The Delayed Doubly Stochastic Linear Quadratic Optimal Control Problem

In this paper, the delayed doubly stochastic linear quadratic optimal control problem is discussed. It deduces the expression of the optimal control for the general delayed doubly stochastic control system which contained time delay both in the state variable and in the control variable at the same time and proves its uniqueness by using the classical parallelogram rule.+e paper is concerned with the generalized matrix value Riccati equation for a special delayed doubly stochastic linear quadratic control system and aims to give the expression of optimal control and value function by the solution of the Riccati equation.


Introduction
As is known to all, the stochastic differential equation and stochastic analysis have developed rapidly. e theory of the stochastic differential equation is widely used in economy, biology, physics, financial mathematics, and other fields. e latest research on the insurance model was given in [1][2][3]. e social optimal mean field control problem was discussed in [4]. In order to provide a probabilistic interpretation for the solution of a kind of partial differential equations, Pardoux and Peng [5] first introduced the backward doubly stochastic differential equations and proved the existence and uniqueness of this kind of differential.
en, people began to study doubly stochastic differential equations. Zhu and Shi [6,7] were concerned with a class of partial information control problems for backward doubly stochastic systems and gave the maximum principle and its applications for the system. Recently, Shi and Zhu [8] studied a type of forwardbackward doubly stochastic differential equations driven by Brownian motions and the Poisson process and applied the result to backward doubly stochastic linear quadratic nonzero sum differential games with random jumps to get the explicit form of the open-loop Nash equilibrium point by the solution of this kind of equation. With the deepening of research, people gradually realized that many problems are not only affected by the current situation, but also by their past history. is kind of problem is called the delay problem. e equation describing this kind of problem is called the delay equation. Due to the fact that time delay widely exists in the practical systems, it will cause the change in system performance. erefore, it can increase the control difficulty of the system. Delayed problems have become the focus of scholar's research studies. Chen and Wu [9] considered the delayed backward system and obtained the maximum principle for these problems. Wu and Wang [10] studied the optimal control problem of the backward stochastic differential delay equation under partial information. Lv et al. [11] considered the maximum principle for optimal control of anticipated forwardbackward stochastic delayed systems with regime switching. Wang and Wu [12] were concerned with the optimal control problems of forward-backward delay systems involving impulse controls and established the stochastic maximum principle for this kind of systems. Yu [13] investigated the maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls.
Linear quadratic (LQ) optimal control problem is the theoretical basis for many problems. When delay variables exist in the doubly stochastic control system, the LQ problem becomes more complex and interesting. Chen and Wu [14] considered the LQ problem with delay in which the state depended on the past time but not the control in the system. Tang and Wu [15] were concerned with the linear stochastic system with Lévy processes. Huang et al. [16] were concerned with one kind of delayed forward-backward linear quadratic stochastic control problems and derived the explicit form of the optimal control. However, to our best knowledge, there is little work on the doubly stochastic LQ problem with delay. Based on the abundant literature, we want to discuss the delayed doubly stochastic LQ problem. When the LQ control system contains time delay, some important characteristic changes have taken place in research. e system contains a delayed doubly stochastic differential equation and a new kind of equation called the anticipated backward doubly stochastic differential equation which was discussed in [17,18]. Inspired by the idea of the maximum principle for the delayed doubly stochastic control system [19,20], we studied the general LQ system in which both the state variable and the control variable contain time delay at the same time. As is known to all, it is the key to find out the feedback control of the LQ problem. We deduce the explicit expression of the optimal control for the delayed doubly stochastic LQ problem. We consider the matrix Riccati equation for a class of the LQ problem. We deduced the solution of the LQ system by the solution corresponding to the Riccati equation, which was introduced originally by Peng [21]. We hope that our research can better describe the optimal feedback control of the delayed doubly stochastic LQ problem. e rest of our paper is organized as follows. First, we introduce preliminary results and some necessary notations. In Section 3, we give the explicit expression of optimal control and prove its uniqueness by using the classical parallelogram rule. And then, we discuss a special kind of the control system, in which the time delay is contained only in the control variables. We try to introduce the generalized matrix value Riccati equation corresponding to the system. And then, we use the solution of the Riccati equation to show the optimal control for the delayed doubly stochastic LQ problem. At the same time, we indicate the objective function by the solution of the Riccati equation and the initial value of the state variable.

Preliminaries
First, let us introduce the common notations in this paper. Let (Ω, F, P) be a probability space. Assume denotes the conditional expectation under filtration F t . 〈·, ·〉 denotes the scalar product, and ⊤ in the superscripts means the transpose of the matrix.
In this paper, we mainly investigate the delayed doubly stochastic linear quadratic control system: Remark 1. In this delayed doubly stochastic control system, the state and control variables contain time delay at the same time. Time delay exists all the time in the system. But, we do nothing before the initial time. So, we give the assumption that u(t) � 0 when time t belongs to the interval before the control intervenes.
e cost functional is written as Our optimal control problem can be stated as minimizing the cost functional over U[0, T]. For optimal control u * (·) satisfying the corresponding (x * (·), y * (·), u * (·)) is called an optimal triple. e corresponding adjoint equation becomes We assume that the following conditions hold:

Main Results
t ∈ [0, T], is the unique optimal control for the delayed doubly stochastic linear quadratic optimal control problem, where (x * (·), y * (·), p(·), q(·)) is the solution of the following system:

Mathematical Problems in Engineering
Proof.

From the definitions of K(t), R(t), S(t)
, and Q, we know K(t), R(t), and Q are symmetric nonnegative definite and S(t) is symmetric uniformly positive definite. So, we have Applying the Itô-Doeblin formula to and paying attention to the initial condition and the terminal condition, we have In fact, we have So, we have en, So from the definition of u * (t), we have for any v(·) ∈ U[0, T]. is shows that u * (t) is the optimal control. Next, we will prove the uniqueness. Assume that u 1 (·) and u 2 (·) are both optimal controls. (x 1 (·), y 1 (·)) and (x 2 (·), y 2 (·)) are the trajectories corresponding to u 1 (·) and u 2 (·), respectively. Equation (5) is a new type of the anticipated backward doubly stochastic differential equation. e existence and uniqueness of the solution for the equation can be guaranteed by eorem 3.2 in [18]. By the uniqueness of the solution of the equation, we know that ((x 1 (·) + x 2 (·)/2), (y 1 (·) + y 2 (·)/2)) is the trajectory corresponding to (u 1 (·) + u 2 (·)/2). From the definition of K(t), R(t), S(t), and Q, we know J(u 1 (·)) � J(u 2 (·)) � α ≥ 0. en,

Mathematical Problems in Engineering
From the definition of S(t), we have u 1 (·) � u 2 (·).
We complete the proof of eorem 1.

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Next, we study a special class of the delayed doubly stochastic LQ problem. We discuss the case that only the control contained the delayed variable, and the initial value of the state variable η is deterministic. e delayed system can be written as We also consider the optimal control problem (4) under the classical quadratic index (4).
From eorem 1, we can deduce the optimal control directly: Peng [21] gave the solution of a kind of stochastic Hamiltonian systems by the Riccati equation corresponding to the system. Following this way, we want to use the solution of the Riccati equation to introduce the optimal control for the delayed doubly stochastic LQ problem. First, we give the generalized matrix Riccati equation: Theorem 2. Let the assumptions (A1)-(A3) be satisfied. If the Riccati equation has a solution (G(·), M(·), N(·)), then system (17) has a unique solution: where x(t) is solved by Proof. We apply Itô's formula to M(t)x(t) and compare the coefficient with p(t), then we can deduce the conclusion directly. Next, we will prove the uniqueness.