#### Abstract

We study Jensen's inequality for generalized Peng's -expectations and give four equivalent conditions on Jensen's inequality for generalized Peng's -expectations without the assumption that the generator is continuous with respect to . This result includes and extends some existing results. Furthermore, we give some applications of Jensen's inequality for generalized Peng's -expectations.

#### 1. Introduction

By Pardoux and Peng [1], we know that there exists a unique adapted and square integrable solution to a backward stochastic differential equation (BSDE for short) of the type provided that the function is Lipschitz in both variables and , and and are square integrable. is said to be the generator of BSDE (1). We denote the unique adapted and square integrable solution of BSDE (1) by .

Based on such a BSDE, Peng [2] introduced the notion of -expectation. He proved that the -expectation preserves many of properties of the classical mathematical expectation, but not the linearity property, and thus the -expectation is a type of nonlinear mathematical expectation. Indeed, -expectation is a kind of nonlinear expectation, which can be considered as a nonlinear extension of the well-known Girsanov transformations. The original motivation for studying -expectation comes from the theory of expected utility. Since the notion of -expectation was introduced, many properties of -expectation have been investigated by many researchers. In 1997, Peng [3] introduced the notions of conditional -expectation and -martingale. Later, Briand et al. [4] studied Jensen's inequality for -expectations and gave a counter example and a proposition to indicate that even for a linear function, Jensen's inequality might fail for some -expectations. This yields a natural question: under which conditions on in the -expectation does Jensen's inequality hold for any convex function? Under the assumptions that does not depend on and is convex, Chen et al. [5, 6] studied Jensen's inequality for -expectations and gave a necessary and sufficient condition on under which Jensen's inequality holds for convex functions. Provided that only does not depend on , Jiang [7] gave another necessary and sufficient condition on under which Jensen's inequality holds for convex functions. It was an improved result in comparison with the result that Chen et al. yielded. Later, this result was improved by Hu [8] and Jiang [9] showing that, in fact, must be independent of . But these results need the assumption that the generator is continuous with respect to .

In this paper, without the assumption that the generator is continuous with respect to , we study Jensen's inequality for generalized Peng's -expectations and give four equivalent conditions on Jensen's inequality for generalized Peng's -expectations, which generalize the known results on Jensen's inequality for -expectations in Chen et al. [5, 6], Jiang [7, 9], and Hu [8]. Furthermore, we give some applications of Jensen's inequality for generalized Peng's -expectations.

This paper is organized as follows: in Section 2, we introduce some notations, assumptions, notions, and lemmas which will be useful in this paper; in Section 3, we give our main results including the proofs and applications.

#### 2. Preliminaries

Firstly, let us list some notations, assumptions, notions, lemmas, and propositions that are used in this paper. Let be a probability space and let be a -dimensional standard Brownian motion with respect to filtration generated by Brownian motion and all -null subsets, that is, where is the set of all -null subsets. Fix a real number . For any positive integer and , denotes its Euclidean norm.

We define the following usual spaces of processes (random variables):(i)Consider is -measurable random variable such that ;(ii)Consider ;(iii)Consider is a continuous process with ;(iv)Consider ;(v)Consider is  a  progressively measurable process with ;(vi)Consider .

Suppose the generator satisfies the following assumptions: there exists a constant , such that -a.s., we have: , , , ;-a.s., , .

The following lemma is a special case of Theorem 4.2 in Briand et al. [10].

Lemma 1. Suppose satisfies and . Then for each given , where , the BSDE (1) has a unique pair of adapted processes .

From Lemma 1, we have the following.

Remark 2. Suppose satisfies and . Then for each given , the BSDE (1) has a unique pair of adapted processes .

Now, we introduce the notions of generalized Peng's -expectation and generalized conditional Peng's -expectation.

Definition 3 (generalized Peng's -expectation [11]). Suppose satisfies and . For any , let be the solution of BSDE (1). Consider the mapping , denoted by . One calls the generalized Peng's -expectation of .

Definition 4 (generalized Peng's conditional -expectation [11]). Suppose satisfies and . The generalized Peng's conditional -expectation of with respect to is defined by

Then, let us list some basic properties of generalized Peng's -expectation.

Proposition 5 (see [11]). Consider is the unique random variable in such that

Proposition 6 (see [11]). Suppose satisfies and . If does not depend on , that is, , then

Proposition 7 (see [11]). Suppose satisfies and . For , where and , if , then Applying Proposition 7, one can immediately obtain the following.

Remark 8. (i) For any , let , , then , a.s., .
(ii) For any , if a.s. and a.s. with , then , a.s., .

Lemma 9. Suppose satisfies and . Let be a -measurable partition of (i.e., , if and ), where . Then for each , , one has

Proof. We consider the following BSDEs: By the fact that = and from (8), we have Comparing this with (9), it follows that a.s. The proof of Lemma 9 is complete.

Proposition 10. Suppose satisfies and . Then the following two statements are equivalent:(i)consider , a.s.,(ii)consider , a.s.

Proof. It is obvious that (ii) implies (i). We only need to prove that (i) implies (ii). Suppose (i) holds. Let be a -measurable partition of and let (). From Lemma 9 and (i), we deduce that for each , In other words, for any and any simple function , Let Obviously, for each , is a simple function in . From (12), we have On the other hand, , . Thus, from Remark 8 (ii), it follows that (ii) is true. The proof of Proposition 10 is complete.

#### 3. Main Results and Applications

Definition 11. Let : . The function is said to be superhomogeneous if for each and any real number , then , a.s. The function is said to be positively homogeneous if for each and any real number , then , a.s.

Before we give our main results, let us see an example.

Example 12. Fix and . Let , where , .
Obviously, is an increasing function. We can easily get Hence, , but .
Let , . Clearly, for each , . For simplicity, we will write for . From Theorem 1 in Chen and Kulperger’s [12], we know that , where .
By Remark 8(i), we have , as . On the other hand, applying Hölder's inequality and noting that and , we obtain where . It then follows from the monotonic convergence theorem that Thus
Let , where . Obviously, is a convex and increasing function. From this, we know that is an increasing function. In a similar manner of the above, we can deduce that
From (18), (19), and the classical Jensen's inequality, we have
This problem yields a natural question: in general, under which conditions on do generalized Peng's -expectations satisfy Jensen's inequality for convex functions?

The following theorem will answer this question.

Theorem 13. Let satisfy and . Then the following four statements are equivalent.(i)Jensen's inequality for generalized Peng's -expectation holds in general, that is, for each convex function and each , if , then one has (ii)consider , ;(iii)consider , a.s.;(iv)consider is independent of , superhomogeneous, and positively homogeneous with respect to .

Proof. (i)(ii) is obvious.
(ii)(iii): let . By (ii), we have That is, Thus, for each , For each , by (24), we know that for each , Thus, On the other hand, for each , define It is easy to check that and are two -expectations on (the notion of -expectation can be seen in [13]). From (ii), we have if , for each Hence, by Lemma 4.5 in [13], we have Similarly, if , for each Hence, by Lemma 4.5 in [13] again, we have Thus from (29) and (31), we have , From (26) and (32), we have
(iii)(iv): Firstly, we prove that is independent of . From (iii), we can obtain that for each , For each , let be the solution of the following SDE defined on : From (34), we have Let , and be the corresponding part of Itô's integrand. It then follows that Thus, and Then, we can apply Lemma 4.4 in Peng [14] to obtain that for each , Namely, is independent of .
Now we prove that is superhomogeneous with respect to . From (iii), we can obtain that for each , For each , let be the solution of the following SDE defined on : From (40), we have Thus, is an -submartingale. From the decomposition theorem of -supermartingale (see [15]), it follows that there exists an increasing process such that This with yields and
At last, we prove that is positively homogeneous with respect to . From (iii), we can obtain that for each fixed and , that is, Thus, we have Obviously, if , (47) still holds. Thus, for each , For each , let be the solution of SDE (34). From (48), for each , we have This implies that there exists a process such that Comparing this with , it follows that and
(iv)(iii): By comparison theorem (for example, we can see [3]), it is easy to obtain (iii).
(iii)(i): Suppose (iii) holds. From (iii) and by Remark 8 (i), we have From (53), we can deduce that for each bounded variable , In fact, let be a -measurable partition of and let (). By (53), we have In other words, for each and each simple function , Thus, thanks to Remark 8(ii), it follows that (54) is true.
The main idea of the following proof is derived from [7]. Given and convex function such that , we set . Then is -measurable. Since is convex, we have Take , . Then we have For each , we define so we have By the definition of , we know Thus, in view of (52) and from Proposition 10, we can get Moreover, from (54), considering that and is bounded by , we can get Hence, we can deduce that for each , Finally, thanks to Remark 8 (ii) again, we can get Hence, Jensen's inequality for holds in general. The proof of Theorem 13 is complete.

Example 14. Suppose is a bounded, convex, and closed subset of and the set of -valued continuous processes such that for each , a.s.. Define the probability measure by Thus, for any convex function , whenever , .

Proof. Let ess . Obviously, is superhomogeneous and positively homogeneous with respect to . and satisfies and .
From El Karoui and Quenez [16], we have . Now we prove , a.s., . Indeed, for any , there exists such that . Let , . Clearly, for each , , then
Since we have With an approach similar to the one above, we can get easily that Combining (42) with (43), we have By Hölder's inequality and noting that and are both martingales with respect to , we can obtain where . It then follows from Lebesgue's dominated convergence theorem that Hence,
On the other hand, from Remark 8 (i), we have Thus,
Applying Theorem 13, we have

Definition 15. Suppose satisfies and . A process satisfying that for each , is called a generalized Peng's -martingale (resp., generalized Peng's -supermartingale, generalized Peng's -submartingale), if for any , satisfying ,

Applying Theorem 13, immediately we have the following.

Theorem 16. Suppose is independent of , superhomogeneous and positively homogeneous with respect to and satisfies and . If is a generalized Peng's -martingale and is a convex function such that , then is a generalized Peng's -submartingale.

Remark 17. Suppose is independent of , superhomogeneous and positively homogeneous with respect to and satisfies and . Similarly, we can get the following.(i)If is a generalized Peng's -submartingale and is a convex and increasing function such that , then is a generalized Peng's -submartingale.(ii)If is a generalized Peng's -supermartingale and is a convex and decreasing function such that , then is a generalized Peng's -submartingale.

Example 18. (i) Let and where . Obviously, satisfies the assumptions of Remark 17 and is a convex and increasing function. By Remark 17 (i), we have the following: suppose is a -submartingale, then is a -submartingale.
(ii) Let and where . With the similar argument, we have the following: suppose is a -supermartingale, then is a -submartingale.

#### Acknowledgments

The author would like to thank the anonymous referees for their careful reading of this paper and valuable suggestions. The author thanks the partial support from the National Natural Science Foundation of China (Grant no. 11171179), the Doctoral Program Foundation of Ministry of Education of China (Grant no. 20093705110002 and 20123705120005), and the Natural Science Foundation of Shandong Province of China (Grant no. ZR2012AQ009).