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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 564524, 11 pages
Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion
1School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China
Received 15 June 2013; Accepted 19 July 2013
Academic Editor: Massimiliano Ferrara
Copyright © 2013 Zhonghao Zheng et al. 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 consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.
Nonlinear BSDEs in the framework of linear expectation were introduced by Pardoux and Peng  in 1990. Then, a lot of researches were studied by many authors, and they provided various applications of BSDEs in stochastic control, finance, stochastic differential games, and second-order partial differential equations theory; see [2–9].
The notion of sublinear expectation space was introduced by Peng [10–12], which is a generalization of classical probability space. The G-expectation, a type of sublinear expectation, has played an important role in the researches of sublinear expectation space recently. It can be regarded as a counterpart of the Wiener probability space in the linear case. Within this G-expectation framework, the G-Brownian motion is the canonical process. Besides, the notions of the G-martingales and the Itô integral with respect to G-Brownian motion were also derived. There are some new structures in these notions and some new applications in the financial models with volatility uncertainty; see Peng [12, 13].
In the G-expectation framework, thanks to a series of studies [14–17], the complete representation theorem for G-martingales has been obtained by Peng et al. . Due to this contribution, a natural formulation of BSDEs driven by G-Brownian motion was found by Hu et al. . In addition, the existence and uniqueness of the solution to the BSDEs driven by G-Brownian motion have been proved. They also have given the comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion in . So the complete theory of BSDEs driven by G-Brownian motion has been established.
An important application of BSDEs is that we can define the recursive utility functions from BSDEs, which can index scaling risks in the study of economics and finance [21–24]. Based on these results, a type of significant stochastic optimal control problems under linear expectation with a BSDE as cost function was studied [2, 4, 7–9]. Under G-expectation, the similar problems will be useful in the future studies of finance models with volatility uncertainty. So we arise a natural question: can we construct the similar results in G-expectation framework? In [25, 26], Zhang have given the study about the stochastic control problems under G-expectation based on the preliminary theory of BSDEs driven by G-Brownian motion. When the complete results about BSDEs driven by G-Brownian motion were established in [19, 20], we tried to prove the complete results of stochastic optimization theory of BSDEs driven by G-Brownian motion in this paper.
In this paper, we investigate the stochastic optimal control problems with a BSDE driven by G-Brownian motion constructed in [19, 20] as cost function. Based on the results in [19, 20], we obtain the dynamic programming principle under G-expectation. Besides, the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.
The rest of the paper is organized as follows. In Section 2, we recall the G-expectation framework and adapt it according to our objective. Besides, we give the related properties of forward and backward stochastic differential equations driven by G-Brownian motion, which will be needed in the sequel sections. In Section 3, the stochastic optimal control problems with a BSDE driven by G-Brownian motion as cost function are investigated and a dynamic programming principle under G-expectation is obtained. In Section 4, The value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.
In this section, we recall the G-expectation framework established by Peng [10–12, 27]. Besides, we give some results about forward and backward stochastic differential equations driven by G-Brownian motion, which we need in the following sections. Some details can be found in [19, 20].
2.1. G-Expectation and G-Martingales
Definition 1. Let be a given set, and let be a linear space of real valued functions defined on ; namely, for each constant and if . The space can be considered as the space of random variables. A sublinear expectation is a functional satisfying the following properties: for all , we have(i)monotonicity: if ;(ii)constant preservation: , for ;(iii)subadditivity: ;(iv)positive homogeneity: , for .
The triple is called a sublinear expectation space.
Definition 2 (G-normal distribution). A -dimensional random vector on a sublinear expectation space is called G-normally distributed if for each , we have where is an independent copy of ; that is, and are identically distributed, and is independent of . Here, the letter denotes the function where denotes the collection of all symmetric matrices.
Proposition 3. Let be G-normal distributed. The distribution of X is characterized by In particular, , where is the unique viscosity solution of the following parabolic PDE defined on : where is defined by (2).
Remark 4. It is easy to check that is a monotonic sublinear function defined on and implies that there exists a bounded, convex, and closed subset such that where denotes the collection of nonnegative elements in . If there exist some such that for any , we call the G-normal distribution nondegenerate, which is the case we consider throughout this paper.
Definition 5. Let , that is, the space of all -valued continuous paths with . The corresponding canonical process is , . is wiener measure. is the filtration generated by . We let be a linear space of random variables for each fixed , where .
The G-expectation is a sublinear expectation defined by for each , where are identically distributed -dimensional G-normally distributed random vectors in a sublinear expectation space such that is independent of for each . is called G-expectation space and the canonical process in the sublinear space is called a G-Brownian motion.
The conditional G-expectation of is defined by where = , , .
We denote by , , the completion of G-expectation space under the norm . For all , and are continuous mappings on endowed with the norm . Therefore, it can be extended continuously to .
Definition 6. A process is called a G-martingale if for each , , and for each , we have .
Now we introduce the Itô integral and quadratic variation process with respect to G-Brownian motion in G-expectation space.
Definition 7. Let be fixed. For a given partition of , we denote as the collection of the following type of simple processes: where , . We denote by the completion of under the norm .
Definition 8. For each , we define The mapping is continuous and thus can be continuously extended to .
Definition 9. The quadratic variation process of G-Brownian motion is defined by which is a continuous, nondecreasing process.
Definition 10. We now define the integral of a process with respect to as follows: The mapping is continuous and can be extended to uniquely.
Then, we detail some results about the quasianalysis theory constructed in .
Theorem 11. There exists a weakly compact probability measures family on such that is called a set of probability measures that represents .
Definition 12. We define the capacity associated to , which is a weakly compact family of probability measure that represents , as follows: is also called the capacity induced by .
Definition 13. Let be the capacity induced by . A set is polar if . A property holds “quasi-surely” (q.s. for short) if it holds outside a polar set.
Let and be two random variables; we say that is a version of if q.s.
Let for . The completion of and under is the same, and we denote them by .
2.2. Forward and Backward Stochastic Differential Equations Driven by G-Brownian Motion
We consider the following stochastic differential equations driven by dimensional G-Brownian motion (G-SDE): where , the initial condition is a given constant, are given functions satisfying , and , for each and the Lipschitz condition, that is, , for each , , , , , and , respectively. The solution is a process satisfying the G-SDE (14).
Now, we give the results about BSDEs driven by G-Brownian motion in the G-expectation space with and . We consider the following type of G-BSDEs (we always use Einstein convention): where , satisfy the following properties: there exist some such that(H1)for any , , ), ;(H2)for some ,
For simplicity, we denote by the collection of processes such that , ; is a decreasing G-martingale with and . Here, is the completion of under , and is the completion of under .
Definition 15. Let with ; and satisfy (H1) and (H2). A triplet of processes is called a solution of (15) if for some the following properties hold:
We have the following estimates.
Proposition 17 (see ). Let , and , satisfy and for some . For some , is solution of equation (15). Then(i)There exists a constant such thatwhere (ii)For any given , there exists a constant depending on such that
Proposition 18 (see ). Let , , and , satisfy (H1) and (H2) for some . For some , are solutions of (15) corresponding to , , and . Set , , and . (i)There exists a constant such thatwhere , .(ii)For any given with , there exists a constant depending on such that
Theorem 19 (see ). Let , , be the solutions of the following G-BSDEs: where , , satisfy (H1) and (H2) with . If , , , then .
Theorem 20 (see ). Let , , be the solutions of the following G-BSDEs: where , , satisfy and , are RCLL processes such that with . If , , , and is an increasing process, then .
3. A DPP for Stochastic Optimal Control Problems under G-Expectation
Now we introduce the setting for stochastic optimal control problems under -expectation. We suppose that the control state space is a compact metric space. Let the set of admissible control processes for the player be a set of -valued stochastic processes in , . For a given admissible control , the corresponding orbit, which regards as the initial time and as the initial state, is defined by the solution of the following type of G-SDE: where , , are deterministic functions and satisfy the following conditions : (A1) for ;(A2)For every fixed , , , are continuous in ;(A3)There exists a constant , for any , , , , such that
From the assumption (H3), we can get global linear growth conditions for , , ; that is, there exists such that, for , , , . Obviously, under the above assumptions, for any , G-SDE (24) has a unique solution. Moreover, we have the following estimates.
Proposition 21. Let , with , , , , and ; then we have where depends on .
Proof. The proof is similar to the proof of Proposition 4.1 in .
Now we give bounded functions , , that satisfy the following conditions: (H4) (i) for .(ii)For every fixed , and are continuous in , .(iii)There exist a constant , for , , , , , , , , , such that
From (H4), we have that , , and also satisfy global linear growth condition in ; that is, there exists , such that for all , , , For any and , the mappings and , where satisfy the conditions of Theorem 16 on the interval . Therefore, there exists a unique solution for the following G-BSDE: where is introduced by (24).
Proposition 22. For each , with and , , we have where depends on , , , and .
Proof. The proof is similar to the Proposition 4.2 in .
Given a control process , we introduce an associated cost functional where the process is defined by G-BSDE (29). Similar to the proof of Theorem 4.4 in , we have that for , , But we are more interested in the case when .
Now we define the value function as follows:
Proposition 23. is a deterministic function of .
Proof. For a partition of : , , , we denote , by the collection of simple processes , where , , and by the completion of under the norm . Use the similar method in Lemma of ; we can prove that is a -valued process; there exists ; is a -valued process, is a partition of such that under probability measure . When , we note that is a deterministic function of because , , , , , and are deterministic functions, and is a G-Brownian motion. So we need to construct a sequence of admissible controls of the form satisfying , where is a -valued processes and is a partition of . Firstly, there exists , such that . Then we define , Therefore, Set , , . So . Without loss of generality, suppose . We denote where is a -valued process and is a partition of . Then we can suppose for , . From Proposition 22, we have Therefore, . Then, we have Now, we suppose that Because , we have Hence, . We have finished the proof.
Lemma 24. For any , , , we have
Lemma 25. For any , , and is measurable; we have , Conversely, , there exists a , such that
Proof. We already know that is continuous with respect to and is continuous with respect to . We want to prove (49) and only need to discuss the simple random variables of the form and of the form Here, , , , and is a -partition. Then, from the same technique used in the proof of Theorem 4.4 in , we have So we have proved (49). Now we prove (50) in a similar way. We first construct a random variable , where is a -partition and , such that . Then, we have for . Now, we choose a control , such that . Set . Finally, we get So, we have (50).
Now, we give a type of DPP for our stochastic optimal control problems. Firstly, we define a family of backward semigroups associated with the G-BSDE (29). Given the initial data , a positive number , and a random variable with , we set where is the solution of the following G-BSDE with the time horizon : Obviously, for the solution of G-BSDE (29), we have Then, we can obtain the DPP for our stochastic optimal control problems as follows.
Theorem 26. The value function has the following proposition: for every , we have
Proposition 27. is -Hölder continuous in .
Proof. For any given and , from Theorem 26, we know that for , there exists a such that Then, we need to prove We only check the first inequality in (66). The second can be proved similarly. We have , where From Proposition 21, we have By Proposition 22 and Lemma 24, we deduce that Based on the definition of , we get By Proposition 22, we can prove the following inequality easily by the similar method in Proposition 3.5 of  So we have . Hence, by (67) we have Let ; we obtain the first inequality of (66). The proof is completed.
4. Value Function and Viscosity Solution of Fully Nonlinear Second-Order Partial Differential Equation
In this section, we consider the following fully nonlinear second-order partial differential equation where
Remark 28. The definition and uniqueness of the viscosity solution of above second-order partial differential equation can be found in Appendix in Peng . So, we only need to prove that is a viscosity solution of (74). Besides, from the result of Section 3, we can have that is continuous in .
Definition 29. A real-valued continuous function , , for any , is called a viscosity subsolution (super-solution) of (74); if for all functions satisfy and at fixed , we have
In order to prove the Theorem, we need three lemmas. Firstly, we set