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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 342038, 10 pages
Representation Theorem for Generators of BSDEs Driven by -Brownian Motion and Its Applications
1Department of Mathematics, Donghua University, 2999 North Renmin Road, Songjiang, Shanghai 201620, China
2School of Mathematics, Shandong University, 27 Shanda Nanlu, Jinan 250100, China
Received 14 September 2013; Accepted 10 November 2013
Academic Editor: Litan Yan
Copyright © 2013 Kun He and Mingshang Hu. 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 obtain a representation theorem for the generators of BSDEs driven by -Brownian motions and then we use the representation theorem to get a converse comparison theorem for -BSDEs and some equivalent results for nonlinear expectations generated by -BSDEs.
Let be a probability space, and, for fixed , let be a standard Brownian motion and let be the augmentation of . Then Pardoux and Peng  introduced the backward stochastic differential equations (BSDEs) and proved the existence and uniqueness result of the BSDEs. In 1997, Peng  promoted -expectations based on BSDEs. One of the important properties of -expectations is comparison theorem or monotonicity. Chen  first considers a converse result of BSDEs under equal case. After that, Briand et al.  obtained a converse comparison theorem for BSDEs under general case. They also derived a representation theorem for the generator . Following this paper, Jiang  discussed a more general representation theorem then, in his another paper , showed a more general converse comparison theorem. Here the representation theorem is an important method in solving the converse comparison problem and other problems (see Jiang ).
Peng [8–13] defined the -expectations and -Brownian motions (-BMs) and proved the representation theorem of -expectation by a set of singular probabilities, which differs from nonlinear -expectations because -expectations are equivalent with a group of absolutely continuous probabilities with respect to the probability measure . Soner et al.  obtained an existence and uniqueness result of 2 BSDEs. Recently, Hu et al.  proved another existence and uniqueness result on BSDEs driven by -Brownian motions (-BSDEs).
An important advantage of -BSDEs is the easiness to define the nonlinear expectations. Hu et al. in  gave a comparison theorem for -BSDEs and talked about the properties of corresponding nonlinear expectations. In this paper, we consider the representation theorem for generators of -BSDEs and then consider the converse comparison theorem of -BSDEs and some equivalent results for nonlinear expectations generated by -BSDEs. In the following, in Section 2, we review some basic concepts and results about -expectations. We give the representation theorem of -BSDEs in Section 3. In Section 4, we consider the applications of representation theorem of -BSDEs, which contain the converse comparison theorem and some equivalent results for nonlinear expectations generated by -BSDEs.
We review some basic notions and results of -expectation, the related spaces of random variables, and the backward stochastic differential equations driven by a -Brownian motion. The readers may refer to [10, 13, 15, 17–19] for more details.
Definition 1. Let be a given set and let be a vector lattice of real valued functions defined on , namely, for each constant and if . is considered as the space of random variables. A sublinear expectation on is a functional satisfying the following properties: for all , one has (a)monotonicity: if , then ;(b)constant preservation: ;(c)subadditivity: ;(d)positive homogeneity: for each . is called a sublinear expectation space.
Definition 2. Let and be two -dimensional random vectors defined, respectively, in sublinear expectation spaces and . They are called identically distributed, denoted by , if , for all, where denotes the space of bounded and Lipschitz functions on .
Definition 3. In a sublinear expectation space , a random vector , , is said to be independent of another random vector , under , denoted by , if for every test function one has .
Definition 4 (-normal distribution). A -dimensional random vector in a sublinear expectation space is called -normally distributed if for each one has where is an independent copy of ; that is, and . Here, the letter denotes the function where denotes the collection of symmetric matrices.
Peng  showed that is -normally distributed if and only if for each , , , is the solution of the following -heat equation:
The function is a monotonic, sublinear mapping on and implies that there exists a bounded, convex, and closed subset such that where denotes the collection of nonnegative elements in .
In this paper, we only consider nondegenerate -normal distribution; that is, there exists some such that for any .
Definition 5. (i) Let denote the space of -valued continuous functions on with and let be the canonical process. Set
Let be a given monotonic and sublinear function. -expectation is a sublinear expectation defined by
for all , where are identically distributed -dimensional -normally distributed random vectors in a sublinear expectation space such that is independent of for every . The corresponding canonical process is called a -Brownian motion.
(ii) For each fixed , the conditional -expectation for , where without loss of generality we suppose , is defined by where
For each fixed , we set For each , we denote by (resp., ) the completion of (resp., ) under the norm . It is easy to check that for and can be extended continuously to .
For each fixed , is a -dimensional -Brownian motion, where , and . Let , , be a sequence of partitions of such that ; the quadratic variation process of is defined by For each fixed , the mutual variation process of and is defined by
Definition 6. For fixed , let be the collection of processes in the following form: for a given partition of , where , . For , one denotes by , the completion of under the norms , , respectively.
For each , we can define the integrals and for each , . For each with , we can define Itô's integral .
Let . For and , set . Denote by the completion of under the norm .
We consider the following type of -BSDEs (in this paper, we always use Einstein convention): where satisfy the following properties.(H1)There exists some such that for any .(H2)There exists some such that
For simplicity, we denote by the collection of processes such that , , is a decreasing -martingale with and .
Definition 7. Let and and satisfy (H1) and (H2) for some . A triplet of processes is called a solution of (13) if for some the following properties hold:(a);(b).
We have the following estimates.
Theorem 11 (see ). Let , , be the solutions of the following -BSDEs: where , and satisfy (H1) and (H2) for some and are RCLL processes in such that . If is an increasing process, then for .
In this paper, we also need the following assumptions for -BSDE (13).(H3)For each fixed , and are continuous.(H4)For each fixed , , , and (H5)For each , .
Assume that ; and satisfy (H1), (H2), and (H5) for some . Let be the solution of -BSDE (13) corresponding to , , and on . It is easy to check that on for . Following , we can define consistent nonlinear expectation and set .
3. Representation Theorem of Generators of -BSDEs
We consider the following type of -FBSDEs: where and , .
We now give the main result in this section.
Theorem 12. Let , , and be Lipschitz functions and let and satisfy (H1), (H2), (H3), and (H4) for some . Then, for each and , one has
Proof. For each fixed , we write instead of for simplicity. We have for each (see [16, 19]). Thus, by Theorem 8, -BSDE (22) has a unique solution and . We set, for , Applying Itô’s formula to on , it is easy to verify that solves the following -BSDE: From Proposition 9, hold for some constant , only depending on , , , and . By Proposition 10 and the Lipschitz assumption, we obtain where is a constant depending on , , , , , , , and . Noting that (see [16, 19]), where depends on and , and the following inequality holds: Together with assumption (H4), we get where depends on , , , , , , , and . Now, we prove (23). Let us consider where It is easy to check that , where depends on , , and . Thus, by (29), we get which implies . We set By the Lipschitz condition, we can get , where depends on , , , and . Noting that (see [16, 19]), where depends on , , and , we obtain which implies . Now, we set It is easy to deduce that , where depends on . Then, Take limit on both sides of the above inequality and use assumption (H4); then, we have On the other hand, Then, we have The proof is complete.
4. Some Applications
4.1. Converse Comparison Theorem for -BSDEs
We consider the following -BSDEs: where .
We first generalized the comparison theorem in .
Proposition 13. Let and satisfy (H1) and (H2) for some , . If , then, for each , one has for .
Proof. From the above -BSDEs, we have where By the assumption, it is easy to check that is a decreasing process. Thus, using Theorem 11, we obtain for .
Remark 14. Suppose and let , , , and . It is easy to check that . Thus, does not imply and .
Now, we give the converse comparison theorem.
Theorem 15. Let and satisfy (H1), (H2), (H3), (H4), and (H5) for some , . If for each and , then q.s..
Proof. For simplicity, we take the notation , . For each fixed , let us consider where . By Theorem 12, we have, for each , Since , Take a such that . Therefore, q.s. By the assumptions (H2) and (H3), it is easy to deduce that q.s.
In the following, we use the notation , .
Corollary 16. Let and be deterministic functions and satisfy (H1), (H2), (H3), and (H5) for some , . If for each , then .
4.2. Some Equivalent Relations
We consider the following -BSDE: where . We use the notation .
Proposition 17. Let and satisfy (H1), (H2), (H3), (H4), and (H5) for some and fix . Then, one has(1) for , , and if and only if for each , , , ,(2) for , , and if and only if for each , , , , ,(3) for , , , and if and only if for each , , , , , ,(4) for , , and if and only if for each , , , ,
Proof. (1) “” part. For each fixed , , , , we take
where . Then, by Theorem 12 and , we can obtain
We choose such that , which implies (47).
“” part. Let be the solution of -BSDE (46) corresponding to terminal condition . We claim that is the solution of -BSDE (46) corresponding to terminal condition on . For this, we only need to check that, for , By (47) we can get which implies (53). The proof of (1) is complete.
(2) “” part. For each fixed , , , , , we consider and , where and . Then, by Theorem 12 and , we obtain We choose , such that and , which implies (48).
“” part. Let and be the solutions of -BSDE (46) corresponding to terminal condition and , respectively. Then, solves the following -BSDE: where