∈[𝑡,𝑇], 0=−𝑉_𝑠𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)−𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝑏(𝑠)−𝑏_𝑥(𝑠)^⊤𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)−12tr𝜎(𝑠)^⊤𝑉_𝑥𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠)−𝜎_𝑥(𝑠)^⊤𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠)+𝑓_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)−𝑓_𝑦𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)×𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)−𝑓_𝑧𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠×𝑉_𝑥𝑥)𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎_𝑥(𝑠), (35) where tr𝜎^⊤𝑉_𝑥𝑥𝑥𝜎∶=tr𝜎^⊤𝑉_𝑥¹_𝑥𝑥𝜎,…,tr𝜎^⊤𝑉_𝑥^𝑛_𝑥𝑥𝜎^⊤, (36) with ((𝑉_𝑥)¹,…,(𝑉_𝑥)^𝑛)^⊤=𝑉_𝑥.
On the other hand, applying Itô’s formula to 𝑉_𝑥(𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)), we get 𝑑𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)=−𝑏_𝑥(𝑠)^⊤𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝜎_𝑥(𝑠)^⊤×𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠)−𝑓_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)+𝑓_𝑦𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)×𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝑓_𝑧𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)×𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎_𝑥(𝑠)𝑑𝑠+𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝜎(𝑠)𝑑𝑊(𝑠). (37) Note that 𝑉_𝑥(𝑇,𝑋^{𝑡,𝑥;𝑢}(𝑇))=−𝜙_𝑥(𝑋^{𝑡,𝑥;𝑢}(𝑇)). Applying again Itô’s formula to 𝑉_𝑥(𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠))^⊤𝑞(𝑠), we have 𝑑𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝑞(𝑠)=−𝑏_𝑥(𝑠)^⊤𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝑞(𝑠)+𝑓_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)^⊤𝑞(𝑠)−𝜎_𝑥(𝑠)^⊤𝑉_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝜎(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤×𝑓_𝑧𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑞(𝑠+𝑉𝑢𝑠)𝑑𝑠_𝑥𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤𝜎(𝑠)+𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)^⊤×𝑓_𝑧𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠),−𝑉_𝑥𝑠,𝑋^{𝑡,𝑥;𝑢}(𝑠)𝜎(𝑠),𝑢(𝑠)×𝑞(𝑠)𝑑𝑊(𝑠). (38) Hence, by the uniqueness of the solutions to the adjoint equation (17), we obtain (24). The proof is complete.

4. Applications to Financial Portfolio Optimization

In this section, we consider an LQ recursive utility portfolio optimization problem in the financial engineering. In this problem, the optimal portfolio in the state feedback form is obtained by both maximum principle and dynamic programming approaches, and the relations which we obtained in Theorem 5 are illustrated explicitly.

Suppose the investors have two kinds of securities in the market for possible investment choice.(i)A risk-free security (e.g., a bond), where the price 𝑆₀(𝑡) at time 𝑡 is given by 𝑑𝑆₀(𝑡)=𝜌_𝑡𝑆₀(𝑡)𝑑𝑡,𝑆₀(0)>0, (39) here 𝜌_𝑡 is a bounded deterministic function.(ii)A risky security (e.g., a stock), where the price 𝑆₁(𝑡) at time 𝑡 is given by 𝑑𝑆₁(𝑡)=𝜇_𝑡𝑆₁(𝑡)𝑑𝑡+𝜎_𝑡𝑆₁(𝑡)𝑑𝑊(𝑡),𝑆₁(0)>0, (40) here 𝑊(⋅) is a 1-dimensional Brownian motion and 𝜇_𝑡,𝜎_𝑡≠0 are bounded deterministic functions with 𝜇_𝑡>𝜌_𝑡.

Let 𝑢(𝑡) denote the total market value of the investor’s wealth invested in the risky security which we call portfolio. Given the initial wealth 𝑋^𝑢(0)=𝑋₀≥0, combining (39) and (40), we can get the following wealth dynamics: 𝑑𝑋^𝑢(𝑡)=𝜌_𝑡𝑋^𝑢(𝑡)+𝜇_𝑡−𝜌_𝑡𝑢(𝑡)𝑑𝑡+𝜎_𝑡𝑢(𝑡)(𝑡)𝑋𝑑𝑊,𝑡≥0,_𝑢(0)=𝑋₀. (41) We denote by 𝐔_ad the set of admissible portfolios valued in 𝐔=𝐑.

For any given initial wealth 𝑋₀>0, Kohlmann and Zhou [28] discussed a mean-variance portfolio optimization problem. That is, the investor’s object is to find an admissible portfolio 𝑢^∗(𝑡) which minimizes the variance Var[𝑋^𝑢(𝑇)]∶=𝔼[(𝑋^𝑢(𝑇)−𝔼[𝑋^𝑢(𝑇)])²] at some future time 𝑇>0 under the condition that 𝔼[𝑋^𝑢(𝑇)]=𝐴 for some given 𝐴∈𝐑. Using the Lagrange multiplier method, we know that it is equivalent to study the following problem: sup_{𝑢(⋅)∈𝐔ad}𝔼−12𝑋^𝑢(𝑇)−𝑎², (42) where some 𝑎∈𝐑 is given. Using the completion of squares technique, the study of [28] obtained an optimal portfolio in the state feedback form by some stochastic Riccati equation and BSDE. The optimal value function was also obtained.

In this paper, we generalize the above mean-variance portfolio optimization problem to a recursive utility portfolio optimization problem. The recursive utility means that the utility at time 𝑡 is a function of the future utility (in this section, we do not consider the consumption). In fact, in our framework, the recursive utility can be assumed to satisfy some controlled BSDE.

We consider a small investor, endowed with initial wealth 𝑋₀>0, who chooses at each time 𝑡 his/her portfolio 𝑢(𝑡). The investor wants to choose an optimal portfolio 𝑢^∗(⋅)∈𝐔_ad to maximize the following recursive utility functional with generator: 𝑓𝑡,𝑢,𝑥,𝑦=𝜌_𝑡𝑥+𝜇_𝑡−𝜌_𝑡𝑢−𝛽𝑦, (43) where constant 𝛽≥0.

Remark 6. In fact, the recursive utility functional (43) defined above stands for some standard additive utility of recursive type. It is a meaningful and nontrivial generalization of the classical standard additive utility and has many applications in mathematical economics and mathematical finance. For more details about utility functions, see Duffie and Epstein [2], Section 1.4 of El Karoui et al. [3] or Schroder and Skiadas [4].
More precisely, for any 𝑢(⋅)∈𝐔_ad, the investor’s utility functional is defined by 𝐽(𝑢(⋅))∶=𝑌^𝑢(𝑡)_𝑡=0, (44) where 𝑌^𝑢(𝑡)−1∶=𝔼2𝑋^𝑢(𝑇)−𝑎²+^𝑇_𝑡𝜌_𝑠𝑋^𝑢(𝑠)+𝜇_𝑠−𝜌_𝑠𝑢(𝑠)−𝛽𝑌^𝑢(𝑠)𝑑𝑠∣ℱ_𝑡,∀𝑡∈[0,𝑇]. (45)
In fact, in our framework, the wealth process 𝑋^𝑢(⋅) and recursive utility process 𝑌^𝑢(⋅) can be regarded as the solution to the following controlled FBSDE: 𝑑𝑋^𝑢(𝑡)=𝜌_𝑡𝑋^𝑢(𝑡)+𝜇_𝑡−𝜌_𝑡𝑢(𝑡)𝑑𝑡+𝜎_𝑡𝑢(𝑡)𝑑𝑊(𝑡),−𝑑𝑌^𝑢(𝑡)=𝜌_𝑡𝑋^𝑢(𝑡)+𝜇_𝑡−𝜌_𝑡𝑢(𝑡)−𝛽𝑌^𝑢(𝑡)𝑑𝑡−𝑍^𝑢(𝑡)(𝑡)[],𝑋𝑑𝑊,𝑡∈0,𝑇_𝑢(0)=𝑋₀,𝑌^𝑢(𝑇)1=−2𝑋^𝑢(𝑇)−𝑎², (46) and our portfolio optimization problem can be rewritten as (denoting 𝐽=−𝐽) 𝐽𝑢^∗(⋅)=inf_{𝑢(⋅)∈𝐔ad}𝐽(𝑢(⋅)). (47)
Since we are going to involve the dynamic programming in treating the above problem, we will adopt the formulation as in Section 2. Let 𝑇>0 be given. For any (𝑡,𝑥)∈[0,𝑇)×𝐑, consider the following controlled FBSDE: 𝑑𝑋^{𝑡,𝑥;𝑢}(𝑠)=𝜌_𝑠𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝜇_𝑠−𝜌_𝑠𝑢(𝑠)𝑑𝑠+𝜎_𝑠𝑢(𝑠)𝑑𝑊(𝑠),−𝑑𝑌^{𝑡,𝑥;𝑢}(𝑠)=𝜌_𝑠𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝜇_𝑠−𝜌_𝑠𝑢(𝑠)−𝛽𝑌^{𝑡,𝑥;𝑢}(𝑠)𝑑𝑠−𝑍^{𝑡,𝑥;𝑢}(𝑠)(𝑠)[],𝑋𝑑𝑊,𝑠∈𝑡,𝑇_{𝑡,𝑥;𝑢}(𝑡)=𝑥,𝑌^{𝑡,𝑥;𝑢}(𝑇)1=−2𝑋^{𝑡,𝑥;𝑢}(𝑇)−𝑎². (48) And our recursive utility portfolio optimization problem is to find an optimal portfolio 𝑢^∗(⋅)∈𝐔_ad to minimize the recursive utility functional 𝐽(𝑡,𝑥;𝑢(⋅))∶=−𝑌^{𝑡,𝑥;𝑢}(𝑠)|_𝑠=𝑡. We define the value function as 𝑉(𝑡,𝑥)∶=𝐽𝑡,𝑥;𝑢^∗(⋅)=inf_{𝑢(⋅)∈𝐔ad}𝐽(𝑡,𝑥;𝑢(⋅)). (49)
We can check that all the assumptions in Section 2 are satisfied. Then we can use both the dynamic programming (Lemma 2) and maximum principle (Lemmas 3 and 4) approaches to solve the above problem (49).

4.1. Maximum Principle Approach

In this case, the Hamiltonian function (18) reduces to 𝐻𝑠,𝑥,𝑦,𝑧,𝑢,𝑝,𝑞,𝑘=𝑝𝜌_𝑠𝑥+𝜇_𝑠−𝜌_𝑠𝑢−𝑞𝜌_𝑠𝑥+𝜇_𝑠−𝜌_𝑠𝑢−𝛽𝑦+𝑘𝜎_𝑠𝑢, (50) and the adjoint equation (17) writes −𝑑𝑝(𝑠)=𝜌_𝑠𝑝(𝑠)(𝑠(𝑠)(𝑠),(𝑠)(𝑠)[],𝑝(𝑇)=𝑋−𝑞)𝑑𝑠−𝑘𝑑𝑊𝑑𝑞=−𝛽𝑞𝑑𝑠,𝑠∈𝑡,𝑇_{𝑡,𝑥;𝑢}(𝑇)−𝑎𝑞(𝑇),𝑞(𝑡)=1. (51) Noting that in this case the adjoint process 𝑞(⋅) reduced to a deterministic function because our generator 𝑓 does not contain the process 𝑧(⋅). We immediately have 𝑞(𝑠)=𝑒^{−𝛽(𝑠−𝑡)},𝑠∈[𝑡,𝑇]. (52) Due to the terminal condition of (51), we try a process 𝑝(⋅) of the form 𝑝(𝑠)=𝜙_𝑠𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝜓_𝑠𝑒^{−𝛽(𝑠−𝑡)},𝑠∈[𝑡,𝑇], (53) where 𝜙_𝑠, 𝜓_𝑠 are deterministic differentiable functions. Applying Itô’s formula to (53), we have 𝑑𝑝(𝑠)=𝑒^{−𝛽(𝑠−𝑡)}𝜙̇_𝑠+𝜌_𝑠−𝛽𝜙_𝑠𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝜙_𝑠𝜇_𝑠−𝜌_𝑠𝑢(𝑠)+𝜓̇_𝑠−𝛽𝜓_𝑠𝑑𝑠+𝑒^{−𝛽(𝑠−𝑡)}𝜙_𝑠𝜎_𝑠𝑢(𝑠)𝑑𝑊(𝑠). (54) Comparing (51) with (54), noting (52) and (53), we get 𝜙̇_𝑠+𝜌_𝑠−𝛽𝜙_𝑠𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝜙_𝑠𝜇_𝑠−𝜌_𝑠𝑢(𝑠)+𝜓̇_𝑠−𝛽𝜓_𝑠=−𝜌_𝑠𝜙_𝑠𝑋^{𝑡,𝑥;𝑢}(𝑠)+𝜓_𝑠,𝑘(𝑠)−1=𝜙_𝑠𝜎_𝑠𝑢(𝑠)𝑒^{−𝛽(𝑠−𝑡)}. (55)(56)

Let 𝑢^∗(⋅) be a candidate optimal portfolio, (𝑋^{𝑡,𝑥;𝑢∗}(⋅),𝑌^{𝑡,𝑥;𝑢∗}(⋅),𝑍^{𝑡,𝑥;𝑢∗}(⋅)) the corresponding solution to controlled FBSDE (48) with corresponding solution (𝑝^∗(⋅),𝑞^∗(⋅),𝑘^∗(⋅)) to the adjoint equation (51). Now the Hamiltonian function (50) is 𝐻𝑠,𝑋^{𝑡,𝑥;𝑢∗}(𝑠),𝑌^{𝑡,𝑥;𝑢∗}(𝑠),𝑍^{𝑡,𝑥;𝑢∗}(𝑠),𝑢,𝑝^∗(𝑠),𝑞^∗(𝑠),𝑘^∗(𝑠)=𝑝^∗(𝑠)𝜌_𝑠𝑋^{𝑡,𝑥;𝑢∗}(𝑠)+𝜇_𝑠−𝜌_𝑠𝑢−𝑞^∗(𝑠)𝜌_𝑠𝑋^{𝑡,𝑥;𝑢∗}(𝑠)+𝜇_𝑠−𝜌_𝑠𝑢−𝛽𝑌^{𝑡,𝑥;𝑢∗}(𝑠)+𝑘^∗(𝑠)𝜎_𝑠𝑢. (57) Since this is a linear expression of 𝑢, by the maximum condition (19), we have 𝜇_𝑠−𝜌_𝑠𝑝^∗(𝑠)−𝑞^∗(𝑠)+𝜎_𝑠𝑘^∗(𝑠)=0. (58) Substituting (52), (53) and (56) into (58), we can get 𝑢^∗(𝑠)=𝜌_𝑠−𝜇_𝑠𝜙_𝑠𝑋^{𝑡,𝑥;𝑢∗}(𝑠)+𝜓_𝑠−1𝜙_𝑠𝜎²_𝑠. (59)

On the other hand, (55) gives 𝑢^∗(𝑠)=𝜙̇_𝑠+2𝜌_𝑠𝜙_𝑠−𝛽𝜙_𝑠𝑋^{𝑡,𝑥;𝑢∗}(𝑠)+𝜓̇_𝑠+𝜌_𝑠−𝛽𝜓_𝑠−𝜌_𝑠𝜙_𝑠𝜌_𝑠−𝜇_𝑠. (60)

Combining (59) and (60) (noting the terminal condition in (51)), we get 𝜙̇_𝑠=𝜌_𝑠−𝜇_𝑠²𝜎²_𝑠−2𝜌_𝑠+𝛽𝜙_𝑠,𝜙_𝑇𝜓̇=1,_𝑠=𝜌_𝑠−𝜇_𝑠²𝜎²_𝑠−𝜌_𝑠+𝛽𝜓_𝑠−𝜌_𝑠−𝜇_𝑠²𝜎²_𝑠+𝜌_𝑠,𝜓_𝑇=−𝑎. (61)(62) The solutions to these equations are 𝜙_𝑠=𝑒^{−∫𝑇𝑠[(𝜌𝑟−𝜇𝑟)2/𝜎2𝑠−2𝜌𝑟+𝛽]𝑑𝑟},𝜓_𝑠=𝑒^{−∫𝑇𝑠[(𝜌𝑟−𝜇𝑟)2/𝜎2𝑟−𝜌𝑟+𝛽]𝑑𝑟}×^𝑇_𝑠𝜌_𝑟−𝜇_𝑟²𝜎²_𝑟−𝜌_𝑟𝑒^{∫𝑇𝑠[(𝜌𝜆−𝜇𝜆)2/𝜎2𝜆−𝜌𝜆+𝛽]𝑑𝜆}𝑑𝑟−𝑎. (63)(64) With this choice of 𝜙_𝑠 and 𝜓_𝑠, the processes 𝑞^∗(𝑠)=𝑒^{−𝛽(𝑠−𝑡)},𝑝^∗(𝑠)=𝜙_𝑠𝑋^{𝑡,𝑥;𝑢∗}(𝑠)+𝜓_𝑠𝑒^{−𝛽(𝑠−𝑡)},𝑘^∗(𝑠)=𝜙_𝑠𝜎_𝑠𝑢^∗(𝑠)𝑒^{−𝛽(𝑠−𝑡)} (65) satisfying the adjoint equation (51) with 𝑢^∗(𝑠) given by (59). With this choice of 𝑢^∗(𝑠), the maximum condition (19) of Lemma 3 holds. Moreover, we can check that all conditions in Lemma 4 hold, then 𝑢^∗(𝑠) given by (59) is really the optimal control.

Finally, let 𝑡=0, then we can solve the initial problem (47) and give the explicit optimal portfolio in the state feedback form.

Theorem 7. The optimal solution 𝑢^∗(⋅) of our recursive utility portfolio optimization problem (47), when the wealth dynamics obeys (41), is given in the state feedback form by 𝑢^∗𝑠,𝑋^∗=𝜌_𝑠−𝜇_𝑠𝜙_𝑠𝑋^∗+𝜓_𝑠−1𝜙_𝑠𝜎²_𝑠, (66) for 𝑠∈[0,𝑇], where 𝜙_𝑠, 𝜓_𝑠 are given by (63) and (64), respectively.

4.2. Dynamic Programming Approach

In this case, the value function 𝑉 should satisfy the following generalized HJB equation: −𝑉_𝑡(𝑡,𝑥)+sup_𝑢∈𝐔𝐺𝑡,𝑥,−𝑉(𝑡,𝑥),−𝑉_𝑥(𝑡,𝑥),−𝑉_𝑥𝑥()()∈[]𝑡,𝑥,𝑢=0,𝑡,𝑥0,𝑇×𝐑^𝑛,𝑉()=1𝑇,𝑥2(𝑥−𝑎)²,𝑥∈𝐑^𝑛, (67) where the generalized Hamiltonian function (8) is 𝐺𝑡,𝑥,−𝑉(𝑡,𝑥),−𝑉_𝑥(𝑡,𝑥),−𝑉_𝑥𝑥()1𝑡,𝑥,𝑢=−2𝑉_𝑥𝑥(𝑡,𝑥)𝜎²_𝑠𝑢²−𝑉_𝑥(𝑡,𝑥)𝜌_𝑡𝑥+𝜇_𝑡−𝜌_𝑡𝑢+𝜌_𝑡𝑥+𝜇_𝑡−𝜌_𝑡𝑢+𝛽𝑉(𝑡,𝑥). (68)

We conjecture that 𝑉(𝑡,𝑥) is quadratic in 𝑥, namely, 𝑉()=1𝑡,𝑥2𝑃_𝑡𝑥²+𝜑_𝑡𝑥+𝜆_𝑡, (69) for some deterministic differentiable functions 𝑃_𝑡, 𝜑_𝑡, and 𝜆_𝑡 with 𝑃_𝑇=1,𝜑_𝑇=−𝑎,𝜆_𝑇=12𝑎². (70) Substituting (69) into (68) and using completion of squares repeatedly, we get 𝐺𝑡,𝑥,−𝑉(𝑡,𝑥),−𝑉_𝑥(𝑡,𝑥),−𝑉_𝑥𝑥()1𝑡,𝑥,𝑢=−2𝑃_𝑡𝜎²_𝑡𝑢(𝑡)−𝜌_𝑡−𝜇_𝑡𝑃_𝑡𝑥+𝜑_𝑡−1𝑃_𝑡𝜎²_𝑡²+𝜇_𝑡−𝜌_𝑡²𝑃_𝑡2𝜎²_𝑡+12𝛽𝑃_𝑡−𝜌_𝑡𝑃_𝑡𝑥²+𝜌_𝑡−𝜇_𝑡²𝜎²_𝑡𝜑_𝑡−1+𝛽𝜑_𝑡−𝜌_𝑡𝜑_𝑡+𝜌_𝑡𝑥−𝜌_𝑡−𝜇_𝑡²𝜑_𝑡𝑃_𝑡𝜎²_𝑡+𝛽𝜆_𝑡≤𝜇_𝑡−𝜌_𝑡²𝑃_𝑡2𝜎²_𝑡+12𝛽𝑃_𝑡−𝜌_𝑡𝑃_𝑡𝑥²+𝜌_𝑡−𝜇_𝑡²𝜎²_𝑡𝜑_𝑡−1+𝛽𝜑_𝑡−𝜌_𝑡𝜑_𝑡+𝜌_𝑡𝑥−𝜌_𝑡−𝜇_𝑡²𝜑_𝑡𝑃_𝑡𝜎²_𝑡+𝛽𝜆_𝑡, (71) provided that 𝑃_𝑡>0 for all 𝑡∈[0,𝑇], which we will prove later. Then we see that the optimal state feedback portfolio is given by 𝑢^∗𝑡,𝑋^∗=𝜌_𝑡−𝜇_𝑡𝑃_𝑡𝑋^∗+𝜑_𝑡−1𝑃_𝑡𝜎²_𝑡. (72) In addition, the generalized HJB equation (67) now reads 12𝑃̇_𝑡𝑥²+𝜑̇_𝑡𝑥+𝜆̇_𝑡=𝜇_𝑡−𝜌_𝑡²𝑃_𝑡2𝜎²_𝑡+12𝛽𝑃_𝑡−𝜌_𝑡𝑃_𝑡𝑥²+𝜌_𝑡−𝜇_𝑡²𝜎²_𝑡𝜑_𝑡−1+𝛽𝜑_𝑡−𝜌_𝑡𝜑_𝑡+𝜌_𝑡𝑥−𝜌_𝑡−𝜇_𝑡²𝜑_𝑡𝑃_𝑡𝜎²_𝑡+𝛽𝜆_𝑡. (73) Then noting (70), by comparing the quadratic terms and linear terms in 𝑥, we recover (61) and (62), respectively. That is to say, we have that 𝑃_𝑡 coincides with 𝜙_𝑡 and 𝜑_𝑡 coincides with 𝜓_𝑡. Then by (63), we have 𝜙_𝑡>0, for all 𝑡∈[0,𝑇] as expected before. And also we have 𝜆̇_𝑡=−𝜌_𝑡−𝜇_𝑡²𝜓_𝑡𝜙_𝑡𝜎²_𝑡+𝛽𝜆_𝑡,𝜆_𝑇=12𝑎². (74) The solution to (74) is 𝜆_𝑡=𝑒^{−𝛽(𝑇−𝑡)𝑇}_𝑡𝜌_𝑟−𝜇_𝑟²𝜓_𝑟𝜙_𝑟𝜎²_𝑟−𝜌_𝑟𝑒^{𝛽(𝑇−𝑟)}1𝑑𝑟+2𝑎². (75) Then the value function 𝑉()=1𝑡,𝑥2𝜙_𝑡𝑥²+𝜓_𝑡𝑥+𝜆_𝑡, (76) where 𝜙_𝑡, 𝜓_𝑡, and 𝜆_𝑡 are determined by (61), (62), and (75), respectively. By Lemma 2, we have proved the following

Theorem 8. The optimal solution 𝑢^∗(⋅) for our recursive utility portfolio optimization problem (47), when the wealth dynamics obeys (41), is given in the state feedback form by 𝑢^∗𝑡,𝑋^∗=𝜌_𝑡−𝜇_𝑡𝜙_𝑡𝑋^∗+𝜓_𝑡−1𝜙_𝑡𝜎²_𝑡, (77) for 𝑡∈[0,𝑇], and the value function is given by (76), where 𝜙_𝑡, 𝜓_𝑡, and 𝜆_𝑡 are determined by (61), (62), and (75), respectively.

4.3. Relationship

We now can explicitly illustrate the relationships in Theorem 5. In fact, relationship (23) is obvious from (67). And (65) is exactly the relationship given in (25) and (24).

5. Concluding Remarks

In this paper, we have studied the relationship between maximum principle and dynamic programming for stochastic recursive optimal control problems. Under certain differentiability conditions, we give relations among the adjoint processes, the generalized Hamiltonian function, and the value function. A linear quadratic recursive utility portfolio optimization problem in the financial market is discussed as an explicitly illustrated example of our result.

An interesting and challenging problem remains open. For the stochastic recursive optimal control problem, what is the relationship between maximum principle and dynamic programming without the illusory differentiability conditions on the value function? This problem may be solved in the framework of nonsmooth analysis. Viscosity solution theory is certainly a nice tool (e.g., see Yong and Zhou [9]). A new result on stochastic verification theorem for forward-backward controlled system using viscosity solutions has been published very recently by Zhang [29]. However, at this moment, we do not have publishable results for the relationship within the framework of viscosity solutions. We hope to address this problem in the future work.

Acknowledgments

The authors would like to thank the anonymous referees for many constructive comments that led to an improved version of the paper. The authors also thank the Academic Editor for his efficient handling of this paper. Finally, many thanks are devoted to Dr. Qingxin Meng for his suggesting discussion during the revising progress. This work is supported by China Postdoctoral Science Foundation Funded Project (No. 20100481278), Postdoctoral Innovation Foundation Funded Project of Shandong Province (No. 201002026), the National Natural Sciences Foundations of China (No. 11201264 and 11101242) and Shandong Province (No. ZR2011AQ012, ZR2010AQ004), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China.