We study a new type of reflected backward stochastic differential
equations (RBSDEs), where the reflecting process enters the drift in a nonlinear manner. This
type of the reflected BSDEs is based on a variance of the Skorohod problem studied recently by
Bank and El Karoui (2004), and is hence named the “Variant Reflected BSDEs” (VRBSDE) in this paper. The special nature of the Variant Skorohod problem leads to a hidden forward-backward feature of the BSDE, and as a consequence this type of BSDE cannot be treated in a usual way. We shall prove that in a small-time duration most of the well-posedness, comparison, and stability results are still valid, although some extra conditions on the boundary process are needed. We will also provide some possible applications where the VRBSDE can be potentially useful. These applications show that the VRBSDE could become a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable.
1. Introduction
In this paper we study a new type of reflected backward stochastic differential equations based on the notion of variant Skorohod problem introduced recently by Bank and El Karoui [1], as an application of a stochastic representation theorem for an optional process. Roughly speaking, the Variant Skorohod Problem states the following.
For a given optional process of class (D), null at , find an -adapted, right-continuous, and increasing process with , such that
(i), , P-a.s.;(ii)The condition (ii) above is called the flat-off condition. If we assume further that is generated by a Brownian motion , then it is easily seen that the problem is equivalent to:
Finding a pair of processes , where A is increasing and Z is square integrable, such that and that the flat-off condition (ii) holds.
We note that the stochastic representation theorem proposed in [1] has already found interesting applications in various areas, such as nonlinear potential theory [2], optimal stopping, and stochastic finance (see, e.g., [3, 4]). However, to date the extension of the Variant Skorohod Problem to the form of an SDE is essentially open, partly due to the highly technical nature already exhibited in its most primitive form.
In this paper we are interested in the following extension of the Variant Skorohod Problem: Let be an optional process of class (D), and let be a random field satisfying appropriate measurability assumptions. Consider the following backward stochastic differential equation (BSDE for short): for ,
where the solution is defined to be such that
(i), ; ;(ii) is an adapted, increasing process such that , and the flat-off condition holds:
Again, if the filtration is generated by a Brownian motion , then we can consider an even more general form of BSDE as extension of (1.1):
where is an increasing process satisfying the flat-off condition, and is a pair of adapted process satisfying some integrable conditions. Hereafter we will call BSDE (1.2) and (1.4) the Variant Reflected Backward Stochastic Differential Equations (VRBSDEs for short), for the obvious reasons. We remark that although the “flat-off’’ condition (iii) looks very similar to the one in the classic Skorohod problem, there is a fundamental difference. That is, the process cannot be used as a measure to directly “push’’ the process downwards as a reflecting process usually does, but instead it has to act through the drift , in a sense as a “density’’ of a reflecting force. Therefore the problem is beyond all the existing frameworks of the reflected SDEs.
Our first task in this paper is to study the well-posedness of the VRBSDE. It is worth noting that the fundamental building block of the nonlinear Skorohod problem is a representation theorem, which in essence is to find an optional process so that the given optional obstacle process can be written as
for all stopping time taking values in . In fact, the “reflecting’’ process is exactly the running maximum of the process . Consequently, while (1.2) and (1.4) are apparently in the forms of BSDEs, they have a strong nature of a forward-backward SDEs. This brings in some very subtle difficulties, which will be reflected in our results. We would like to mention that the main difficulty here is to find a control for the reflecting process . In fact, unlike the classic Skorohod problem, the characterization of reflecting process is far more complicated, and there is no simple way to link it with the solution process . We will prove, nevertheless, that the SDE is well-posed over a small-time duration, and a certain continuous dependance and comparison theorems are still valid.
The second goal of this paper is to present some possible applications where the VRBSDE could play a role that no existing methods are amenable. In fact, the form of the VRBSDE (1.2) suggests that the process can be viewed as a stochastic recursive intertemporal utility (see, e.g., [5]). We will show that if we consider the utility optimization problem with Hindy-Kreps-Huang type preference (see, e.g., [1, 6, 7]), and the goal is minimizing such a utility while trying to keep it aloft, then the optimal solution will be given by solving a VRBSDE with the given lower boundary. To our best knowledge, such a result is novel. Another possible application of the VRBSDE that will be explored in the paper is a class of optimal stopping problems. We show that the solution to our VRBSDE can be used to describe the value function of a family of optimal stopping problems, and the corresponding reflecting process can be used as a universal signal of exercise time, which extends a result of Bank-Föllmer [3] to an SDE setting.
The rest of the paper is organized as follows. In Section 2 we revisit the stochastic representation theorem, and give the detailed formulation of the VRBSDE. In Section 3 we study the well-posedness of the equation. In Sections 4 and 5 we study the comparison theorem and the continuous dependence results. Finally we present some possible applications of VRBSDEs in the utility minimization problems and a class of optimal stopping problems in Section 6.
2. Formulation of the Variant RBSDE
Throughout this paper we assume that is a filtered probability space, where is a filtration that satisfies the usual hypothses. For simplicity we assume that . In the case when the filtration is generated by a standard Brownian motion on the space , we say that is “Brownian’’ and denote it by . We will always assume that is augmented by all the -null sets in .
We will frequently make use the following notations. Let
(i) be the space of all measurable bounded random variables,(ii) the space of all -valued, progressively measurable, bounded processes, (iii) the space of all -valued, progressively measurable process , such that ,(iv) the set of all the stopping times taking values in Similar to the Variant Skorohod Problem, a VRBSDE involves two basic elements: a boundary process which is assumed to be an optional process of class (D) (A process is said to belong to Class (D) on if the family of random variables is uniformly integrable), and is lower-semicontinuous in expectation; and a drift coefficient . In this paper we will focus only on the case where is independent of , and we assume that it satisfies the following Standing Assumptions:
(H1) the coefficient enjoys the following properties:
(i)for fixed , , and , the function is continuous and strictly decreasing from to (ii)for fixed , the process is progressively measurable with
(iii)there exists a constant , such that for all fixed it holds that
(iv) there exist two constants and , such that for all fixed it holds that
We remark that the assumption (iv) in (H1) amounts to saying that the derivative of with respect to , if exists, should be bounded from below. While this is merely technical, it also indicates that we require a certain sensitivity of the solution process with respect to the reflection process . This is largely due to the nonlinearity between the solution and the reflecting process, which was not an issue in the classical Skorohod problem.
We now introduce our variant reflected BSDE. Note that we do not assume that the filtration is Brownian at this point.
Definition 2.1. Let and the boundary process be given. A pair of processes is called a solution of Variant Reflected BSDE with terminal value and boundary if (i) and are -adapted processes with càdlàg paths;(ii);(ii), ; ;(iv) the process is -adapted, increasing, càdlàg, and , such that the “flat-off’’ condition holds:
Remark 2.2. The assumption has an important implication: the solution must satisfy . This can be deduced from the flat of condition (2.4), and the fact that always holds. Such a fact was implicitly, but frequently, used in [1], and will be crucial in some of our arguments below.
We note that if we denote , then is a martingale on , and the VRBSDE will read
Thus if we assume further that the filtration is Brownian, than we can consider the more general form of VRBSDE.
Definition 2.3. Assume that the filtration , that is, it is generated by a standard Brownian motion , with the usual augmentation. Let and the boundary process be given. A triplet of processes is called a solution of Variant Reflected BSDE with terminal value and boundary if(i), (ii), (iii), ; (iv) the process is -adapted, increasing, càdlàg, and , such that the flat-off condition holds:
Our study of VRBSDE is based on a Stochastic Representation Theorem of Bank and El Karoui [1]. We summarize the stochastic representation and some related fact in the following theorem, which is slightly modified to suit our situation.
Theorem 2.4 (see, Bank-El Karoui [1]). Assume (H1)-(i), (ii). Then every optional process of class (D) which is lower semicontinuous in expectation admits a representation of the form
for any stopping time , where is an optional process taking values in , and it can be characterized as follows: (i) for any stopping time ,(ii), where the “’’ is taken over all stopping times such that , a.s.; and is the unique -measurable random variable satisfying:
(iii) (Gittin Index) if , , is the value functions of a family of optimal stopping problems indexed by , then
We should note here, unlike the original stochastic representation theorem in [1] where it assumed that , we allow arbitrary terminal value for . This can be obtained easily by considering a new process , . A direct consequence of the stochastic representation theorem is the following Variant Skorohod Problem, which is again slightly adjusted to our non-zero terminal value case.
Theorem 2.5. Assume (H1)-(i), (ii). Then for every optional process of class (D) which is lower semicontinuous in expectation, there exists a unique pair of adapted processes , where is continuous and is increasing, such that
Furthermore, the process can be expressed as , where is the process in Theorem 2.4.
We conclude this section by making following observations. First, the random variable , defined by (2.7) is -measrable for any stopping time , thus the process is -adapted. However, the running maximum process depends on the whole path of process , whence . Thus, although the variant Skorohod problem (2.9) looks quite similar to a standard backward stochastic differential equation, it contains a strong “ forward-backward’’ nature. These facts will be important in our future discussions.
3. Existence and Uniqueness
In this section we study the well-posedness of the VRBSDE (2.4). We note that in this case we do not make any restriction on the filtration, as long as it satisfies the usual hypotheses.
We will follow the usual technique, namely the contraction mapping theorem, to attack the existence and uniqueness of the solution. It is worth noting that due to the strong forward-backward structure as well as the fundamental non-Markovian nature of the problem, a general result with arbitrary duration is not clear at this point. The results presented in this section will provide the first look at some basic features of such an equation.
We will make use of the following extra assumptions on the boundary process and the drift coefficient :
(H2) there exists a constant , such that
(i) for any , it holds that
(ii). Remark 3.1. The assumption (3.1) is merely technical. It is motivated by the “Gittin indices’’ studied in [8], and it essentially requires a certain “path regularity’’ on the boundary process . However, one should note that it by no means implies the continuity of the paths of (!). In fact, a semimartingale with absolutely continuous bounded variation part can easily satisfy (3.1), but this does not prevent jumps from the martingale part.
We begin by considering the following mapping on : for a given process we define , , where is the unique solution of the Variant Skorohod problem:
We are to prove that the mapping is a contraction from to itself. It is not hard to see, by virtue of Theorems 2.4 and 2.5, that the reflecting process is determined by in the following way: , and is the solution to the Stochastic Representation:
We should note, however, that the contraction mapping argument does not completely solve the existence and uniqueness issue for the Variant BSDE. In fact, it only gives the existence of the fixed point , and we will have to argue the uniqueness of the process separately.
We now establish some a priori estimates that will be useful in our discussion. To begin with, let us consider the stochastic representation
Denote . We have the following estimate for .
Lemma 3.2. Assume (H1) and (H2). Then it holds that , where and are the constants appearing in (H1) and (H2).
Proof. For fixed and any stopping time , let be the measurable random variable such that
Then by Theorem 2.4 we have , and .
Now consider the set . Since is decreasing, we have
In other words we have
Similarly, one can show that on the set it holds that
Consequently, we have
Now note that
we derive from (3.9) and (H2) that
proving the lemma.
Clearly, a main task in proving that is a contraction mapping is to find the control on the difference of two reflecting processes. To see this let be given, and consider the two solutions of the variant Skorohod problem: and . We would like to control in terms of . The following lemma is crucial.
Lemma 3.3. Assume (H1) and (H2). Then, for any , it holds almost surely that
Proof. Again, we fix and let be such that , a.s. Recalling Theorem 2.4, we let and be two -measurable random variables such that
Define , then , for any stopping time .
Now, from (3.13) and noting that is -measurable, we deduce that
Now, by (H1)-(iv), the left-hand side of (3.14) satisfies
On the other hand, by (H1)-(iii) we see that the right-hand side of (3.14) satisfies
Combining above we obtain that
Thus , on , since , a.s.
Similarly, one shows that the inequality holds on the complement of as well. It follows that
Next, recall from Theorem 2.4 that , , , and . We conclude from (3.18) that, for any ,
The proof is now complete.
Remark 3.4. We observe that the step from (3.16) to (3.17) is seemingly rough. It would be more desirable if some more delicate estimates, such as
could hold for some constant , so that one can at least remove the boundedness requirement on the solution. But unfortunately (3.20) is not true in general, unless some conditional independence is assumed. Here is a quick example: let and let be a binomial random variable that takes value with probability and with probability . Define two processes: , ; , ; and define with . Then is an -stopping time and is an -adapted continuous process.
It is easy to check that and . Thus if we choose , , and a constant such that
then (3.20) will fail at , with .
We are now ready to prove the main result of this section, the existence and uniqueness of the solution to the Variant RBSDE.
Theorem 3.5. Assume (H1) and (H2). Assume further that , then the Variant reflected BSDE (1.2) admits a unique solution .
Proof. We first show that the mapping defined by (3.2) is from to itself. To see this, we note that by using assumption (H1) and Lemmas 3.2 and 3.3, one has
Since by assumption, we can then easily deduce that .
To prove that is a contraction, we take , and denote and . Then, for any , applying Lemma 3.3 we have
Since by assumption, we see that is a contraction.
Now, let be the (unique) fixed point of , and let be the corresponding reflecting process defined by , where satisfies the representation
We now show that is the solution to the Variant RBSDE (1.2). To see this, note that (3.24), the definition of , and the monotonicity of the function (on the variable ) tell us that, for ,
Thus it remains to show that the flat-off condition holds. But by the properties of optional projections and definition of and , we have
here the last equality follows from the Fubini theorem and the fact that the Lebesgues measure does not charge the discontinuities of the paths , which are only countably many.
Finally, note that on the set , must be a point of increase of . Since is the running supreme of we conclude that , for all . This yields that
Thus the right side of (3.26) is identically zero, and the flat-off condition holds. This proves the existence of the solution .
The uniqueness of the solution can be argued as follows. Suppose that there is another solution to the VRBSDE such that , , , and