On Variant Reflected Backward SDEs, with Applications

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={𝐴_𝑡}_𝑡≥0  with  𝐴₀₋=−∞, such that

(i)𝑌=_𝑡Δ𝐸{∫^𝑇𝑡𝑓(𝑠,𝐴_𝑠)𝑑𝑠|ℱ_𝑡}≤𝑋_𝑡, 𝑡∈[0,𝑇], P-a.s.;(ii)𝐸∫^𝑇0|𝑌_𝑡−𝑋_𝑡|𝑑𝐴_𝑡=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 𝑌_𝑡=^𝑇𝑡𝑓𝑠,𝐴_𝑠𝑑𝑠−^𝑇𝑡𝑍_𝑠𝑑𝐵_𝑠≤𝑋_𝑡,0≤𝑡≤𝑇,(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={𝑋_𝑡}_𝑡≥0 be an optional process of class (D), and let 𝑓∶Ω×[0,𝑇]×ℝ×ℝ×ℝ↦ℝ be a random field satisfying appropriate measurability assumptions. Consider the following backward stochastic differential equation (BSDE for short): for 𝑡∈[0,𝑇], 𝑌_𝑡=𝐸𝑋_𝑇+^𝑇𝑡𝑓𝑠,𝑌_𝑠,𝐴_𝑠𝑑𝑠∣ℱ_𝑡,(1.2) where the solution (𝑌,𝐴) is defined to be such that

(i)𝑌_𝑡≤𝑋_𝑡, 0≤𝑡≤𝑇; 𝑌_𝑇=𝑋_𝑇;(ii)𝐴={𝐴_𝑡} is an adapted, increasing process such that 𝐴_Δ0−=−∞, and the flat-off condition holds: 𝐸^𝑇0||𝑌_𝑡−𝑋_𝑡||𝑑𝐴_𝑡=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): 𝑑𝑌_𝑡=−𝑓𝑡,𝑌_𝑡,𝑍_𝑡,𝐴_𝑡𝑑𝑡+𝑍_𝑡𝑑𝑊_𝑡,𝑌_𝑡≤𝑋_𝑡,𝑡∈[0,𝑇],𝑌_𝑇=𝑋_𝑇,(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 𝑋_𝑆=𝐸^𝑇𝑆𝑓𝑢,sup_{𝑆≤𝑣≤𝑢}𝐿_𝑣𝑑𝑢∣ℱ^𝑆,(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,ℱ,𝑃;𝔽) is a filtered probability space, where 𝔽=^Δ{ℱ_𝑡}_{0≤𝑡≤𝑇} is a filtration that satisfies the usual hypothses. For simplicity we assume that ℱ=ℱ_𝑇. In the case when the filtration 𝔽 is generated by a standard Brownian motion 𝐵 on the space (Ω,ℱ,𝑃), we say that 𝔽 is “Brownian’’ and denote it by 𝔽=𝔽^𝐵. We will always assume that 𝔽^𝐵 is augmented by all the 𝑃-null sets in ℱ.

We will frequently make use the following notations. Let

(i)𝕃^∞𝑇 be the space of all ℱ_𝑇 measurable bounded random variables,(ii)ℍ^∞𝑇 the space of all ℝ-valued, progressively measurable, bounded processes, (iii)ℍ^2𝑇 the space of all ℝ^𝑑-valued, progressively measurable process 𝑍, such that 𝐸∫^𝑇0|𝑍^2𝑠|𝑑𝑠<∞,(iv)ℳ_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={𝑋_𝑡,0≤𝑡≤𝑇} 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 {𝑋_𝜏∶𝜏∈ℳ_0,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×ℝ×ℝ×ℝ↦ℝ enjoys the following properties:

(i)for fixed 𝜔∈Ω, 𝑡∈[0,𝑇], and 𝑦∈ℝ, the function 𝑓(𝜔,𝑡,𝑦,⋅)∶ℝ↦ℝ is continuous and strictly decreasing from +∞ to −∞,(ii)for fixed 𝑦,𝑙∈ℝ³, the process 𝑓(⋅,⋅,𝑦,𝑙) is progressively measurable with 𝐸^𝑇0||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 ||𝑓𝑡,𝜔,𝑦^,𝑙−𝑓(𝑡,𝜔,𝑦,𝑙)||≤𝐿||𝑦^−𝑦||,∀𝑦^,𝑦∈ℝ,(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^−𝑙||≤||𝑓𝑡,𝑦,𝑙^−𝑓(𝑡,𝑦,𝑙)||≤𝐾||𝑙^−𝑙||,∀𝑙^,𝑙∈ℝ.(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 𝜉∈𝕃^∞𝑇 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)𝑌_𝑡=𝐸{𝜉+∫^𝑇𝑡𝑓(𝑠,𝑌_𝑠,𝐴_𝑠)𝑑𝑡∣ℱ_𝑡};(ii)𝑌_𝑡≤𝑋_𝑡, 0≤𝑡≤𝑇; 𝑌_𝑇=𝑋_𝑇=𝜉;(iv) the process 𝐴 is 𝔽-adapted, increasing, càdlàg, and 𝐴_Δ0−=−∞, such that the “flat-off’’ condition holds: 𝐸^𝑇0||𝑌_𝑡−𝑋_𝑡||𝑑𝐴_𝑡=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 𝑀_𝑡=𝐸{∫^𝑇0𝑓(𝑡,𝑌_𝑡,𝐴_𝑡)𝑑𝑡∣ℱ_𝑡}, 𝑡∈[0,𝑇] then 𝑀 is a martingale on [0,𝑇], and the VRBSDE will read 𝑌_𝑡=𝜉+^𝑇𝑡𝑓𝑠,𝑌_𝑠,𝐴_𝑠𝑑𝑡−𝑀_𝑇−𝑀_𝑡,0≤𝑡≤𝑇.(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=𝔽^𝐵, that is, it is generated by a standard Brownian motion 𝐵, with the usual augmentation. Let 𝜉∈𝕃^∞𝑇 and the boundary process 𝑋 be given. A triplet of processes {(𝑌_𝑡,𝑍_𝑡,𝐴_𝑡),0≤𝑡≤𝑇} is called a solution of Variant Reflected BSDE with terminal value 𝜉 and boundary 𝑋 if(i)𝑌∈ℍ^∞𝑇, 𝑍∈ℍ^2𝑇,(ii)𝑌_𝑡=𝜉+∫^𝑇𝑡𝑓(𝑠,𝑌_𝑠,𝑍_𝑠,𝐴_𝑠)𝑑𝑠−∫^𝑇𝑡𝑍_𝑠𝑑𝐵_𝑠, 0≤𝑡≤𝑇,(iii)𝑌_𝑡≤𝑋_𝑡, 0≤𝑡≤𝑇; 𝑌_𝑇=𝑋_𝑇=𝜉,(iv) the process {𝐴_𝑡} is 𝔽-adapted, increasing, càdlàg, and 𝐴₀₋=−∞, such that the flat-off condition holds: 𝐸∫^𝑇0|𝑌_𝑡−𝑋_𝑡|𝑑𝐴_𝑡=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 𝑋_𝑆=𝐸𝑋_𝑇+^𝑇𝑆𝑓𝑢,sup_{𝑆≤𝑣≤𝑢}𝐿_𝑣𝑑𝑢∣ℱ^𝑆(2.6) for any stopping time 𝑆∈ℳ_0,𝑇, where 𝐿 is an optional process taking values in ℝ∪{−∞}, and it can be characterized as follows: (i)𝑓(𝑢,sup_{𝑆≤𝑣≤𝑢}𝐿_𝑣)∈𝐿¹(ℙ⊗𝑑𝑡) for any stopping time 𝑆,(ii)𝐿_𝑆=essinf_𝜏>𝑆𝑙_𝑆,𝜏, where the “essinf’’ is taken over all stopping times 𝑆∈ℳ_0,𝑇 such that 𝑆<𝑇, a.s.; and 𝑙_𝑆,𝜏 is the unique ℱ_𝑆-measurable random variable satisfying: 𝐸𝑋_𝑆−𝑋_𝜏∣ℱ_𝑆=𝐸^𝜏𝑆𝑓𝑢,𝑙_𝑆,𝜏𝑑𝑢∣ℱ_𝑆,(2.7)(iii) (Gittin Index) if 𝑉(𝑡,𝑙)=^Δessinf_𝜏≥𝑡𝐸{𝐸∫^𝜏𝑡𝑓(𝑢,𝑙)𝑑𝑢+𝑋_𝜏|ℱ_𝑡}, 𝑡∈[0,𝑇], is the value functions of a family of optimal stopping problems indexed by 𝑙∈ℝ, then 𝐿_𝑡=sup𝑙∶𝑉(𝑡,𝑙)=𝑋_𝑡,𝑡∈[0,𝑇].(2.8)

We should note here, unlike the original stochastic representation theorem in [1] where it assumed that 𝑋_𝑇=0, we allow arbitrary terminal value for 𝑋_𝑇. This can be obtained easily by considering a new process 𝑋=_𝑡Δ𝑋_𝑡−𝐸[𝜉∣ℱ_𝑡], 𝑡≥0. 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 𝑌_𝑡=𝐸𝑋_𝑇+^𝑇𝑡𝑓𝑠,𝐴_𝑠𝑑𝑠∣ℱ_𝑡,𝑡∈[0,𝑇].(2.9) Furthermore, the process 𝐴 can be expressed as 𝐴_𝑡=sup_{0≤𝑠≤𝑡+}𝐿_𝑠, where 𝐿 is the process in Theorem 2.4.

We conclude this section by making following observations. First, the random variable 𝑙_𝑆,𝜏, defined by (2.7) is ℱ_𝑆-measrable for any stopping time 𝜏>𝑆, thus the process 𝑠↦𝐿_𝑠 is 𝔽-adapted. However, the running maximum process 𝐴=_𝑡Δsup_{0≤𝑢≤𝑡+}𝐿_𝑢 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 𝑆∈ℳ_0,𝑇, it holds that esssup_{𝜏>𝑆𝜏∈ℳ0,𝑇}||||𝐸𝑋_𝜏−𝑋_𝑆∣ℱ_𝑆𝐸𝜏−𝑆∣ℱ_𝑆||||≤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 ℍ^∞𝑇: for a given process 𝑦 we define 𝒯(𝑦)=_𝑡Δ𝑌_𝑡, 𝑡∈[0,𝑇], where 𝑌 is the unique solution of the Variant Skorohod problem: 𝑌_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑠,𝑦_𝑠,𝐴_𝑠𝑑𝑠∣ℱ_𝑠𝐸,𝑡∈[0,𝑇_𝑇0],𝑋_𝑡−𝑌_𝑡𝑑𝐴_𝑡=0,𝑡∈[0,𝑇].(3.2) We are to prove that the mapping 𝒯 is a contraction from ℍ^∞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: 𝐴_𝑡=sup_{0≤𝑣≤𝑡+}𝐿_𝑣, and 𝐿 is the solution to the Stochastic Representation: 𝑋_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑠,𝑦_𝑠,sup_{𝑡≤𝑣≤𝑠}𝐿_𝑣𝑑𝑠∣ℱ^𝑡,𝑡∈[0,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 𝑋_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑠,0,sup_{𝑡≤𝑣≤𝑠}𝐿^0𝑣𝑑𝑠∣ℱ^𝑡.(3.4) Denote 𝐴=^0𝑡Δsup_{0≤𝑠≤𝑡+}𝐿^0𝑡. We have the following estimate for 𝐴⁰.

Lemma 3.2. Assume (H1) and (H2). Then it holds that ‖𝐴⁰‖_∞≤2Γ/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 𝑙^0𝑠,𝜏 be the ℱ_𝑠 measurable random variable such that 𝐸𝑋_𝑠−𝑋_𝜏∣ℱ_𝑠=𝐸^𝜏𝑠𝑓𝑡,0,𝑙^0𝑠,𝜏𝑑𝑡∣ℱ_𝑠.(3.5) Then by Theorem 2.4 we have 𝐿^0𝑠=essinf_𝜏>𝑠𝑙^0𝑠,𝜏, and 𝐴^0𝑡=sup_{0≤𝑠≤𝑡+}𝐿^0𝑠.

Now consider the set {𝜔∶𝑙^0𝑠,𝜏(𝜔)<0}. Since 𝑓(𝑡,0,⋅) is decreasing, we have 𝐸𝑋_𝑠−𝑋_𝜏∣ℱ_𝑠−𝐸^𝜏𝑠𝑓(𝑡,0,0)𝑑𝑡∣ℱ_𝑠=𝐸^𝜏𝑠𝑓𝑡,0,𝑙^0𝑠,𝜏−𝑓(𝑡,0,0)𝑑𝑡∣ℱ_𝑠≥𝐸_𝜏𝑠𝑘||𝑙^0𝑠,𝜏||𝑑𝑡∣ℱ_𝑠≥𝑘||𝑙^0𝑠,𝜏||𝐸𝜏−𝑠∣ℱ_𝑠.(3.6) In other words we have ||𝑙^0𝑠,𝜏||≤1𝑘𝐸𝑋^𝑠−𝑋_𝜏∣ℱ_𝑠𝐸𝜏−𝑠∣ℱ_𝑠−𝐸∫^𝜏𝑠𝑓(𝑡,0,0)𝑑𝑡∣ℱ_𝑠𝐸𝜏−𝑠∣ℱ_𝑠,onl^0s,𝜏<0.(3.7)

Similarly, one can show that on the set {𝑙^0𝑠,𝜏≥0} it holds that 𝑙^0𝑠,𝜏≤1𝑘−𝐸𝑋^𝑠−𝑋_𝜏∣ℱ_𝑠𝐸𝜏−𝑠∣ℱ_𝑠+𝐸∫^𝜏𝑠𝑓(𝑡,0,0)𝑑𝑡∣ℱ_𝑠𝐸𝜏−𝑠∣ℱ_𝑠.(3.8)

Consequently, we have ||𝑙^0𝑠,𝜏||≤1𝑘||||𝐸𝑋^𝜏−𝑋_𝑠∣ℱ_𝑠𝐸𝜏−𝑠∣ℱ_𝑠||||+𝐸∫^𝜏𝑠||𝑓(𝑡,0,0)||𝑑𝑡∣ℱ_𝑠𝐸𝜏−𝑠∣ℱ_𝑠.(3.9)

Now note that ||𝐴^0𝑡||=||||sup_{0≤𝑠≤𝑡+}𝐿^0𝑠||||≤sup_{0≤𝑠≤𝑡+}||𝐿^0𝑠||=sup_{0≤𝑠≤𝑡+}essinf_𝜏>𝑠||𝑙^0𝑠,𝜏||,(3.10) we derive from (3.9) and (H2) that ||𝐴^0𝑡||≤sup_{0≤𝑠≤𝑡+}esssup_𝜏>𝑠||𝑙^0𝑠,𝜏||≤sup_{0≤𝑠≤𝑡+}Γ+𝑘Γ=2𝑘Γ,(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,𝑦^∈ℍ^∞𝑇 be given, and consider the two solutions of the variant Skorohod problem: (𝑌,𝐴) and (𝑌^,𝐴^). We would like to control |𝐴^𝑠−𝐴_𝑠| in terms of |𝑦^_s−𝑦_𝑠|. The following lemma is crucial.

Lemma 3.3. Assume (H1) and (H2). Then, for any 𝑡∈[0,𝑇], it holds almost surely that ||A^𝑡−𝐴_𝑡||≤𝐿𝑘‖‖𝑦^−𝑦‖‖_∞.(3.12)

Proof. Again, we fix 𝑠 and let 𝜏∈ℳ_0,𝑇 be such that 𝜏>𝑠, a.s. Recalling Theorem 2.4, we let 𝑙_𝑠,𝜏 and 𝑙^𝑠,𝜏 be two ℱ_𝑠-measurable random variables such that 𝐸𝑋_𝑠−𝑋_𝜏∣ℱ_𝑠=𝐸^𝜏𝑠𝑓𝑢,𝑦_𝑢,𝑙_𝑠,𝜏𝑑𝑢∣ℱ_𝑠=𝐸^𝜏𝑠𝑓𝑢,𝑦^𝑢,𝑙^𝑠,𝜏𝑑𝑢∣ℱ_𝑠.(3.13) Define 𝐷^𝜏𝑠={𝜔|𝑙^𝑠,𝜏(𝜔)>𝑙_𝑠,𝜏(𝜔)}, then 𝐷^𝜏𝑠∈ℱ_𝑠, for any stopping time 𝜏>𝑠.
Now, from (3.13) and noting that 1_𝐷^𝜏𝑠 is ℱ_𝑠-measurable, we deduce that 𝐸^𝜏𝑠𝑓𝑢,𝑦_𝑢,𝑙_𝑠,𝜏−𝑓𝑢,𝑦_𝑢,𝑙^_s1_𝐷^𝜏𝑠𝑑𝑢∣ℱ_𝑠=𝐸_𝜏𝑠𝑓𝑢,𝑦^𝑢,𝑙^𝑠,𝜏−𝑓𝑢,𝑦_𝑢,𝑙^𝑠,𝜏1_𝐷^𝜏𝑠𝑑𝑢∣ℱ_𝑠.(3.14)
Now, by (H1)-(iv), the left-hand side of (3.14) satisfies 𝐸^𝜏𝑠𝑓𝑢,𝑦_𝑢,𝑙_𝑠,𝜏−𝑓𝑢,𝑦_𝑢,𝑙^𝑠,𝜏1_𝐷^𝜏𝑠𝑑𝑢∣ℱ_𝑠≥𝑘||𝑙^𝑠,𝜏−𝑙^𝑠,𝜏||𝐸𝜏−𝑠∣ℱ_𝑠1_𝐷^𝜏𝑠.(3.15) On the other hand, by (H1)-(iii) we see that the right-hand side of (3.14) satisfies 𝐸^𝜏𝑠𝑓𝑢,𝑦^_u,𝑙^𝑠,𝜏−𝑓𝑢,𝑦_𝑢,𝑙^𝑠,𝜏1_𝐷^𝜏𝑠𝑑𝑢∣ℱ_𝑠≤𝐸_𝜏𝑠||𝑓𝑢,𝑦^𝑢,𝑙^𝑠,𝜏−𝑓𝑢,𝑦_𝑢,𝑙^𝑠,𝜏||1_𝐷^𝜏𝑠𝑑𝑢∣ℱ_𝑠≤𝐿𝐸‖‖𝑦^−𝑦‖‖_∞(𝜏−𝑠)∣ℱ_𝑠1_𝐷^𝜏𝑠.(3.16) Combining above we obtain that 𝑘||𝑙_𝑠,𝜏−𝑙^𝑠,𝜏||𝐸𝜏−𝑠∣ℱ_𝑠≤𝐿‖‖𝑦^−𝑦‖‖_∞𝐸𝜏−𝑠∣ℱ_𝑠,on𝐷^𝜏𝑠.(3.17) Thus |𝑙_𝑠,𝜏−𝑙^𝑠,𝜏|≤(𝐿/𝑘)‖𝑦^−𝑦‖_∞, on 𝐷^𝜏𝑠, since 𝜏>𝑠, a.s.
Similarly, one shows that the inequality holds on the complement of 𝐷^𝜏𝑠 as well. It follows that ||𝑙_𝑠,𝜏−𝑙^𝑠,𝜏||≤𝐿𝑘‖‖𝑦^−𝑦‖‖_∞.(3.18)
Next, recall from Theorem 2.4 that 𝐿_𝑠=essinf_𝜏>𝑠𝑙_𝑠,𝜏, 𝐿^_s=essinf_𝜏>𝑠𝑙^𝑠,𝜏, 𝐴_𝑡=sup_{0≤𝑠≤𝑡}𝐿_𝑠, and 𝐴^𝑡=sup_{0≤𝑠≤𝑡}𝐿^𝑠. We conclude from (3.18) that, for any 𝑡∈[0,𝑇], ||𝐴^𝑡−𝐴_𝑡||=||||sup_{0≤𝑠≤𝑡+}𝐿^_s−sup_{0≤𝑠≤𝑡+}𝐿_𝑠||||≤sup_{0≤𝑠≤𝑡+}|||essinf_𝜏>𝑠𝑙^𝑠,𝜏−essinf_𝜏>𝑠𝑙_𝑠,𝜏≤sup_{0≤𝑠≤𝑡+}|||esssup_𝜏>𝑠||𝑙^𝑠,𝜏−𝑙_𝑠,𝜏||≤𝐿𝑘‖‖𝑦^−𝑦‖‖_∞,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 𝐸^𝜏𝑠||𝑦_𝑢−𝑦^_u||𝑑𝑢∣ℱ_𝑠≤𝐶𝐸𝜏−𝑠∣ℱ^𝑠𝐸sup_{0≤𝑢≤𝑇}||𝑦_𝑢−𝑦^_u||∣ℱ_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 1−𝑝. Define two processes: 𝑦_𝑡=1_{𝜏=1}, 𝑡∈[0,1]; ℎ_𝑡=1_{{𝜏≤𝑡}}, 𝑡∈[0,1]; and define ℱ_𝑡=𝜎{(𝑦_𝑢,ℎ_𝑢)∶0≤𝑢≤𝑡} with 𝔽={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|𝑦_𝑢|𝑑𝑢}=𝑝 and 𝐸{𝜏}𝐸{sup_0≤𝑢≤1|𝑦_𝑢|}=(𝑝+(1/𝑛)(1−𝑝))𝑝. Thus if we choose 𝑝, 𝑛, and a constant 𝑐≥1 such that 𝑝<𝑛−𝑐(𝑛−1)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 ℍ^∞𝑡 to itself. To see this, we note that by using assumption (H1) and Lemmas 3.2 and 3.3, one has ||𝑓𝑠,𝑦_𝑠,𝐴_𝑠||≤≤||||𝑓𝑓(𝑠𝑠,,00,,0𝐴)||^0𝑠+||+𝐾𝐿||𝐴||_0𝑠𝑦_𝑠||+𝐾||𝐴_𝑠−𝐴^0𝑠||||+𝐿‖𝑦‖_∞+𝐾𝐿𝑘‖𝑦‖^∞≤Γ+𝐾2Γ𝑘+𝐿‖𝑦‖^∞+𝐾𝐿𝑘‖𝑦‖^∞.(3.22) Since 𝜉∈𝐿^∞ by assumption, we can then easily deduce that 𝑌=𝒯(𝑦)∈ℍ^∞𝑇.
To prove that 𝒯 is a contraction, we take 𝑦,𝑦^∈ℍ^∞𝑇, and denote 𝒯(𝑦)=𝑌 and 𝒯(𝑦^)=𝑌^. Then, for any 𝑡∈[0,𝑇], applying Lemma 3.3 we have ||𝒯(𝑦)_𝑡−𝒯(𝑦^)_𝑡||≤||||𝐸^𝑇𝑡𝑓𝑠,𝑦_𝑠,𝐴_𝑠−𝑓𝑠,𝑦^_s,𝐴_𝑠𝑑𝑠∣ℱ_𝑡≤𝑇𝐿‖‖𝑦−𝑦_||||‖‖_∞+𝐾‖‖𝐴−𝐴^‖‖_∞≤𝑇𝐿+𝐾𝐿𝑘‖‖𝑦−𝑦^‖‖_∞.(3.23) Since 𝑇(𝐿+𝐾(𝐿/𝑘))<1 by assumption, we see that 𝒯 is a contraction.
Now, let 𝑌∈ℍ^∞𝑇 be the (unique) fixed point of 𝒯, and let 𝐴 be the corresponding reflecting process defined by 𝐴_𝑡=sup_{0≤𝑣≤𝑡+}𝐿_𝑣, where 𝐿 satisfies the representation 𝑋_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑠,𝑌_𝑠,sup_{𝑡≤𝑣≤𝑠}𝐿_𝑣𝑑𝑠∣ℱ^𝑡.(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], 𝑌_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑠,𝑌_𝑠,𝐴_𝑠𝑑𝑠∣ℱ_𝑡≤𝐸𝜉+^𝑇𝑡𝑓𝑠,𝑌_𝑠,sup_{𝑡≤𝑣≤𝑠}𝐿_𝑣𝑑𝑠∣ℱ^𝑡=𝑋^𝑡.(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 𝐸^𝑇0𝑋_𝑡−𝑌_𝑡𝑑𝐴_𝑡=𝐸^𝑇0^𝑇𝑡𝑓𝑢,𝑌^𝑢,sup_{𝑡≤𝑣≤𝑢}𝐿_𝑣−𝑓𝑢,𝑌^𝑢,sup_{0≤𝑣≤𝑢+}𝐿_𝑣𝑑𝑢𝑑𝐴^𝑡=𝐸^𝑇0^𝑢0𝑓𝑢,𝑌^𝑢,sup_{𝑡≤𝑣≤𝑢+}𝐿_𝑣−𝑓𝑢,𝑌^𝑢,sup_{0≤𝑣≤𝑢+}𝐿_𝑣𝑑𝐴^𝑡𝑑𝑢,(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 𝑢↦sup_{𝑡≤𝑣≤𝑢}𝐿_𝑣, which are only countably many.
Finally, note that on the set {(𝑡,𝜔)∶𝑑𝐴_𝑡(𝜔)>0}, 𝑡 must be a point of increase of 𝐴⋅(𝜔). Since 𝐴 is the running supreme of 𝐿 we conclude that sup_{0≤𝑣≤𝑡+𝛿}𝐿_𝑣>sup_{0≤𝑣≤𝑡−}𝐿_𝑣, for all 𝛿>0. This yields that sup_{𝑡≤𝑣≤𝑢+}𝐿_𝑣=sup_{0≤𝑣≤𝑢+}𝐿_𝑣,on(t,𝜔)∶dA_t(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 𝑌_𝑡≤𝑋_𝑡, 𝑌^𝑡≤𝑋_𝑡, 𝑡∈[0,𝑇], and 𝑌_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢∣ℱ_𝑡,𝐸^𝑇0𝑌||𝑋_𝑢−𝑌_𝑢||𝑑𝐴_𝑢=0;_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑢,𝑌^𝑢,𝐴^𝑢𝑑𝑢∣ℱ_𝑡,𝐸^𝑇0||𝑋_𝑢−𝑌^𝑢||𝑑𝐴^𝑢=0.(3.28)

Since both 𝑌 and 𝑌^ are the fixed points of the mapping 𝒯, it follows that 𝑌_𝑡=𝑌^𝑢, 𝑡∈[0,𝑇], 𝑃-a.s. Now consider the Variant Skorohod Problem 𝑌_𝑡=𝐸𝜉+^𝑇𝑡𝑓^𝑌𝑌_𝑡𝑢,𝐴^𝑢𝑑𝑢∣ℱ^𝑡,𝐸≤𝑋_𝑡,𝑌_𝑇=𝑋_𝑇=𝜉,_𝑇0||𝑌_𝑡−𝑋_𝑡||𝑑𝐴_𝑡=0,(3.29) where 𝑓^𝑌(𝑢,𝑙)=^Δ𝑓(𝑢,𝑌_𝑢,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 𝑌_𝑡=𝑌_𝑡 and 𝐴_𝑡=𝐴^𝑡=𝐴_𝑡, 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=𝐸𝜉+^𝑇0𝑓𝑠,𝑌_𝑠,sup_{0≤𝑣≤𝑠}𝐿_𝑣𝑌𝑑𝑠₀,=𝐸𝜉+^𝑇0𝑓𝑠,𝑌_𝑠,𝐴_𝑠𝑑𝑠=𝐸𝜉+^𝑇0𝑓𝑠,𝑌_𝑠,sup_{0≤𝑣≤𝑠+}𝐿_𝑣𝑑𝑠,(3.30) but as we argued before that the paths of the increasing process 𝑢↦sup_{0≤𝑣≤𝑢}𝐿_𝑣 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, 𝑌^𝑖𝑡=𝐸𝜉^𝑖+^𝑇𝑡𝑓^𝑖𝑢,𝑌^𝑖𝑢,𝐴^𝑖𝑢𝑑𝑢∣ℱ_𝑡𝑌_𝑖𝑡,𝐸≤𝑋^𝑖𝑡_𝑇0,𝑌^𝑖𝑇=𝑋^𝑖𝑇=𝜉^𝑖,||𝑌^𝑖𝑡−𝑋^𝑖𝑡||𝑑𝐴^𝑖𝑡=0.(4.1) In what follows we call (𝑓^𝑖,𝑋^𝑖), 𝑖=1,2, the “parameters’’ of the VRBSDE (4.1), 𝑖=1,2, respectively. Define two stopping times: 𝜏𝑠=^Δinf𝑡∈[0,𝑇)∣𝐴^2𝑡>𝐴^1𝑡+𝜀∧𝑇;_Δ=inf𝑡∈[𝑠,𝑇)∣𝐴^1𝑡>𝐴^2𝑡−𝜀2∧𝑇.(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 𝐴^2𝑠−<𝐴^2𝑠+𝛿 and 𝐴^1𝜏−<𝐴^1𝜏+𝛿, (ii)𝑃{𝑠<𝜏}=1; and 𝐴^1𝑡≤𝐴^2𝑡−𝜀/2, for all 𝑡∈[𝑠,𝜏], 𝑃-a.s., (iii) it holds that 𝑌^2𝑠=𝑋^2𝑠 and 𝑌^1𝜏=𝑋^1𝜏, 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 {𝑠_𝑛} such that 𝑠_𝑛↘𝑠, and 𝐴^2𝑠_𝑛>𝐴^1𝑠_𝑛+𝜀, for 𝑛 sufficiently large (may assume for all 𝑛). Since 𝐴¹ is increasing, we have 𝐴^2𝑠_𝑛>𝐴^1𝑠_𝑛+𝜀≥𝐴^1𝑠+𝜀≥𝐴^1𝑠−+𝜀.(4.3) Note that 𝑠 is the first time 𝐴² goes above 𝐴¹+𝜀, one has 𝐴^2𝑠−≤𝐴^1𝑠−+𝜀. Thus, 𝐴^2𝑠_𝑛>𝐴^2𝑠−, for all 𝑛. Now for any 𝛿>0, one can choose 𝑛 large enough such that 𝑠_𝑛<𝑠+𝛿 and it follows that 𝐴^2𝑠+𝛿≥𝐴^2𝑠_𝑛>𝐴^2𝑠−, 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 (𝑌^𝑖,𝐴^𝑖), 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 𝑌^2𝑠=𝑋^2𝑠 and 𝑌^1𝜏=𝑋^1𝜏. To simplify notations let us denote 𝛿Θ=Θ¹−Θ², Θ=𝑋,𝑌,𝐴, and 𝜉. Furthermore, let us define two martingales 𝑀=^𝑖𝑡Δ𝐸{∫^𝑇0𝑓^𝑖(𝑢,𝑌^1𝑢,𝐴^1𝑢)𝑑𝑢∣ℱ_𝑡}, 𝑡∈[0,𝑇], 𝑖=1,2, then on the set {𝑠<𝑇} we can write 𝛿𝑌_𝑠==𝛿𝛿𝑌𝑌_𝜏𝜏+^𝜏𝑠𝑓¹𝑢,𝑌^1𝑢,𝐴^1𝑢−𝑓²𝑢,𝑌^2𝑢,𝐴^2𝑢𝑑𝑢+𝛿𝑀_𝑇−𝛿𝑀_𝑠,+^𝜏𝑠∇_𝑦𝑓^1𝑢𝛿𝑌_𝑢𝑑𝑢+^𝜏𝑠𝛿_𝑎𝑓^1𝑢+𝛿₂𝑓_𝑢𝑑𝑢+𝛿𝑀_𝑇−𝛿𝑀_𝑠,(4.4) where 𝛿𝑀=^Δ𝑀¹−𝑀², and ∇_𝑦𝑓=^1𝑢Δ𝑓¹𝑢,𝑌^1𝑢,𝐴^1𝑢−𝑓¹𝑢,𝑌^2𝑢,𝐴^1𝑢𝑌^1𝑢−𝑌^2𝑢𝛿_𝑎1_{𝑌^1𝑢_≠𝑌^2𝑢_},𝛿₂𝑓=^1𝑢Δ𝑓¹𝑢,𝑌^2𝑢,𝐴^1𝑢−𝑓¹𝑢,𝑌^2𝑢,𝐴^2𝑢,𝑓=_𝑢Δ𝑓¹𝑢,𝑌^2𝑢,𝐴^2𝑢−𝑓²𝑢,𝑌^2𝑢,𝐴^2𝑢.(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 𝛿_𝑎𝑓¹>0 on the interval [𝑠,𝜏]. As usual, we now define Γ_𝑡=𝑒^{∫𝑡0∇𝑦𝑓1𝑢𝑑𝑢}, 𝑡∈[0,𝑇], and apply Itô's formula to obtain that Γ_𝑠𝛿𝑌_𝑠−Γ_𝜏𝛿𝑌_𝜏=^𝜏𝑠Γ_𝑢𝛿_𝑎𝑓^1𝑢+𝛿₂𝑓_𝑢𝑑𝑢−^𝜏𝑠Γ_𝑢𝑑𝛿𝑀_𝑢.(4.6)

Therefore, if we assume that 𝑓¹≥𝑓², then 𝛿₂𝑓≥0, 𝑑𝑃⊗𝑑𝑡-a.s., and consequently, taking conditional expectation on both sides of (4.6) we have 𝐸Γ_𝑠𝛿𝑌_𝑠−Γ_𝜏𝛿𝑌_𝜏∣ℱ_𝑠=𝐸^𝜏𝑠Γ_𝑢𝛿_𝑎𝑓^1𝑢+𝛿₂𝑓_𝑢𝑑𝑢∣ℱ_𝑠>0.(4.7)

On the other hand by the flat-off condition and Lemma 4.1-(iii), one can check that 𝑌^1𝑠−𝑌^2𝑠≤𝑋^1𝑠−𝑋^2𝑠 and 𝑌^1𝜏−𝑌^2𝜏≥𝑋^1𝜏−𝑋^2𝜏, 𝐸Γ_𝑠𝛿𝑌_𝑠−Γ_𝜏𝛿𝑌_𝜏∣ℱ_𝑠≤𝐸Γ_𝑠𝛿𝑋_𝑠−Γ_𝜏𝛿𝑋_𝜏∣ℱ_𝑠.(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, 𝐴^2𝑡≤𝐴^1𝑡+𝜀, for all 𝑡∈[0,𝑇], 𝑃-a.s. Since 𝜀 is arbitrary, this would entail that 𝐴^1𝑡≥𝐴^2𝑡,𝑡∈[0,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) (𝑓^𝑖,𝑋^𝑖), 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)𝑋^1𝑡≤𝑋^2𝑡, 0≤𝑡≤𝑇, a.s., (iii)𝛿𝑋_𝑠≤𝐸[𝑒^{𝐿(𝑡−𝑠)}𝛿𝑋_𝑡∣ℱ_𝑠] a.s. for all 𝑠 and 𝑡 such that 𝑠<𝑡. Then it holds that 𝐴^1𝑡≥𝐴^2𝑡, 𝑡∈[0,𝑇], 𝑃-a.s.

We remark that the assumption (iii) in Theorem 4.2 amounts to saying that the process 𝑒^𝐿𝑠𝛿𝑋_𝑠 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 𝛿𝑋_𝜏≤0 by assumption (ii), we derive from (4.8) that 𝐸Γ_𝑠𝛿𝑌_𝑠−Γ_𝜏𝛿𝑌_𝜏∣ℱ_𝑠≤≤ΓΓ_𝑠𝑠𝐸𝛿𝑋_𝑠−𝑒^{∫𝜏𝑠∇𝑦𝑓1𝑢𝑑𝑢}𝛿𝑋_𝜏∣ℱ_𝑠𝐸𝛿𝑋_𝑠−𝑒^{𝐿(𝜏−𝑠)}𝛿𝑋_𝜏∣ℱ_𝑠≤0.(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 𝑓^𝑖’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 𝑌^1𝑡≤𝑌^2𝑡, for all 𝑡∈[0,𝑇], 𝑃-a.s.

Proof. Let 𝑓=𝑓¹=𝑓². Define two random functions: 𝑓^𝑖(𝑡,𝜔,𝑦)=^Δ𝑓(𝑡,𝜔,𝑦,𝐴^𝑖𝑡(𝜔)), for (𝑡,𝜔,𝑦)∈[0,𝑇]×Ω×ℝ. Then, 𝑌¹ and 𝑌² can be viewed as the solutions of BSDEs 𝑌^𝑖𝑡=𝐸𝜉^𝑖+^𝑇𝑡𝑓^𝑖𝑠,𝑌^𝑖𝑢𝑑𝑢∣ℱ_𝑡,𝑡∈[0,𝑇],𝑖=1,2.(4.11)
Note that 𝑓¹(𝑡,𝜔,𝑦)=𝑓(𝑡,𝜔,𝑦,𝐴^1𝑡(𝜔))≤𝑓(𝑡,𝜔,𝑦,𝐴^2𝑡(𝜔))=𝑓²(𝑡,𝜔,𝑦), here the inequality holds due to the fact 𝐴¹≥𝐴². Since 𝜉¹=𝑋^1𝑇≤𝑋^2𝑇=𝜉², by the comparison theorem of BSDEs, we have 𝑌^1𝑡≤𝑌^2𝑡, 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, 𝑚_𝑠,𝜏(𝑋)=𝐸𝑋^𝜏−𝑋_𝑠∣ℱ_𝑠𝐸𝜏−𝑠∣ℱ_𝑠.(5.1) As we pointed out in Remark 3.1, the random variable 𝑚_𝑠,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 {𝑋^𝑛}^∞𝑛=1 be a sequence optional processes satisfying (H2). We assume that {𝑋^𝑛} converge to 𝑋^0𝑡 in ℍ^∞𝑇, and that that 𝑋⁰ satisfies (H2) as well.

Let (𝑌^𝑛,𝐴^𝑛) be the solutions to the VRBSDE's with parameters (𝑓,𝑋^𝑛), for 𝑛=0,1,2,…. To be more precise, for 𝑖=0,1,2,…, we have 𝑋^𝑛𝑡=𝐸𝜉^𝑛+^𝑇𝑡𝑓𝑠,𝑌^𝑛𝑠,Sup_{𝑡≤𝑣≤𝑠}𝐿^𝑛𝑣𝐴_𝑛𝑡𝑑𝑠∣ℱ^𝑡,𝑌_𝑛𝑡=sup_{0≤𝑣≤𝑡+}𝐿^𝑛𝑣,=𝐸𝜉^𝑛+^𝑇𝑡𝑓𝑠,𝑌^𝑛𝑠,𝐴^𝑛𝑠𝑑𝑠∣ℱ_𝑡.(5.2)

We now follow the similar arguments as in Theorem 3.5 to obtain the following obvious estimate: ||𝑌^𝑛𝑡−𝑌^0𝑡||≤‖‖𝜉^𝑛−𝜉⁰‖‖_∞+𝑇𝐿‖‖𝑌^𝑛−𝑌⁰‖‖_∞+𝐾‖‖𝐴^𝑛𝑢−𝐴^0𝑢‖‖.(5.3)

Again, we need the following lemma that provides the control of |𝐴^𝑛𝑢−𝐴^0𝑢|.

Lemma 5.1. Assume (H1) and (H2). Then for all 𝑡∈[0,𝑇], it holds that ||𝐴^𝑛𝑡−𝐴^0𝑡||≤sup_{𝑠∈[0,𝑇]}esssup_𝜏>𝑠1𝑘||𝑚^{𝑛𝑠,𝜏}−𝑚^0𝑠,𝜏||+𝐿𝑘‖‖𝑌^𝑛−𝑌⁰‖‖_∞,(5.4) where 𝑚^𝑛=𝑚(𝑋^𝑛), for 𝑛=0,1,2,….

Proof. The proof is very similar to that of Lemma 3.3. Let 𝑙^{𝑛𝑠,𝜏}, 𝑛=0,1,2,… be the ℱ_𝑠 random variables such that 𝐸𝑋^𝑛𝑠−𝑋^𝑛𝜏∣ℱ_𝑠=𝐸^𝜏𝑠𝑓𝑢,𝑌^𝑛𝑢,𝑙^{𝑛𝑠,𝜏}𝑑𝑢∣ℱ_𝑠.(5.5) Then 𝐸^𝜏𝑠𝑓𝑢,𝑌^𝑛𝑢,𝑙^{𝑛𝑠,𝜏}−𝑓𝑢,𝑌^0𝑢,𝑙^0𝑠,𝜏𝑑𝑢∣ℱ_𝑠=𝐸𝑋^𝑛𝑠−𝑋^𝑛𝜏∣ℱ_𝑠−𝐸𝑋^0𝑠−𝑋^0𝜏∣ℱ_𝑠.(5.6)
Then on the set 𝐷^𝜏𝑠={𝑙^{𝑛𝑠,𝜏}<𝑙^0𝑠,𝜏}∈ℱ_𝑠 we have 1_𝐷^𝜏𝑠=𝐸𝐸𝑋^𝑛𝑠1−_𝐷𝜏𝑠𝑋^𝑛𝜏∣ℱ_𝑠−𝐸𝑋^0𝑠−𝑋^0𝜏∣ℱ_𝑠^𝜏𝑠𝑓𝑢,𝑌^𝑛𝑢,𝑙^{𝑛𝑠,𝜏}−𝑓𝑠,𝑌^0𝑢,𝑙^{𝑛𝑠,𝜏}+𝑓𝑠,𝑌^0𝑢,𝑙^{𝑛𝑠,𝜏}−𝑓𝑢,𝑌^0𝑢,𝑙^0𝑠,𝜏𝑑𝑢∣ℱ_𝑠.(5.7) Since 𝑓(𝑠,𝑌^0𝑢,𝑙^{𝑛𝑠,𝜏})>𝑓(𝑢,𝑌^0𝑢,𝑙^0𝑠,𝜏) on 𝐷^𝜏𝑠, we have by (H1) that 𝑓(𝑠,𝑌^0𝑢,𝑙^{𝑛𝑠,𝜏})−𝑓(𝑢,𝑌^0𝑢,𝑙^0𝑠,𝜏)≥𝑘|𝑙^{𝑛𝑠,𝜏}−𝑙^0𝑠,𝜏| on 𝐷^𝜏𝑠 and hence 1_𝐷^𝜏𝑠𝑘||𝑙^{𝑛𝑠,𝜏}−𝑙^0𝑠,𝜏||𝐸𝜏−𝑠∣ℱ_𝑠≤+11_𝐷^𝜏𝑠_𝐷𝜏𝑠𝐸𝑋^𝑛𝑠−𝑋^𝑛𝜏∣ℱ_𝑠−𝐸𝑋^0𝑠−𝑋^0𝜏∣ℱ_𝑠𝐸^𝜏𝑠𝐿||𝑌^𝑛𝑢−𝑌^0𝑢||𝑑𝑢∣ℱ_𝑠.(5.8) We thus conclude that ||𝑙^{𝑛𝑠,𝜏}−𝑙^0𝑠,𝜏||≤1𝑘||𝑚^{𝑛𝑠,𝜏}−𝑚^0𝑠,𝜏||+𝐿𝑘‖‖𝑌^𝑛−𝑌⁰‖‖_∞,𝑃-a.s.on𝐷^𝜏𝑠.(5.9) A similar argument also shows that (5.9) holds on (𝐷^𝜏𝑠)^𝑐. Hence (5.9) holds almost surely.
Finally, using the facts that |𝐿^𝑛𝑠−𝐿^0𝑠|=|essinf_𝜏>𝑠𝑙^{𝑛𝑠,𝜏}−essinf_𝜏>𝑠𝑙^0𝑠,𝜏|≤esssup_𝜏>𝑠|𝑙^{𝑛𝑠,𝜏}−𝑙⁰_s,𝜏|, we conclude that, for any 𝑡∈[0,𝑇], it holds 𝑃-almost surely that ||𝐴^𝑛𝑡−𝐴^0𝑡||=||||sup_{0≤𝑠≤𝑡+}𝐿^𝑛𝑠−sup_{0≤𝑠≤𝑡+}𝐿^0𝑠||||≤sup_{0≤𝑠≤𝑇}≤sup_{0≤𝑠≤𝑇}||𝐿^𝑛𝑠−𝐿^0𝑠||esssup_𝜏>𝑠1𝑘||𝑚^{𝑛𝑠,𝜏}−𝑚^0𝑠,𝜏||+𝐿𝑘‖‖𝑌^𝑛−𝑌⁰‖‖_∞,(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 ||𝑌^𝑛𝑡−𝑌^0𝑡||≤11−(1+(𝐾/𝑘))𝐿𝑇‖‖𝜉^𝑛−𝜉⁰‖‖_∞+𝐾𝑇𝑘‖‖‖sup_{𝑠∈[0,𝑇]}esssup_𝜏>𝑠|𝑚^{𝑛𝑠,𝜏}−𝑚^0𝑠,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: 𝐴_𝑡=𝐴(𝐶)=_t^Δ𝜂_𝑡+^𝑡0𝜃(𝑡,𝑠)𝑑𝐶_𝑠,𝑡∈[0,𝑇],(6.1) where 𝜂∶[0,𝑇]↦ℝ represents the exogenously given level of satisfaction at time 𝑡; 𝜃∶[0,𝑇]²↦ℝ are the instantaneous weights assigned to consumptions made up to time 𝑡; and 𝑡↦𝐶_𝑡 is the accumulated consumption up to time 𝑡 (hence 𝐶={𝐶_𝑡∶𝑡≥0} is an increasing process, called a consumption plan). The Hindy-Huang-Kreps utility is then defined by (cf. [7]) 𝐸𝑈(𝐶)=^Δ𝐸𝑉𝐶_𝑇+^𝑇0𝑢𝑡,𝐴(𝐶)_𝑡𝑑𝑡,(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 𝑈_𝑡(𝐶)=𝐸𝑉_𝑇+^𝑇𝑡𝑢𝑟,𝑈_𝑟(𝐶),𝐴(𝐶)_𝑟𝑑𝑟∣ℱ_𝑡,𝑡∈[0,𝑇](6.3) is well-defined for each consumption plan 𝐶. Here 𝑢(𝑟,𝑦,𝑎)∶[0,𝑇]×ℝ×ℝ↦ℝ 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 𝑈_𝑡≥𝑋_𝑡 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 𝑈_𝑡=𝑈_𝑡(𝐶)≥𝑋_𝑡, 𝑡∈[0,𝑇], 𝑃-a.s. Our goal is then to find 𝐶^∗∈𝒜 that minimizes the expected utility (or cost) 𝐸𝑈=_0Δ𝐸𝑋_𝑇+^𝑇0𝑢𝑟,𝑈_𝑟,𝐴_𝑟𝑑𝑟,(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 𝐸𝑈^∗0=min_𝐶∈𝒜𝐸{𝑈₀(𝐶)}.

We remark that the set of admissible consumption plans 𝒜 is not empty. In fact, let 𝑌_𝑡=−𝑈_𝑡, 𝑍_𝑡=−𝑍_𝑡, 𝑋_𝑡=−𝑋_𝑡 and define 𝑓(𝑡,𝑦,𝑙)=^Δ−𝑢(𝑡,−𝑦,𝑙). Then we can write the recursive utility as 𝑌_𝑡=𝐸𝑌_𝑇+^𝑇𝑡𝑓𝑠,𝑌_𝑠,𝐴_𝑠𝑑𝑠∣ℱ_𝑡,𝑡∈[0,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: 𝑈^0𝑡=𝐸𝑋_𝑇+^𝑇𝑡𝑢𝑟,𝑈^0𝑟,𝐴^0𝑟𝑑𝑟∣ℱ_𝑡,𝑈^0𝑡𝐸≥𝑋_𝑡,𝑡∈[0,𝑇_𝑇0],||𝑈^0𝑡−𝑋_𝑡||𝑑𝐴^0𝑡=0.(6.6) Clearly, this implies that 𝐴⁰∈𝒜. Furthermore, for any 𝜀>0, define 𝐴^𝜀𝑡=𝐴^0𝑡+𝜀, and let 𝑈^𝜀 be the solution to the BSDE 𝑈^𝜀𝑡=𝐸{𝑋_𝑇+∫^𝑇𝑡𝑢(𝑟,𝑈^𝜀𝑟,𝐴^𝜀𝑟)𝑑𝑟∣ℱ_𝑇}. By the comparison theorem of BSDEs, the utility 𝑈^𝜀𝑡≥𝑈^0𝑡≥𝑋_𝑡, thus 𝐴^𝜀∈𝒜 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=𝑋₀≤𝑈(𝐶), 𝑃-a.s., for all 𝐶∈𝒜. 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 𝐶∈𝒜, it holds that 𝑈⁰⁰≤𝑈₀(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 𝑋^𝑖, 𝑖=1,2 are two lower utility boundaries satisfying the conditions in Theorem 4.2, and 𝑈^𝑖, 𝑖=1,2 are the corresponding minimal recursive utilities satisfying (6.6), then 𝑋^1𝑡≥𝑋^2𝑡, 0≤𝑡≤𝑇, a.s., implies that 𝑈^1𝑡≥𝑈^2𝑡 and 𝐴^1𝑡≥𝐴^2𝑡, 0≤𝑡≤𝑇, 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 𝑌∈ℍ^∞𝑇: 𝑉(𝑡,𝑙;𝑌)=^Δessinf_𝜏≥𝑡𝐸^𝜏𝑡𝑓𝑢,𝑌_𝑢,𝑙𝑑𝑢+𝑋_𝜏∣ℱ_𝑡.(6.7) Here 𝑙 could be either a constant or a random variable. We note that by choosing the stopping time 𝜏≡𝑡, we deduce the natural upper boundary of the value function 𝑉(𝑡,𝑙;𝑌)≤𝑋_𝑡,𝑡∈[0,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)𝑌_𝑡=𝑉(𝑡,𝐴_𝑡;𝑌), 0≤𝑡≤𝑇,(ii)𝐴_𝑡=sup_{0≤𝑠≤𝑡+}𝐿_𝑠 and 𝐿_𝑠=sup{𝑙∈ℝ∶𝑉(𝑠,𝑙;𝑌)=𝑋_𝑠},(iii) it holds that 𝑌_𝑡=essinf_𝜏≥𝑡𝐸^𝜏𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑋_𝜏∣ℱ_𝑡,𝑡∈[0,𝑇].(6.9)Furthermore, the stopping time 𝜏^∗𝑡=inf{𝑡≤𝑢≤𝑇∶𝑌_𝑢=𝑋_𝑢} 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 𝜏≥𝑡, we have 𝑌_𝑡=𝐸𝑌_𝜏+^𝜏𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢∣ℱ_𝑡,𝑡∈[0,𝑇].(6.10) Since 𝐴 is increasing, we have 𝐴_𝑢≥𝐴_𝑡, for all 𝑢∈[𝑡,𝜏]. Thus by using the monotonicity of 𝑓 one has 𝑌_𝑡≤𝐸𝑋_𝜏+^𝜏𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑡𝑑𝑢∣ℱ_𝑡.(6.11) Note that this holds for all stopping times 𝜏≥𝑡, we conclude that 𝑌_𝑡≤essinf_𝜏≥𝑡𝐸𝑋_𝜏+^𝜏𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑡𝑑𝑢∣ℱ_𝑡=𝑉𝑡,𝐴^𝑡;𝑌,𝑃-a.s.(6.12)
Next, define 𝜏=^∗𝑡Δinf{𝑡≤𝑢≤𝑇;𝑌_𝑢=𝑋_𝑢}⋀𝑇. Then 𝜏^∗_t is a stopping time, and the flat-off condition implies that 𝐸∫^𝜏∗𝑡_𝑡|𝑌_𝑢−𝑋_𝑢|𝑑𝐴_𝑢=0, and therefore 𝐴_𝑢=𝐴_𝑡, for all 𝑢∈[𝑡,𝜏^∗𝑡). Consequently, 𝑌_𝑡=𝐸𝑌_𝜏^∗𝑡+^𝜏∗𝑡_𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢∣ℱ_𝑡=𝐸𝑋_𝜏∗𝑡+^𝜏∗𝑡_𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑡𝑑𝑢∣ℱ_𝑡≥𝑉𝑡,𝐴_𝑡;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 𝑋_𝑡=𝐸𝜉+^𝑇𝑡𝑓𝑠,𝑌_𝑠,sup_{𝜏≤𝑣≤𝑠}𝐿_𝑣𝐴_𝑡𝑑𝑠∣ℱ^𝑡,=sup_{0≤𝑣≤𝑡+}𝐿_𝑣.(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 𝐿_𝑡=sup𝑙∈ℝ∣𝑉(𝑡,𝑙;𝑌)=𝑋_𝑡,𝑃-a.s.,(6.15) hence (𝑌,𝐴) is the solution to (i)–(iii).
We now prove the converse, that is, any solution (𝑌,𝐴) of (i)–(iii) must be the solution to the VRBSDE (1.2) with parameters (𝑓,𝑋). The uniqueness of the solution to problem (i)–(iii) will then follow from Theorem 3.5.
To see this, let (𝑌,𝐴) be the solution to (i)–(iii). By using the Stochastic Representation of [1], one can check that 𝑋_𝜏=𝐸𝜉+^𝑇𝜏𝑓𝑢,𝑌_𝑢,sup_{𝜏≤𝑣≤𝑢}𝐿_𝑣𝑑𝑢∣ℱ^𝜏,(6.16) for any stopping time 𝜏≥𝑡.
Next, we define 𝑈=_𝑡Δ𝑌_𝑡+∫^𝑡0𝑓(𝑢,𝑌_𝑢,𝐴_𝑢)𝑑𝑢. Then by definition of the optimal stopping problem we see that 𝑈_𝑡 is the value function of an optimal stopping problem with payoff 𝐻_𝑡=∫^𝑡0𝑓(𝑢,𝑌_𝑢,𝐴_𝑢)𝑑𝑢+𝑋_𝑡, that is, 𝑈_𝑡=essinf_𝜏≥𝑡𝐸[𝐻_𝜏∣ℱ_𝑡]. It then follows that −𝑈 is the Snell envelope of −𝐻, that is, −𝑈 is the smallest supermartingale that dominates −𝐻.
Now denote 𝜏=^∗𝑡Δinf𝑡≤𝑠≤𝑇∶−𝑈_𝑠=−𝐻_𝑠∧𝑇=inf𝑡≤𝑡≤𝑇∶𝑌_𝑠=𝑋_𝑠∧𝑇.(6.17) By the theory of Snell envelope (cf., e.g., [11]), we know that −𝑈_𝑡=𝐸{−𝐻_𝜏^∗𝑡∣ℱ_𝑡}, or equivalently 𝑌_𝑡=𝐸^𝜏∗𝑡_𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑋_𝜏^∗𝑡∣ℱ_𝑡=𝐸𝜉+_{𝜏∗𝑡𝑡}𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+^𝑇𝜏_∗𝑡𝑓𝑢,𝑌_𝑢,∗ŝup_{𝑡≤𝑣≤𝑢}𝐿_𝑢𝑑𝑢∣ℱ^𝑡.(6.18) The last equality is due to the Stochastic representation (6.16). From definition (ii) we see that 𝐴 is the running supreme of 𝐿 and by assumption the mapping 𝑙↦𝑓(𝑢,𝑌_𝑢,𝑙) is decreasing, we have 𝑌_𝑡≥𝐸𝜉+^𝑇𝑡𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢∣ℱ_𝑡.(6.19) But on the other hand the definition (iii) implies that the reverse direction of the above inequality also holds, thus (𝑌,𝐴) satisfies (1.2). Finally, following the same argument as that in Theorem 3.5 by using the definition (ii) it's easy to check that the flat-off condition holds. Namly (𝑌,𝐴) is a solution to the VRBSDE (1.2). The proof is now complete.

We now consider a special case where VRBSDE is linear, in the sense that 𝑓(𝑡,𝑦,𝑎)=𝜑_𝑡+𝛽_𝑡𝑦+𝛾_𝑡𝑎, where 𝜑, 𝛽, and 𝛾 are bounded, adapted processes. In particular, let us assume that |𝛽_𝑡|,|𝜑_𝑡|≤𝐿 and −𝐾≤𝛾_𝑡≤−𝑘<0, for all 𝑡∈[0,𝑇], 𝑃-a.s. Here 𝑘, 𝐾, and 𝐿 are some given positive constants.

Suppose that the linear VRBSDE (𝑓,𝑋) has a solution (𝑌,𝐴). Then, we define a martingale 𝑀_𝑡=𝐸{∫^𝑇0𝑓(𝑠,𝑌_𝑠,𝐴_𝑠)𝑑𝑠∣ℱ_𝑡}, 𝑡∈[0,𝑇] and write the VRBSDE as 𝑌_𝑡=𝑋_𝑇+^𝑇𝑡𝑓𝑠,𝑌_𝑠,𝐴_𝑠𝑑𝑠−𝑀_𝑇−𝑀_𝑡,𝑡∈[0,𝑇].(6.20) Next, we define Γ=_𝑡Δ𝑒^{∫𝑡0𝛽𝑠𝑑𝑠}, and denote 𝜉̃_𝑡=Γ_𝑡𝜉_𝑡, for 𝜉=𝑋,𝑌,𝜑,𝛾, respectively. An easy application of Itô's formula then leads to that 𝑌_𝑡=𝐸𝑋_𝑇+^𝑇𝑡𝜑_𝑠+𝛾̃_𝑠𝐴_𝑠𝑑𝑠∣ℱ_𝑡,𝑡∈[0,𝑇].(6.21) Furthermore, one also has 𝑌_𝑡≤𝑋_𝑡, 𝑡∈[0,𝑇]; and 𝐸^𝑇0||𝑌_𝑡−𝑋_𝑡||𝑑𝐴_𝑡≤‖Γ‖^∞𝐸^𝑇0||𝑌_𝑡−𝑋_𝑡||𝑑𝐴_𝑡=0.(6.22) Namely, the flat-off condition holds.

Summarizing, if we define 𝑉(𝑡,𝑙)=^Δessinf_𝜏≥𝑡𝐸{∫^𝜏𝑡[𝜑_𝑠+𝛾̃_𝑠𝑙]𝑑𝑠+𝑋_𝜏∣ℱ_𝑡}. We then have the following corollary of Theorem 6.2.

Corollary 6.3. The linear variant RBSDE has unique solution of the form 𝑌_𝑡=Γ^𝑡−1essinf_𝜏≥𝑡𝐸^𝜏𝑡Γ_𝑠𝜑_𝑠+Γ_𝑠𝛾_𝑠𝐴_𝑡𝑑𝑠+Γ_𝜏𝑋_𝜏∣ℱ_𝑡𝐴_𝑡,𝐿_𝑡=sup_{0≤𝑠≤𝑡+}𝐿_𝑡,=sup𝑙∣𝑉(𝑡,𝑙)=Γ_𝑡𝑋_𝑡.(6.23)

6.3. Universal Signal for a Family of Optimal Stopping Problems

Continuing from the previous subsection, we conclude by considering the so-called universal exercise signal for a family of optimal stopping problems, in the spirit of the “universal exercise time’’ for the family of American options proposed by Bank-Föllmer [3]. To be more precise, let (𝑌,𝐴) be the solution to our VRBSDE with generator 𝑓 and lower bound 𝑋, consider the following family of optimal stopping problems indexed by 𝑙: min_{𝜏∈𝒮[0,𝑇]}𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝑙𝑑𝑢+𝑋_𝜏,𝑙∈ℝ.(6.24)

A standard approach for solving such a problem could be to find the Snell envelope for each 𝑙. But this is obviously tedious, and often becomes unpractical when 𝑙 ranges in a large family. Instead, in [3] it was noted that a universal exercise signal for the whole family of optimal stopping problems (6.24) could be determined by the process 𝐴, which we present in the following theorem.Theorem 6.4. Suppose that (𝑌,𝐴) is a solution to the VRBSDE (1.2). For each 𝑙∈ℝ, define 𝜏=^∗𝑙Δinf𝑢≥0∣𝐴_𝑢>𝑙∧𝑇.(6.25) Then 𝜏^∗𝑙 is the optimal stopping time for the problem (𝑙) in (6.24). Namely, it holds that 𝐸^𝜏∗𝑙₀𝑓𝑢,𝑌_𝑢,𝑙𝑑𝑢+𝑋_𝜏^∗𝑙=inf^{𝜏∈𝒮[0,𝑇]}𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝑙𝑑𝑢+𝑋_𝜏,𝑙∈ℝ.(6.26)

Proof. Let 𝜏 be any stopping time in 𝒮[0,𝑇]. By the definition of 𝐴 we have 𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝑙𝑑𝑢+𝑋_𝜏=𝐸^𝜏0𝑓𝑢,𝑌_𝑢+𝐸_𝜏0,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑋_𝜏=𝐼¹+𝐼₂,(6.27) where 𝐼₁ and 𝐼₂ are the two integrals, respectively. Note that we can further decompose 𝐼₁ as follows 𝐼₁=𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢1_{𝜏≤𝜏^∗𝑙_}+𝐸_𝜏0𝑓𝑢,𝑌_𝑢,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢1_{𝜏>𝜏^∗𝑙_}=𝐼¹¹+𝐼²¹.(6.28) Since on the set {𝜏≤𝜏^∗𝑙}, we have 𝐴_𝑢≤𝑙, for all 𝑢∈[𝜏,𝜏^∗𝑙], almost surely. The monotonicity of 𝑓 then yields that 𝐼¹¹=𝐸^𝜏∗𝑙₀−^𝜏∗𝑙_𝜏𝑓𝑢,𝑌^𝑢,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢1_{𝜏≤𝜏^∗𝑙_}≥𝐸_𝜏∗𝑙0𝑓𝑢,𝑌_𝑢,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢1_{𝜏≤𝜏^∗𝑙_}.(6.29) On the other hand, since 𝐴 is an increasing process, thus 𝐴_𝑢≥𝑙 for all 𝑢≥𝜏^∗𝑙, In particular, on the set {𝜏>𝜏^∗𝑙}, it must hold that 𝑓(𝑢,𝑌_𝑢,𝑙)−𝑓(𝑢,𝑌_𝑢,𝐴_𝑢)≥0 for all 𝑢∈[𝜏^∗𝑙,𝜏]. In other words, we have 𝐼²¹=𝐸^𝜏∗𝑙₀+^𝜏𝜏_∗𝑙𝑓𝑢,𝑌^𝑢,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢1_{𝜏>𝜏^∗𝑙_}≥𝐸_𝜏∗𝑙0𝑓𝑢,𝑌_𝑢,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢1_{𝜏>𝜏^∗𝑙_}≥0,(6.30) Combining (6.29) and (6.30) we obtain that 𝐼₁≥𝐸^𝜏∗𝑙₀𝑓𝑢,𝑌_𝑢,𝑙−𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢.(6.31) We now analyze 𝐼₂. First note that since 𝑋 is the upper boundary, one must have 𝐼₂=𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑋_𝜏≥𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑌_𝜏.(6.32) But the right hand side above is equal to 𝐸𝑌₀, since (𝑌,𝐴) solve the VRBSDE (1.2), and for the same reason we can deduce (replacing 𝜏 by 𝜏^∗𝜆) that 𝐼₂≥𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑌_𝜏=𝐸𝑌⁰=𝐸^𝜏∗𝑙₀𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑌_𝜏^∗𝑙.(6.33)
We now claim that, 𝑃-almost surely, 𝜏^∗𝑙 is either a point of increase of 𝐴 or 𝜏^∗𝑙=𝑇. Indeed, for each fixed 𝜔, let us assume without loss of generality that 𝜏^∗𝑙(𝜔)<𝑇. Then, we show that 𝐴_𝜏^∗𝑙₋<𝐴_𝜏^∗𝑙_+𝜀 for any 𝜀>0 as long as 𝜏^∗𝑙+𝜀≤𝑇. To see this we first recall that by definition of 𝜏^∗𝑙, and the fact that 𝐴 is an increasing process we must have 𝐴_𝑢≥𝑙 for all 𝑢∈[𝜏^∗𝑙,𝑇]. We are to show that for any given 𝜀>0, there exists 𝑡₀=𝑡₀(𝜀)∈[𝜏^∗𝑙,𝜏^∗𝑙+𝜀] such that 𝐴_𝑡0>𝑙. In fact, if not, then 𝐴_𝑢=𝑙 for all 𝑢∈[𝜏^∗𝑙,𝜏^∗𝑙+𝜀], and this will easily lead to a contradiction to the definition of 𝜏^∗𝑙. It then follows that 𝐴_𝜏^∗𝑙_+𝜀≥𝐴_𝑡0>𝑙≤𝐴_𝜏^∗𝑙₋, proving the claim.
The direct consequence of the above claim is that 𝑌_𝜏^∗𝑙=X_𝜏^∗𝑙, thanks to the flat-off and the terminal conditions. We then derive from (6.33) that 𝐼₂≥𝐸^𝜏∗𝑙₀𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑌_𝜏^∗𝑙=𝐸^𝜏∗𝑙₀𝑓𝑢,𝑌_𝑢,𝐴_𝑢𝑑𝑢+𝑋_𝜏^∗𝑙.(6.34) This, together with (6.27) and (6.31), shows that 𝐸^𝜏0𝑓𝑢,𝑌_𝑢,𝑙𝑑𝑢+𝑋_𝜏≥𝐸^𝜏∗𝑙₀𝑓𝑢,𝑌_𝑢,𝑙𝑑𝑢+𝑋_𝜏^∗𝑙.(6.35) Namely, 𝜏^∗𝑙 it the optimal stopping time, proving the theorem.

Theorem 6.4 shows that the “reflecting process’’ in the solution of VRBSDE can be used as a universal signal for exercise, and the optimal exercise time for each problem (𝑙) is exactly the time when process 𝐴 crosses level 𝑙. A further extension of such an idea is to consider a combination of Variant Reflected BSDE with a traditional reflecting boundary, which would have the potential to be applied to study the family of callable and convertible bonds with different interest rates. We hope to address this issue in our future publications.

Acknowledgment

This author is supported in part by NSF grants DMS no. 0505472 and no. 0806017.