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Maximum Principle for Forward-Backward Stochastic Control System Driven by Lévy Process
We study a stochastic optimal control problem where the controlled system is described by a forward-backward stochastic differential equation driven by Lévy process. In order to get our main result of this paper, the maximum principle, we prove the continuity result depending on parameters about fully coupled forward-backward stochastic differential equations driven by Lévy process. Under some additional convexity conditions, the maximum principle is also proved to be sufficient. Finally, the result is applied to the linear quadratic problem.
The stochastic optimal control problem is one of the central themes of modern control science. Forward-backward stochastic control systems which the controlled systems described by forward-backward stochastic differential equations (FBSDEs) are widely used in mathematics and finance. Peng and Wu  firstly used a probabilistic method to get the existence and uniqueness results of fully coupled FBSDEs; then Peng  considered one kind of forward-backward stochastic control systems with economic background when the control domain is convex and obtained the maximum principle; since then a number of developments in this direction were reported in Wu  and Shi and Wu . Wu  firstly proved the existence and uniqueness results of the solutions to fully coupled FBSDEs with Brownian motion and Poisson process; then Shi and Wu  got the stochastic maximum principle for fully coupled FBSDEs with random jumps. More conclusions about stochastic maximum principle about forward-backward stochastic control systems driven by Brownian motion and Poisson process can be seen in [7–9].
It is natural to extend the stochastic differential equations (SDEs) with Brownian motion and Poisson process to the case of Lévy process with independent and stationary increments. Baghery et al.  firstly considered the following fully coupled forward-backward stochastic differential equation driven by Lévy process (FBSDEL):and, under some monotonicity assumptions, they got the existence and uniqueness of solutions for this equation. Zhu  had proposed the asymptotic stability in the th moment for SDE with Lévy noise. Nualart and Schoutens  constructed a set of pairwise strongly orthonormal martingales called Teugels martingale and they also proved a martingale representation theorem for Lévy processes satisfying some exponential moment condition. Using the martingale representation theorem they  had proved the existence and uniqueness of a solution for backward stochastic differential equations (BSDEs) driven by Teugels martingale. Bahlali et al.  extended this conclusion to the BSDEs driven by Teugels martingale and an independent Brownian motion; they got the existence, uniqueness, and comparison of solutions for these equations under Lipschitz and locally Lipschitz conditions on the coefficient. Based on these consequences, Mitsui and Tabata  established the closeness property of the solution of the multidimensional backward stochastic Riccati differential equation with Lévy process; then they used this solution to study a linear quadratic regulation problem with Lévy process. After the foundation of the existence and uniqueness of the solutions of SDEs and multidimensional BSDEs driven by Lévy process, Tang and Wu  proceed to study a stochastic linear quadratic optimal control problem with a Lévy process, where the cost weighting matrices of the state and control were allowed to be indefinite.
These consequences are important for the researching of maximum principle for forward stochastic control system driven by Lévy process, as the adjoint equation for forward stochastic control system is a BSDE. Meng and Tang  firstly were concerned with optimal control for forward stochastic control system driven by Teugels martingale; they got the maximum principle and verification theorem for this system. In 2012, Tang and Zhang  were concerned with optimal control of BSDE driven by Teugels martingale and an independent multidimensional Brownian motion; they derived the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. When the control domain was nonconcave and the control variable was allowed to enter the coefficients of the Teugels martingales, Lin  got the necessary maximum principle for optimal control of stochastic system driven by multidimensional Teugels martingales. Zhang et al.  firstly studied the forward-backward stochastic control system where the system was driven by Teugels martingale and an independent multidimensional Brownian motion as follows:and they had got the maximum principle and verification theorem in the condition of the SDE part did not contain the backward state variables; the forward-backward stochastic control system they studied was not fully coupled.
In this paper, we extend the result of Zhang et al.  to the fully coupled forward-backward stochastic control system. Here the state variables are described by fully coupled FBSDEs driven by Brownian motion and an independent Teugels martingale. Before applying the convex variation and duality technique to obtain the stochastic maximum principle, we use the same method in  to get the continuity result depending on parameters, as the continuity result is not only important for us to get the stochastic maximum principle but also important property of FBSDEL especially in practice. Different from the Wu  and Shi and Wu  about maximum principles to Brownian motions and Poisson process, we also need more general Ito’s formula about semimartingale.
This paper is organized as follows. In Section 2, we will give some preliminaries used in this paper. Section 3 presents the continuity result depending on parameters about fully coupled FBSDEs driven by Lévy process. In Section 4, we obtain the main result of this paper, the maximum principle. We also prove that, under some additional convexity conditions, the maximum principle can be a sufficient condition for optimal control. And, in Section 5, an application of our stochastic maximum principle to the linear quadratic problem which the linear control system described by fully coupled FBSDEL is proved.
2. Preliminaries and Notations
Let be a complete space driving by Brownian motion and Lévy process in , with Lévy measure ; that is, is a standard Brownian motion. is -valued Lévy process of the form independent of , corresponding to a standard Lévy measure satisfying the following conditions:(i),(ii), for every and for some , and Here is the totality of -null sets and denotes the -field generated by .
Let be a Lévy process and denote the left limit process by and the jump size at time by . Setand we denote the compensated power jump process of order by ; then Teugels martingale can be defined as follows:Here the coefficients correspond to orthonormalization of the polynomials with respect to the measure .
Now we introduce some notations adopted in this paper as follows:(1) : Hilbert space,(2) : the inner product in , ,(3) : the norm in , ,(4) : the inner product in , ,(5) : the inner product in , ,(6) : the space of real valued sequences such that (7) : the space of -valued sequences such that (8) : the corresponding spaces of valued -measurable processes equipped with the norm(9) : the space of -valued random variable with the norm (10) : the space of -valued -measurable process with the norm (11) : the space of -valued -measurable process with the norm (12) for notational brevity:
Let us recall more general Ito’s formula about semimartingales which is important for us to get the maximum principle. Let be semimartingales, and we denoted the quadratic variation by ; is a real valued function; then is also semimartingales and following Ito’s formula holds:where is the continuous part of the quadratic variation .
When and , where , are two semimartingales, we getHere is the quadratic covariation of , . We can refer to Protter  for a complete survey in this topic.
Next, we introduce the existence and uniqueness results for fully coupled FBSDEL (1):where
For a given full rank matrix , we set
Assumption 1. Assume the following.(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. We also assume thatwhere , , , , , , , and , and are nonnegative constants with , . Moreover, we have , (resp., ) when (resp., ). Under Assumptions 1 and 2, in , they have got the following lemma.
3. Continuity Result Depending on Parameters about FBSDEL
Next, we are going to get the continuity result depending on parameters about FBSDEL.
Let , be a family of FBSDEL:
Assumption 4. (i) The family , are equi-Lipschitz with respect to and separately.
(ii) The function is continuous in their existing space norm sense, respectively.
Then we can get the following continuity result depending on parameters of forward-backward stochastic differential equation driven by Lévy processes.
Proof. For notational brevity, we only prove the continuity of FBSDEL (19) at . Set and . From Assumptions 1, 2, and 4, applying the usual technique to of Itô’s SDE with Lévy process, we can get Applying the same technique to of BSDE with Lévy process, then Here , depend on the Lipschitz constants and , and Set and applying Itô’s formula to yields From the above three estimates, we get where the constant depends on the Lipschitz constants and . When , , and , then If , , and , then Thus, it is clear whatever , , and or , , and we always haveThe proof is completed.
4. Maximum Principle
Let us consider the following full coupled forward-backward stochastic control system: where take values in ; is given.
Let be a nonempty convex subset of . We define the admissible control set and the cost functional: The optimal control problem is to find , such that
Assumption 6. Now we introduce the basic assumptions of this section as follows.(i), , and are continuously differentiable with respect to ; is continuously differentiable with respect to ; is continuously differentiable with respect to ; and are continuously differentiable with respect to ; is continuously differentiable with respect to . And the derivatives of each function are all bounded.(ii)For each , there exists a constant , such that (iii)For any given admissible control , (30) satisfies Assumptions 1 and 2.
Let be an optimal control and let be the corresponding trajectory. For any given admissible control and , we define Since is convex, then is in ; that is, is an admissible control and is the corresponding trajectory.
We introduce the following variational equation:
From Assumption 6, we can verify that variational equation (35) satisfies Lemma 3. Thus, there exists a unique solution satisfying variational equation. In order to get the maximum principle, we also need the following lemma.
Proof. Set and thenWe can transform (38) intowhere for , respectively, andwhere From the continuity result depending on parameters we have got in Section 3, we know thatthat is, in , as . Together with Assumption 6, we can get As FBSDEL (35) has a unique solution under Assumption 6, from the continuity and uniqueness result by Lemma 3, we know that converges to in as .
The proof is complete.
Notice that and we can get the following variational inequality.
Lemma 8. Assume that Assumption 6 holds; then
We define the Hamiltonian function as follows:and the following adjoint forward-backward equation to variational equation (35):
Then we have the main result of this paper which is the following theorem.
Theorem 9. Supposing that Assumptions 1, 2, 4, and 6 hold, is an optimal pair for our optimal control problem and is the solution to corresponding adjoint equation (49). Then for each admissible control we havewhere is defined by (48).
Proof. Applying Ito’s formula to , we can obtainThe variational inequality implies for each that The proof is completed.
Next, under some additional convexity conditions, we prove that the maximum principle can be a sufficient condition for optimal control.
Theorem 10. For stochastic control system (30) and the cost functional , if Assumptions 1, 2, 4, and 6 hold, and , , is convex in , and is convex in . Let be an admissible control and let be the corresponding trajectory. Let be the solution of corresponding adjoint equation (49). Suppose that the Hamiltonian function is convex in and inequality (50) holds; then is an optimal control.
Proof. Let be an arbitrary admissible control and the corresponding trajectory is ; thenwhere From the definition of Hamiltonian function , we getBy convexity of and using Itô’s formula to we can get