International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 854768 | 26 pages | https://doi.org/10.1155/2009/854768

On Variant Reflected Backward SDEs, with Applications

Academic Editor: Vo Anh
Received11 Dec 2008
Accepted11 Mar 2009
Published18 Jun 2009

Abstract

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  𝑋of class (D), null at  𝑇, find an  𝔽-adapted, right-continuous, and increasing process  𝐴={𝐴𝑡}𝑡0  with  𝐴0=, such that

(i)𝑌𝑡Δ=𝐸{𝑇𝑡𝑓(𝑠,𝐴𝑠)𝑑𝑠|𝑡}𝑋𝑡, 𝑡[0,𝑇], P-a.s.;(ii)𝐸𝑇0|𝑌𝑡𝑋𝑡|𝑑𝐴𝑡=0.

The condition (ii) above is called the flat-off condition. If we assume further that 𝔽 is generated by a Brownian motion 𝐵, then it is easily seen that the problem is equivalent to:

Finding a pair of processes  (𝐴,𝑍),   where  A  is increasing and Z is square integrable, such that 𝑌𝑡=𝑇𝑡𝑓𝑠,𝐴𝑠𝑑𝑠𝑇𝑡𝑍𝑠𝑑𝐵𝑠𝑋𝑡,0𝑡𝑇,(1.1)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 𝑋={𝑋𝑡}𝑡0 be an optional process of class (D), and let 𝑓Ω×[0,𝑇]××× be a random field satisfying appropriate measurability assumptions. Consider the following backward stochastic differential equation (BSDE for short): for 𝑡[0,𝑇], 𝑌𝑡𝑋=𝐸𝑇+𝑇𝑡𝑓𝑠,𝑌𝑠,𝐴𝑠𝑑𝑠𝑡,(1.2) where the solution (𝑌,𝐴) is defined to be such that

(i)𝑌𝑡𝑋𝑡, 0𝑡𝑇; 𝑌𝑇=𝑋𝑇;(ii)𝐴={𝐴𝑡} is an adapted, increasing process such that 𝐴Δ0=, and the flat-off condition holds: 𝐸𝑇0||𝑌𝑡𝑋𝑡||𝑑𝐴𝑡=0.(1.3)

Again, if the filtration 𝔽 is generated by a Brownian motion 𝐵, then we can consider an even more general form of BSDE as extension of (1.1): 𝑑𝑌𝑡=𝑓𝑡,𝑌𝑡,𝑍𝑡,𝐴𝑡𝑑𝑡+𝑍𝑡𝑑𝑊𝑡,𝑌𝑡𝑋𝑡[],𝑡0,𝑇,𝑌𝑇=𝑋𝑇,(1.4) where 𝐴 is an increasing process satisfying the flat-off condition, and (𝑌,𝑍) 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 𝐴 cannot be used as a measure to directly “push’’ the process 𝑌 downwards as a reflecting process usually does, but instead it has to act through the drift 𝑓, 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 𝐿 so that the given optional obstacle process 𝑋 can be written as 𝑋𝑆=𝐸𝑇𝑆𝑓𝑢,sup𝑆𝑣𝑢𝐿𝑣𝑑𝑢𝑆,(1.5) for all stopping time 𝑆 taking values in [0,𝑇]. In fact, the “reflecting’’ process 𝐴 is exactly the running maximum of the process 𝐿. 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 𝐴. In fact, unlike the classic Skorohod problem, the characterization of reflecting process 𝐴 is far more complicated, and there is no simple way to link it with the solution process 𝑌. 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 𝑌 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 (Ω,,𝑃;𝔽) is a filtered probability space, where 𝔽Δ={𝑡}0𝑡𝑇 is a filtration that satisfies the usual hypothses. For simplicity we assume that =𝑇. In the case when the filtration 𝔽 is generated by a standard Brownian motion 𝐵 on the space (Ω,,𝑃), we say that 𝔽 is “Brownian’’ and denote it by 𝔽=𝔽𝐵. We will always assume that 𝔽𝐵 is augmented by all the 𝑃-null sets in .

We will frequently make use the following notations. Let

(i)𝕃𝑇 be the space of all 𝑇 measurable bounded random variables,(ii)𝑇 the space of all -valued, progressively measurable, bounded processes, (iii)2𝑇 the space of all 𝑑-valued, progressively measurable process 𝑍, such that 𝐸𝑇0|𝑍2𝑠|𝑑𝑠<,(iv)0,𝑇 the set of all the stopping times taking values in [0,𝑇].

Similar to the Variant Skorohod Problem, a VRBSDE involves two basic elements: (1) a boundary process 𝑋={𝑋𝑡,0𝑡𝑇} which is assumed to be an optional process of class (D) (A process 𝑋 is said to belong to Class (D) on [0,𝑇] if the family of random variables {𝑋𝜏𝜏0,𝑇} is uniformly integrable), and is lower-semicontinuous in expectation; and (2) a drift coefficient 𝑓. In this paper we will focus only on the case where 𝑓 is independent of 𝑧, and we assume that it satisfies the following Standing Assumptions:

(H1) the coefficient 𝑓[0,𝑇]×Ω××× enjoys the following properties:

(i)for fixed 𝜔Ω, 𝑡[0,𝑇], and 𝑦, the function 𝑓(𝜔,𝑡,𝑦,) is continuous and strictly decreasing from + to ,(ii)for fixed 𝑦,𝑙3, the process 𝑓(,,𝑦,𝑙) is progressively measurable with 𝐸𝑇0||||𝑓(𝑡,𝑦,𝑙)𝑑𝑡+,(2.1)(iii)there exists a constant 𝐿>0, such that for all fixed 𝑡,𝜔,𝑙 it holds that ||𝑓𝑡,𝜔,𝑦||||𝑦,𝑙𝑓(𝑡,𝜔,𝑦,𝑙)𝐿||𝑦,𝑦,𝑦,(2.2)(iv) there exist two constants 𝑘>0 and 𝐾>0, such that for all fixed 𝑡,𝜔,𝑦 it holds that 𝑘||𝑙||||𝑓𝑙𝑡,𝑦,𝑙||||𝑙𝑓(𝑡,𝑦,𝑙)𝐾||𝑙,𝑙,𝑙.(2.3)

We remark that the assumption (iv) in (H1) amounts to saying that the derivative of 𝑓 with respect to 𝑙, 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 𝑌 with respect to the reflection process 𝐴. 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 𝔽 is Brownian at this point.

Definition 2.1. Let 𝜉𝕃𝑇 and the boundary process 𝑋 be given. A pair of processes (𝑌,𝐴) is called a solution of Variant Reflected BSDE with terminal value 𝜉 and boundary 𝑋 if (i)𝑌 and 𝐴 are 𝔽-adapted processes with càdlàg paths;(ii)𝑌𝑡=𝐸{𝜉+𝑇𝑡𝑓(𝑠,𝑌𝑠,𝐴𝑠)𝑑𝑡𝑡};(ii)𝑌𝑡𝑋𝑡, 0𝑡𝑇; 𝑌𝑇=𝑋𝑇=𝜉;(iv) the process 𝐴 is 𝔽-adapted, increasing, càdlàg, and 𝐴Δ0=, such that the “flat-off’’ condition holds: 𝐸𝑇0||𝑌𝑡𝑋𝑡||𝑑𝐴𝑡=0.(2.4)

Remark 2.2. The assumption 𝐴0= has an important implication: the solution 𝑌 must satisfy 𝑌0=𝑋0. This can be deduced from the flat of condition (2.4), and the fact that 𝑑𝐴0>0 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 𝑀𝑡=𝐸{𝑇0𝑓(𝑡,𝑌𝑡,𝐴𝑡)𝑑𝑡𝑡}, 𝑡[0,𝑇] then 𝑀 is a martingale on [0,𝑇], and the VRBSDE will read 𝑌𝑡=𝜉+𝑇𝑡𝑓𝑠,𝑌𝑠,𝐴𝑠𝑀𝑑𝑡𝑇𝑀𝑡,0𝑡𝑇.(2.5) 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 𝔽=𝔽𝐵, that is, it is generated by a standard Brownian motion 𝐵, with the usual augmentation. Let 𝜉𝕃𝑇 and the boundary process 𝑋 be given. A triplet of processes {(𝑌𝑡,𝑍𝑡,𝐴𝑡),0𝑡𝑇} is called a solution of Variant Reflected BSDE with terminal value 𝜉 and boundary 𝑋 if(i)𝑌𝑇, 𝑍2𝑇,(ii)𝑌𝑡=𝜉+𝑇𝑡𝑓(𝑠,𝑌𝑠,𝑍𝑠,𝐴𝑠)𝑑𝑠𝑇𝑡𝑍𝑠𝑑𝐵𝑠, 0𝑡𝑇,(iii)𝑌𝑡𝑋𝑡, 0𝑡𝑇; 𝑌𝑇=𝑋𝑇=𝜉,(iv) the process {𝐴𝑡} is 𝔽-adapted, increasing, càdlàg, and 𝐴0=, such that the flat-off condition holds: 𝐸𝑇0|𝑌𝑡𝑋𝑡|𝑑𝐴𝑡=0.

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 𝑋 of class (D) which is lower semicontinuous in expectation admits a representation of the form 𝑋𝑆𝑋=𝐸𝑇+𝑇𝑆𝑓𝑢,sup𝑆𝑣𝑢𝐿𝑣𝑑𝑢𝑆(2.6) for any stopping time 𝑆0,𝑇, where 𝐿 is an optional process taking values in {}, and it can be characterized as follows: (i)𝑓(𝑢,sup𝑆𝑣𝑢𝐿𝑣)𝐿1(𝑑𝑡) for any stopping time 𝑆,(ii)𝐿𝑆=essinf𝜏>𝑆𝑙𝑆,𝜏, where the “essinf’’ is taken over all stopping times 𝑆0,𝑇 such that 𝑆<𝑇, a.s.; and 𝑙𝑆,𝜏 is the unique 𝑆-measurable random variable satisfying: 𝐸𝑋𝑆𝑋𝜏𝑆=𝐸𝜏𝑆𝑓𝑢,𝑙𝑆,𝜏𝑑𝑢𝑆,(2.7)(iii) (Gittin Index) if 𝑉(𝑡,𝑙)Δ=essinf𝜏𝑡𝐸{𝐸𝜏𝑡𝑓(𝑢,𝑙)𝑑𝑢+𝑋𝜏|𝑡}, 𝑡[0,𝑇], is the value functions of a family of optimal stopping problems indexed by 𝑙, then 𝐿𝑡=sup𝑙𝑉(𝑡,𝑙)=𝑋𝑡[],𝑡0,𝑇.(2.8)

We should note here, unlike the original stochastic representation theorem in [1] where it assumed that 𝑋𝑇=0, we allow arbitrary terminal value for 𝑋𝑇. This can be obtained easily by considering a new process 𝑋𝑡Δ=𝑋𝑡𝐸[𝜉𝑡], 𝑡0. 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 𝑋 of class (D) which is lower semicontinuous in expectation, there exists a unique pair of adapted processes (𝑌,𝐴), where 𝑌 is continuous and 𝐴 is increasing, such that 𝑌𝑡𝑋=𝐸𝑇+𝑇𝑡𝑓𝑠,𝐴𝑠𝑑𝑠𝑡[],𝑡0,𝑇.(2.9) Furthermore, the process 𝐴 can be expressed as 𝐴𝑡=sup0𝑠𝑡+𝐿𝑠, where 𝐿 is the process in Theorem 2.4.

We conclude this section by making following observations. First, the random variable 𝑙𝑆,𝜏, defined by (2.7) is 𝑆-measrable for any stopping time 𝜏>𝑆, thus the process 𝑠𝐿𝑠 is 𝔽-adapted. However, the running maximum process 𝐴𝑡Δ=sup0𝑢𝑡+𝐿𝑢 depends on the whole path of process 𝐿, whence 𝑋. 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 𝑋 and the drift coefficient 𝑓:

(H2) there exists a constant Γ>0, such that

(i) for any 𝑆0,𝑇, it holds that esssup𝜏>𝑆𝜏0,𝑇||||𝐸𝑋𝜏𝑋𝑆𝑆𝐸𝜏𝑆𝑆||||Γ,a.s.(3.1)(ii)|𝑓(𝑡,0,0)|Γ,𝑡[0,𝑇].

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 𝑋. However, one should note that it by no means implies the continuity of the paths of 𝑋(!). 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 𝒯 on 𝑇: for a given process 𝑦 we define 𝒯(𝑦)𝑡Δ=𝑌𝑡, 𝑡[0,𝑇], where 𝑌 is the unique solution of the Variant Skorohod problem: 𝑌𝑡=𝐸𝜉+𝑇𝑡𝑓𝑠,𝑦𝑠,𝐴𝑠𝑑𝑠𝑠[],𝐸,𝑡0,𝑇𝑇0𝑋𝑡𝑌𝑡𝑑𝐴𝑡[].=0,𝑡0,𝑇(3.2) We are to prove that the mapping 𝒯 is a contraction from 𝑇 to itself. It is not hard to see, by virtue of Theorems 2.4 and 2.5, that the reflecting process 𝐴 is determined by 𝑦 in the following way: 𝐴𝑡=sup0𝑣𝑡+𝐿𝑣, and 𝐿 is the solution to the Stochastic Representation: 𝑋𝑡=𝐸𝜉+𝑇𝑡𝑓𝑠,𝑦𝑠,sup𝑡𝑣𝑠𝐿𝑣𝑑𝑠𝑡[],𝑡0,𝑇.(3.3) 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 𝑌, and we will have to argue the uniqueness of the process 𝐴 separately.

We now establish some a priori estimates that will be useful in our discussion. To begin with, let us consider the stochastic representation 𝑋𝑡=𝐸𝜉+𝑇𝑡𝑓𝑠,0,sup𝑡𝑣𝑠𝐿0𝑣𝑑𝑠𝑡.(3.4) Denote 𝐴0𝑡Δ=sup0𝑠𝑡+𝐿0𝑡. We have the following estimate for 𝐴0.

Lemma 3.2. Assume (H1) and (H2). Then it holds that 𝐴02Γ/𝑘, where 𝑘 and Γ are the constants appearing in (H1) and (H2).

Proof. For fixed 𝑠[0,𝑇] and any stopping time 𝜏>𝑠, let 𝑙0𝑠,𝜏 be the 𝑠 measurable random variable such that 𝐸𝑋𝑠𝑋𝜏𝑠=𝐸𝜏𝑠𝑓𝑡,0,𝑙0𝑠,𝜏𝑑𝑡𝑠.(3.5) Then by Theorem 2.4 we have 𝐿0𝑠=essinf𝜏>𝑠𝑙0𝑠,𝜏, and 𝐴0𝑡=sup0𝑠𝑡+𝐿0𝑠.

Now consider the set {𝜔𝑙0𝑠,𝜏(𝜔)<0}. Since 𝑓(𝑡,0,) is decreasing, we have 𝐸𝑋𝑠𝑋𝜏𝑠𝐸𝜏𝑠𝑓(𝑡,0,0)𝑑𝑡𝑠=𝐸𝜏𝑠𝑓𝑡,0,𝑙0𝑠,𝜏𝑓(𝑡,0,0)𝑑𝑡𝑠𝐸𝜏𝑠𝑘||𝑙0𝑠,𝜏||𝑑𝑡𝑠||𝑙𝑘0𝑠,𝜏||𝐸𝜏𝑠𝑠.(3.6) In other words we have ||𝑙0𝑠,𝜏||1𝑘𝐸𝑋𝑠𝑋𝜏𝑠𝐸𝜏𝑠𝑠𝐸𝜏𝑠𝑓(𝑡,0,0)𝑑𝑡𝑠𝐸𝜏𝑠𝑠l,on0s,𝜏<0.(3.7)

Similarly, one can show that on the set {𝑙0𝑠,𝜏0} it holds that 𝑙0𝑠,𝜏1𝑘𝐸𝑋𝑠𝑋𝜏𝑠𝐸𝜏𝑠𝑠+𝐸𝜏𝑠𝑓(𝑡,0,0)𝑑𝑡𝑠𝐸𝜏𝑠𝑠.(3.8)

Consequently, we have ||𝑙0𝑠,𝜏||1𝑘||||𝐸𝑋𝜏𝑋𝑠𝑠𝐸𝜏𝑠𝑠||||+𝐸𝜏𝑠||||𝑓(𝑡,0,0)𝑑𝑡𝑠𝐸𝜏𝑠𝑠.(3.9)

Now note that ||𝐴0𝑡||=||||sup0𝑠𝑡+𝐿0𝑠||||sup0𝑠𝑡+||𝐿0𝑠||=sup0𝑠𝑡+essinf𝜏>𝑠||𝑙0𝑠,𝜏||,(3.10) we derive from (3.9) and (H2) that ||𝐴0𝑡||sup0𝑠𝑡+esssup𝜏>𝑠||𝑙0𝑠,𝜏||sup0𝑠𝑡+Γ+Γ𝑘=2Γ𝑘,(3.11) proving the lemma.

Clearly, a main task in proving that 𝒯 is a contraction mapping is to find the control on the difference of two reflecting processes. To see this let 𝑦,𝑦𝑇 be given, and consider the two solutions of the variant Skorohod problem: (𝑌,𝐴) and (𝑌,𝐴). We would like to control |𝐴𝑠𝐴𝑠| in terms of |𝑦s𝑦𝑠|. The following lemma is crucial.

Lemma 3.3. Assume (H1) and (H2). Then, for any 𝑡[0,𝑇], it holds almost surely that ||A𝑡𝐴𝑡||𝐿𝑘𝑦𝑦.(3.12)

Proof. Again, we fix 𝑠 and let 𝜏0,𝑇 be such that 𝜏>𝑠, a.s. Recalling Theorem 2.4, we let 𝑙𝑠,𝜏 and 𝑙𝑠,𝜏 be two 𝑠-measurable random variables such that 𝐸𝑋𝑠𝑋𝜏𝑠=𝐸𝜏𝑠𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏𝑑𝑢𝑠=𝐸𝜏𝑠𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏𝑑𝑢𝑠.(3.13) Define 𝐷𝜏𝑠={𝜔|𝑙𝑠,𝜏(𝜔)>𝑙𝑠,𝜏(𝜔)}, then 𝐷𝜏𝑠𝑠, for any stopping time 𝜏>𝑠.
Now, from (3.13) and noting that 1𝐷𝜏𝑠 is 𝑠-measurable, we deduce that 𝐸𝜏𝑠𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏𝑓𝑢,𝑦𝑢,𝑙s1𝐷𝜏𝑠𝑑𝑢𝑠=𝐸𝜏𝑠𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏1𝐷𝜏𝑠𝑑𝑢𝑠.(3.14)
Now, by (H1)-(iv), the left-hand side of (3.14) satisfies 𝐸𝜏𝑠𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏1𝐷𝜏𝑠𝑑𝑢𝑠||𝑙𝑘𝑠,𝜏𝑙𝑠,𝜏||𝐸𝜏𝑠𝑠1𝐷𝜏𝑠.(3.15) On the other hand, by (H1)-(iii) we see that the right-hand side of (3.14) satisfies 𝐸𝜏𝑠𝑓𝑢,𝑦u,𝑙𝑠,𝜏𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏1𝐷𝜏𝑠𝑑𝑢𝑠𝐸𝜏𝑠||𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏𝑓𝑢,𝑦𝑢,𝑙𝑠,𝜏||1𝐷𝜏𝑠𝑑𝑢𝑠𝑦𝐿𝐸𝑦(𝜏𝑠)𝑠1𝐷𝜏𝑠.(3.16) Combining above we obtain that 𝑘||𝑙𝑠,𝜏𝑙𝑠,𝜏||𝐸𝜏𝑠𝑠𝑦𝐿𝑦𝐸𝜏𝑠𝑠,on𝐷𝜏𝑠.(3.17) Thus |𝑙𝑠,𝜏𝑙𝑠,𝜏|(𝐿/𝑘)𝑦𝑦, on 𝐷𝜏𝑠, since 𝜏>𝑠, a.s.
Similarly, one shows that the inequality holds on the complement of 𝐷𝜏𝑠 as well. It follows that ||𝑙𝑠,𝜏𝑙𝑠,𝜏||𝐿𝑘𝑦𝑦.(3.18)
Next, recall from Theorem 2.4 that 𝐿𝑠=essinf𝜏>𝑠𝑙𝑠,𝜏, 𝐿s=essinf𝜏>𝑠𝑙𝑠,𝜏, 𝐴𝑡=sup0𝑠𝑡𝐿𝑠, and 𝐴𝑡=sup0𝑠𝑡𝐿𝑠. We conclude from (3.18) that, for any 𝑡[0,𝑇], ||𝐴𝑡𝐴𝑡||=||||sup0𝑠𝑡+𝐿ssup0𝑠𝑡+𝐿𝑠||||sup0𝑠𝑡+|||essinf𝜏>𝑠𝑙𝑠,𝜏essinf𝜏>𝑠𝑙𝑠,𝜏|||sup0𝑠𝑡+esssup𝜏>𝑠||𝑙𝑠,𝜏𝑙𝑠,𝜏||𝐿𝑘𝑦𝑦,𝑃-a.s.(3.19) 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 𝐸𝜏𝑠||𝑦𝑢𝑦u||𝑑𝑢𝑠𝐶𝐸𝜏𝑠𝑠𝐸sup0𝑢𝑇||𝑦𝑢𝑦u||𝑠(3.20) could hold for some constant 𝐶, 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 𝑇=1 and let 𝜏 be a binomial random variable that takes value 1 with probability 𝑝 and 1/𝑛 with probability 1𝑝. Define two processes: 𝑦𝑡=1{𝜏=1}, 𝑡[0,1]; 𝑡=1{𝜏𝑡}, 𝑡[0,1]; and define 𝑡=𝜎{(𝑦𝑢,𝑢)0𝑢𝑡} with 𝔽={𝑡}𝑡[0,1]. Then 𝜏 is an 𝔽-stopping time and 𝑦 is an 𝔽-adapted continuous process.
It is easy to check that 𝐸{𝜏0|𝑦𝑢|𝑑𝑢}=𝑝 and 𝐸{𝜏}𝐸{sup0𝑢1|𝑦𝑢|}=(𝑝+(1/𝑛)(1𝑝))𝑝. Thus if we choose 𝑝, 𝑛, and a constant 𝑐1 such that 𝑝<𝑛𝑐(𝑛1)𝑐<1,(3.21) then (3.20) will fail at 𝑠=0, with 𝐶=𝑐.

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 (𝐿+𝐾(𝐿/𝑘)𝑇<1, then the Variant reflected BSDE (1.2) admits a unique solution (𝑌,𝐴).

Proof. We first show that the mapping 𝒯 defined by (3.2) is from 𝑡 to itself. To see this, we note that by using assumption (H1) and Lemmas 3.2 and 3.3, one has ||𝑓𝑠,𝑦𝑠,𝐴𝑠||||𝑓𝑠,0,𝐴0𝑠||||𝑦+𝐿𝑠||||𝐴+𝐾𝑠𝐴0𝑠||||||||𝐴𝑓(𝑠,0,0)+𝐾0𝑠||+𝐿𝑦𝐿+𝐾𝑘𝑦Γ+𝐾2Γ𝑘+𝐿𝑦𝐿+𝐾𝑘𝑦.(3.22) Since 𝜉𝐿 by assumption, we can then easily deduce that 𝑌=𝒯(𝑦)𝑇.
To prove that 𝒯 is a contraction, we take 𝑦,𝑦𝑇, and denote 𝒯(𝑦)=𝑌 and 𝒯(𝑦)=𝑌. Then, for any 𝑡[0,𝑇], applying Lemma 3.3 we have ||𝒯(𝑦)𝑡𝒯(𝑦)𝑡||||||𝐸𝑇𝑡𝑓𝑠,𝑦𝑠,𝐴𝑠𝑓𝑠,𝑦s,𝐴𝑠𝑑𝑠𝑡||||𝐿𝑇𝑦𝑦+𝐾𝐴𝐴𝐿𝑇𝐿+𝐾𝑘𝑦𝑦.(3.23) Since 𝑇(𝐿+𝐾(𝐿/𝑘))<1 by assumption, we see that 𝒯 is a contraction.
Now, let 𝑌𝑇 be the (unique) fixed point of 𝒯, and let 𝐴 be the corresponding reflecting process defined by 𝐴𝑡=sup0𝑣𝑡+𝐿𝑣, where 𝐿 satisfies the representation 𝑋𝑡=𝐸𝜉+𝑇𝑡𝑓𝑠,𝑌𝑠,sup𝑡𝑣𝑠𝐿𝑣𝑑𝑠𝑡.(3.24)
We now show that (𝑌,𝐴) is the solution to the Variant RBSDE (1.2). To see this, note that (3.24), the definition of 𝐴, and the monotonicity of the function 𝑓 (on the variable 𝑙) tell us that, for 𝑡[0,𝑇], 𝑌𝑡=𝐸𝜉+𝑇𝑡𝑓𝑠,𝑌𝑠,𝐴𝑠𝑑𝑠𝑡𝐸𝜉+𝑇𝑡𝑓𝑠,𝑌𝑠,sup𝑡𝑣𝑠𝐿𝑣𝑑𝑠𝑡=𝑋𝑡.(3.25) Thus it remains to show that the flat-off condition holds. But by the properties of optional projections and definition of 𝐿 and 𝐴, we have 𝐸𝑇0𝑋𝑡𝑌𝑡𝑑𝐴𝑡=𝐸𝑇0𝑇𝑡𝑓𝑢,𝑌𝑢,sup𝑡𝑣𝑢𝐿𝑣𝑓𝑢,𝑌𝑢,sup0𝑣𝑢+𝐿𝑣𝑑𝑢𝑑𝐴𝑡=𝐸𝑇0𝑢0𝑓𝑢,𝑌𝑢,sup𝑡𝑣𝑢+𝐿𝑣𝑓𝑢,𝑌𝑢,sup0𝑣𝑢+𝐿𝑣𝑑𝐴𝑡𝑑𝑢,(3.26) here the last equality follows from the Fubini theorem and the fact that the Lebesgues measure does not charge the discontinuities of the paths 𝑢sup𝑡𝑣𝑢𝐿𝑣, which are only countably many.
Finally, note that on the set {(𝑡,𝜔)𝑑𝐴𝑡(𝜔)>0}, 𝑡 must be a point of increase of 𝐴(𝜔). Since 𝐴 is the running supreme of 𝐿 we conclude that sup0𝑣𝑡+𝛿𝐿𝑣>sup0𝑣𝑡𝐿𝑣, for all 𝛿>0. This yields that sup𝑡𝑣𝑢+𝐿𝑣=sup0𝑣𝑢+𝐿𝑣,on(t,𝜔)dAt(𝜔)>0.(3.27) Thus the right side of (3.26) is identically zero, and the flat-off condition holds. This proves the existence of the solution (𝑌,𝐴).
The uniqueness of the solution can be argued as follows. Suppose that there is another solution (𝑌,𝐴) to the VRBSDE such that 𝑌𝑡𝑋𝑡, 𝑌𝑡𝑋𝑡, 𝑡[0,𝑇], and 𝑌𝑡=𝐸𝜉+𝑇𝑡𝑓𝑢,𝑌𝑢,𝐴𝑢𝑑𝑢𝑡,𝐸𝑇0||𝑋𝑢𝑌𝑢||𝑑𝐴𝑢𝑌=0;𝑡=𝐸𝜉+𝑇𝑡𝑓𝑢,𝑌𝑢,𝐴𝑢𝑑𝑢𝑡,𝐸𝑇0||𝑋𝑢𝑌𝑢||𝑑𝐴𝑢=0.(3.28)

Since both 𝑌 and 𝑌 are the fixed points of the mapping 𝒯, it follows that 𝑌𝑡=𝑌𝑢, 𝑡[0,𝑇], 𝑃-a.s. Now consider the Variant Skorohod Problem 𝑌𝑡=𝐸𝜉+𝑇𝑡𝑓𝑌𝐴𝑢,𝑢𝑑𝑢𝑡,𝑌𝑡𝑋𝑡,𝑌𝑇=𝑋𝑇𝐸=𝜉,𝑇0||𝑌𝑡𝑋𝑡||𝑑𝐴𝑡=0,(3.29) where 𝑓𝑌(𝑢,𝑙)Δ=𝑓(𝑢,𝑌𝑢,𝑙). Then there exists a unique pair of process (𝑌,𝐴) that solves the Variant Skorohold problem, thanks to Theorem 2.5. But since both (𝑌,𝐴) and (𝑌,𝐴) are the solutions to the Variant RBSDE (3.29), it follows that 𝑌𝑡=𝑌𝑡 and 𝐴𝑡=𝐴𝑡=𝐴𝑡, 𝑡[0,𝑇], a.s., proving the uniqueness, whence the theorem.

We remark that our existence and uniqueness proof depends heavily on the well-posedness result of the stochastic representation theorem in [1], which requires that 𝐴0= so that 𝑡=0 must be a point of increase of process 𝐴. A direct consequence is then 𝑌0=𝑋0, by the flat-off condition, as we pointed out in Remark 2.2. The following corollary shows that this is not the only reason that solution of VRBSDE is actually a “bridge’’ with respect to the boundary process 𝑋.

Corollary 3.6. Suppose that 𝑌 is a solution to VRBSDE with generator 𝑓 and upper boundary 𝑋. Then 𝑌0=𝑋0.

Proof. Since 𝑌 is a fixed point of the mapping 𝒯 defined by (3.2), we see that 𝑌0 and 𝑋0 satisfy the following equalities: 𝑋0=𝐸𝜉+𝑇0𝑓𝑠,𝑌𝑠,sup0𝑣𝑠𝐿𝑣,𝑌𝑑𝑠0=𝐸𝜉+𝑇0𝑓𝑠,𝑌𝑠,𝐴𝑠𝑑𝑠=𝐸𝜉+𝑇0𝑓𝑠,𝑌𝑠,sup0𝑣𝑠+𝐿𝑣,𝑑𝑠(3.30) but as we argued before that the paths of the increasing process 𝑢sup0𝑣𝑢𝐿𝑣 has only countably many discontinuities, which are negligible under the Lebesgue measure, we conclude that 𝑌0=𝑋0.

4. Comparison Theorems

In this section we study the comparison theorem of the Variant RBSDE, one of the most useful tools in the theory of the BSDEs. We should note that the method that we will employ below follows closely to the uniqueness argument used in [1], which was more or less hidden in the proof of Theorem 3.5 as we applied the uniqueness of the Variant Skorohod problem. As we will see below, such a method is quite different from all the existing arguments in the BSDE context.

We begin by considering two VRBSDEs for 𝑖=1,2, 𝑌𝑖𝑡𝜉=𝐸𝑖+𝑇𝑡𝑓𝑖𝑢,𝑌𝑖𝑢,𝐴𝑖𝑢𝑑𝑢𝑡,𝑌𝑖𝑡𝑋𝑖𝑡,𝑌𝑖𝑇=𝑋𝑖𝑇=𝜉𝑖,𝐸𝑇0||𝑌𝑖𝑡𝑋𝑖𝑡||𝑑𝐴𝑖𝑡=0.(4.1) In what follows we call (𝑓𝑖,𝑋𝑖), 𝑖=1,2, the “parameters’’ of the VRBSDE (4.1), 𝑖=1,2, respectively. Define two stopping times: 𝑠Δ[=inf𝑡0,𝑇)𝐴2𝑡>𝐴1𝑡𝜏+𝜀𝑇;Δ[=inf𝑡𝑠,𝑇)𝐴1𝑡>𝐴2𝑡𝜀2𝑇.(4.2)

The following statements are similar to the solutions to Variant Skorohod problems (see [1]). We provide a sketch for completeness.

Lemma 4.1. The stopping times 𝑠 and 𝜏 defined by (4.2) have the following properties: (i)𝑠,𝜏 are points of increase for 𝐴2 and 𝐴1, respectively. In other words, for any 𝛿>0, it holds that 𝐴2𝑠<𝐴2𝑠+𝛿 and 𝐴1𝜏<𝐴1𝜏+𝛿, (ii)𝑃{𝑠<𝜏}=1; and 𝐴1𝑡𝐴2𝑡𝜀/2, for all 𝑡[𝑠,𝜏], 𝑃-a.s., (iii) it holds that 𝑌2𝑠=𝑋2𝑠 and 𝑌1𝜏=𝑋1𝜏, 𝑃-a.s.

Proof. Since (ii) is obvious by the definition of 𝑠 and 𝜏 and (iii) is a direct consequence of (i) and the flat-off condition, we need only check property (i).
Let 𝜔 be fixed. By the right continuity of 𝐴2 and 𝐴1, as well as the definition of 𝑠, we can find a decreasing sequence of stopping times {𝑠𝑛} such that 𝑠𝑛𝑠, and 𝐴2𝑠𝑛>𝐴1𝑠𝑛+𝜀, for 𝑛 sufficiently large (may assume for all 𝑛). Since 𝐴1 is increasing, we have 𝐴2𝑠𝑛>𝐴1𝑠𝑛+𝜀𝐴1𝑠+𝜀𝐴1𝑠+𝜀.(4.3) Note that 𝑠 is the first time 𝐴2 goes above 𝐴1+𝜀, one has 𝐴2𝑠𝐴1𝑠+𝜀. Thus, 𝐴2𝑠𝑛>𝐴2𝑠, for all 𝑛. Now for any 𝛿>0, one can choose 𝑛 large enough such that 𝑠𝑛<𝑠+𝛿 and it follows that 𝐴2𝑠+𝛿𝐴2𝑠𝑛>𝐴2𝑠, that is, 𝑠 is a point of increase of 𝐴2.
That 𝜏 is a point of increase of 𝐴1 can be proved using a similar argument.

We now give a simple analysis that would lead to the comparison theorem. Let (𝑌𝑖,𝐴𝑖), 𝑖=1,2 be the solutions to two VRBSDEs with boundaries 𝑋1 and 𝑋2, respectively. Define 𝑠 and 𝜏 as in (4.2). By Lemma 4.1, 𝑠<𝜏, 𝑃-a.s., with 𝑌2𝑠=𝑋2𝑠 and 𝑌1𝜏=𝑋1𝜏. To simplify notations let us denote 𝛿Θ=Θ1Θ2, Θ=𝑋,𝑌,𝐴, and 𝜉. Furthermore, let us define two martingales 𝑀𝑖𝑡Δ=𝐸{𝑇0𝑓𝑖(𝑢,𝑌1𝑢,𝐴1𝑢)𝑑𝑢𝑡}, 𝑡[0,𝑇], 𝑖=1,2, then on the set {𝑠<𝑇} we can write 𝛿𝑌𝑠=𝛿𝑌𝜏+𝜏𝑠𝑓1𝑢,𝑌1𝑢,𝐴1𝑢𝑓2𝑢,𝑌2𝑢,𝐴2𝑢𝑑𝑢+𝛿𝑀𝑇𝛿𝑀𝑠,=𝛿𝑌𝜏+𝜏𝑠𝑦𝑓1𝑢𝛿𝑌𝑢𝑑𝑢+𝜏𝑠𝛿𝑎𝑓1𝑢+𝛿2𝑓𝑢𝑑𝑢+𝛿𝑀𝑇𝛿𝑀𝑠,(4.4) where 𝛿𝑀Δ=𝑀1𝑀2, and 𝑦𝑓1𝑢Δ=𝑓1𝑢,𝑌1𝑢,𝐴1𝑢𝑓1𝑢,𝑌2𝑢,𝐴1𝑢𝑌1𝑢𝑌2𝑢1{𝑌1𝑢𝑌2𝑢},𝛿𝑎𝑓1𝑢Δ=𝑓1𝑢,𝑌2𝑢,𝐴1𝑢𝑓1𝑢,𝑌2𝑢,𝐴2𝑢,𝛿2𝑓𝑢Δ=𝑓1𝑢,𝑌2𝑢,𝐴2𝑢𝑓2𝑢,𝑌2𝑢,𝐴2𝑢.(4.5)

Now, by (H1) we see that 𝑦𝑓1 is a bounded process, and by the definition of 𝑠, 𝜏, and the monotonicity of 𝑓 in the variable 𝑙, we have 𝛿𝑎𝑓1>0 on the interval [𝑠,𝜏]. As usual, we now define Γ𝑡=𝑒𝑡0𝑦𝑓1𝑢𝑑𝑢, 𝑡[0,𝑇], and apply Itô's formula to obtain that Γ𝑠𝛿𝑌𝑠Γ𝜏𝛿𝑌𝜏=𝜏𝑠Γ𝑢𝛿𝑎𝑓1𝑢+𝛿2𝑓𝑢𝑑𝑢𝜏𝑠Γ𝑢𝑑𝛿𝑀𝑢.(4.6)

Therefore, if we assume that 𝑓1𝑓2, then 𝛿2𝑓0, 𝑑𝑃𝑑𝑡-a.s., and consequently, taking conditional expectation on both sides of (4.6) we have 𝐸Γ𝑠𝛿𝑌𝑠Γ𝜏𝛿𝑌𝜏𝑠=𝐸𝜏𝑠Γ𝑢𝛿𝑎𝑓1𝑢+𝛿2𝑓𝑢𝑑𝑢𝑠>0.(4.7)

On the other hand by the flat-off condition and Lemma 4.1-(iii), one can check that 𝑌1𝑠𝑌2𝑠𝑋1𝑠𝑋2𝑠 and 𝑌1𝜏𝑌2𝜏𝑋1𝜏𝑋2𝜏, 𝐸Γ𝑠𝛿𝑌𝑠Γ𝜏𝛿𝑌𝜏𝑠Γ𝐸𝑠𝛿𝑋𝑠Γ𝜏𝛿𝑋𝜏𝑠.(4.8) It is now clear that if the right hand above is nonpositive, then (4.8) contradicts (4.7), and consequently one must have 𝑃{𝑠<𝑇}=0. In other words, 𝐴2𝑡𝐴1𝑡+𝜀, for all 𝑡[0,𝑇], 𝑃-a.s. Since 𝜀 is arbitrary, this would entail that 𝐴1𝑡𝐴2𝑡[],𝑡0,𝑇,𝑃-a.s.(4.9)

We summarize the arguments into the following comparison theorem.

Theorem 4.2. Suppose that the parameters of the VRBSDEs (4.1) (𝑓𝑖,𝑋𝑖), 𝑖=1,2, satisfy (H1) and (H2). Suppose further that (i)𝑓1(𝑡,𝑦,𝑎)𝑓2(𝑡,𝑦,𝑎)0, 𝑑𝑃×𝑑𝑡 a.s., (ii)𝑋1𝑡𝑋2𝑡, 0𝑡𝑇, a.s., (iii)𝛿𝑋𝑠𝐸[𝑒𝐿(𝑡𝑠)𝛿𝑋𝑡𝑠] a.s. for all 𝑠 and 𝑡 such that 𝑠<𝑡. Then it holds that 𝐴1𝑡𝐴2𝑡, 𝑡[0,𝑇], 𝑃-a.s.

We remark that the assumption (iii) in Theorem 4.2 amounts to saying that the process 𝑒𝐿𝑠𝛿𝑋𝑠 is a submartingale. This is a merely technical condition required for the comparison theorem, and it does not add restriction on the regularity of the boundary processes 𝑋1 and 𝑋2 themselves, which are only required to be optional processes satisfying (H2).

Proof of Theorem 4.2. We need only show that the right hand side of (4.8) is nonpositive. To see this, note that since 𝛿𝑋𝜏0 by assumption (ii), we derive from (4.8) that 𝐸Γ𝑠𝛿𝑌𝑠Γ𝜏𝛿𝑌𝜏𝑠Γ𝑠𝐸𝛿𝑋𝑠𝑒𝜏𝑠𝑦𝑓1𝑢𝑑𝑢𝛿𝑋𝜏𝑠Γ𝑠𝐸𝛿𝑋𝑠𝑒𝐿(𝜏𝑠)𝛿𝑋𝜏𝑠0.(4.10) The last inequality is due to Assumption 3(iii) and optional sampling. This proves the theorem.

We should point out that Theorem 4.2 only gives the comparison between the reflecting processes 𝐴1 and 𝐴2, thus it is still one step away from the comparison between 𝑌1 and 𝑌2, which is much desirable for obvious reasons. Unfortunately, the latter is not necessarily true in general, due to the “opposite’’ monotonicity on 𝑓𝑖’s on the variable 𝑙. We nevertheless have the following corollaries of Theorem 4.2.

Corollary 4.3. Suppose that all the assumptions of Theorem 4.2 hold. Assume further that 𝑓1=𝑓2, then 𝑌1𝑡𝑌2𝑡, for all 𝑡[0,𝑇], 𝑃-a.s.

Proof. Let 𝑓=𝑓1=𝑓2. Define two random functions: 𝑓𝑖(𝑡,𝜔,𝑦)Δ=𝑓(𝑡,𝜔,𝑦,𝐴𝑖𝑡(𝜔)), for (𝑡,𝜔,𝑦)[0,𝑇]×Ω×. Then, 𝑌1 and 𝑌2 can be viewed as the solutions of BSDEs 𝑌𝑖𝑡𝜉=𝐸𝑖+𝑇𝑡𝑓𝑖𝑠,𝑌𝑖𝑢𝑑𝑢𝑡[],𝑡0,𝑇,𝑖=1,2.(4.11)
Note that 𝑓1(𝑡,𝜔,𝑦)=𝑓(𝑡,𝜔,𝑦,𝐴1𝑡(𝜔))𝑓(𝑡,𝜔,𝑦,𝐴2𝑡𝑓(𝜔))=2(𝑡,𝜔,𝑦), here the inequality holds due to the fact 𝐴1𝐴2. Since 𝜉1=𝑋1𝑇𝑋2𝑇=𝜉2, by the comparison theorem of BSDEs, we have 𝑌1𝑡𝑌2𝑡, for all 𝑡[0,𝑇], 𝑃-a.s.

Finally, we point out that Theorem 4.2 and Corollary 4.3 provide another proof of the uniqueness of VRBSDE. Namely, 𝑓1=𝑓2 and 𝑋1=𝑋2 imply 𝐴1=𝐴2 and 𝑌1=𝑌2.

5. Continuous Dependence Theorems

In this section we study another important aspect of well-posedness of the VRBSDE, namely the continuous dependence of the solution on the boundary process (whence the terminal as well).

To begin with, let us denote, for any optional process 𝑋 and any stopping time 𝑠 and 𝜏 such that 𝑠<𝜏, 𝑚𝑠,𝜏𝐸𝑋(𝑋)=𝜏𝑋𝑠𝑠𝐸𝜏𝑠𝑠.(5.1) As we pointed out in Remark 3.1, the random variable 𝑚𝑠,𝜏(𝑋) in a sense measures the path regularity of the “nonmartingale’’ part of the boundary process 𝑋. We will show that this will be a major measurement for the “closeness’’ of the boundary processes, as far as the continuous dependence is concerned.

Let {𝑋𝑛}𝑛=1 be a sequence optional processes satisfying (H2). We assume that {𝑋𝑛} converge to 𝑋0𝑡 in 𝑇, and that that 𝑋0 satisfies (H2) as well.

Let (𝑌𝑛,𝐴𝑛) be the solutions to the VRBSDE's with parameters (𝑓,𝑋𝑛), for 𝑛=0,1,2,. To be more precise, for 𝑖=0,1,2,, we have 𝑋𝑛𝑡𝜉=𝐸𝑛+𝑇𝑡𝑓𝑠,𝑌𝑛𝑠,Sup𝑡𝑣𝑠𝐿𝑛𝑣𝑑𝑠𝑡,𝐴𝑛𝑡=sup0𝑣𝑡+𝐿𝑛𝑣,𝑌𝑛𝑡𝜉=𝐸𝑛+𝑇𝑡𝑓𝑠,𝑌𝑛𝑠,𝐴𝑛𝑠𝑑𝑠𝑡.(5.2)

We now follow the similar arguments as in Theorem 3.5 to obtain the following obvious estimate: ||𝑌𝑛𝑡𝑌0𝑡||𝜉𝑛𝜉0𝐿𝑌+𝑇𝑛𝑌0𝐴+𝐾𝑛𝑢𝐴0𝑢.(5.3)

Again, we need the following lemma that provides the control of |𝐴𝑛𝑢𝐴0𝑢|.

Lemma 5.1. Assume (H1) and (H2). Then for all 𝑡[0,𝑇], it holds that ||𝐴𝑛𝑡𝐴0𝑡||sup𝑠[0,𝑇]esssup𝜏>𝑠1𝑘||𝑚𝑛𝑠,𝜏𝑚0𝑠,𝜏||+𝐿𝑘𝑌𝑛𝑌0,(5.4) where 𝑚𝑛=𝑚(𝑋𝑛), for 𝑛=0,1,2,.

Proof. The proof is very similar to that of Lemma 3.3. Let 𝑙𝑛𝑠,𝜏, 𝑛=0,1,2, be the 𝑠 random variables such that 𝐸𝑋𝑛𝑠𝑋𝑛𝜏𝑠=𝐸𝜏𝑠𝑓𝑢,𝑌𝑛𝑢,𝑙𝑛𝑠,𝜏𝑑𝑢𝑠.(5.5) Then 𝐸𝜏𝑠𝑓𝑢,𝑌𝑛𝑢,𝑙𝑛𝑠,𝜏𝑓𝑢,𝑌0𝑢,𝑙0𝑠,𝜏𝑑𝑢𝑠𝑋=𝐸𝑛𝑠𝑋𝑛𝜏𝑠𝑋𝐸0𝑠𝑋0𝜏𝑠.(5.6)
Then on the set 𝐷𝜏𝑠={𝑙𝑛𝑠,𝜏<𝑙0𝑠,𝜏}𝑠 we have 1𝐷𝜏𝑠𝐸𝑋𝑛𝑠𝑋𝑛𝜏𝑠𝑋𝐸0𝑠𝑋0𝜏𝑠1=𝐸𝐷𝜏𝑠𝜏𝑠𝑓𝑢,𝑌𝑛𝑢,𝑙𝑛𝑠,𝜏𝑓𝑠,𝑌0𝑢,𝑙𝑛𝑠,𝜏+𝑓𝑠,𝑌0𝑢,𝑙𝑛𝑠,𝜏𝑓𝑢,𝑌0𝑢,𝑙0𝑠,𝜏𝑑𝑢𝑠.(5.7) Since 𝑓(𝑠,𝑌0𝑢,𝑙𝑛𝑠,𝜏)>𝑓(𝑢,𝑌0𝑢,𝑙0𝑠,𝜏) on 𝐷𝜏𝑠, we have by (H1) that 𝑓(𝑠,𝑌0𝑢,𝑙𝑛𝑠,𝜏)𝑓(𝑢,𝑌0𝑢,𝑙0𝑠,𝜏)𝑘|𝑙𝑛𝑠,𝜏𝑙0𝑠,𝜏| on 𝐷𝜏𝑠 and hence 1𝐷𝜏𝑠𝑘||𝑙𝑛𝑠,𝜏𝑙0𝑠,𝜏||𝐸𝜏𝑠𝑠1𝐷𝜏𝑠𝐸𝑋𝑛𝑠𝑋𝑛𝜏𝑠𝑋𝐸0𝑠𝑋0𝜏𝑠+1𝐷𝜏𝑠𝐸𝜏𝑠𝐿||𝑌𝑛𝑢𝑌0𝑢||𝑑𝑢𝑠.(5.8) We thus conclude that ||𝑙𝑛𝑠,𝜏𝑙0𝑠,𝜏||1𝑘||𝑚𝑛𝑠,𝜏𝑚0𝑠,𝜏||+𝐿𝑘𝑌𝑛𝑌0,𝑃-a.s.on𝐷𝜏𝑠.(5.9) A similar argument also shows that (5.9) holds on (𝐷𝜏𝑠)𝑐. Hence (5.9) holds almost surely.
Finally, using the facts that |𝐿𝑛𝑠𝐿0𝑠|=|essinf𝜏>𝑠𝑙𝑛𝑠,𝜏essinf𝜏>𝑠𝑙0𝑠,𝜏|esssup𝜏>𝑠|𝑙𝑛𝑠,𝜏𝑙0s,𝜏|, we conclude that, for any 𝑡[0,𝑇], it holds 𝑃-almost surely that ||𝐴𝑛𝑡𝐴0𝑡||=||||sup0𝑠𝑡+𝐿𝑛𝑠sup0𝑠𝑡+𝐿0𝑠||||sup0𝑠𝑇||𝐿𝑛𝑠𝐿0𝑠||sup0𝑠𝑇esssup𝜏>𝑠1𝑘||𝑚𝑛𝑠,𝜏𝑚0𝑠,𝜏||+𝐿𝑘𝑌𝑛𝑌0,(5.10) proving the lemma.

Combining (5.3) and Lemma 5.1 we have the following theorem.

Theorem 5.2. Assume (H1) and (H2). Assume further that (𝐿+𝐾(𝐿/𝑘))𝑇<1. Then it holds that ||𝑌𝑛𝑡𝑌0𝑡||1𝜉1(1+(𝐾/𝑘))𝐿𝑇𝑛𝜉0+𝐾𝑇𝑘sup𝑠[0,𝑇]esssup𝜏>𝑠|𝑚𝑛𝑠,𝜏𝑚0𝑠,𝜏|.(5.11)

6. Applications of Variant Reflected BSDEs

In this section we consider some possible applications of VRBSDEs. We should note that while these problems are more or less ad hoc, we nevertheless believe that they are novel in that they cannot be solved by standard (or “classical’’) techniques, and the theory of Variant RBSDEs seems to provide exactly the right solution.

6.1. A Recursive Intertemporal Utility Minization Problem

As one of the main applications of the stochastic representation theorem, Bank and Riedel studied both utility maximization problems and stochastic equilibrium problems with Hindy-Huang-Kreps type of preferences (cf. [6, 9]). We will consider a slight variation of these problems, and show that the VRBSDE is the natural solution.

The main idea of Hindy-Huang-Kreps utility functional is as follows. Instead of considering utility functionals depending directly on the consumption rate, one assumes that that the utilities are derived from the current level of satisfaction, defined as a weighted average of the accumulated consumptions: 𝐴𝑡=𝐴(𝐶)tΔ=𝜂𝑡+𝑡0𝜃(𝑡,𝑠)𝑑𝐶𝑠[],𝑡0,𝑇,(6.1) where 𝜂[0,𝑇] represents the exogenously given level of satisfaction at time 𝑡; 𝜃[0,𝑇]2 are the instantaneous weights assigned to consumptions made up to time 𝑡; and 𝑡𝐶𝑡 is the accumulated consumption up to time 𝑡 (hence 𝐶={𝐶𝑡𝑡0} is an increasing process, called a consumption plan). The Hindy-Huang-Kreps utility is then defined by (cf. [7]) 𝐸𝑈(𝐶)Δ𝑉𝐶=𝐸𝑇+𝑇0𝑢𝑡,𝐴(𝐶)𝑡𝑑𝑡,(6.2) here both 𝑉() and 𝑢(𝑡,) are concave and increasing (utility) functions.

It is now natural to extend the problem to the recursive utility setting. In fact, in [9] it was indicated that, following the similar argument of Duffie-Epstein [5], the recursive utility 𝑈𝑡𝑉(𝐶)=𝐸𝑇+𝑇𝑡𝑢𝑟,𝑈𝑟(𝐶),𝐴(𝐶)𝑟𝑑𝑟𝑡[],𝑡0,𝑇(6.3) is well-defined for each consumption plan 𝐶. Here 𝑢(𝑟,𝑦,𝑎)[0,𝑇]×× denotes a felicity function which is continuous, increasing and concave in 𝑎; and 𝐴(𝐶) is the corresponding level of satisfaction defined by (6.1). In what follows we will denote 𝑈=𝑈(𝐶) and 𝐴=𝐴(𝐶) for simplicity.

Let us now consider the following optimization problem. Let us assume that 𝜂 and 𝜃 in (6.1) are chosen so that for any consumption plan 𝐶, 𝐴(𝐶) is an increasing process, and that for a given increasing process 𝐴, there is a unique consumption plan 𝐶 satisfying (6.1). Furthermore, we assume that there is an exogenous lower bound of the utility at each time 𝑡 (e.g., the minimum cost to execute any consumption plan). We denote it by 𝑋, and assume that it is an optional process of Class (D) so that 𝑈𝑡𝑋𝑡 at each time 𝑡. Let us define the set of admissible consumption plans, denoted by 𝒜, to be the set of all right-continuous increasing processes 𝐶, such that the corresponding recursive utility 𝑈𝑡=𝑈𝑡𝑋(𝐶)𝑡, 𝑡[0,𝑇], 𝑃-a.s. Our goal is then to find 𝐶𝒜 that minimizes the expected utility (or cost) 𝐸𝑈0Δ𝑋=𝐸𝑇+𝑇0𝑢𝑟,𝑈𝑟,𝐴𝑟𝑑𝑟,(6.4) where 𝐴=𝐴(𝐶) is determined by 𝐶 via (6.1). A consumption plan 𝐶 is optimal if the associated recursive utility 𝑈 satisfies 𝐸𝑈0=min𝐶𝒜𝐸{𝑈0(𝐶)}.

We remark that the set of admissible consumption plans 𝒜 is not empty. In fact, let 𝑌𝑡=𝑈𝑡, 𝑍𝑡𝑍=𝑡, 𝑋𝑡𝑋=𝑡 and define 𝑓(𝑡,𝑦,𝑙)Δ=𝑢(𝑡,𝑦,𝑙). Then we can write the recursive utility as 𝑌𝑡𝑌=𝐸𝑇+𝑇𝑡𝑓𝑠,𝑌𝑠,𝐴𝑠𝑑𝑠𝑡[],𝑡0,𝑇.(6.5)

Let us now assume further that the function 𝑓 and the process 𝑋 satisfy (H1) and (H2), then we can solve the VRBSDE with parameters (𝑓,𝑋), to obtain a unique solution (𝑌0,𝐴0). Rewriting 𝑈0=𝑌0, then (𝑈0,𝐴0) satisfies the following VRBSDE: 𝑈0𝑡𝑋=𝐸𝑇+𝑇𝑡𝑢𝑟,𝑈0𝑟,𝐴0𝑟𝑑𝑟𝑡,𝑈0𝑡𝑋𝑡[],𝐸,𝑡0,𝑇𝑇0||𝑈0𝑡𝑋𝑡||𝑑𝐴0𝑡=0.(6.6) Clearly, this implies that 𝐴0𝒜. Furthermore, for any 𝜀>0, define 𝐴𝜀𝑡=𝐴0𝑡+𝜀, and let 𝑈𝜀 be the solution to the BSDE 𝑈𝜀𝑡𝑋=𝐸{𝑇+𝑇𝑡𝑢(𝑟,𝑈𝜀𝑟,𝐴𝜀𝑟)𝑑𝑟𝑇}. By the comparison theorem of BSDEs, the utility 𝑈𝜀𝑡𝑈0𝑡𝑋𝑡, thus 𝐴𝜀𝒜 as well. In other words, the set 𝒜 contains infinitely many elements if it is not empty.

Intuitively, the best choice of the consumption plan would be the one whose corresponding level of satisfaction 𝐴 is such that the associated utility 𝑈 coincides with the lower boundary 𝑋. But this amounts to saying that the boundary process 𝑋 must satisfy a backward SDE, which is clearly not necessarily true in general.

The second best guess is then that the optimal level 𝐴 allows its associated recursive utility 𝑈 follow the VRBSDE with the exogenous lower bound 𝑋. This turns out to be exactly the case: recall from Corollary 3.6 that the solution 𝑈0=𝑌0 of the VRBSDE (6.6) must satisfy 𝑈00=𝑌0=𝑋0=𝑋0𝑈(𝐶), 𝑃-a.s., for all 𝐶𝒜. Thus 𝐴0 is indeed the optimal level of satisfaction. The following theorem is thus essentially trivial.

Theorem 6.1. Assume that (𝑈0,𝐴0) is the solution to VRBSDE (6.6), then for any admissible consumption plan 𝐶𝒜, it holds that 𝑈00𝑈0(𝐶) almost surely. Consequently, 𝐴0 is the optimal level of satisfaction.

Finally, we note that the Theorem 4.2 also leads to the comparison between different recursive utilities corresponding to different lower boundaries. Namely, if 𝑋𝑖, 𝑖=1,2 are two lower utility boundaries satisfying the conditions in Theorem 4.2, and 𝑈𝑖, 𝑖=1,2 are the corresponding minimal recursive utilities satisfying (6.6), then 𝑋1𝑡𝑋2𝑡, 0𝑡𝑇, a.s., implies that 𝑈1𝑡𝑈2𝑡 and 𝐴1𝑡𝐴2𝑡, 0𝑡𝑇, a.s. In particular, it holds that 𝐸[𝑈10]𝐸[𝑈20].

6.2. VRBSDE and Optimal Stopping Problems

We now look at a possible extension of the so-called multiarmed bandits problem proposed by El Karoui and Karatzas [10]. To be more precise, let us consider a family of optimal stopping problems, parameterized by a given process 𝑌𝑇: 𝑉(𝑡,𝑙;𝑌)Δ=essinf𝜏𝑡𝐸𝜏𝑡𝑓𝑢,𝑌𝑢,𝑙𝑑𝑢+𝑋𝜏𝑡.(6.7) Here 𝑙 could be either a constant or a random variable. We note that by choosing the stopping time 𝜏𝑡, we deduce the natural upper boundary of the value function 𝑉(𝑡,𝑙;𝑌)𝑋𝑡[],𝑡0,T,𝑃-a.s.(6.8) The following result characterize the relation between the VRBSDE and the value of the optimal stopping problem.

Theorem 6.2. Assume that the parameters (𝑓,𝑋) in (6.7) satisfies (H1) and (H2). Then a pair of processes (𝑌,𝐴) is a solution to the VRBSDE (1.2) if and only if they solve the following optimal stopping problems: (i)𝑌𝑡=𝑉(𝑡,𝐴𝑡;𝑌), 0𝑡𝑇,(ii)𝐴𝑡=sup0𝑠𝑡+𝐿𝑠 and 𝐿𝑠=sup{𝑙𝑉(𝑠,𝑙;𝑌)=𝑋𝑠},(iii) it holds that 𝑌𝑡=essinf𝜏𝑡𝐸𝜏𝑡𝑓𝑢,𝑌𝑢,𝐴𝑢𝑑𝑢+𝑋𝜏𝑡[],𝑡0,𝑇.(6.9)Furthermore, the stopping time 𝜏𝑡=inf{𝑡𝑢𝑇𝑌𝑢=𝑋𝑢} is optimal.

Proof. We first asssume that (𝑌,𝐴) is a solution to the variant RBSDE with parameter (𝑓,𝑋). Note that for any stopping time 𝜏𝑡, we have 𝑌𝑡𝑌=𝐸𝜏+𝜏𝑡𝑓𝑢,𝑌𝑢,𝐴𝑢𝑑𝑢𝑡[],𝑡0,𝑇.(6.10) Since 𝐴 is increasing, we have 𝐴𝑢𝐴𝑡, for all 𝑢[𝑡,𝜏]. Thus by using the monotonicity of 𝑓 one has 𝑌𝑡𝑋𝐸𝜏+𝜏𝑡𝑓𝑢,𝑌𝑢,𝐴𝑡𝑑𝑢𝑡.(6.11) Note that this holds for all stopping times 𝜏𝑡, we conclude that 𝑌𝑡essinf𝜏𝑡𝐸𝑋𝜏+𝜏𝑡𝑓𝑢,𝑌𝑢,𝐴𝑡𝑑𝑢𝑡=𝑉𝑡,𝐴𝑡;𝑌,𝑃-a.s.(6.12)
Next, define 𝜏𝑡Δ=inf{𝑡𝑢𝑇;𝑌𝑢=𝑋𝑢}𝑇. Then 𝜏t is a stopping time, and the flat-off condition implies that 𝐸𝜏𝑡𝑡|𝑌𝑢𝑋𝑢|𝑑𝐴𝑢=0, and therefore 𝐴𝑢=𝐴𝑡, for all 𝑢[𝑡,𝜏𝑡). Consequently, 𝑌𝑡𝑌=𝐸𝜏𝑡+𝜏𝑡𝑡𝑓𝑢,𝑌𝑢,𝐴𝑢𝑑𝑢𝑡𝑋=𝐸𝜏𝑡+𝜏𝑡𝑡𝑓𝑢,𝑌𝑢,𝐴𝑡𝑑𝑢𝑡𝑉𝑡,𝐴𝑡;𝑌,𝑃-a.s.(6.13) Combining (6.12) and (6.13) we obtain (i) and (iii).
To prove (ii), we note that by the uniqueness the VRBSDE, we have the solution (𝑌,𝐴) of VRBSDE must satisfy 𝑋𝑡=𝐸𝜉+𝑇𝑡𝑓𝑠,𝑌𝑠,sup𝜏𝑣𝑠𝐿𝑣𝑑𝑠𝑡,𝐴𝑡=sup0𝑣𝑡+𝐿𝑣.(6.14)
As Bank and El Karoui have shown in [1], if we define 𝑉(𝑡,𝑙;𝑌) as (6.7), then the level process 𝐿 in the stochastic representation in (6.14) satisfies 𝐿𝑡=sup𝑙𝑉(𝑡,𝑙;𝑌)=𝑋𝑡,𝑃-a.s.,(6.15