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Linear Quadratic Stochastic Optimal Control of Forward Backward Stochastic Control System Associated with Lévy Process
This paper analyzes one kind of linear quadratic (LQ) stochastic control problem of forward backward stochastic control system associated with Lévy process. We obtain the explicit form of the optimal control, then prove it to be unique, and get the linear feedback regulator by introducing one kind of generalized Riccati equation. Finally, we discuss the solvability of the generalized Riccati equation, and its existence and uniqueness of the solutions are proved in a special case.
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  firstly established a forward random stochastic LQ model with a dynamic programming method and Wonham  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 , Ma and Hou , Liu et al. , Wang et al. , and Shen and Wang . In 2003, Wang et al.  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  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 ; 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  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  extended Wu and Wang’s results in  to a fully coupled LQ stochastic control problem of forward backward stochastic control system with Poisson jumps. Moreover, Lin and Zhang  considered the control problem for linear stochastic systems driven by both Brownian motion and Poisson jumps. In 2016, Li et al.  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  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  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.  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  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 , Ren and Fan , Tang and Zhang , and Huang and Wang . On the basis of these results, in 2008, Mitsui and Tabata  studied a LQ regulation stochastic control problem with Lévy process and obtained the optimal control for the nonhomogeneous case. In , Tang and Wu considered the following LQ stochastic control problem in a given finite horizon with Lévy process:and the cost function wasThey 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 adapted stochastic processes taking values in and is 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 isAnd the optimal control problem is to find , such that
Note that (3) is a fully coupled FBSDEL. In 2012, Pereira and Shamarova  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.  proved the existence and uniqueness of solutions of fully coupled FBSDEL and then obtained the existence of an open-loop Nash equilibrium point for nonzero sum stochastic differential games by using this result. Based on , Wang and Huang  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  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.
2. Preliminaries and Notations
Let be a complete probability space satisfying the usual conditions; is a right continuous increasing family of complete sub-algebra which is generated by the following two mutually independent processes: a one-dimensional standard Brownian motion and an valued Lévy process with a standard Lévy measure satisfy(i),(ii), for every and for some .
Naluart and Schoutens denoted Teugels martingale associated with the Lévy process by , and is given bywhere is the compensated power-jump process of order and is power-jump processes: Coefficients correspond to orthonormalization of the polynomials with respect the measure . Please refer to Naluart and Schoutens  for more details about Teugels martingale.
Introduce the following notations adopted in this paper: : the inner product in , : the norm in , : the space of valued measurable random variable satisfies : the space of valued measurable process satisfies : the space of valued satisfies : the space of valued measurable processes satisfies : the space of valued measurable càdlàg process satisfies
For notational brevity, we set
Next, consider the following fully coupled FBSDELwhere , , ,
For a given full rank matrix , set
Assumption 1. (i) , and are uniformly Lipschitz continuous with respect to .
(ii) For each , and , where , respectively.
(iii) is uniformly Lipschitz continuous with respect to and , .
Assumption 2. where , , , , , , and ,, are nonnegative constants with and . Moreover, we have (resp., ) when (resp., ).
In the following sections we also need the more general Ito’s formula about a càdlàg semimartingales.
Lemma 4 (Ito’s formula ). Let be càdlàg semimartingales, denote as the quadratic variation process, is a real valued function, then is also a semimartingales, and the following Ito’s formula holds where is the continuous part of .
In particular, when and , where are two càdlàg semimartingales, we getHere is the quadratic covariation of .
3. Linear Quadratic Stochastic Optimal Control Problem
Let us consider the LQ stochastic optimal control problem (6). First of all, we give some necessary explanations for the coefficients in the system:
, , and are all bounded progressively measurable matrix-valued processes. are nonnegative symmetric bounded progressively measurable matrix-valued processes, and is a positive bounded progressively measurable matrix-valued process; the inverse is , which is also bounded. is a adapted nonnegative symmetric bounded matrix-valued random variable.
For a given admissible control , under assumptions of the coefficients above, we can verify that FBSDEL (3) satisfies Assumptions 1 and 2. Therefore, there exists a unique solution satisfying the control system (3) from Lemma 3.
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.
Proof. As we know, for a given admissible control , the control system (15) has a unique solution .
Existence. For any admissible control , assume the corresponding trajectory is ; thenApplying Ito’s formula to 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).
Unique. Assume admissible control is an optimal control; the corresponding trajectories are and is another optimal control; the corresponding trajectories are . So the trajectories corresponding to areand the trajectories corresponding to areSince and are both optimal controls, is positive, and , are nonnegative, we haveHere is a constant and ; thenhence, in .
Assume , , and are all deterministic matrices, denoted as , and for convenience. Introducing the following generalized matrix-valued Riccati equation (23),
Then we can get the following conclusions.
Theorem 6. Suppose the generalized matrix-valued Riccati equation (23) has solution for all ; 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) satisfiesAs we have proved that the optimal control has the form of (15), take (26) into (15); then the optimal control can be written byFor 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 , and the control system is reduced toThe cost functional now is
Remark 7. Comparing the LQ stochastic optimal control system (32) and control system (1) which was considered in  by Tang and Wu, we know that control system (1) is a special case of control system (32) when .
4. 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 , we only discuss a special case: ; in this case Riccati equation (23) becomesEquivalently, consider the following equation:
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 is a Banach space of valued continuous functions on . We have the following uniqueness result.
Theorem 10. The Riccati equation (40) admits at most one solution .
Proof. Suppose satisfying is another solution of (40). Let ; thenwhere and are uniformly bounded as they are continuously in ; apply Gronwall’s inequality; we can get, for all , . Then we prove the uniqueness of solution.
For the existence part, first of all, if we let then from the conventional Riccati equation theory, for all , the following conventional Riccati equation