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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 216053, 12 pages
The Optimal Control Problem with State Constraints for Fully Coupled Forward-Backward Stochastic Systems with Jumps
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Received 26 November 2013; Accepted 22 December 2013; Published 13 February 2014
Academic Editor: Litan Yan
Copyright © 2014 Qingmeng Wei. 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.
We focus on the fully coupled forward-backward stochastic differential equations with jumps and investigate the associated stochastic optimal control problem (with the nonconvex control and the convex state constraint) along with stochastic maximum principle. To derive the necessary condition (i.e., stochastic maximum principle) for the optimal control, first we transform the fully coupled forward-backward stochastic control system into a fully coupled backward one; then, by using the terminal perturbation method, we obtain the stochastic maximum principle. Finally, we study a linear quadratic model.
In finance, various stochastic models are applied to simulate the price movements of financial instruments. For example, in the Black-Scholes model for pricing options, the price process of financial instrument is described by a geometrical Brownian movement which is a continuous-time stochastic process. However, the sudden world events would cause larger fluctuation in asset prices, which is intuitive in financial markets. In this situation, the price process usually follows a stochastic equation with jumps; that is, the uncertainty of the model comes from not only a Brownian motion but also a Possion random measure. Accordingly, for this case, there are also many applications in finance which produce the associated optimal control problem, such as risk management and option prices. In this paper, we focus on the optimal control problem for the discontinuous case.
There are two important approaches in control theory, namely, stochastic maximum principle and dynamic programming principle, where the stochastic maximum principle presents the necessary condition for the solution of the controlled system. In stochastic control theory, the maximum principle for the optimal control has been developed rapidly, especially, along with the appearance of nonlinear backward stochastic differential equations (BSDEs, for short) introduced by Pardoux and Peng , by Kushner , Bismut , Bensoussan , Haussmann , Hu , Peng [7, 8], and Øksendal and Sulem  (see  for the complete bibliography). In 1993, Peng  researched the decoupled forward-backward stochastic control system with the control domain being convex and deduced the maximum principle. Since then, under different assumptions, a lot of works about this topic have been springing up (refer to [9, 11–20]).
For the models with jumps, Situ  first got the maximum principle for forward stochastic controlled system with the jump diffusion coefficient independent of the control variable. Tang and Li  obtained the maximum principle in global form, when both diffusion and jump coefficients depend on the control variable and some state constraints are imposed. For the control problem of decoupled forward-backward stochastic differential equations (FBSDEs, for short) with jumps controlled system, the readers may refer to Øksendal and Sulem , Shi and Wu , and the references therein.
For the optimal control problem of the fully coupled FBSDEs with jumps, Shi and Wu  studied one kind of control problems with the control domain nonconvex, but the control variable does not appear in the diffusion and the jump coefficients of the forward equation. In , Shi discussed a general optimal control problem and derived Pontraygin’s maximum principle when the control domain is nonconvex, and the control variable appears in both diffusion and jump coefficients of the forward equation.
This paper is concerned with the optimal control problem of the fully coupled FBSDE with jumps (9), where we restrict the forward terminal state in a convex set , the control variables are non-convex, and the controlled model is not continuous. Moreover, we allow the diffusion and jump coefficients , in the forward equation to depend on the two different control variables, which is mainly because two diffusion terms are needed when applying martingale representation theorem to determine the solution of the BSDE with jumps. The control variables , associate with the diffusion term and the jump diffusion , respectively.
In order to deduce the stochastic maximum principle, we adopt the dual method (developed in ) and terminal perturbation method which are used to solve optimization problems with state constraints (refer to [12–14, 17, 25–28]). Recently, the dual approach is applied to utility optimization problem with volatility and ambiguity (see [29, 30]). First we transform the FBSDE with jumps (9) into a pure backward controlled one with jumps, in which the terminal state is regarded as the control variable which is much more easier to be dealt with than the initial state constraint. Meanwhile, the initial condition of the forward equation becomes an additional constraint. Fortunately, this constraint can be solved by applying Ekeland’s variational principle.
The paper is structured as follows. In Section 2, for fully coupled FBSDEs with jumps, we present some preliminaries. In Subsection 2.1, we first formulate the optimal control problem. By proving the variational inequality, we derive a stochastic maximum principle which presents the necessary condition for the optimal terminal state. Section 3 studies an application in stochastic linear quadratic control problem.
Let be a complete probability space equipped with a natural filtration which is generated by the following two mutually independent processes and completed by all -null sets:(i)a -dimensional standard Brownian motion ;(ii)a Poisson random measure on , where is equipped with its Borel -field , with the compensator such that is a martingale for all satisfying . Here is assumed to be a -finite Lévy measure on with the property that .
For any , denotes the Euclidean norm of . is a fixed time horizon. Let us present the following spaces of processes which will be used later:(i)(ii)(iii)(iv)(v)where denotes the -field of -predictable subsets of .
First we recall some results about fully coupled FBSDEs with jumps. Consider the following fully coupled FBSDEs with jumps: where , ,:, : ,:, : ,:
are -progressively measurable processes.
Here we adopt the monotonic condition for fully coupled FBSDEs with jumps developed by Wu .
Given an full-rank matrix , we define where is the transposed matrix of .
We assume the following.(i) , (ii) , , , , , , where are nonnegative constants with , . Moreover, we have , , when (resp., ).For every , , are progressively measurable and
Lemma 1. Under assumptions , , for any , FBSDE with jumps (6) has a unique adapted solution .
For the proof, the readers may refer to Wu .
2.1. Problem Formulation
Denote , by the set of admissible controls , respectively.
For the given admissible control processes , , the dynamic of control system is described by the following fully coupled FBSDEs with jumps: with , , and:,:,:,:: .
(i) , and are continuous in their arguments and continuously differentiable in ; (ii) the derivatives of in are bounded; (iii) the derivatives of in are bounded by , and the derivatives of and in are bounded by .
Under and , for any , , from the existence and uniqueness theorem of fully coupled FBSDEs with jumps (refer to ), we know that (9) has a unique adapted solution so that the following cost functional is well defined: where , , .
Our optimal control problem with state constraints can be stated as follows.
Problem I. Minimize subject to , , and , where is convex.
Remark 2. In our optimal control problem, the control variables are not convex. However, the terminal state is constrained in a convex set . For the non-convex state constraint case, it is still an open problem.
2.2. Backward Formulation
In the above optimal control problem, the state constraint is new for fully coupled FBSDEs with jumps control system. To solve the state constraints, we adopt the dual method and terminal perturbation method. First we need to transform the discontinuous fully coupled forward-backward control system into a discontinuous backward form. In addition, the following assumption is necessary:there exists , such that and , for all , , , , and , .
We point out that under and , for any , and , and are one-to-one correspondences. By letting , , we get the existence of the inverse functions , satisfying , , respectively. So, (9) is rewritten as where
Then, combined with the existence and uniqueness theorem of fully coupled FBSDEs with jumps (refer to Lemma 1), we know it is equivalent to choose , and select the terminal state . Thus, we obtain the following backward control system: which associates with the following cost functional: where is the control variable, , and , , , .
Then, the optimal control problem is reformulated as follows.
Problem II. Minimize subject to and .
We propose to consider Problem II, though the terminal state becomes a control variable and the initial condition (which is the solution of (13) at time under ) is regarded as a constraint. Even so, it is more relaxing to study Problem II than Problem I. Moreover, , , also satisfy the similar conditions in .
Henceforth, we convert to Problem II to prove the maximum principle for the optimal control .
2.3. Variational Equation
In this subsection, we will present the estimates for the variational equations to serve for the variational inequality.
First we define a metric in , for . Obviously, is a complete metric space.
Suppose that is an optimal control to Problem II and is the associated state processes of (13) (with ). For each , , we know (note is convex). And denotes the state processes of (13) corresponding to .
Consider the following BSDEs with jumps: where , , , , , , respectively. Equation (15) is regarded as the variational equation along with the optimal control and the state .
We point that (15) is composed of two fully coupled linear BSDEs with jumps. In fact, under and , from the existence and uniqueness theorem for BSDE with jumps (Lemma 1 in ), we know that (15) has the unique adapted solution , .
For convenience, we denote
Then, we get the following results.
Lemma 3. Under , , and , one has
Proof. From (13), (15), and the notations (16), we have
Applying Itô’s formula to , from , , we obtain
with . Furthermore, from Gronwall’s inequality, we have
From Lebesgue’s dominated convergence theorem, we know . Also, it is clear that .
So, letting in (21), we complete the proof.
2.4. Variational Inequality
In this section we are going to explore the variational inequality for the variational equation (15). Let us start with the following Ekeland’s variational principle  used to deal with the initial constraint .
Lemma 4 (Ekeland’s variational principle). Suppose that is a complete metric space and is a proper lower semicontinuous function bounded from below. If, for some , there exists satisfying , then there exists such that(i),(ii),(iii), .
Given the optimal control , for any , we introduce the penalty function as where , , , , , , .
Remark 5. Note that under the mappings , , , defined on are continuous mainly due to the continuous dependence of solutions of BSDE with jumps.
Theorem 6. Suppose , hold. If is optimal to Problem II, then there exist and , with and , such that the following variational inequality holds: where , , respectively, and , is the solution of (15).
Proof. Clearly, the following properties hold for the penalty function :
Combined with Ekeland’s variational principle (refer to Lemma 4), we obtain the existence of satisfying (i);(ii);(iii), .
For , , we have . The solution of (13) with (resp., is denoted by (resp., . And denotes the solution of (15) with . So, Similar to the proof of Lemma 3, we have It follows that Furthermore, we get the following expansions: where
Given , we discuss the following cases.
Case 1. There exists such that, for all , In this case, Setting and combining with (25), we get Case 2. There exists a positive sequence satisfying , such that In this case, from the definition of , . Due to being continuous, we have .
Thus, Similarly, from (25), where , , , .
We point out that, for the other six cases, the similar (33) also holds.
So, in any way, we always have that (i) (33) holds, (ii) , , , and (iii) .
Therefore, we obtain the existence of a convergent subsequence of whose limit is denoted by . Because of , we know in , as . According to the regularity of the solutions of BSDEs with jumps, as , we have
2.5. Maximum Principle
In this subsection, we will prove the stochastic maximum principle. First, the adjoint equation is introduced as follows: