Linear Quadratic Control of Backward Stochastic Differential Equation with Partial Information ^††thanks: This work supported by the National Natural Science Foundations of China under Grants 61821004, 61633015, 61877062, and 61977043.

A BSDE is an Itô SDE for which a random terminal rather than an initial condition on state has been specified. Bismut [1] first introduced a linear BSDE, which is an adjoint equation of stochastic optimal control problem. Pardoux and Peng [2] extended the linear BSDE to a general case. Since then, there has been considerable attention on related topics and their applications among researchers in mathematical finance and stochastic optimal control. See for example, El Karoui et al. [3], Ma and Yong [4], Kohlmann and Zhou [5].

Since BSDE stems from stochastic control theory, it is very natural and appealing to investigate optimal control problem of BSDE. Moreover, controlled BSDE is expected to have wide and important applications in various fields, especially in mathematical finance. In financial investment, a European contingent claim $\xi$, which is a random variable, can be thought as a contract to be guaranteed at maturity $T$. Peng [6] and Dokuchaev and Zhou [7] derived local and global stochastic maximum principles of optimality for BSDEs, respectively. Linear quadratic (LQ) optimal control problems of BSDEs have also been investigated. Lim and Zhou [8] discussed an LQ control problem of BSDE with a general setting and gave a feedback representation of the optimal control. Li et al. [9] extended the results in [8] to the case with mean-field term. Huang et al. [10] and Du et al. [11] considered LQ backward mean-field games. Du and Wu [12] concerned a stackelberg game for mean field linear BSDE with quadratic cost functionals.

In this paper, we investigate an LQ control problem of BSDE with partial information, which will be referred as a stochastic backward LQ control problem. We are devoted to deriving the optimal control with a feedback representation and establishing an explicit formula for the corresponding optimal cost. Note that the mentioned papers above are concentrated on the complete information case. The motivation of studying stochastic control problems with partial information arises naturally from the area of financial economics. In a portfolio and consumption problem, let $\mathbb{F}\equiv\{\mathcal{F}_{t}\}_{t\geq 0}$ denote the flow of information generated by all market noises. In reality, the information available to an agent maybe less than the one produced by the market noises, that is, $\mathcal{G}_{t}\subseteq\mathcal{F}_{t}$, where $\mathbb{G}\equiv\{\mathcal{G}_{t}\}_{t\geq 0}$ is the information available to the agent. There are considerable literatures on related topics [13, 14, 15, 16, 17, 18]. In particular, Huang et al. [15] derived a necessary condition for optimality of BSDE with partial information and applied their results to two classes of LQ problems. Wang et al. [16] and Wang et al. [17] concerned LQ problems with partially observable information driven by FBSDE and mean field FBSDE, respectively. LQ non-zero sum stochastic differential game of BSDE is considered in Wang et al. [18]. They obtained feedback Nash equilibrium points by FBSDE and Riccati equation under asymmetric information.

Our work distinguishes itself from existing literatures in the following aspects. (i) Both the generator of dynamic system and the cost functional contain diffusion terms $Z_{1}$ and $Z_{2}$. Moreover, our results are obtained under some usual conditions (see Assumptions $A1$ and $A2$ in section 2). In the literatures on this topic, diffusion terms $Z_{1}$ and $Z_{2}$ are usually assumed not to be contained into the generator (see [15], [17]), or there are some additional conditions to ensure the solvability of Riccati equation (see [16], [18]). (ii) Sufficient and necessary conditions of optimality are established, which provide an expression for optimal control via the solution of stochastic Hamiltonian system. (iii) Explicit representations for optimal control in terms of three Riccati equations, a BSDE with filtering, and an SDE with filtering are obtained, as well as the associated optimal cost. The derivation of associated Riccati equations is extremely different from [8] and [9], since the stochastic Hamiltonian system is an FBSDE with filtering. Moreover, the uniqueness and existence of solution to BSDE (3.5) is first obtained, which is important in deriving explicit representations for optimal control and associated optimal cost. (iv) Last but not least, we consider two special scalar-valued control problems of BSDEs with partial information. In the case of $H=N_{1}=0$, we obtain explicit solutions of the stochastic Hamiltonian system, as well as related Riccati equations. In the case of $C_{2}=0$, we give some numerical simulations to illustrate our theoretical results.

The rest of this paper is organized as follows. In Section 2, we formulate the stochastic backward LQ control problem and give some preliminary results. Section 3 aims to decouple the associated stochastic Hamiltonian system and derive some Riccati equations. In Section 4, we give explicit representations of optimal control and the associated optimal cost. Section 5 is devoted to solving two special scalar-valued control problems and giving some numerical simulations. Finally, we conclude this paper.

Let $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ be a complete filtered probability space and let $T>0$ be a fixed time horizon. Let $\{(W_{1t},W_{2t}):0\leq t\leq T\}$ be a $\mathbb{R}^{2}$-valued standard Wiener process, defined on $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. $\mathbb{F}\equiv\{\mathcal{F}_{t}\}_{t\geq 0}$ is a natural filtration of $(W_{1},W_{2})$ augmented by all $\mathbb{P}$-null sets. Let $\mathcal{F}_{t}^{\beta}=\sigma\{\beta_{s},\ 0\leq s\leq t\}$ be the filtration generated by a stochastic process $\beta$. Let $\mathbb{R}^{n\times m}$ be the set of all $n\times m$ matrices and $\mathbb{S}^{n}$ be the set of all $n\times n$ symmetric matrices. For a matrix $M\in\mathbb{R}^{n}$, let $M^{\top}$ be its transpose. The inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{R}^{n\times m}$ is defined by $\langle M,N\rangle\mapsto tr(M^{\top}N)$ with an induced norm $|M|=\sqrt{tr(M^{\top}M)}$. In particular, we denote by $\mathbb{S}_{+}^{n}$ ($\widehat{\mathbb{S}}_{+}^{n}$) the set of all $n\times n$ (uniformly) positive definite matrices. For any Euclidean space $M$, we adopt the following notations:
$\mathcal{L}_{\mathcal{F}_{T}}^{2}(\Omega;M)=\Big{\{}\zeta:\Omega\to M|\zeta$ is an $\mathcal{F}_{T}$-measurable random variable, $\mathbb{E}[|\zeta|^{2}]<\infty$};
$\mathcal{L}^{\infty}(0,T;M)=\Big{\{}v:[0,T]\to M|v$ is a bounded function};
$\mathcal{L}_{\mathcal{F}}^{2}(0,T;M)=\Big{\{}v:[0,T]\times\Omega\to M|v$ is an $\{\mathcal{F}_{t}\}_{t\geq 0}$-adapted stochastic process, $\mathbb{E}\left[\int_{0}^{T}|v_{t}|^{2}dt\right]<\infty\Big{\}};$
$\mathcal{S}_{\mathcal{F}}^{2}(0,T;M)=\Big{\{}v:[0,T]\times\Omega\to M|v$ is an $\{\mathcal{F}_{t}\}_{t\geq 0}$-adapted stochastic process and has continuous paths, $\mathbb{E}\left[\sup_{t\in[0,T]}|v_{t}|^{2}\right]<\infty\Big{\}}$.

Consider a controlled linear BSDE

$$\left\{\begin{aligned} dY_{t}=&\big{(}A_{t}Y_{t}+B_{t}v_{t}+C_{1t}Z_{1t}+C_{2t}Z_{2t}\big{)}dt+Z_{1t}dW_{1t}+Z_{2t}dW_{2t},\quad t\in{[0,T]},\\ Y_{T}=&\ \zeta,\end{aligned}\right.$$

(2.1)

where $\zeta\in L_{\mathcal{F}_{T}}^{2}(\Omega;\mathbb{R}^{n})$ and $v$, valued in $\mathbb{R}^{m}$, is a control process. Introduce an admissible control set
$\mathcal{V}_{ad}[0,T]=\Big{\{}v:[0,T]\times\Omega\to\mathbb{R}^{m}|v\ is\ \{\mathcal{F}_{t}^{W_{1}}\}_{t\geq 0}-$ adapted, $\mathbb{E}\left[\int_{0}^{T}|v_{t}|^{2}dt\right]<\infty\Big{\}}.$
Any $v\in\mathcal{V}_{ad}[0,T]$ is called an admissible control.
Assumption $A1$: The coefficients of dynamic system satisfy

$$A,C_{1},C_{2}\in\mathcal{L}^{\infty}(0,T;\mathbb{R}^{n\times n}),\ B\in\mathcal{L}^{\infty}(0,T;\mathbb{R}^{n\times m}).$$

Under Assumption $A1$, dynamic system (2.1) admits a unique solution pair $(Y,Z_{1},Z_{2})\in\mathcal{S}_{\mathcal{F}}^{2}(0,T;\mathbb{R}^{n})\times\mathcal{L}_{\mathcal{F}}^{2}(0,T;\mathbb{R}^{n})\times\mathcal{L}_{\mathcal{F}}^{2}(0,T;\mathbb{R}^{n})$, which is called the corresponding state process, for any $v\in\mathcal{V}_{ad}[0,T]$ (see Pardoux and Peng [2], Yong and Zhou [19]). We introduce a quadratic cost functional

$\displaystyle J(v)=$

$\displaystyle\frac{1}{2}\mathbb{E}\Bigg{[}Y_{0}^{\top}GY_{0}+\int_{0}^{T}\Big{(}Y_{t}^{\top}H_{t}Y_{t}+v_{t}^{\top}R_{t}v_{t}+Z_{1t}^{\top}N_{1t}Z_{1t}+Z_{2t}^{\top}N_{2t}Z_{2t}\Big{)}dt\Bigg{]}.$

(2.2)

Assumption $A2$: The weighting matrices in cost functional satisfy

$\displaystyle H,N_{1},N_{2}\in\mathcal{L}^{\infty}(0,T;\mathbb{S}_{+}^{n}),\ R\in\mathcal{L}^{\infty}(0,T;\widehat{\mathbb{S}}_{+}^{m}),G\in\mathbb{S}_{+}^{n}.$

Our stochastic backward LQ control problem can be stated as follows.
Problem BLQ. Find a $v^{*}\in\mathcal{V}_{ad}[0,T]$ such that

$$J(v^{*})=\inf_{v\in\mathcal{V}_{ad}[0,T]}J(v).$$

(2.3)

Any $v^{*}\in\mathcal{V}_{ad}[0,T]$ satisfying (2.3) is called an optimal control, and the state process $(Y^{*},Z_{1}^{*},Z_{2}^{*})$ is called an optimal state process. Under Assumptions $A1$ and $A2$, Problem BLQ is uniquely solvable for any terminal state $\zeta\in\mathcal{L}_{\mathcal{F}_{T}}^{2}(\Omega;\mathbb{R}^{n})$ (see Li et al. [9]). We suppressed the time argument in the sequel of this paper wherever necessary, for the sake of notation simplicity. The following theorem is a necessary condition of optimality, which is easy to be obtained from Theorem 3.1 in Huang et al. [15].

According to the above analysis, we end up with a stochastic Hamiltonian system

$$\left\{\begin{aligned} &dY=\left(AY+Bv+C_{1}Z_{1}+C_{2}Z_{2}\right)dt+Z_{1}dW_{1}+Z_{2}dW_{2},\\ &dX=-\left(A^{\top}X+HY\right)dt-\left(C_{1}^{\top}X+N_{1}Z_{1}\right)dW_{1}-\left(C_{2}^{\top}X+N_{2}Z_{2}\right)dW_{2},\\ &Y=\zeta,\ \ \ \ X_{0}=-GY_{0},\\ &\mathbb{E}[R_{t}v_{t}+B_{t}^{\top}X_{t}|\mathcal{F}_{t}^{W_{1}}]=0.\end{aligned}\right.$$

(2.5)

This is a coupled FBSDE with filtering. Note that the coupling comes from the last equation in (2.5), which is also called a stationarity condition. We point out that in our setting, the stationarity condition involves a conditional expectation, which makes the decoupling of this stochastic Hamiltonian system different and difficult. We now prove the sufficiency of the above result.

In this section, we use the decoupling method for general FBSDE introduced in [4] to solve stochastic Hamiltonian system (2.5), which is an FBSDE with filtering. Different from the results in [8], we obtain three Riccati equations, an BSDE with filtering and an SDE with filtering. For simplicity of notation, we denote $\widehat{\beta}_{t}=\mathbb{E}[\beta_{t}|\mathcal{F}_{t}^{W_{1}}]$. To be precise, we assume that

$$Y=\Upsilon\widehat{X}+\varphi,$$

(3.1)

where $\Upsilon$ is a differential and deterministic matrix-valued function with a terminal condition $\Upsilon_{T}=0$, and $\varphi$ is a stochastic process satisfying the BSDE

$$\left\{\begin{aligned} d\varphi=&\ \lambda dt+\eta_{1}dW_{1}+\eta_{2}dW_{2},\\ \varphi_{T}=&\ \zeta,\end{aligned}\right.$$

for $\{\mathcal{F}_{t}\}_{t\geq 0}$-adapted processes $\lambda$, $\eta_{1}$ and $\eta_{2}$. According to Theorem 2.1 in Wang et al. [20] (see also Theorem 5.7 in Xiong [21] and Theorem 8.1 in Liptser and Shiryayev [22] ), we have

$$\left\{\begin{aligned} d\widehat{X}=&-\left(A^{\top}\widehat{X}+H\widehat{Y}\right)dt-\left(C_{1}^{\top}\widehat{X}+N_{1}\widehat{Z}_{1}\right)dW_{1},\\ \widehat{X}_{0}=&-G\widehat{Y}_{0}.\end{aligned}\right.$$

Applying Itô formula to (3.1), we get

	$\displaystyle 0=$	$\displaystyle\ dY-\dot{\Upsilon}\widehat{X}dt-\Upsilon d\widehat{X}-d\varphi$
	$\displaystyle=$	$\displaystyle\left(AY+Bv+C_{1}Z_{1}+C_{2}Z_{2}\right)dt+Z_{1}dW_{1}+Z_{2}dW_{2}-\dot{\Upsilon}\widehat{X}dt+\Upsilon\left(A^{\top}\widehat{X}+H\widehat{Y}\right)dt$
		$\displaystyle+\Upsilon(C_{1}^{\top}\widehat{X}+N_{1}\widehat{Z}_{1})dW_{1}-\lambda dt-\eta_{1}dW_{1}-\eta_{2}dW_{2}.$

This implies

$$\left\{\begin{aligned} &AY-BR^{-1}B^{\top}\widehat{X}+C_{1}Z_{1}+C_{2}Z_{2}-\dot{\Upsilon}\widehat{X}+\Upsilon\left(A^{\top}\widehat{X}+H\widehat{Y}\right)-\lambda=0,\\ &Z_{1}+\Upsilon(C_{1}^{\top}\widehat{X}+N_{1}\widehat{Z_{1}})-\eta_{1}=0,\\ &Z_{2}-\eta_{2}=0.\end{aligned}\right.$$

(3.2)

Assuming that $I+\Upsilon N_{1}$ is invertible, we have

$$\left\{\begin{aligned} &Z_{1}=\eta_{1}-\widehat{\eta}_{1}+(I+\Upsilon N_{1})^{-1}(\widehat{\eta}_{1}-\Upsilon C_{1}^{\top}\widehat{X}),\\ &Z_{2}=\eta_{2}.\end{aligned}\right.$$

(3.3)

Substituting (3.1) and (3.3) into the first equation in (3.2), we obtain

	$\displaystyle\ A(\Upsilon\widehat{X}+\varphi)-BR^{-1}B^{\top}\widehat{X}+C_{1}(\eta_{1}-\widehat{\eta}_{1})+C_{1}(I+\Upsilon N_{1})^{-1}(\widehat{\eta}_{1}-\Upsilon C_{1}^{\top}\widehat{X})$
	$\displaystyle+C_{2}\eta_{2}-\dot{\Upsilon}\widehat{X}+\Upsilon A^{\top}\widehat{X}+\Upsilon H(\Upsilon\widehat{X}+\widehat{\varphi})-\lambda=0.$

Then $\Upsilon$ satisfies a Riccati equation

$$\left\{\begin{aligned} &\dot{\Upsilon}-\Upsilon A^{\top}-A\Upsilon-\Upsilon H\Upsilon+BR^{-1}B^{\top}+C_{1}(I+\Upsilon N_{1})^{-1}\Upsilon C_{1}^{\top}=0,\\ &\Upsilon_{T}=0,\end{aligned}\right.$$

(3.4)

and $\varphi$ satisfies a BSDE

$$\left\{\begin{aligned} d\varphi=&\Big{[}A\varphi+\Upsilon H\widehat{\varphi}+C_{1}(\eta_{1}-\widehat{\eta}_{1})+C_{1}(I+\Upsilon N_{1})^{-1}\widehat{\eta}_{1}+C_{2}\eta_{2}\Big{]}dt\\ &+\eta_{1}dW_{1}+\eta_{2}dW_{2},\\ \varphi_{T}=&\ \zeta.\end{aligned}\right.$$

(3.5)

Riccati equation (3.4) admits a unique solution $\Upsilon\in\mathcal{L}^{\infty}(0,T;\mathbb{S}_{+}^{n})$ under Assumptions $A1-A2$ (see [9, 8]). Note that (3.5) is a BSDE with filtering, for which the solvability has not been given in literatures before. We will specified this problem in Section 4. In order to give the optimal control with a feedback representation, we conjecture that

$$X=-\Gamma_{1}(Y-\widehat{Y})-\Gamma_{2}\widehat{Y}-\psi,$$

(3.6)

where $\Gamma_{1}$ and $\Gamma_{2}$ are differential and deterministic matrix-valued functions with initial conditions $\Gamma_{10}=G$ and $\Gamma_{20}=G$, respectively; $\psi$ is a stochastic process satisfying an SDE

$$\left\{\begin{aligned} d\psi=&\ \alpha_{0}dt+\alpha_{1}dW_{1}+\alpha_{2}dW_{2},\\ \psi_{0}=&\ 0,\end{aligned}\right.$$

where $\alpha_{0}$, $\alpha_{1}$ and $\alpha_{2}$ are $\{\mathcal{F}_{t}\}_{t\geq 0}$-adapted processes. Note that

$$\left\{\begin{aligned} d\widehat{Y}=&\left(A\widehat{Y}-BR^{-1}B^{\top}\widehat{X}+C_{1}\widehat{Z}_{1}+C_{2}\widehat{Z}_{2}\right)dt+\widehat{Z}_{1}dW_{1},\\ \widehat{Y}_{T}=&\ \widehat{\zeta},\end{aligned}\right.$$

where $\widehat{\zeta}=\mathbb{E}[\zeta|\mathcal{F}_{T}^{W_{1}}]$. Hence,

$$\left\{\begin{aligned} d(Y-\widehat{Y})=&\left[A(Y-\widehat{Y})+C_{1}(Z_{1}-\widehat{Z}_{1})+C_{2}(Z_{2}-\widehat{Z}_{2})\right]dt+(Z_{1}-\widehat{Z}_{1})dW_{1}+Z_{2}dW_{2},\\ Y_{T}-\widehat{Y}_{T}=&\ \zeta-\widehat{\zeta}.\end{aligned}\right.$$

Applying Itô formula to (3.6), we obtain

	$\displaystyle 0=$	$\displaystyle\ dX+\dot{\Gamma}_{1}(Y-\widehat{Y})dt+\Gamma_{1}d(Y-\widehat{Y})+\dot{\Gamma}_{2}\widehat{Y}dt+\Gamma_{2}d\widehat{Y}+d\psi$
	$\displaystyle=$	$\displaystyle-\left(A^{\top}X+HY\right)dt-\left(C_{1}^{\top}X+N_{1}Z_{1}\right)dW_{1}-\left(C_{2}^{\top}X+N_{2}Z_{2}\right)dW_{2}$
		$\displaystyle+\dot{\Gamma}_{1}(Y-\widehat{Y})dt+\Gamma_{1}\Big{[}A(Y-\widehat{Y})+C_{1}(Z_{1}-\widehat{Z}_{1})+C_{2}(Z_{2}-\widehat{Z}_{2})\Big{]}dt+\Gamma_{1}(Z_{1}-\widehat{Z}_{1})dW_{1}$
		$\displaystyle+\Gamma_{1}Z_{2}dW_{2}+\dot{\Gamma}_{2}\widehat{Y}dt+\Gamma_{2}\Big{(}A\widehat{Y}-BR^{-1}B^{\top}\widehat{X}+C_{1}\widehat{Z}_{1}+C_{2}\widehat{Z}_{2}\Big{)}dt+\Gamma_{2}\widehat{Z}_{1}dW_{1}$
		$\displaystyle+\alpha_{0}dt+\alpha_{1}dW_{1}+\alpha_{2}dW_{2}.$

It yields

$$\left\{\begin{aligned} &-\left(A^{\top}X+HY\right)+\dot{\Gamma}_{1}(Y-\widehat{Y})+\Gamma_{1}\Big{[}A(Y-\widehat{Y})+C_{1}(Z_{1}-\widehat{Z}_{1})+C_{2}(Z_{2}-\widehat{Z}_{2})\Big{]}\\ &+\dot{\Gamma}_{2}\widehat{Y}+\Gamma_{2}\Big{(}A\widehat{Y}-BR^{-1}B^{\top}\widehat{X}+C_{1}\widehat{Z}_{1}+C_{2}\widehat{Z}_{2}\Big{)}+\alpha_{0}=0,\\ &-(C_{1}^{\top}X+N_{1}Z_{1})+\Gamma_{1}(Z_{1}-\widehat{Z}_{1})+\Gamma_{2}\widehat{Z}_{1}+\alpha_{1}=0,\\ &-(C_{2}^{\top}X+N_{2}Z_{2})+\Gamma_{1}Z_{2}+\alpha_{2}=0.\end{aligned}\right.$$

Assuming that $I+\Gamma_{2}\Upsilon$ is invertible, we arrive at

$$\left\{\begin{aligned} \alpha_{1}=&\ (N_{1}-\Gamma_{1})(\eta_{1}-\widehat{\eta}_{1})+(N_{1}-\Gamma_{2})(I+\Upsilon N_{1})^{-1}\left[\widehat{\eta}_{1}+\Upsilon C_{1}^{\top}(I+\Gamma_{2}\Upsilon)^{-1}(\Gamma_{2}\widehat{\varphi}+\widehat{\psi})\right]\\ &-C_{1}^{\top}\Gamma_{1}(\varphi-\widehat{\varphi})-C_{1}^{\top}(\psi-\widehat{\psi})-C_{1}^{\top}(I+\Gamma_{2}\Upsilon)^{-1}(\Gamma_{2}\widehat{\varphi}+\widehat{\psi}),\\ \alpha_{2}=&\ (N_{2}-\Gamma_{1})\eta_{2}-C_{2}^{\top}\Gamma_{1}(\varphi-\widehat{\varphi})-C_{2}^{\top}(\psi-\widehat{\psi})-C_{2}^{\top}(I+\Gamma_{2}\Upsilon)^{-1}(\Gamma_{2}\widehat{\varphi}+\widehat{\psi}).\end{aligned}\right.$$

Further, it follows from (3.3) and (3.6) that

	$\displaystyle A^{\top}\Gamma_{1}(Y-\widehat{Y})+A^{\top}\Gamma_{2}\widehat{Y}+A^{\top}\psi-HY+\dot{\Gamma}_{1}(Y-\widehat{Y})+\Gamma_{1}A(Y-\widehat{Y})$
	$\displaystyle+\Gamma_{1}C_{1}(\eta_{1}-\widehat{\eta}_{1})+\Gamma_{1}C_{2}(\eta_{2}-\widehat{\eta}_{2})+\dot{\Gamma}_{2}\widehat{Y}+\Gamma_{2}A\widehat{Y}+\Gamma_{2}BR^{-1}B^{\top}(\Gamma_{2}\widehat{Y}+\widehat{\psi})$
	$\displaystyle+\Gamma_{2}C_{1}(I+\Upsilon N_{1})^{-1}\left[\widehat{\eta}_{1}+\Upsilon C_{1}^{\top}(\Gamma_{2}\widehat{Y}+\widehat{\psi})\right]+\Gamma_{2}C_{2}\widehat{\eta}_{2}+\alpha_{0}=0.$

Introduce

$$\left\{\begin{aligned} &\dot{\Gamma}_{1}+\Gamma_{1}A+A^{\top}\Gamma_{1}-H=0,\\ &\Gamma_{10}=G,\end{aligned}\right.$$

(3.7)

$$\left\{\begin{aligned} &\dot{\Gamma}_{2}+\Gamma_{2}A+A^{\top}\Gamma_{2}+\Gamma_{2}BR^{-1}B^{\top}\Gamma_{2}+\Gamma_{2}C_{1}(I+\Upsilon N_{1})^{-1}\Upsilon C_{1}^{\top}\Gamma_{2}-H=0,\\ &\Gamma_{20}=G,\end{aligned}\right.$$

(3.8)

and

$$\left\{\begin{aligned} d\psi=&-\Big{[}A^{\top}\psi+\Gamma_{2}BR^{-1}B^{\top}\widehat{\psi}+\Gamma_{2}C_{1}(I+\Upsilon N_{1})^{-1}\left(\widehat{\eta}_{1}+\Upsilon C_{1}^{\top}\widehat{\psi}\right)\\ &+\Gamma_{2}C_{2}\widehat{\eta}_{2}+\Gamma_{1}C_{1}(\eta_{1}-\widehat{\eta}_{1})+\Gamma_{1}C_{2}(\eta_{2}-\widehat{\eta}_{2})\Big{]}dt\\ &+\Big{\{}(N_{1}-\Gamma_{1})(\eta_{1}-\widehat{\eta}_{1})-C_{1}^{\top}\Gamma_{1}(\varphi-\widehat{\varphi})-C_{1}^{\top}(\psi-\widehat{\psi})\\ &+(N_{1}-\Gamma_{2})(I+\Upsilon N_{1})^{-1}\left[\widehat{\eta}_{1}+\Upsilon C_{1}^{\top}(I+\Gamma_{2}\Upsilon)^{-1}(\Gamma_{2}\widehat{\varphi}+\widehat{\psi})\right]\\ &-C_{1}^{\top}(I+\Gamma_{2}\Upsilon)^{-1}(\Gamma_{2}\widehat{\varphi}+\widehat{\psi})\Big{\}}dW_{1}\\ &+\Big{\{}(N_{2}-\Gamma_{1})\eta_{2}-C_{2}^{\top}\left[\Gamma_{1}(\varphi-\widehat{\varphi})+(\psi-\widehat{\psi})\right]-C_{2}^{\top}(I+\Gamma_{2}\Upsilon)^{-1}(\Gamma_{2}\widehat{\varphi}+\widehat{\psi})\Big{\}}dW_{2},\\ \psi_{0}=&\ 0.\end{aligned}\right.$$

(3.9)

There is a unique solution $\Gamma_{1}\in\mathcal{L}^{\infty}(0,T;\mathbb{S}_{+}^{n})$ to Riccati equation (3.7), since Assumptions $A1$ and $A2$ hold (see Yong and Zhou [19]). Corollary 4.6 in Lim and Zhou [8] implies that (3.8) admits a unique solution $\Gamma_{2}\in\mathcal{L}^{\infty}(0,T;\mathbb{S}_{+}^{n})$. Once $\Upsilon$, $\Gamma_{1}$, $\Gamma_{2}$ and the solution $(\varphi,\eta_{1},\eta_{2})$ of (3.5) are known, the solvability of (3.9) will be obtained immediately.

Now we would like to give explicit formulas of optimal control and associated optimal cost in terms of Riccati equations (3.4), (3.7), (3.8), BSDE (3.5) and SDE (3.9). We first prove that (3.5) admits a unique solution. Consider a BSDE

$$\left\{\begin{aligned} dP=&\ g(t,P,Q_{1},Q_{2},\widehat{P},\widehat{Q_{1}},\widehat{Q_{2}})dt+Q_{1}dW_{1}+Q_{2}dW_{2},\\ P_{T}=&\ \zeta,\end{aligned}\right.$$

(4.1)

where $\widehat{P}_{t}=\mathbb{E}[P_{t}|\mathcal{F}_{t}^{W_{1}}],\widehat{Q}_{1t}=\mathbb{E}[Q_{1t}|\mathcal{F}_{t}^{W_{1}}],\widehat{Q}_{2t}=\mathbb{E}[Q_{2t}|\mathcal{F}_{t}^{W_{1}}]$.
We assume that
Assumption $A3$: There exists a constant $L$, such that, $\mathbb{P}$-a.s., for all $t\in[0,T]$, $p,q_{1},q_{2},\bar{p},\bar{q_{1}},\bar{q_{2}}$, $p^{\prime},q_{1}^{\prime},q_{2}^{\prime},\bar{p}^{\prime},\bar{q_{1}}^{\prime},\bar{q_{2}}^{\prime}\in\mathbb{R}^{n}$,

		$\displaystyle\left|g(t,p,q_{1},q_{2},\bar{p},\bar{q_{1}},\bar{q_{2}})-g(t,p^{\prime},q_{1}^{\prime},q_{2}^{\prime},\bar{p}^{\prime},\bar{q_{1}}^{\prime},\bar{q_{2}}^{\prime})\right|$
	$\displaystyle\leq$	$\displaystyle\ L\big{(}|p-p^{\prime}|+|q_{1}-q_{1}^{\prime}|+|q_{2}-q_{2}^{\prime}|+|\bar{p}-\bar{p}^{\prime}|+|\bar{q_{1}}-\bar{q_{1}}^{\prime}|+|\bar{q_{2}}-\bar{q_{2}}^{\prime}|\big{)}.$

Assumption $A4$: $g(\cdot,0,0,0,0,0,0)\in\mathcal{L}_{\mathcal{F}}^{2}(0,T;\mathbb{R}^{n})$.