Near Optimality of Linear Delayed Doubly Stochastic Control Problem

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic diﬀerential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are diﬀerent. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.


Introduction
As known to all, stochastic differential equations and stochastic analysis develop rapidly. e theory of stochastic differential equations is widely used in economy, biology, physics, financial mathematics, and other fields. In order to give the probabilistic expression of stochastic partial differential equations, Pardoux and Peng [1] gave a class of double stochastic differential equations. Due to the wide applications of this kind of equation in many fields, more and more people pay attention to it. Han et al. [2] deduced the maximum principle for the backward doubly stochastic control system. Zhu and Shi [3] discussed the optimal control problem of the backward doubly stochastic system with partial information. And then they studied a type of forward-backward doubly stochastic differential equations with random jumps and applied their results to related games [4]. Many scholars have discussed the maximum principle of optimal control for different control systems [5].
With the further exploration of stochastic problems, we find that many problems in the objective world are not only affected by the current state but also influenced by the past history. is kind of problem is called time delay problem. Time delay exists in many fields such as the latent period of infectious diseases, genetic problems, advertising effects, network transmission, and so on. e equation describing this kind of problem is called delay equation. Because of the importance of time delay, people try to study this kind of problem. Chen and Wu [6] considered the delayed backward stochastic system and obtained the maximum principle for this problem. Wu and Wang [7] studied the optimal control problem of the backward stochastic differential delay equation under partial information. Lv et al. [8] considered the maximum principle for optimal control of the anticipated forward-backward stochastic delayed system with regime switching. Wang and Wu [9] concerned with the optimal control problems of the forward-backward delay system involving impulse controls and established the stochastic maximum principle for this kind of system. Zhou [10] investigated the maximum principle for stochastic optimal control problems of the delay system with random coefficients involving both continuous and impulse controls. In previous work, we mainly studied the theory of doubly stochastic differential equations with time delay. We deduced the maximum principle for the double stochastic control system when all variables contain time delay variables [11]. And we concerned the expression of optimal control and value function by the solution of the Riccati equation for a special delayed doubly stochastic linear quadratic control system [12].
When we study the control problems, we usually focus on finding optimal control. However, in practice, the optimal control may not exist or be difficult to obtain. Whether in theoretical analysis or numerical calculation, it is easier to obtain near optimal control than optimal control. Moreover, near optimal control has its unique advantages both in theory and practice. In order to solve the problem better, we need to pay attention to the research of near optimal control. Ekeland [13] discussed the necessary conditions for near optimality of the control system driven by ordinary differential equations. Zhou [14][15][16] discussed the dynamical system and gave the necessary and sufficient conditions for the existence of near optimal solutions for a kind of stochastic control problem. Bahlali et al. [17] considered a class of nonlinear forward-backward stochastic differential equations and gave the necessary conditions for near optimal control. Hafayed et al. [18] concerned with the stochastic maximum principle for near optimal control of nonlinear controlled mean-field forward-backward stochastic systems driven by Brownian motions and random Poisson martingale measure. Wang and Wu [19] and Zhang [20] discussed near optimal problem for the stochastic system with time delay, respectively. Li and Hu [21] concerned with a near optimal control problem for systems governed by mean-field forward-backward stochastic differential equations with mixed initial-terminal conditions. By consulting some literatures, we find that the near optimal control problem of the deterministic control system and stochastic system has relatively complete conclusions. However, the similar results about the doubly stochastic system are relatively few. Inspired by such problems, we try to study near optimal control problem of delayed doubly stochastic linear quadratic optimal control problem. We deduce the necessary condition of the near optimal control problem for the delayed system, which is similar to the maximum principle for the optimal control problem. e rest of our paper is organized as follows. In this section, we introduce the elementary introduction. In Section 2, we give some common notations and necessary formulas, as well as the main conclusions to be used later. In Section 3, we give our main results of this paper. When we deal with the time delay problem, how to deal with the delay term reasonably is the key to our research. At the same time, different time delay variables will make our research more difficult. We define a function H which is similar to the Hamiltonian function and discuss some estimates for the solution of the adjoint equations. en, we deduce the conclusions according to Ekeland's variational principle.

Preliminaries
Let us give some notations used in this paper. Set (Ω, F, P) as a probability space and T > 0 as fixed throughout our paper.
} are two mutually independent standard Brownian motions which are defined on (Ω, F, P). e integral with respect to W(t) { } is defined to be the forward Itô's integral, and its value is in R m . Note that the integral with respect to B(t) { } is defined to be the backward Itô's integral, and its value is in R d . Let N denote the class of P-null sets of F.
Note that the collection F t : t ∈ [0, T] is neither increasing nor decreasing, so it does not constitute a filtration.
Let M 2 (0, T; R n ) denote the set of all classes of (dt × dP a.e. equal) F t measurable stochastic process φ(t) satisfying E T 0 |φ(t)| 2 dt < +∞. Similarly, S 2 (0, T; R n ) denote the set of continuous n-dimensional F t measurable stochastic process φ(t) satisfying Esup t∈[0,T] |φ(t)| 2 < + ∞. 〈·, ·〉 denotes the inner product. And ⊤ in the superscripts of the matrix means the transpose of the matrix. Moreover, In general, the delayed doubly stochastic systems can be defined as follows: Functions f and g can be defined in different forms according to different problems. In this paper, we mainly investigate the delayed doubly stochastic linear quadratic control system, that is, where the delayed variables δ 1 , δ 2 , and δ 3 are not equal.

Remark 1.
In this delayed doubly stochastic control system, the state variables and the control variables contain time delay at the same time, and the three delay variables are different. Time delay exists all the time in the system. However, we do nothing before the initial time. So, we give the assumption that u(t) � 0 when the time t belongs to the interval before the intervention of the control variable. e cost functional can be written as For better analysis and research, we give some definitions similar to these in reference [16].

Definition 1.
e optimal control problem of the delayed doubly stochastic system can be described as minimizing the cost functional over U[0, T] to obtain the optimal control u * (·) satisfying and the corresponding (x * (·), y * (·), u * (·)) is called an optimal triple.
We assume that the following conditions hold: e function l is continuously differentiable in (x, y, u), and every partial derivative is bounded Corresponding to the delayed doubly stochastic control system, the adjoint equation can be written as Remark 2. According to eorem 3.1 in [11], the delayed doubly stochastic differential equation (2) admits a unique solution.

Lemma 1. Under the assumption (A1), the adjoint equation (5) admits a unique solution (p(t), q(t)) for any u ∈ U[0, T].
And there exists a positive constant C > 0 such that Proof. Adjoint equation (5) is a new kind of equation which is similar to the anticipated backward stochastic differential equation in [22]. We call it anticipated backward doubly stochastic differential equation. eorem 2.2 in reference [23] introduces the conditions for the existence and Mathematical Problems in Engineering uniqueness of solution of general anticipated backward doubly stochastic differential equation. In this paper, we only discuss the linear system, which is a special case in reference [23]. Characteristics of the linear system and boundedness of coefficient from assumption (A1) satisfy the condition of eorem 2.3. We can directly deduce the existence and uniqueness of the solution from this theorem.
Obviously, (U, d) is a complete metric space. Next, we will discuss the relation by using the metric d.
Proof. Applying Itô's formula and Jensen inequality for the general delayed doubly stochastic system (1), we have For the liner system (2), we can deal with the first term in (8) as the following: Using variable substitution and paying attention to the initial conditions, we can get the following conclusions: Similarly, en, substitute inequalities (10)-(12) into (9). Under the assumption (A1), there is a constant C > 0 such that Similarly, for the second term in (8), we have Using Gronwall's inequality and Lemma 3.1 in [9], we can deduce conclusion (7) directly.

□
Similarly, using the same method and Proposition 2.5 in reference [24], we can deduce the following conclusion directly.

Proof. From (3) and the elementary inequality, we have
From condition (A2), Lemma 2, and Cauchy-Schwartz inequality, we find that

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For the convenience of proof, we denote symbol en, we have By using the same method, from the assumption (A3), Lemmas 2 and 3, and the Definition 4, we can deduce that Combining inequalities (18) and (21), we can prove the conclusion directly. □ Ekeland's variational principle is an important tool for our study which can be seen in [25].
From inequality (24), we have at is, Let us introduce variational equations.

Main Results
Theorem 1. Let (A1)-(A3) hold. en, there exists a constant β > 0 independent of ε, such that Proof. From the definition of function J(·) and inequality (26), we have Next, we will deal with the term 〈Φ x (x ε (T)), x 1 (T)〉. We connect it with the solution of the adjoint equation. Using the Itô-Doeblin formula, we have Let us deal with the first term.
Mathematical Problems in Engineering 7 From the definition of adjoint equation (5) and variation equation (10), we have Combining equalities (31)-(33), we deduce the following equality: Similarly, we have In the same way, we have Substituting (34)-(37) into equality (30), we have Let us deal with delayed control variables.

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From the remark, we know that u(t) � 0 when −δ 3 ≤ t ≤ 0. From the adjoint equation (5), we have the terminal condition that p(t) � 0 for T ≤ t ≤ T + δ, δ � max δ 1 , δ 2 , δ 3 . en, we have Equation (38) can be written as According to inequality (29) and equation (41), we can deduce that We know that u is a variable such that u ε + u ∈ U. Assume that u ε + u � v ∈ U, then the desired conclusion (28) is deduced directly, that is, Mathematical Problems in Engineering 9 eorem 1 is proved.

□
Next, we will show the necessary condition for the near optimal control of the delayed doubly stochastic control system.
First, we give the definition of the Hamiltonian function of general delayed doubly stochastic system (1). (44) For linear system (2), we have Assume that en, we have the following conclusion.
Proof. From the definition of function H, we have en, inequality (47) can be written as We find that inequalities (28) and (49) are very similar. We need to focus on the differences between them.
We denote where 10 Mathematical Problems in Engineering (51) Next, we will deal with these two terms. From the assumption (A1), Lemma 3, and the bounded of the control domain, there exist a series of constants C ′ , C ″ , C 1 , C 11 , C 12 , C 2 , . . ., which are all independent of ε. We have And then from the assumption (A1), Lemmas 1 and 3, and the Cauchy-Schwartz inequality, we can deduce that Combining (52) and (53), we have , where C 1 � max C 11 , C 12 . (54) We denote Δ 2 and prove it like Δ 1 . en, we have Set Using variable substitution, we can deduce that Similar to the proof of Δ 1 , the results can be obtained by using the boundedness of control domain and coefficients. We can deduce the result directly, that is, Similarly, we have en, we have Mathematical Problems in Engineering