Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 702802, 12 pages
http://dx.doi.org/10.1155/2015/702802

Research Article

Maximum Principle for Forward-Backward Stochastic Control System Driven by Lévy Process

Xiangrong Wang¹ and Hong Huang²

¹College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
²College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China

Received 4 February 2015; Revised 26 April 2015; Accepted 12 May 2015

Academic Editor: Son Nguyen

Copyright © 2015 Xiangrong Wang and Hong Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1] firstly used a probabilistic method to get the existence and uniqueness results of fully coupled FBSDEs; then Peng [2] 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 [3] and Shi and Wu [4]. Wu [5] firstly proved the existence and uniqueness results of the solutions to fully coupled FBSDEs with Brownian motion and Poisson process; then Shi and Wu [6] 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. [10] 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 [11] had proposed the asymptotic stability in the th moment for SDE with Lévy noise. Nualart and Schoutens [12] 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 [13] had proved the existence and uniqueness of a solution for backward stochastic differential equations (BSDEs) driven by Teugels martingale. Bahlali et al. [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] got the necessary maximum principle for optimal control of stochastic system driven by multidimensional Teugels martingales. Zhang et al. [20] 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. [20] 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 [5] 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 [3] and Shi and Wu [6] 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 . Set

and 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 get

Here is the quadratic covariation of , . We can refer to Protter [21] 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 that

where , , , , , , , and , and are nonnegative constants with , . Moreover, we have , (resp., ) when (resp., ). Under Assumptions 1 and 2, in [10], they have got the following lemma.

Lemma 3 (existence and uniqueness theorem of FBSDEL [10]). Under Assumptions 1 and 2, FBSDEL (15) has a unique solution.

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.

Theorem 5. Let , be a family of FBSDEL satisfying Assumptions 1, 2, and 4 with solutions denoted by . Thus, the function

is continuous.

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 have

The 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.

Then, for a given admissible control, from Lemma 3, there exists a unique solution satisfying control system (30).

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.

Lemma 7. Assume that Assumption 6 holds. We have

where is the solution of variational equation (35).

Proof. Set

and then

We can transform (38) into

where

for , respectively, and

where

From the continuity result depending on parameters we have got in Section 3, we know that

that 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

Proof. First we have

Under Assumption 6, from Lemma 7, we can get

Then (45) is proved.
The proof is complete.

We define the Hamiltonian function as follows:

and the following adjoint forward-backward equation to variational equation (35):

It is easy to verify that (49) satisfies Assumptions 1 and 2; then there exists a unique quarter satisfying (49).

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 have

where is defined by (48).

Proof. Applying Ito’s formula to , we can obtain

The 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 ; then

where

From the definition of Hamiltonian function , we get

By convexity of and using Itô’s formula to we can get

and from (53) to (56) we have

Moreover, as the Hamiltonian function is convex in , the following inequality holds:

and from (57) to (58) and together with (50) for arbitrary admissible control we have

then admissible control is an optimal control.
The proof is completed.

5. Applications in Linear Quadratic Problem

In this section, we will apply our stochastic maximum principle to the linear quadratic problem which the linear control system described by fully coupled forward-backward stochastic differential equation driven by Lévy processes:

and the cost functional:

where the -predictable matrix processes

the -predictable random matrix and the -predictable stochastic matrix are all uniformly bounded. And , , and .

To study this problem, we need the assumptions on the coefficients as follows.

Assumption 11. The state weighting matrix processes , , , and , , and the control weighting matrix process and random matrixes and are almost everywhere almost surely symmetric and nonnegative. Furthermore, is almost everywhere almost surely uniformly positive; that is, , for some positive constant almost everywhere almost surely.

Assumption 12. For the control processes there is no further constraint:

If we denote the norm of by

then is a Hilbert space. And by Lemma 3 we also know that the linear FBSDEL (60) has a unique solution; that is, the linear quadratic problem is well defined.

Theorem 13. Under Assumptions 11 and 12, LQ problems (60) and (61) have a unique optimal control, and the optimal control is

here , , , and are the solution of the following adjoint FBSDEL:

Proof. From Assumptions 11 and 12 and inequality (29), we can verify that the cost function (61) is strictly convex and continuous over and

then, from Lemma in [18], the cost function has a unique minimal value and, together with Lemma 3 (existence and uniqueness theorem of FBSDEL), the LQ problem has a unique optimal control. Next, we will prove that the optimal control has an expression as (65).
Let be the optimal state process corresponding to the optimal control and let be the unique solution to adjoint equation (66) corresponding to the optimal pair ; then the Hamiltonian function

From Theorem 9 and Assumption 12 we have

that is

then (65) holds.
The proof is completed.

6. Conclusion

In this paper, the continuity result depending on parameters of forward-backward stochastic differential equation driven by Lévy process is proved. Based on this result, we get the stochastic maximum principle for fully coupled forward-backward stochastic control system driven by Lévy process. And then we use the stochastic maximum principle to solve LQ problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11271007), the SDUET Research Fund (no. 2012kytd105), and the Doctoral Fund of the Ministry of Education of China (no. 20123718110010).

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