International Journal of Stochastic AnalysisVolume 2012 (2012), Article ID 258674, 50 pagesdoi:10.1155/2012/258674

Research Article

Necessary Conditions for Optimal Control of Forward-Backward Stochastic Systems with Random Jumps

Jingtao Shi

School of Mathematics, Shandong University, Jinan 250100, China

Received 28 September 2011; Revised 28 December 2011; Accepted 3 January 2012

Academic Editor: Jiongmin Yong

Copyright © 2012 Jingtao Shi. 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.

This paper deals with the general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Necessary conditions of Pontraygin's type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.

1. Introduction

1.1. Basic Notations

Throughout this paper, we denote by the space of -dimensional Euclidean space, by the space of matrices, and by the space of symmetric matrices. and denote the scalar product and norm in the Euclidean space, respectively. appearing in the superscripts denotes the transpose of a matrix.

Let be a complete filtered probability space satisfying the usual conditions, where the filtration is generated by the following two mutually independent processes:(i)a one-dimensional standard Brownian motion ;(ii)a Poisson random measure on , where is a nonempty open set equipped with its Borel field , with compensator , such that is a martingale for all satisfying .   is assumed to be a -finite measure on and is called the characteristic measure.

Let be fixed and be a nonempty subset of . We denote . Any generic point in is denoted by . Let be the set of all -predictable processes such that . Any is called an admissible control process. We denote by or the set of integrable functions with norm . We define We denote Clearly, is a Banach space. Any process in is denoted by , whose norm is defined by

1.2. Formulation of the Optimal Control Problem and Basic Assumptions

For any and , we consider the following fully coupled forward-backward stochastic control system: with the cost functional given by Here, takes value in and

For any and , we refer to as the state process corresponding to the admissible control if FBSDEJ (1.4) admits a unique adapted solution. For controlled FBSDEJ (1.4) with cost functional (1.5), we consider the following.

Problem C. Find , such that

Any satisfying (1.7) is called an optimal control process of Problem C, and the corresponding state process, denoted by , is called optimal state process. We also refer to as an optimal 5-tuple of Problem C.

Our main goal in this paper is to derive some necessary conditions for the optimal control of Problem C, which is called the stochastic maximum principle of Pontraygin’s type. For this target, we first introduce the following basic assumption throughout this paper. (H0) For any and , FBSDEJ (1.4) admits a unique adapted solution . Moreover, the following estimate holds: Further, if   is the unique adapted solution of (1.4) with   and   replaced by   and , respectively, then the following stability estimate holds: By adopting the ideas from Wu [1], we know that under certain -monotonicity conditions for the coefficients, the existence and uniqueness of FBSDEJ (1.4) guaranteed which leads to hypothesis (H0). Our main goal in this paper is to derive some necessary conditions for the optimal control of Problem C. Hence, we impose the well-posedness of the state equation (1.4) as an assumption to avoid some technicalities not closely to our main results.

1.3. Developments of Stochastic Optimal Control and Contributions of the Paper

It is well known that the optimal control problem is one of the central themes of modern control sciences. Necessary conditions for the optimal control of the (forward) continuous stochastic control system, that is, the so-called stochastic maximum principle of Pontraygin’s type, were extensively studied since early 1960s. When Brownian motion is the unique noise source, Peng [2] (see also Yong and Zhou [3]) obtained the maximum principle for the general case, that is, the control variable entering the diffusion coefficient and control domain being not convex.

Forward-backward stochastic control systems where the controlled systems are described by forward-backward stochastic differential equations (FBSDEs) are widely used in mathematical economics and mathematical finance, which includes the usual forward SDEs as a special case. They are encountered in stochastic recursive utility optimization problems (see [4–8]) and principal-agent problems (see [9, 10]). Peng [11] first obtained the necessary conditions for optimal control for the partially coupling case when the control domain is convex. And then Xu [12] studied the nonconvex control domain case and obtained the corresponding necessary conditions. But he needs to assume that the diffusion coefficient in the forward control system does not contain the control variable. Ji and Zhou [8] applied the Ekeland variational principle to establish a maximum principle for a partially coupled forward-backward stochastic control system, while the forward state is constrained in a convex set at the terminal time. Wu [13] recently established a general maximum principle for optimal control problems derived by forward-backward stochastic systems, where control domains are nonconvex and forward diffusion coefficients explicitly depend on control variables. Moreover, some financial optimization problems for large investors (see [14–16]) and some asset pricing problems with forward-backward differential utility (see [7]) directly lead to fully coupled FBSDEs. Wu [17] first (see also Meng [18]) obtained the necessary conditions for optimal control of fully coupled forward-backward stochastic control systems when the control domain is convex. And then Shi and Wu [19] studied the nonconvex control domain case and obtained the corresponding necessary conditions under some -monotonic assumptions. But they also (similar to Xu [12]) need to assume that the control variable does not appear in the diffusion coefficient of the forward equation. Very recently, Yong [20] completely solved the problem of finding necessary conditions for optimal control of fully coupled FBSDEs. He considered an optimal control problem for general coupled FBSDEs with mixed initial-terminal conditions and derived the necessary conditions for the optimal controls when the control domain is not assumed to be convex and the control variable appears in the diffusion coefficient of the forward equation.

However, recently more and more research attentions are drawn towards the optimal control problem for discontinuous stochastic systems or stochastic systems with random jumps. The reason is clear for its applicable aspect. For example, there is compelling evidence that the dynamics of prices of financial instruments exhibit jumps that cannot be adequately captured solely by diffusion processes (i.e., processes satisfying some Itô-type stochastic differential equations (SDE for short), see Merton [21] and Cont and Tankov [22]. Several empirical studies demonstrate the existence of jumps in stock markets, the foreign exchange market, and bond markets. Jumps constitute also a key feature in the description of credit risk sensitive instruments. Therefore, models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering, this leads to paying more attention to stochastic differential equations with jumps (SDEJs for short). As consequences, optimal control problems involving systems of SDEJs are widely studied. Situ [23] first obtained the maximum principle for (forward) stochastic control system with jumps but the jump coefficient does not contain the control variable. Tang and Li [24] discussed a more general case, where the control is allowed into both diffusion and jump coefficients, and the control domain is not necessarily convex; also some general state constraints are imposed. Maximum principles for forward-backward stochastic control systems with random jumps are studied in Øksendal and Sulem [25], Shi and Wu [26], where the FBSDEJs are partially coupled and the control domains are convex. Recently, Shi [27] obtained the necessary condition of optimal control as well as a sufficient condition of optimality under the assumption that the diffusion and jump coefficients do not contain the control variable and the control domain need not be convex. Necessary conditions for fully coupled forward-backward stochastic control systems with random jumps were studied in Shi and Wu [28] (see also Meng and Sun [29]) where the control domains are convex.

In this paper, we consider the general optimal control problem for fully coupled FBSDEJ (1.4). Here, by the word “general” we mean the allowance of the control variable into both diffusion and jump coefficients of the forward equation and the control domain is not assumed to be convex. It is well worth mentioning that the idea of second-order spike variation technique, developed by Peng [2], plays an important role in deriving the necessary conditions of general stochastic optimal control of jump-diffusion process in Tang and Li [24]. Following the standard idea of deriving necessary conditions for optimal control processes, due to the fact the control domain is not assumed to be convex, one needs to use spike variation for the control process and then to try obtaining a Taylor-type expansion for the state process and the cost functional (1.5) with respect to the spike variation of the control process, followed by some suitable duality relations to get a maximum principle of Pontryagin-type. However, the derivation of Taylor expansion of the state process with respect to the spike variation of the control process is technically difficult. The main reasons are that both and appear in the diffusion and jump coefficients of the forward equation of (1.4) and that the regularity/integrability of the continuous martingale process and the discontinuous martingale process (as part of the state process) seems to be not enough in the case when a second-order expansion is necessary. Note that in [25–29], due to the special structure of the problems, the second-order expansion is not necessary when one tries to derive the necessary conditions for optimal controls.

We overcome the above difficulty by the newly developed reduction method by Wu [13] and Yong [20] in the continuous case, independently. In fact, some ideas have been proposed early in Kohlmann and Zhou [30] and Ma and Yong [31]. Let us make it more precise. We first introduce the controlled initial value problem for a system of SDEJs, where is regarded as the state process and is regarded as the control process. Next, we regard the original terminal condition as terminal state constraint and then we translate Problem C into a high-dimensional reduced optimal control problem described by standard SDEJ with state constraint (see Problem in Section 3). The advantage of this reduced optimal control problem is that one needs not much regularity/integrability of processes since it is treated as a control process. We apply the Ekeland variational principle to deal with this high-dimensional reduced optimal control problem with state constraint. Finally, necessary conditions for the optimal control of Problem C are derived by the equivalence of Problems C and .

The rest of this paper is organized as follows. In Section 2, under some suitable assumptions, we give the main result of this paper, together with some discussions on special cases. In Section 3, after we make the reduction of our optimal control problems for FBSDEJs Problem C, a proof of our main result will be presented. Section 4 is devoted to a linear quadratic stochastic optimal control problem as an illustrating example. Finally, in Section 5 we give the concluding remarks and compare our theorem with some existing results.

2. The Main Result and Some Special Cases

In this section, under some suitable assumptions, we will state the necessary conditions for the optimal control of our Problem C. Also, some interesting special cases will be discussed.

Let us introduce the following further assumptions beyond (H0).(H1)For any , maps is   measurable and there exists a constant such that (H2)Maps is twice continuously differential, with the (partial) derivatives up to the second order being uniformly bounded, Lipschitz continuous in , and continuous in .(H3)The map is twice differentiable with the derivatives up to the second order being uniformly bounded and uniformly Lipschitz continuous; the maps are twice differentiable with the derivatives up to the second order being Lipschitz continuous; the map is twice differentiable with the derivatives up to the second order being Lipschitz continuous in , and continuous in .

Next, to simplify our presentation, we now introduce some abbreviation notations. First, we make the following convention: for any differentiable map , let In particular, for is an -dimensional row vector. Also, for any twice differentiable function , the Hessian matrix is given by hereafter, stands for the set of all real symmetric matrices. On the other hand, for a twice differentiable function (denoting ), we denote

Now, let be an optimal 5-tuple of Problem C. For , we denote and let We also introduce For , let and let

Our main result in this paper is the following theorem for Problem C.

Theorem 2.1. Suppose that (H0)–(H3) hold, and let be an optimal 5-tuple of Problem C. Then there exists a unique adapted solution to the following FBSDEJ: Let be the unique adapted solution to the following matrix-valued BSDEJ: where , and the Hamiltonian function is defined as follows: Then, with , and .

Remark 2.2. In fact, the second-order adjoint equation (2.11) can be split into the following three BSDEJs if we add that :

Note that this kind of three second-order adjoint equations appears in Wu [13] whereas not in Yong [20].

Let us now look at some special cases. It can be seen that our theorem recovers the known results.

A Classical Stochastic Optimal Control Problem with Random JumpsConsider a controlled SDEJ: with the cost functional In this case, Hence, we have, by some direct computation/observation, Consequently, (2.10) and (2.11) (or equivalently, (2.14)) become where the Hamiltonian function is defined as The maximum condition (2.13) is reduced to the following: These are the necessary conditions for the stochastic optimal control problem with random jumps (see (2.37) of Tang and Li [24]). When the jump coefficient is independent of , our result is reduced to Situ [23]. Where there are no random jumps, our result recovers the well-known maximum principle for the classical stochastic optimal control problem of Peng [2] and Yong and Zhou [3]. And when is convex, the classical result of Bensoussan [32] is recovered.

A Stochastic Optimal Control Problem for BSDEJs with Random JumpsConsider a controlled BSDEJ: with the cost functional In this case, Hence, we have Consequently, (2.10) and (2.11) (or equivalently, (2.16)) become The maximum condition (2.13) is reduced to the following: As we know, this new result has not been published elsewhere. When there are no random jumps, our result partially recovers that of Dokuchaev and Zhou [33].

A Stochastic Optimal Control Problem for FBSDEJs with Random JumpsConsider a controlled FBSDEJ: with the cost functional In this case, we have Consequently, (2.10) becomes and (2.14), (2.15), (2.16) become where the Hamiltonian function is defined as follows: The maximum condition (2.13) remains the same. When is convex, we essentially recover the results in Shi and Wu [26] and Øksendal and Sulem [25] (partial information case), and when are independent of and is not assumed to be convex, our result is reduced to that of Shi [27]. Note that in both of these cases, our second adjoint equations are new.When there are no random jumps, our result partially recovers that of Wu [13]. Note that in (3.27) of [13], some additional parameters have to be involved and to be determined. And when is convex, our result is reduced to that of Peng [11] and Wang and Wu [34] (partial observation case). Result of Xu [12] is recovered when is independent of and is not assumed to be convex.

A Stochastic Optimal Control Problem for Fully Coupled FBSDEJs with Random JumpsConsider the controlled fully coupled FBSDEJ (1.4) with the cost functional (1.5). When is convex, we essentially recover the results in Shi and Wu [28] and Meng and Sun [29] (partial information case). When there are no random jumps, our result becomes a special case of Yong [20], because in [20] the author considered the mixed initial-terminal conditions, and some additional necessary conditions for the optimal control are derived. When is independent of , our result recovers those of Shi and Wu [19, 35] (partial observation case). And when is convex, our result is reduced to that of Wu [17].

3. Problem Reduction and the Proof of the Main Theorem

This section is devoted to the proof of our main theorem. The proof is lengthy and technical. Therefore, we divide it into several steps to make the idea clear.

Step 1 (problem reduction). Consider the following initial value problem for a control system of SDEJs: where is regarded as the state process and as the control process.It is standard that (see [36]), for any and , there exists a unique strong solution to (3.1), depending on the 4-tuple . Next, we regard the original terminal condition as the terminal state constraint:

Let be the set of all 4-tuples such that the unique corresponding state process satisfies the constraint (3.3). Note that hypothesis (H0) implies that, for any , there exists a unique 4-tuple such that state equation (3.1) admits a unique state process satisfying the state constraint (3.3). Hence, (H0) and (H1) imply . We rewrite the cost functional (1.5) as follows:

Now, we can pose the following.

Problem . Find , such that

We, respectively, refer to as an optimal control process, to as the corresponding optimal state process, and to as an optimal 4-tuple of Problem .

It is clear that Problems C and are equivalent. The advantage of Problem is that one does not need much regularity/integrability on since it is treated as part of a control process; the disadvantage is that one has to treat terminal constraint (3.3).

Step 2 (applying Ekeland’s variational principle). For convenience, let us cite Ekeland’s variational principle, whose proof can be found in Ekeland [37] or Yong and Zhou [3].

Lemma 3.1. Let be a complete metric space, and let be lower-semicontinuous, bounded below. If for all there exists satisfying then there exists , satisfying the following: (i), (ii), (iii).

Let be an optimal 4-tuple of Problem , with the corresponding optimal state process . For any , we define

If we define the Ekeland’s distance on bywith being the Lebesgue measure of set , then is a complete metric space under the following metric:where , , and is continuous. Also it is clear that Hence, by Lemma 3.1, there exists a 4-tuple such that Thus, is a global minimum point of the following penalized cost functional In other words, if we pose a penalized optimal control problem with the state constraint (3.3) and the cost functional (3.12), then is an optimal 4-tuple of the problem. Note that this problem does not have state constraints, and the optimal 4-tuple approaches as .

Step 3 (nontriviality of the multiplier). Note that the state process corresponding to the 4-tuple satisfies the following SDEJ: with

We now regard as the unique solution to the following FBSDEJ: with

Next, for any , with by (H0), the following FBSDEJ admits a unique adapted solution . Note that FBSDEJ (3.18) is nothing but FBSDEJ (3.15) with only replaced by , and remains unchanged. Thus, by (1.9), we have which implies Hence, with , by taking in the last relation in (3.11), we have where stands for certain scalars or vectors that go to 0 as ,

We point out that is independent of , and Then there is a subsequence of , still denoted by , such that

We claim that To show this, let us observe that by (3.19), one has Hence, dividing by in (3.21) and then letting yields

with the constant independent of . Now, if , then the above leads to

with , as , uniformly in . Then,

Thus, for small enough, we must have It is a contradiction which proves (3.26).

We refer to as the Lagrange multiplier of the corresponding optimal 4-tuple . Equation (3.26) shows the nontriviality of the Lagrange multiplier .

Step 4 (spike variations). For notational simplicity, we now denote Consequently,

Note that and are Hilbert spaces. We identify and . Also, from the above, We denote the gradient and the Hessian of as follows: where is the set of all linear bounded operators from to , and is the set of all linear bounded self-adjoint operators from to itself. Clearly,

Also, by the Fréchet differentiability, for , we have

To make the above more precise, let us take any . Then,

Thus,

with

with

Next, for fixed and any , let for some measurable set with . It is clear that .

Let be the state process of (3.1) corresponding to . Further, let and be, respectively, the solutions to the following SDEJs: where denotes the indicator function of the set , and for any ,

We point out that everything is reduced to the classical optimal control problems for SDEJs. Then by Lemma  2.1 of Tang and Li [24], the following estimates hold for any integers :

Now, from the last relation in (3.11), noting (3.19), we obtain where is defined in (3.25) and is defined similarly to (3.22), with replaced by (3.42). We have shown that along a sequence,

Note that

We first deal with term . We have with

Next we deal with term . Similarly, we have with

Also, we have with

Under assumptions (H2), (H3), we have

Hence, by (3.47), we have

Consequently, combining (3.50) with (3.55), from (3.48), we can derive that

Step 5 (duality). Let be the adapted solution to the following BSDEJ: where . Then, as , goes to , which is the adapted solution to the following BSDEJ: where is determined by (3.25).

Applying Itô’s formula to , we can get Then by (3.59), we have Note that satisfies Now, let be the adapted solution to the following BSDEJ: where with being the Hessian of the following Applying Itô’s formula to , we have Consequently, from (3.64), we have

Step 6 (variational inequality). In (3.70), dividing and then sending followed by sending , we get which implies

Using a standard argument of Tang and Li [24] in (3.70), we have the following variational inequality:

Step 7 (finalizing the proof). Since , by rescaling, we let . Then the first-order adjoint equation (3.61) reduces to Let in (3.66), then which is the adapted solution to the following second-order adjoint equation:

where with being the Hessian of the following Also, from (3.72), we have

Note that On the other hand, since we have

Let Then it follows from (3.74), (3.78), and (3.82) that By (3.82), we can rewrite the equations in (3.74) as and the variational inequality (3.73) now takes the form: Thus, taking , and , then dividing by and sending , we get Hence, Combining (3.83), (3.84) with (3.87), we arrive at (2.10).

Next, we have Then (2.11) follows from (3.75).

Further, by taking in the variational inequality (3.85), we obtain which gives (2.13). This completes the proof of our main result—Theorem 2.1.

4. A Linear Quadratic Example

In this section, we briefly discuss a linear quadratic case, which serves as an illustrating example of our main result. Consider the following linear controlled FBSDEJ: with the quadratic cost functional given by In the above, , and are all real constants of . The control domain could be very arbitrary; in particular, does not have to be a convex set. We let be the set of all -predictable processes such that .

For any , we denote and assume that (H4.1) where , and are given constants with . Then by Wu [1], there exists a unique adapted solution to (4.1).

Now, suppose that the corresponding stochastic control problem admits an optimal 5-tuple . Now let us look at Theorem 2.1. The first-order adjoint equation (2.10) reads Similarly, we denote Since controlled FBSDEJ satisfies (H4.1), we can easily check that the above FBSDEJ (4.4) satisfies the following condition (H4.2): where , and are the same as in (H4.1). Then by Wu [38], FBSDEJ (4.4) admits a unique adapted solution .

The matrix-valued second-order adjoint equation (2.11) becomes where

As before, (4.5) can be split into the following three scalar BSDEJs:

Note that since all the coefficients in (4.5) are constants and the terminal condition is deterministic, we must have and is deterministic. Consequently, And at the same time (4.7)–(4.10) reduce to and . Clearly, linear ODE (4.11) admits a unique explicit solution and then ODEs (4.12) do.

Since the Hamiltonian function (2.12) reads the maximum condition (2.13) now takes the form (noting that ) This implies that

Now, let us look at an interesting special case. Noting that some related problems without random jumps were discussed by Shi and Wu [39] by a purely completion-of-squares technique.

Suppose that It is very easy to check that the assumptions (H4.1) and (H4.2) hold. Now the first-order adjoint equation (4.4) reads and the second-order adjoint ODEs (4.12) reduce to

If, in addition, (noting that this does not mean that the state equation (4.1) is decoupled), then (4.21) further reduce to We can first solve explicitly from (4.22): and then solve explicitly from (4.24). Finally, explicit can be obtained from (4.23). The maximum condition (4.18) remains the same.

However, we cannot obtain the explicit optimal control from the maximum condition (4.18) in general, since this depends heavily on the adjoint processes . The explicit, adapted solution to fully coupled FBSDEJ such as (4.20) is an interesting and challenging open problem. And the explicit optimal control in its state feedback form is rather difficult to obtain even in some very simple cases when there are no random jumps. We will consider some relevant issues in the future work.

5. Concluding Remarks

In this paper, we have discussed a general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Enlightened by Tang and Li [24], Wu [13], Yong [20], necessary conditions of Pontraygin’s type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. And the general maximum principle for forward stochastic control systems with random jumps [24] and the maximum principles for forward-backward stochastic control systems with random jumps [25, 26] are recovered in this paper. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.

Acknowledgments

This work was supported by China Postdoctoral Science Foundation Funded Project (no. 20100481278), Postdoctoral Innovation Foundation Funded Project of Shandong Province (no. 201002026), National Natural Sciences Foundations of China (no. 11126209) and Shandong Province (no. ZR2011AQ012), and Independent Innovation Foundation of Shandong University (IIFSDU, no. 2010TS060).

References

Z. Wu, “Forward-backward stochastic differential equations with Brownian motion and Poisson process,” Acta Mathematicae Applicatae Sinica, vol. 15, no. 4, pp. 433–443, 1999.

S. G. Peng, “A general stochastic maximum principle for optimal control problems,” SIAM Journal on Control and Optimization, vol. 28, no. 4, pp. 966–979, 1990.

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43, Springer, New York, NY, USA, 1999.

N. El Karoui, S. G. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Mathematical Finance, vol. 7, no. 1, pp. 1–71, 1997.

N. El Karoui, S. G. Peng, and M. C. Quenez, “A dynamic maximum principle for the optimization of recursive utilities under constraints,” The Annals of Applied Probability, vol. 11, no. 3, pp. 664–693, 2001.

M. Schroder and C. Skiadas, “Optimal consumption and portfolio selection with stochastic differential utility,” Journal of Economic Theory, vol. 89, no. 1, pp. 68–126, 1999.

F. Antonelli, E. Barucci, and M. E. Mancino, “Asset pricing with a forward-backward stochastic differential utility,” Economics Letters, vol. 72, no. 2, pp. 151–157, 2001.

S. L. Ji and X. Y. Zhou, “A maximum principle for stochastic optimal control with terminal state constraints, and its applications,” Communications in Information and Systems, vol. 6, no. 4, pp. 321–337, 2006.

N. Williams, “On dynamic principal-agent problems in continuous time,” Working paper, 2008, http://www.ssc.wisc.edu/~nwilliam/dynamic-pa1.pdf

J. Cvitanić, X. H Wan, and J. F. Zhang, “Optimal contracts in continuous-time models,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 95203, 27 pages, 2006.

S. G. Peng, “Backward stochastic differential equations and applications to optimal control,” Applied Mathematics and Optimization, vol. 27, no. 2, pp. 125–144, 1993.

W. S. Xu, “Stochastic maximum principle for optimal control problem of forward and backward system,” Journal of the Australian Mathematical Society Series B, vol. 37, no. 2, pp. 172–185, 1995.

Z. Wu, “A general maximum principle for optimal control problems of forward-backward stochastic control systems,” Working paper, Shandong University, China, 2009

J. Cvitanić and J. Ma, “Hedging options for a large investor and forward-backward SDE's,” The Annals of Applied Probability, vol. 6, no. 2, pp. 370–398, 1996.

D. Cuoco and J. Cvitanić, “Optimal consumption choices for a “large” investor,” Journal of Economic Dynamics & Control, vol. 22, no. 3, pp. 401–436, 1998.

R. Buckdahn and Y. Hu, “Hedging contingent claims for a large investor in an incomplete market,” Advances in Applied Probability, vol. 30, no. 1, pp. 239–255, 1998.

Z. Wu, “Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems,” Systems Science and Mathematical Sciences, vol. 11, no. 3, pp. 249–259, 1998.

Q. X. Meng, “A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information,” Science in China. Series A, vol. 52, no. 7, pp. 1579–1588, 2009.

J. T. Shi and Z. Wu, “The maximum principle for fully coupled forward-backward stochastic control system,” Acta Automatica Sinica, vol. 32, no. 2, pp. 161–169, 2006.

J. M. Yong, “Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions,” SIAM Journal on Control and Optimization, vol. 48, no. 6, pp. 4119–4156, 2010.

R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” Journal of Financial Economics, vol. 3, no. 1-2, pp. 125–144, 1976.

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.

R. Situ, “A maximum principle for optimal controls of stochastic systems with random jumps,” in Proceedings of the National Conference on Control Theory and Applications, Qingdao, China, 1991.

S. J. Tang and X. J. Li, “Necessary conditions for optimal control of stochastic systems with random jumps,” SIAM Journal on Control and Optimization, vol. 32, no. 5, pp. 1447–1475, 1994.

B. Øksendal and A. Sulem, “Maximum principles for optimal control of forward-backward stochastic differential equations with jumps,” SIAM Journal on Control and Optimization, vol. 48, no. 5, pp. 2945–2976, 2009.

J. T. Shi and Z. Wu, “Maximum principle for forward-backward stochastic control system with random jumps and applications to finance,” Journal of Systems Science & Complexity, vol. 23, no. 2, pp. 219–231, 2010.

J. T. Shi, “Global maximum principle for the forward-backward stochastic optimal control problem with Poisson jumps,” Early View in Asian Journal of Control, vol. 16, no. 1, pp. 1–11, 2014.

J. T. Shi and Z. Wu, “Maximum principle for fully coupled forward-backward stochastic control system with random jumps,” in Proceedings of the 26th Chinese Control Conference, pp. 375–380, Zhangjiajie, China, 2007.

Q. X. Meng and Y. Z. Sun, “Optimal control problem of fully coupled forward-backward stochastic systems with Poisson jumps under partial information,” in Proceedings of the 30th Chinese Control Conference (CCC '11), pp. 1317–1322, Yantai, China, 2011.

M. Kohlmann and X. Y. Zhou, “Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1392–1407, 2000.

J. Ma and J. M. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, vol. 1702 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1999.

A. Bensoussan, “Lectures on stochastic control,” in Nonlinear Filtering and Stochastic Control, vol. 972 of Lecture Notes in Mathematics, pp. 1–62, Springer, Berlin, Germany, 1982.

N. Dokuchaev and X. Y. Zhou, “Stochastic controls with terminal contingent conditions,” Journal of Mathematical Analysis and Applications, vol. 238, no. 1, pp. 143–165, 1999.

G. C. Wang and Z. Wu, “The maximum principles for stochastic recursive optimal control problems under partial information,” IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1230–1242, 2009.

J. T. Shi and Z. Wu, “Maximum principle for partially-observed optimal control of fully-coupled forward-backward stochastic systems,” Journal of Optimization Theory and Applications, vol. 145, no. 3, pp. 543–578, 2010.

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, vol. 24, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1989.

I. Ekeland, “Nonconvex minimization problems,” Bulletin of The American Mathematical Society, vol. 1, no. 3, pp. 443–474, 1979.

Z. Wu, “Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration,” Journal of the Australian Mathematical Society, vol. 74, no. 2, pp. 249–266, 2003.

J. T. Shi and Z. Wu, “The LQ problem for fully coupled forward-backward stochastic control system,” Journal of Shandong University, vol. 40, no. 1, pp. 11–17, 2005.