This paper investigates the monotonic limit properties for the minimal and maximal solutions of certain one-dimensional backward stochastic differential equations with continuous coefficients.

1. Introduction

Let be a probability space carrying a standard -dimensional Brownian motion . Fix a terminal time , let be the natural -algebra generated by , and assume . For every positive integer , we use to denote norm of Euclidean space . For , let denote the set of all -measurable random variables such that . Let denote the set of -progressively measurable -valued processes such that

This paper is concerned with the following one-dimensional BSDE:

where the random function is progressively measurable for each in , termed the generator of the BSDE(), and is an -measurable random variables termed the terminal condition. The triple is called the parameters of the BSDE(). In this paper, for each , by solution to the BSDE() we mean a pair of processes in which satisfies the BSDE() and is a continuous process. Such equations, in nonlinear case, have been introduced by Pardoux and Peng [1]; they established an existence and uniqueness result of solutions of the BSDE() under the Lipschitz assumption of the generator . Since then, these equations and their generalizations have been the subject of a great number of investigations, such as [2, 3]. Particularly, Lepeltier and San Martin [4] obtained the following result when the generator is only continuous with a linear growth.

Proposition 1.1 (see [4,Theorem ]). Assume that the generator satisfies(H1) linear growth: there existsfor all(H2) for fixed , is continuous.
Then, if , the BSDE(1) has a unique minimal solution and a unique maximal solution , which means that both and are the solution of (1.2), and for any other solution of (1.2) one has For convenience, for each , one denoted by , and by .

This paper will work on the assumptions (H1) and (H2) and investigate the monotonic limit properties on the operators and .

2. Main Results

In this section, we always assume that the generator satisfies assumptions (H1) and (H2). The following Theorem 2.1 and Remark 2.2 are the main results of this paper.

Theorem 2.1. Assume that the generator satisfies assumptions (H1) and (H2). Let , , , and .
If , then for all ,
If , then for all ,

Remark 2.2. If the condition “” in Theorem 2.1 is replaced by “”, the conclusion of the first part of Theorem 2.1 does not hold in general. Similarly, the condition “” in Theorem 2.1 cannot be replaced by “” in general. For example, we consider the BSDE with It is easy to see that both are solutions of BSDE For each , set , then , and However, one can verify that for each , Consequently, So, the conclusion of the first part of Theorem 2.1 does not hold.

In order to prove Theorem 2.1, we need the following lemmas. Lemma 2.3 is actually a direct corollary of Theorem in [5].

Lemma 2.3. Assume that the generator satisfies assumptions (H1) and (H2). Let and . If , then

From the procedure of the proof of Theorem 2.1 in [4], we can obtain the following Lemma 2.4.

Lemma 2.4. If the function satisfies (H1) and (H2), and one sets then for any , and are Lipschitz functions with constant , that is, for any and ,

Moreover, let and , and let and be the unique solutions of the BSDEs with parameters and , respectively. For convenience, from now on, we denoted by , and by for each , then for each , we have

Finally, the following Lemma 2.5 can be easily obtained by [6,Lemma ].

Lemma 2.5. Let , , , and , and let the generators and be defined as that in Lemma 2.4.
If , then for all and each ,
If , then for all and each ,

Now, we are in the position to prove Theorem 2.1.

Proof of Theorem 2.1. We only prove the first part of this theorem, in the same way, one can complete the proof of the second part.
Since and , one knows that . Thus, by , we have , then by Proposition 1.1, for each , both and are well defined. Moreover, according to Lemma 2.3, is nondecreasing with respect to and bounded by from above, so in the sense of “almost surely,” the limit of the sequence must exist. Thus, in order to complete the proof of Theorem 2.1, we need only to prove that this limit is just .
Let functions and be defined for each as that in Lemma 2.4, then from Lemmas 2.3 and 2.4 one deduce that for each , and , Letting in (2.14), from Lemma 2.5 we get that for each , Furthermore, letting in (2.15), from Lemma 2.4 we can easily deduce that for each , The proof of Theorem 2.1 is completed.

According to Theorem 2.1, we can obtain the following theorem.

Theorem 2.6. Assume that the generator satisfies assumptions (H1) and (H2). Let and , , Let and .
If with , then for all ,
If with , then for all ,

Proof. We only prove the first part of this theorem, the proof of the second part is similar.
Let us fix . Since , then , and by the assumption of this theorem one knows that thus by Proposition 1.1, is well defined.
We set , then So for each , Thus, then is also well defined. Since , by Lemma 2.3 we know that , then applying Theorem 2.1 to the random variable sequence , we get The proof is completed.


This work is supported by the National Natural Science Foundation of China (no. 10971220) and Youth Foundation of China University of Mining and Technology (no. 2007A029).