Linear Quadratic Stochastic Optimal Control of Forward Backward Stochastic Control System Associated with Lévy Process

1 Institute of Financial Engineering, College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China 2Institute of Financial Engineering, Shandong Women’s University, Jinan 250300, China 3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China 4School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433, China


Introduction
LQ stochastic optimal control is a kind of special optimal control problem, which not only can be used to model many linear optimal problems practically, but also can reasonably be used to approach and solve many nonlinear problems.In 1962, Kushner [1] firstly established a forward random stochastic LQ model with a dynamic programming method and Wonham [2] firstly studied a LQ stochastic optimal control problem by introducing a Riccati equation in 1968.Then a lot of works have been done for forward or backward stochastic LQ control problems, the corresponding Riccati equation, and its application in finance, such as Li and Zhang [3], Ma and Hou [4], Liu et al. [5], Wang et al. [6], and Shen and Wang [7].In 2003, Wang et al. [8] discussed a special kind of forward backward stochastic LQ problem and got the existence and uniqueness of the optimal control for the control system.Subsequently, Wu [9] extended this conclusion to the fully coupled forward backward stochastic LQ problem.
The optimal control problem with random jumps was first considered by Boel and Varaiya [10]; in this case, the control system is often described by Brownian motion and Poisson processes.On the basis of proving the existence and uniqueness of solutions of a kind of forward backward stochastic differential equation with Poisson jumps (FBSDEP), Wu and Wang [11] got the explicit form of the optimal control for LQ stochastic control problem where the state variable was described by a stochastic differential equation with a Poisson process (SDEP).In 2009, Shi and Wu [12] extended Wu and Wang's results in [11] to a fully coupled LQ stochastic control problem of forward backward stochastic control system with Poisson jumps.Moreover, Lin and Zhang [13] considered the  ∞ control problem for linear stochastic systems driven by both Brownian motion and Poisson jumps.In 2016, Li et al. [14] studied a stochastic differential equations driven by G-Brownian motion and got the existence and uniqueness of the solution for these equations.
In 2000, Nualart and Schoutens [15] introduced a class of Lévy processes with exponential moments satisfying some conditions.Using these exponential moments and the standard orthogonalization process, they constructed a series of orthogonal normal martingales called Teugels martingale.And they also proved a martingale representation theorem associated with Teugels martingale.In the next year, Nualart and Schoutens [16] considered a backward stochastic differential equation (BSDE) driven by Teugels martingale and proved the existence and uniqueness theory of this BSDE.In 2003, Bahlali et al. [17] studied a BSDE driven by Teugels martingale and an independent Brownian motion; they got the existence, uniqueness, and comparison of solutions for these equations, having a Lipschitz or locally Lipschitz coefficient.El Otmani [18] considered a kind of generalized BSDE (GBSDE) associated with Teugels martingale and Brownian motion associated with a pure jump-independent Lévy process.They got the existence and uniqueness theory of this GBSDE when the coefficient verifies some conditions of Lipschitz.More results about BSDE associated with Teugels martingale can be found in the theses of El Otmani [19], Ren and Fan [20], Tang and Zhang [21], and Huang and Wang [22].On the basis of these results, in 2008, Mitsui and Tabata [23] studied a LQ regulation stochastic control problem with Lévy process and obtained the optimal control for the nonhomogeneous case.In [24], Tang and Wu considered the following LQ stochastic control problem in a given finite horizon [, ] with Lévy process: (2) They show that the solvability of one kind of generalized Riccati equation is sufficient to the well-posedness of this LQ problem and proved the existence of the optimal control.
In this paper, we consider one kind of LQ stochastic control problem where the controlled system is driven by a fully coupled linear forward backward stochastic differential equation associated with Lévy process (FBSDEL).
where (  ,   ,   ,    ) are F  -adapted stochastic processes taking values in   ×   ×   ×  2 (  ) and (⋅) is F  -adapted stochastic process called admissible control process.Assume the control process set  =   and define the admissible control set as follows: The cost functional we considered is And the optimal control problem is to find Note that (3) is a fully coupled FBSDEL.In 2012, Pereira and Shamarova [25] firstly considered this kind of FBSDEL, obtained a solution to this FBSDEL via a partial integrodifferential equation, and proved the uniqueness.Under some monotonicity assumptions, Baghery et al. [26] proved the existence and uniqueness of solutions of fully coupled FBSDEL and then obtained the existence of an openloop Nash equilibrium point for nonzero sum stochastic differential games by using this result.Based on [25], Wang and Huang [27] got the maximum principle for forward backward stochastic control system driven by Lévy process; then they discussed a kind of LQ stochastic control problem of forward backward stochastic control system and got a necessary condition for the optimal control.
We extend the result of Shi and Wu [12] to the fully coupled linear forward backward stochastic control system driven by Brownian motion and an independent Teugels martingale.Since Teugels martingale is more complex than the Poisson process, we also need more general formula about càdlàg semimartingale.The rest of this paper is organized as follows.In Section 2, we provide a list of notations and results of the existence and uniqueness of solutions of fully coupled FBSDEL.In Section 3, we prove the existence and uniqueness of the optimal control of LQ stochastic control problem (6) and give the linear feedback regulator for the optimal control by the solution of a kind of generalized matrix-valued Riccati equation when assuming the coefficient matrices are deterministic.In Section 4, the solvability of this kind of matrix-valued Riccati equation is discussed.
Introduce the following notations adopted in this paper: For notational brevity, we set Next, consider the following fully coupled FBSDEL where ) . ( Assumption 1. (i) , , , and  are uniformly Lipschitz continuous with respect to (, , , ).
In the following sections we also need the more general Ito's formula about a càdlàg semimartingales.
Then we get the explicit form of the optimal control   for the LQ stochastic optimal control problem (6).Theorem 5.There exists a unique optimal control   for LQ stochastic optimal control problem (6), and   is given by the following equation.
Existence.For any admissible control V  , assume the corresponding trajectory is Applying Ito's formula to ⟨ V  −   ,   ⟩ we have Since (, ), (, ), and () are nonnegative and (, ) is positive, we can get Then the admissible control   defined by ( 15) is the optimal control of LQ stochastic control problem (6).
Then we can get the following conclusions.
Theorem 6. Suppose the generalized matrix-valued Riccati equation (23) has solution (  ,   ,    ) for all  ∈ [0, ]; then the optimal linear feedback regulator for LQ stochastic optimal control problem ( 6) is and the optimal value function is Proof.If (  ,   ,    ) is the solution of the matrix-valued Riccati equation ( 23), then we can check that the solution of (6) (  ,   ,   ,    ) satisfies As we have proved that the optimal control has the form of (15), take ( 26) into (15); then the optimal control can be written by For the optimal value function, using Ito's formula to ⟨  ,   ⟩, then On the other hand, from the relationship of  and (  ,   ,   ,    ), we can verify that and then By the definition of cost function (⋅) (5), we prove that the optimal value function is Now consider a special case of stochastic LQ control problem when (, ) = 0, and the control system is reduced to The cost functional now is Remark 7. Comparing the LQ stochastic optimal control system (32) and control system (1) which was considered in [22] by Tang and Wu, we know that control system (1) is a special case of control system (32) when (, ) = (, ) = 0.
We can get the following Corollary 8 easily from Theorem 5.

Corollary 8. There exists a unique optimal control for LQ stochastic optimal control problem (32)-(33), and
where the (  ,   ,   ) is the solution of the following BSDE driven by Lévy process.

Solvability of the Generalized Riccati Equation
From the discussion of the previous section, we can see that the key to get the optimal linear feedback regulator for LQ stochastic optimal control problem is the solvability of the generalized Riccati equation (23).But ( 23) is so complicated that we cannot prove its existence and uniqueness at this moment.Using technique introduced by Shi and Wu [12], we only discuss a special case:   = 0; in this case Riccati equation (23) becomes Equivalently, consider the following equation: Compare (39) and (40); we can find that if we can prove   the solution of (40), then is the solution of the Riccati equation (39).
In the following, we will focus on the existence and uniqueness of solutions of (40).Firstly, let   + denote the space of all  ×  nonnegative symmetric matrices, and ([0, ];   + ) is a Banach space of   + -valued continuous functions on [0, ].We have the following uniqueness result.
Proof.For given Φ + , Φ − which satisfied (49), define the sequences Φ +  , Φ −  ,  +  ,  −  as follows: From (50) and Lemma 11, by induction, we obtain and Φ +  , Φ −  ∈    ; we have lim So  + is a solution of (45) corresponding to Φ = Φ + ; then where () is the unique solution of the following equation: It is easy to known that when matrix   is invertible and  = , Assumption 13 is satisfied.Then we get the main result of this section.At last, we give a simple example of the Riccati equation which has a unique solution.

Conclusion
In this paper, we discussed one kind of LQ stochastic control problem with Lévy process as noise source where the control system is described by a linear FBSDEL.Explicit form of optimal control is obtained, and it can be proved to be unique.When assuming that all the coefficient matrices in this control problem are deterministic, it has been shown that the linear feedback regulator for this LQ problem has a close relation to the solutions of a kind of generalized Riccati equation.Finally, we discuss the solvability of the generalized Riccati equation and prove the existence and uniqueness of the solution for it in a special case.