# On Variant Reflected Backward SDEs, with Applications

Authors: Jin Ma; Yusun Wang
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2009)

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## 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.


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## Body

## 1. Introduction

In this paper we study a new type of reflected backward stochastic differential equations based on the notion ofvariant 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}t≥0  with  A0-=-∞, such that(i)
Yt=ΔE{∫tTf(s,As)ds|ℱt}≤Xt, t∈[0,T],  P-a.s.;(ii)
E∫0T|Yt-Xt|dAt=0.The condition (ii) above is called theflat-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(1.1)Yt=∫tTf(s,As)ds-∫tTZsdBs≤Xt,0≤t≤T,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: LetX={Xt}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],
(1.2)Yt=E{XT+∫tTf(s,Ys,As)ds∣ℱt},
where the solution (Y,A) is defined to be such that(i)
Yt≤Xt, 0≤t≤T; YT=XT;(ii)
A={At} is an adapted, increasing process such that A0-=Δ-∞, and the flat-off condition holds:
(1.3)E∫0T|Yt-Xt|dAt=0.Again, if the filtration𝔽 is generated by a Brownian motion B, then we can consider an even more general form of BSDE as extension of (1.1):
(1.4)dYt=-f(t,Yt,Zt,At)dt+ZtdWt,Yt≤Xt,t∈[0,T],YT=XT,
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 processL so that the given optional obstacle process X can be written as
(1.5)XS=E{∫STf(u,supS≤v≤uLv)du∣ℱS},
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 Section2 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{ℱt}0≤t≤T is a filtration that satisfies the usual hypothses. For simplicity we assume that ℱ=ℱT. 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=𝔽B. We will always assume that 𝔽B is augmented by all the P-null sets in ℱ.We will frequently make use the following notations. Let(i)
𝕃T∞ be the space of all ℱT measurable bounded random variables,(ii)
ℍT∞ the space of all ℝ-valued, progressively measurable, bounded processes,(iii)
ℍT2 the space of all ℝd-valued, progressively measurable process Z, such that E∫0T|Zs2|ds<∞,(iv)
ℳ0,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,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 {Xτ:τ∈ℳ0,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 coefficientf:[0,T]×Ω×ℝ×ℝ×ℝ↦ℝ enjoys the following properties:(i)
for fixedω∈Ω, t∈[0,T], and y∈ℝ, the function f(ω,t,y,·):ℝ↦ℝ is continuous and strictly decreasing from +∞ to -∞,(ii)
for fixedy,l∈ℝ3, the process f(·,·,y,l) is progressively measurable with
(2.1)E∫0T|f(t,y,l)|dt≤+∞,(iii)
there exists a constantL>0, such that for all fixed t,ω,l it holds that
(2.2)|f(t,ω,y′,l)-f(t,ω,y,l)|≤L|y′-y|,∀y′,y∈ℝ,(iv)
there exist two constantsk>0 and K>0, such that for all fixed t,ω,y it holds that
(2.3)k|l′-l|≤|f(t,y,l′)-f(t,y,l)|≤K|l′-l|,∀l′,l∈ℝ.We remark that the assumption (iv) in (H1) amounts to saying that the derivative off 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 filtration𝔽 is Brownian at this point.Definition 2.1.
Letξ∈𝕃T∞ 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)dt∣ℱt};(ii)
Yt≤Xt, 0≤t≤T; YT=XT=ξ;(iv)
the processA is 𝔽-adapted, increasing,  càdlàg, and A0-=Δ-∞, such that the “flat-off’’ condition holds:
(2.4)E∫0T|Yt-Xt|dAt=0.Remark 2.2.
The assumptionA0-=-∞ 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 denoteMt=E{∫0Tf(t,Yt,At)dt∣ℱt}, t∈[0,T] then M is a martingale on [0,T], and the VRBSDE will read
(2.5)Yt=ξ+∫tTf(s,Ys,As)dt-(MT-Mt),0≤t≤T.
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 filtration𝔽=𝔽B, that is, it is generated by a standard Brownian motion B, with the usual augmentation. Let ξ∈𝕃T∞ and the boundary process X be given. A triplet of processes {(Yt,Zt,At),0≤t≤T} is called a solution of Variant Reflected BSDE with terminal value ξ and  boundary X if(i)
Y∈ℍT∞, Z∈ℍT2,(ii)
Yt=ξ+∫tTf(s,Ys,Zs,As)ds-∫tTZsdBs, 0≤t≤T,(iii)
Yt≤Xt, 0≤t≤T; YT=XT=ξ,(iv)
the process{At} is 𝔽-adapted, increasing,  càdlàg, and A0-=-∞, such that the flat-off condition holds: E∫0T|Yt-Xt|dAt=0.Our study of VRBSDE is based on aStochastic 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 processX of class (D) which is lower semicontinuous in expectation admits a representation of the form
(2.6)XS=E{XT+∫STf(u,supS≤v≤uLv)du∣ℱS}
for any stopping time S∈ℳ0,T, where L is an optional process taking values in ℝ∪{-∞}, and it can be characterized as follows:(i)
f(u,supS≤v≤uLv)∈L1(ℙ⊗dt) for any stopping time S,(ii)
LS=essinfτ>SlS,τ, where the “essinf’’ is taken over all stopping times S∈ℳ0,T such that S<T, a.s.; and lS,τ is the unique ℱS-measurable random variable satisfying:
(2.7)E{XS-Xτ∣ℱS}=E{∫Sτf(u,lS,τ)du∣ℱS},(iii)
(Gittin Index) ifV(t,l)=Δessinfτ≥tE{E∫tτf(u,l)du+Xτ|ℱt}, t∈[0,T], is the value functions of a family of optimal stopping problems indexed by l∈ℝ, then
(2.8)Lt=sup{l:V(t,l)=Xt},t∈[0,T].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 X̃t=ΔXt-E[ξ∣ℱt], 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 processX 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
(2.9)Yt=E{XT+∫tTf(s,As)ds∣ℱt},t∈[0,T].
Furthermore, the process A can be expressed as At=sup0≤s≤t+Ls, where L is the process in Theorem 2.4.We conclude this section by making following observations. First, the random variablelS,τ, defined by (2.7) is ℱS-measrable for any stopping time τ>S, thus the process s↦Ls is 𝔽-adapted. However, the running maximum process At=Δsup0≤u≤t+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 processX and the drift coefficient f:(H2)  there exists a constantΓ>0, such  that(i)
for anyS∈ℳ0,T, it holds that
(3.1)esssupτ>Sτ∈ℳ0,T|E{Xτ-XS∣ℱS}E{τ-S∣ℱS}|≤Γ,a.s.(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 mapping𝒯 on ℍT∞: for a given process y we define 𝒯(y)t=ΔYt, t∈[0,T], where Y is the unique solution of the Variant Skorohod problem:
(3.2)Yt=E{ξ+∫tTf(s,ys,As)ds∣ℱs},t∈[0,T],E∫0T[Xt-Yt]dAt=0,t∈[0,T].
We are to prove that the mapping 𝒯 is a contraction from ℍT∞ 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=sup0≤v≤t+Lv, and L is the solution to the Stochastic Representation:
(3.3)Xt=E{ξ+∫tTf(s,ys,supt≤v≤sLv)ds∣ℱt},t∈[0,T].
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(3.4)Xt=E{ξ+∫tTf(s,0,supt≤v≤sLv0)ds∣ℱt}.
Denote At0=Δsup0≤s≤t+Lt0. 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 fixeds∈[0,T] and any stopping time τ>s, let ls,τ0 be the ℱs measurable random variable such that
(3.5)E{Xs-Xτ∣ℱs}=E{∫sτf(t,0,ls,τ0)dt∣ℱs}.
Then by Theorem 2.4 we have Ls0=essinfτ>sls,τ0, and At0=sup0≤s≤t+Ls0.Now consider the set{ω:ls,τ0(ω)<0}. Since f(t,0,·) is decreasing, we have
(3.6)E{Xs-Xτ∣ℱs}-E{∫sτf(t,0,0)dt∣ℱs}=E{∫sτf(t,0,ls,τ0)-f(t,0,0)dt∣ℱs}≥E{∫sτk|ls,τ0|dt∣ℱs}≥k|ls,τ0|E{τ-s∣ℱs}.
In other words we have
(3.7)|ls,τ0|≤1k{E{Xs-Xτ∣ℱs}E{τ-s∣ℱs}-E{∫sτf(t,0,0)dt∣ℱs}E{τ-s∣ℱs}},on{ls,τ0<0}.Similarly, one can show that on the set{ls,τ0≥0} it holds that
(3.8)ls,τ0≤1k{-E{Xs-Xτ∣ℱs}E{τ-s∣ℱs}+E{∫sτf(t,0,0)dt∣ℱs}E{τ-s∣ℱs}}.Consequently, we have(3.9)|ls,τ0|≤1k{|E{Xτ-Xs∣ℱs}E{τ-s∣ℱs}|+E{∫sτ|f(t,0,0)|dt∣ℱs}E{τ-s∣ℱs}}.Now note that(3.10)|At0|=|sup0≤s≤t+Ls0|≤sup0≤s≤t+|Ls0|=sup0≤s≤t+{essinfτ>s|ls,τ0|},
we derive from (3.9) and (H2) that
(3.11)|At0|≤sup0≤s≤t+{esssupτ>s|ls,τ0|}≤sup0≤s≤t+{Γ+Γk}=2Γk,
proving the lemma.Clearly, a main task in proving that𝒯 is a contraction mapping is to find the control on the difference of two reflecting processes. To see this let y,y′∈ℍT∞ be given, and consider the two solutions of the variant Skorohod problem: (Y,A) and (Y′,A′). We would like to control |As′-As| in terms of |ys′-ys|. The following lemma is crucial.Lemma 3.3.
Assume (H1) and (H2). Then, for anyt∈[0,T], it holds almost surely that
(3.12)|At′-At|≤Lk∥y′-y∥∞.Proof.
Again, we fixs and let τ∈ℳ0,T be such that τ>s, a.s. Recalling Theorem 2.4, we let ls,τ and ls,τ′ be two ℱs-measurable random variables such that
(3.13)E{Xs-Xτ∣ℱs}=E{∫sτf(u,yu,ls,τ)du∣ℱs}=E{∫sτf(u,yu′,ls,τ′)du∣ℱs}.
Define Dsτ={ω|ls,τ′(ω)>ls,τ(ω)}, then Dsτ∈ℱs, for any stopping time τ>s.
Now, from (3.13) and noting that  1Dsτ is ℱs-measurable, we deduce that
(3.14)E{∫sτ[f(u,yu,ls,τ)-f(u,yu,ls′)]1Dsτdu∣ℱs}=E{∫sτ[f(u,yu′,ls,τ′)-f(u,yu,ls,τ′)]1Dsτdu∣ℱs}.
Now, by (H1)-(iv), the left-hand side of (3.14) satisfies
(3.15)E{∫sτ[f(u,yu,ls,τ)-f(u,yu,ls,τ′)]1Dsτdu∣ℱs}≥k|ls,τ-ls,τ′|E{τ-s∣ℱs}1Dsτ.
On the other hand, by (H1)-(iii) we see that the right-hand side of (3.14) satisfies
(3.16)E{∫sτ[f(u,yu′,ls,τ′)-f(u,yu,ls,τ′)]1Dsτdu∣ℱs}≤E{∫sτ|f(u,yu′,ls,τ′)-f(u,yu,ls,τ′)|1Dsτdu∣ℱs}≤LE{∥y′-y∥∞(τ-s)∣ℱs}1Dsτ.
Combining above we obtain that
(3.17)k|ls,τ-ls,τ′|E{τ-s∣ℱs}≤L∥y′-y∥∞E{τ-s∣ℱs},onDsτ.
Thus |ls,τ-ls,τ′|≤(L/k)∥y′-y∥∞, on Dsτ, since τ>s, a.s.
Similarly, one shows that the inequality holds on the complement ofDsτ as well. It follows that
(3.18)|ls,τ-ls,τ′|≤Lk∥y′-y∥∞.
Next, recall from Theorem2.4 that Ls=essinfτ>sls,τ, Ls′=essinfτ>sls,τ′, At=sup0≤s≤tLs, and At′=sup0≤s≤tLs′. We conclude from (3.18) that, for any t∈[0,T],
(3.19)|At′-At|=|sup0≤s≤t+Ls′-sup0≤s≤t+Ls|≤sup0≤s≤t+|essinfτ>sls,τ′-essinfτ>sls,τ|≤sup0≤s≤t+esssupτ>s|ls,τ′-ls,τ|≤Lk∥y′-y∥∞,P-a.s.
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
(3.20)E{∫sτ|yu-yu′|du∣ℱs}≤CE{τ-s∣ℱs}E{sup0≤u≤T|yu-yu′|∣ℱs}
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: yt=1{τ=1}, t∈[0,1]; ht=1{τ≤t}, t∈[0,1]; and define ℱt=σ{(yu,hu):0≤u≤t} with 𝔽={ℱt}t∈[0,1]. Then τ is an 𝔽-stopping time and y is an 𝔽-adapted continuous process.
It is easy to check thatE{∫0τ|yu|du}=p and E{τ}E{sup0≤u≤1|yu|}=(p+(1/n)(1-p))p. Thus if we choose p, n, and a constant c≥1 such that
(3.21)p<n-c(n-1)c<1,
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 mapping𝒯 defined by (3.2) is from ℍt∞ to itself. To see this, we note that by using  assumption (H1) and Lemmas 3.2 and 3.3, one has
(3.22)|f(s,ys,As)|≤|f(s,0,As0)|+L|ys|+K|As-As0|≤|f(s,0,0)|+K|As0|+L∥y∥∞+KLk∥y∥∞≤Γ+K2Γk+L∥y∥∞+KLk∥y∥∞.
Since ξ∈L∞ by assumption, we can then easily deduce that Y=𝒯(y)∈ℍT∞.
To prove that𝒯 is a contraction, we take y,y′∈ℍT∞, and denote 𝒯(y)=Y and 𝒯(y′)=Y′. Then, for any t∈[0,T], applying Lemma 3.3 we have
(3.23)|𝒯(y)t-𝒯(y′)t|≤|E{∫tT[f(s,ys,As)-f(s,ys′,As)]ds∣ℱt}|≤T(L∥y-y′∥∞+K∥A-A′∥∞)≤T(L+KLk)∥y-y′∥∞.
Since T(L+K(L/k))<1 by assumption, we see that 𝒯 is a contraction.
Now, letY∈ℍT∞ be the (unique) fixed point of 𝒯, and let A be the corresponding reflecting process defined by At=sup0≤v≤t+Lv, where L satisfies the representation
(3.24)Xt=E{ξ+∫tTf(s,Ys,supt≤v≤sLv)ds∣ℱt}.
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],
(3.25)Yt=E{ξ+∫tTf(s,Ys,As)ds∣ℱt}≤E{ξ+∫tTf(s,Ys,supt≤v≤sLv)ds∣ℱt}=Xt.
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
(3.26)E∫0T[Xt-Yt]dAt=E∫0T{∫tT[f(u,Yu,supt≤v≤uLv)-f(u,Yu,sup0≤v≤u+Lv)]du}dAt=E∫0T{∫0u[f(u,Yu,supt≤v≤u+Lv)-f(u,Yu,sup0≤v≤u+Lv)]dAt}du,
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↦supt≤v≤uLv, 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 sup0≤v≤t+δLv>sup0≤v≤t-Lv, for all δ>0. This yields that
(3.27)supt≤v≤u+Lv=sup0≤v≤u+Lv,on{(t,ω):dAt(ω)>0}.
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 Yt≤Xt, Yt′≤Xt, t∈[0,T], and
(3.28)Yt=E{ξ+∫tTf(u,Yu,Au)du∣ℱt},E∫0T|Xu-Yu|dAu=0;Yt′=E{ξ+∫tTf(u,Yu′,Au′)du∣ℱt},E∫0T|Xu-Yu′|dAu′=0.
Since bothY 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
(3.29)Ỹt=E{ξ+∫tTfY(u,Ãu)du∣ℱt},Ỹt≤Xt,ỸT=XT=ξ,E∫0T|Ỹt-Xt|dÃt=0,
where fY(u,l)=Δf(u,Yu,l). Then there exists a unique pair of process (Ỹ,Ã) 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=Ỹt and At=At′=Ãt, 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 thatY is a solution to VRBSDE with generator f and upper boundary X. Then Y0=X0.Proof.
SinceY is a fixed point of the mapping 𝒯 defined by (3.2), we see that Y0 and X0 satisfy the following equalities:
(3.30)X0=E{ξ+∫0Tf(s,Ys,sup0≤v≤sLv)ds},Y0=E{ξ+∫0Tf(s,Ys,As)ds}=E{ξ+∫0Tf(s,Ys,sup0≤v≤s+Lv)ds},
but as we argued before that the paths of the increasing process u↦sup0≤v≤uLv 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 fori=1,2,
(4.1)Yti=E{ξi+∫tTfi(u,Yui,Aui)du∣ℱt},Yti≤Xti,YTi=XTi=ξi,E∫0T|Yti-Xti|dAti=0.
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:
(4.2)s=Δinf{t∈[0,T)∣At2>At1+ε}⋀T;τ=Δinf{t∈[s,T)∣At1>At2-ε2}⋀T.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 timess 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 As-2<As+δ2 and Aτ-1<Aτ+δ1,(ii)
P{s<τ}=1; and At1≤At2-ε/2, for all t∈[s,τ], P-a.s.,(iii)
it holds thatYs2=Xs2 and Yτ1=Xτ1, P-a.s.Proof.
Since (ii) is obvious by the definition ofs and τ and (iii) is a direct consequence of (i) and the flat-off condition, we need only check property (i).
Letω 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 sn↘s, and Asn2>Asn1+ε, for n sufficiently large (may assume for all n). Since A1 is increasing, we have
(4.3)Asn2>Asn1+ε≥As1+ε≥As-1+ε.
Note that s is the first time A2 goes above A1+ε, one has As-2≤As-1+ε. Thus, Asn2>As-2, for all n. Now for any δ>0, one can choose n large enough such that sn<s+δ and it follows that As+δ2≥Asn2>As-2, that is, s is a point of increase of A2.
Thatτ 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 Ys2=Xs2 and Yτ1=Xτ1. To simplify notations let us denote δΘ=Θ1-Θ2, Θ=X,Y,A, and ξ. Furthermore, let us define two martingales Mti=ΔE{∫0Tfi(u,Yu1,Au1)du∣ℱt}, t∈[0,T], i=1,2, then on the set {s<T} we can write
(4.4)δYs=δYτ+∫sτ[f1(u,Yu1,Au1)-f2(u,Yu2,Au2)]du+(δMT-δMs),=δYτ+∫sτ∇yfu1δYudu+∫sτ[δafu1+δ2fu]du+(δMT-δMs),
where δM=ΔM1-M2, and
(4.5)∇yfu1=Δf1(u,Yu1,Au1)-f1(u,Yu2,Au1)Yu1-Yu21{Yu1≠Yu2},δafu1=Δf1(u,Yu2,Au1)-f1(u,Yu2,Au2),δ2fu=Δf1(u,Yu2,Au2)-f2(u,Yu2,Au2).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=e∫0t∇yfu1du, t∈[0,T], and apply Itô's formula to obtain that
(4.6)ΓsδYs-ΓτδYτ=∫sτΓu(δafu1+δ2fu)du-∫sτΓud(δMu).Therefore, if we assume thatf1≥f2, then δ2f≥0, dP⊗dt-a.s., and consequently, taking conditional expectation on both sides of (4.6) we have
(4.7)E{ΓsδYs-ΓτδYτ∣ℱs}=E{∫sτΓu(δafu1+δ2fu)du∣ℱs}>0.On the other hand by the flat-off condition and Lemma4.1-(iii), one can check that Ys1-Ys2≤Xs1-Xs2 and Yτ1-Yτ2≥Xτ1-Xτ2,
(4.8)E{ΓsδYs-ΓτδYτ∣ℱs}≤E{ΓsδXs-ΓτδXτ∣ℱs}.
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, At2≤At1+ε, for all t∈[0,T], P-a.s. Since ε is arbitrary, this would entail that
(4.9)At1≥At2,t∈[0,T],P-a.s.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)
Xt1≤Xt2, 0≤t≤T, a.s.,(iii)
δXs≤E[eL(t-s)δXt∣ℱs] a.s. for all s and t such that s<t. Then it holds that At1≥At2, t∈[0,T], P-a.s.We remark that the assumption (iii) in Theorem4.2 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 Theorem4.2.
We need only show that the right hand side of (4.8) is nonpositive. To see this, note that since δXτ≤0 by assumption (ii), we derive from (4.8) that
(4.10)E{ΓsδYs-ΓτδYτ∣ℱs}≤ΓsE{δXs-e∫sτ∇yfu1duδXτ∣ℱs}≤ΓsE[δXs-eL(τ-s)δXτ∣ℱs]≤0.
The last inequality is due to Assumption 3(iii) and optional sampling. This proves the theorem.We should point out that Theorem4.2 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 Theorem4.2 hold. Assume further that f1=f2, then Yt1≤Yt2, for all t∈[0,T], P-a.s.Proof.
Letf=f1=f2. Define two random functions: f̃i(t,ω,y)=Δf(t,ω,y,Ati(ω)), for (t,ω,y)∈[0,T]×Ω×ℝ. Then, Y1 and Y2 can be viewed as the solutions of BSDEs
(4.11)Yti=E{ξi+∫tTf̃i(s,Yui)du∣ℱt},t∈[0,T],i=1,2.
Note thatf̃1(t,ω,y)=f(t,ω,y,At1(ω))≤f(t,ω,y,At2(ω))=f̃2(t,ω,y), here the inequality holds due to the fact A1≥A2. Since ξ1=XT1≤XT2=ξ2, by the comparison theorem of BSDEs, we have Yt1≤Yt2, for all t∈[0,T], P-a.s.Finally, we point out that Theorem4.2 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 processX and any stopping time s and τ such that s<τ,
(5.1)ms,τ(X)=E{Xτ-Xs∣ℱs}E{τ-s∣ℱs}.
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 Xt0 in ℍT∞, 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
(5.2)Xtn=E{ξn+∫tTf(s,Ysn,Supt≤v≤sLvn)ds∣ℱt},Atn=sup0≤v≤t+Lvn,Ytn=E{ξn+∫tTf(s,Ysn,Asn)ds∣ℱt}.We now follow the similar arguments as in Theorem3.5 to obtain the following obvious estimate:
(5.3)|Ytn-Yt0|≤∥ξn-ξ0∥∞+T(L∥Yn-Y0∥∞+K∥Aun-Au0∥).Again, we need the following lemma that provides the control of|Aun-Au0|.Lemma 5.1.
Assume (H1) and (H2). Then for allt∈[0,T], it holds that
(5.4)|Atn-At0|≤sups∈[0,T]esssupτ>s1k|ms,τn-ms,τ0|+Lk∥Yn-Y0∥∞,
where mn=m(Xn), for n=0,1,2,….Proof.
The proof is very similar to that of Lemma3.3. Let ls,τn, n=0,1,2,… be the ℱs random variables such that
(5.5)E{Xsn-Xτn∣ℱs}=E{∫sτf(u,Yun,ls,τn)du∣ℱs}.
Then
(5.6)E{∫sτf(u,Yun,ls,τn)-f(u,Yu0,ls,τ0)du∣ℱs}=E{Xsn-Xτn∣ℱs}-E{Xs0-Xτ0∣ℱs}.
Then on the setDsτ={ls,τn<ls,τ0}∈ℱs we have
(5.7)1Dsτ{E{Xsn-Xτn∣ℱs}-E{Xs0-Xτ0∣ℱs}}=E{1Dsτ∫sτ[f(u,Yun,ls,τn)-f(s,Yu0,ls,τn)+f(s,Yu0,ls,τn)-f(u,Yu0,ls,τ0)]du∣ℱs}.
Since f(s,Yu0,ls,τn)>f(u,Yu0,ls,τ0) on Dsτ, we have by (H1) that f(s,Yu0,ls,τn)-f(u,Yu0,ls,τ0)≥k|ls,τn-ls,τ0| on Dsτ and hence
(5.8)1Dsτk|ls,τn-ls,τ0|E{τ-s∣ℱs}≤1Dsτ{E{Xsn-Xτn∣ℱs}-E{Xs0-Xτ0∣ℱs}}+1DsτE{∫sτL|Yun-Yu0|du∣ℱs}.
We thus conclude that
(5.9)|ls,τn-ls,τ0|≤1k|ms,τn-ms,τ0|+Lk∥Yn-Y0∥∞,P-a.s.onDsτ.
A similar argument also shows that (5.9) holds on (Dsτ)c. Hence (5.9) holds almost surely.
Finally, using the facts that|Lsn-Ls0|=|essinfτ>sls,τn-essinfτ>sls,τ0|≤esssupτ>s|ls,τn-ls,τ0|, we conclude that, for any t∈[0,T], it holds P-almost surely that
(5.10)|Atn-At0|=|sup0≤s≤t+Lsn-sup0≤s≤t+Ls0|≤sup0≤s≤T|Lsn-Ls0|≤sup0≤s≤Tesssupτ>s1k|ms,τn-ms,τ0|+Lk∥Yn-Y0∥∞,
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
(5.11)|Ytn-Yt0|≤11-(1+(K/k))LT{∥ξn-ξ0∥∞+KTk∥sups∈[0,T]esssupτ>s|ms,τn-ms,τ0|∥∞}.


## 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 currentlevel of satisfaction, defined as a weighted average of the accumulated consumptions:
(6.1)At=A(C)t=Δηt+∫0tθ(t,s)dCs,t∈[0,T],
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↦Ct is the accumulated consumption up to time t (hence C={Ct:t≥0} is an increasing process, called a consumption plan). The Hindy-Huang-Kreps utility is then defined by (cf. [7])
(6.2)EU(C)=ΔE{V(CT)+∫0Tu(t,A(C)t)dt},
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
(6.3)Ut(C)=E{VT+∫tTu(r,Ur(C),A(C)r)dr∣ℱt},t∈[0,T]
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 thatη 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 X̃, and assume that it is an optional process of Class (D) so that Ut≥X̃t 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=Ut(C)≥X̃t, t∈[0,T], P-a.s. Our goal is then to find C*∈𝒜 that minimizes the expected utility (or cost)
(6.4)EU0=ΔE{X̃T+∫0Tu(r,Ur,Ar)dr},
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*=minC∈𝒜E{U0(C)}.We remark that the set of admissible consumption plans𝒜 is not empty. In fact, let Yt=-Ut, Zt=-Z̃t, Xt=-X̃t and define f(t,y,l)=Δ-u(t,-y,l). Then we can write the recursive utility as
(6.5)Yt=E{YT+∫tTf(s,Ys,As)ds∣ℱt},t∈[0,T].Let us now assume further that the functionf 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:
(6.6)Ut0=E{X̃T+∫tTu(r,Ur0,Ar0)dr∣ℱt},Ut0≥X̃t,t∈[0,T],E∫0T|Ut0-X̃t|dAt0=0.
Clearly, this implies that A0∈𝒜. Furthermore, for any ε>0, define Atε=At0+ε, and let Uε be the solution to the BSDE Utε=E{X̃T+∫tTu(r,Urε,Arε)dr∣ℱT}. By the comparison theorem of BSDEs, the utility Utε≥Ut0≥X̃t, thus Aε∈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 satisfactionA is such that the associated utility U coincides with the lower boundary X̃. 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 levelA* allows its associated recursive utility U* follow the VRBSDE with the exogenous lower bound X̃. 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=X̃0≤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 Theorem4.2 also leads to the comparison between different recursive utilities corresponding to different lower boundaries. Namely, if X̃i, 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 X̃t1≥X̃t2, 0≤t≤T, a.s., implies that Ut1≥Ut2 and At1≥At2, 0≤t≤T, a.s. In particular, it holds that E[U01]≥E[U02].


### 6.2. VRBSDE and Optimal Stopping Problems

We now look at a possible extension of the so-calledmultiarmed 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∈ℍT∞:
(6.7)V(t,l;Y)=Δessinfτ≥tE{∫tτf(u,Yu,l)du+Xτ∣ℱt}.
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
(6.8)V(t,l;Y)≤Xt,t∈[0,T],P-a.s.
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), 0≤t≤T,(ii)
At=sup0≤s≤t+Ls and Ls=sup{l∈ℝ:V(s,l;Y)=Xs},(iii)
it holds that(6.9)Yt=essinfτ≥tE{∫tτf(u,Yu,Au)du+Xτ∣ℱt},t∈[0,T].Furthermore, the stopping time τt*=inf{t≤u≤T:Yu=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 τ≥t, we have
(6.10)Yt=E{Yτ+∫tτf(u,Yu,Au)du∣ℱt},t∈[0,T].
Since A is increasing, we have Au≥At, for all u∈[t,τ]. Thus by using the monotonicity of f one has
(6.11)Yt≤E{Xτ+∫tτf(u,Yu,At)du∣ℱt}.
Note that this holds for all stopping times τ≥t, we conclude that
(6.12)Yt≤essinfτ≥tE{Xτ+∫tτf(u,Yu,At)du∣ℱt}=V(t,At;Y),P-a.s.
Next, defineτt*=Δinf{t≤u≤T;Yu=Xu}⋀T. Then τt* is a stopping time, and the flat-off condition implies that E∫tτt*|Yu-Xu|dAu=0, and therefore Au=At, for all u∈[t,τt*). Consequently,
(6.13)Yt=E{Yτt*+∫tτt*f(u,Yu,Au)du∣ℱt}=E{Xτt*+∫tτt*f(u,Yu,At)du∣ℱt}≥V(t,At;Y),P-a.s.
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
(6.14)Xt=E{ξ+∫tTf(s,Ys,supτ≤v≤sLv)ds∣ℱt},At=sup0≤v≤t+Lv.
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
(6.15)Lt=sup{l∈ℝ∣V(t,l;Y)=Xt},P-a.s.,
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
(6.16)Xτ=E{ξ+∫τTf(u,Yu,supτ≤v≤uLv)du∣ℱτ},
for any stopping time τ≥t.
Next, we defineUt=ΔYt+∫0tf(u,Yu,Au)du. Then by definition of the optimal stopping problem we see that Ut is the value function of an optimal stopping problem with payoff Ht=∫0tf(u,Yu,Au)du+Xt, that is, Ut=essinfτ≥tE[Hτ∣ℱt]. It then follows that -U is the Snell envelope of -H, that is, -U is the smallest supermartingale that dominates -H.
Now denote(6.17)τt*=Δinf{t≤s≤T:-Us=-Hs}⋀T=inf{t≤t≤T:Ys=Xs}⋀T.
By the theory of Snell envelope (cf., e.g., [11]), we know that -Ut=E{-Hτt*∣ℱt}, or equivalently
(6.18)Yt=E{∫tτt*f(u,Yu,Au)du+Xτt*∣ℱt}=E{ξ+∫tτt*f(u,Yu,Au)du+∫τt*Tf(u,Yu,sup*̂t≤v≤uLu)du∣ℱt}.
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,Yu,l) is decreasing, we have
(6.19)Yt≥E{ξ+∫tTf(u,Yu,Au)du∣ℱt}.
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 islinear, in the sense that f(t,y,a)=φt+βty+γta, where φ, β, and γ are bounded, adapted processes. In particular, let us assume that |βt|,|φt|≤L and -K≤γt≤-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 Mt=E{∫0Tf(s,Ys,As)ds∣ℱt}, t∈[0,T] and write the VRBSDE as
(6.20)Yt=XT+∫tTf(s,Ys,As)ds-(MT-Mt),t∈[0,T].
Next, we define Γt=Δe∫0tβsds, and denote ξ̃t=Γtξt, for ξ=X,Y,φ,γ, respectively. An easy application of Itô's formula then leads to that
(6.21)Ỹt=E{X̃T+∫tT[φ̃s+γ̃sAs]ds∣ℱt},t∈[0,T].
Furthermore, one also has Ỹt≤X̃t, t∈[0,T]; and
(6.22)E{∫0T|Ỹt-X̃t|dAt}≤∥Γ∥∞E{∫0T|Yt-Xt|dAt}=0.
Namely, the flat-off condition holds.Summarizing, if we defineṼ(t,l)=Δessinfτ≥tE{∫tτ[φ̃s+γ̃sl]ds+X̃τ∣ℱt}. We then have the following corollary of Theorem 6.2.Corollary 6.3.
The linear variant RBSDE has unique solution of the form(6.23)Yt=Γt-1essinfτ≥tE{∫tτΓsφs+ΓsγsAtds+ΓτXτ∣ℱt},At=sup0≤s≤t+Lt,Lt=sup{l∣Ṽ(t,l)=ΓtXt}.


### 6.3. Universal Signal for a Family of Optimal Stopping Problems

Continuing from the previous subsection, we conclude by considering the so-calleduniversal 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:
(6.24)minτ∈𝒮[0,T]E{∫0τf(u,Yu,l)du+Xτ},l∈ℝ.A standard approach for solving such a problem could be to find the Snell envelope for eachl. 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
(6.25)τl*=Δinf{u≥0∣Au>l}⋀T.
Then τl* is the optimal stopping time for the problem (l) in (6.24). Namely, it holds that
(6.26)E{∫0τl*f(u,Yu,l)du+Xτl*}=infτ∈𝒮[0,T]E{∫0τf(u,Yu,l)du+Xτ},l∈ℝ.Proof.
Letτ be any stopping time in 𝒮[0,T]. By the definition of A we have
(6.27)E{∫0τf(u,Yu,l)du+Xτ}=E{∫0τ[f(u,Yu,l)-f(u,Yu,Au)]du}+E{∫0τf(u,Yu,Au)du+Xτ}=I1+I2,
where I1 and I2 are the two integrals, respectively. Note that we can further decompose I1 as follows
(6.28)I1=E{∫0τ[f(u,Yu,l)-f(u,Yu,Au)]du1{τ≤τl*}}+E{∫0τ[f(u,Yu,l)-f(u,Yu,Au)]du1{τ>τl*}}=I11+I12.
Since on the set {τ≤τl*}, we have Au≤l, for all u∈[τ,τl*], almost surely. The monotonicity of f then yields that
(6.29)I11=E{(∫0τl*-∫ττl*)[f(u,Yu,l)-f(u,Yu,Au)]du1{τ≤τl*}}≥E{∫0τl*[f(u,Yu,l)-f(u,Yu,Au)]du1{τ≤τl*}}.
On the other hand, since A is an increasing process, thus Au≥l for all u≥τl*, In particular, on the set {τ>τl*}, it must hold that f(u,Yu,l)-f(u,Yu,Au)≥0 for all u∈[τl*,τ]. In other words, we have
(6.30)I12=E{(∫0τl*+∫τl*τ)[f(u,Yu,l)-f(u,Yu,Au)]du1{τ>τl*}}≥E{∫0τl*[f(u,Yu,l)-f(u,Yu,Au)]du1{τ>τl*}}≥0,
Combining (6.29) and (6.30) we obtain that
(6.31)I1≥E{∫0τl*[f(u,Yu,l)-f(u,Yu,Au)]du}.
We now analyze I2. First note that since X is the upper boundary, one must have
(6.32)I2=E{∫0τf(u,Yu,Au)du+Xτ}≥E{∫0τf(u,Yu,Au)du+Yτ}.
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 τλ*) that
(6.33)I2≥E{∫0τf(u,Yu,Au)du+Yτ}=EY0=E{∫0τl*f(u,Yu,Au)du+Yτl*}.
We now claim that,P-almost surely, τl* is either a point of increase of A or τl*=T. Indeed, for each fixed ω, let us assume without loss of generality that τl*(ω)<T. Then, we show that Aτl*-<Aτl*+ε for any ε>0 as long as τl*+ε≤T. To see this we first recall that by definition of τl*, and the fact that A is an increasing process we must have Au≥l for all u∈[τl*,T]. We are to show that for any given ε>0, there exists t0=t0(ε)∈[τl*,τl*+ε] such that At0>l. In fact, if not, then Au=l for all u∈[τl*,τl*+ε], and this will easily lead to a contradiction to the definition of τl*. It then follows that Aτl*+ε≥At0>l≤Aτl*-, proving the claim.
The direct consequence of the above claim is thatYτl*=Xτl*, thanks to the flat-off and the terminal conditions. We then derive from (6.33) that
(6.34)I2≥E{∫0τl*f(u,Yu,Au)du+Yτl*}=E{∫0τl*f(u,Yu,Au)du+Xτl*}.
This, together with (6.27) and (6.31), shows that
(6.35)E{∫0τf(u,Yu,l)du+Xτ}≥E{∫0τl*f(u,Yu,l)du+Xτl*}.
Namely, τl* it the optimal stopping time, proving the theorem.Theorem6.4 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.


## 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 currentlevel of satisfaction, defined as a weighted average of the accumulated consumptions:
(6.1)At=A(C)t=Δηt+∫0tθ(t,s)dCs,t∈[0,T],
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↦Ct is the accumulated consumption up to time t (hence C={Ct:t≥0} is an increasing process, called a consumption plan). The Hindy-Huang-Kreps utility is then defined by (cf. [7])
(6.2)EU(C)=ΔE{V(CT)+∫0Tu(t,A(C)t)dt},
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
(6.3)Ut(C)=E{VT+∫tTu(r,Ur(C),A(C)r)dr∣ℱt},t∈[0,T]
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 thatη 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 X̃, and assume that it is an optional process of Class (D) so that Ut≥X̃t 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=Ut(C)≥X̃t, t∈[0,T], P-a.s. Our goal is then to find C*∈𝒜 that minimizes the expected utility (or cost)
(6.4)EU0=ΔE{X̃T+∫0Tu(r,Ur,Ar)dr},
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*=minC∈𝒜E{U0(C)}.We remark that the set of admissible consumption plans𝒜 is not empty. In fact, let Yt=-Ut, Zt=-Z̃t, Xt=-X̃t and define f(t,y,l)=Δ-u(t,-y,l). Then we can write the recursive utility as
(6.5)Yt=E{YT+∫tTf(s,Ys,As)ds∣ℱt},t∈[0,T].Let us now assume further that the functionf 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:
(6.6)Ut0=E{X̃T+∫tTu(r,Ur0,Ar0)dr∣ℱt},Ut0≥X̃t,t∈[0,T],E∫0T|Ut0-X̃t|dAt0=0.
Clearly, this implies that A0∈𝒜. Furthermore, for any ε>0, define Atε=At0+ε, and let Uε be the solution to the BSDE Utε=E{X̃T+∫tTu(r,Urε,Arε)dr∣ℱT}. By the comparison theorem of BSDEs, the utility Utε≥Ut0≥X̃t, thus Aε∈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 satisfactionA is such that the associated utility U coincides with the lower boundary X̃. 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 levelA* allows its associated recursive utility U* follow the VRBSDE with the exogenous lower bound X̃. 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=X̃0≤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 Theorem4.2 also leads to the comparison between different recursive utilities corresponding to different lower boundaries. Namely, if X̃i, 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 X̃t1≥X̃t2, 0≤t≤T, a.s., implies that Ut1≥Ut2 and At1≥At2, 0≤t≤T, a.s. In particular, it holds that E[U01]≥E[U02].


## 6.2. VRBSDE and Optimal Stopping Problems

We now look at a possible extension of the so-calledmultiarmed 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∈ℍT∞:
(6.7)V(t,l;Y)=Δessinfτ≥tE{∫tτf(u,Yu,l)du+Xτ∣ℱt}.
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
(6.8)V(t,l;Y)≤Xt,t∈[0,T],P-a.s.
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), 0≤t≤T,(ii)
At=sup0≤s≤t+Ls and Ls=sup{l∈ℝ:V(s,l;Y)=Xs},(iii)
it holds that(6.9)Yt=essinfτ≥tE{∫tτf(u,Yu,Au)du+Xτ∣ℱt},t∈[0,T].Furthermore, the stopping time τt*=inf{t≤u≤T:Yu=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 τ≥t, we have
(6.10)Yt=E{Yτ+∫tτf(u,Yu,Au)du∣ℱt},t∈[0,T].
Since A is increasing, we have Au≥At, for all u∈[t,τ]. Thus by using the monotonicity of f one has
(6.11)Yt≤E{Xτ+∫tτf(u,Yu,At)du∣ℱt}.
Note that this holds for all stopping times τ≥t, we conclude that
(6.12)Yt≤essinfτ≥tE{Xτ+∫tτf(u,Yu,At)du∣ℱt}=V(t,At;Y),P-a.s.
Next, defineτt*=Δinf{t≤u≤T;Yu=Xu}⋀T. Then τt* is a stopping time, and the flat-off condition implies that E∫tτt*|Yu-Xu|dAu=0, and therefore Au=At, for all u∈[t,τt*). Consequently,
(6.13)Yt=E{Yτt*+∫tτt*f(u,Yu,Au)du∣ℱt}=E{Xτt*+∫tτt*f(u,Yu,At)du∣ℱt}≥V(t,At;Y),P-a.s.
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
(6.14)Xt=E{ξ+∫tTf(s,Ys,supτ≤v≤sLv)ds∣ℱt},At=sup0≤v≤t+Lv.
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
(6.15)Lt=sup{l∈ℝ∣V(t,l;Y)=Xt},P-a.s.,
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
(6.16)Xτ=E{ξ+∫τTf(u,Yu,supτ≤v≤uLv)du∣ℱτ},
for any stopping time τ≥t.
Next, we defineUt=ΔYt+∫0tf(u,Yu,Au)du. Then by definition of the optimal stopping problem we see that Ut is the value function of an optimal stopping problem with payoff Ht=∫0tf(u,Yu,Au)du+Xt, that is, Ut=essinfτ≥tE[Hτ∣ℱt]. It then follows that -U is the Snell envelope of -H, that is, -U is the smallest supermartingale that dominates -H.
Now denote(6.17)τt*=Δinf{t≤s≤T:-Us=-Hs}⋀T=inf{t≤t≤T:Ys=Xs}⋀T.
By the theory of Snell envelope (cf., e.g., [11]), we know that -Ut=E{-Hτt*∣ℱt}, or equivalently
(6.18)Yt=E{∫tτt*f(u,Yu,Au)du+Xτt*∣ℱt}=E{ξ+∫tτt*f(u,Yu,Au)du+∫τt*Tf(u,Yu,sup*̂t≤v≤uLu)du∣ℱt}.
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,Yu,l) is decreasing, we have
(6.19)Yt≥E{ξ+∫tTf(u,Yu,Au)du∣ℱt}.
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 islinear, in the sense that f(t,y,a)=φt+βty+γta, where φ, β, and γ are bounded, adapted processes. In particular, let us assume that |βt|,|φt|≤L and -K≤γt≤-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 Mt=E{∫0Tf(s,Ys,As)ds∣ℱt}, t∈[0,T] and write the VRBSDE as
(6.20)Yt=XT+∫tTf(s,Ys,As)ds-(MT-Mt),t∈[0,T].
Next, we define Γt=Δe∫0tβsds, and denote ξ̃t=Γtξt, for ξ=X,Y,φ,γ, respectively. An easy application of Itô's formula then leads to that
(6.21)Ỹt=E{X̃T+∫tT[φ̃s+γ̃sAs]ds∣ℱt},t∈[0,T].
Furthermore, one also has Ỹt≤X̃t, t∈[0,T]; and
(6.22)E{∫0T|Ỹt-X̃t|dAt}≤∥Γ∥∞E{∫0T|Yt-Xt|dAt}=0.
Namely, the flat-off condition holds.Summarizing, if we defineṼ(t,l)=Δessinfτ≥tE{∫tτ[φ̃s+γ̃sl]ds+X̃τ∣ℱt}. We then have the following corollary of Theorem 6.2.Corollary 6.3.
The linear variant RBSDE has unique solution of the form(6.23)Yt=Γt-1essinfτ≥tE{∫tτΓsφs+ΓsγsAtds+ΓτXτ∣ℱt},At=sup0≤s≤t+Lt,Lt=sup{l∣Ṽ(t,l)=ΓtXt}.


## 6.3. Universal Signal for a Family of Optimal Stopping Problems

Continuing from the previous subsection, we conclude by considering the so-calleduniversal 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:
(6.24)minτ∈𝒮[0,T]E{∫0τf(u,Yu,l)du+Xτ},l∈ℝ.A standard approach for solving such a problem could be to find the Snell envelope for eachl. 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
(6.25)τl*=Δinf{u≥0∣Au>l}⋀T.
Then τl* is the optimal stopping time for the problem (l) in (6.24). Namely, it holds that
(6.26)E{∫0τl*f(u,Yu,l)du+Xτl*}=infτ∈𝒮[0,T]E{∫0τf(u,Yu,l)du+Xτ},l∈ℝ.Proof.
Letτ be any stopping time in 𝒮[0,T]. By the definition of A we have
(6.27)E{∫0τf(u,Yu,l)du+Xτ}=E{∫0τ[f(u,Yu,l)-f(u,Yu,Au)]du}+E{∫0τf(u,Yu,Au)du+Xτ}=I1+I2,
where I1 and I2 are the two integrals, respectively. Note that we can further decompose I1 as follows
(6.28)I1=E{∫0τ[f(u,Yu,l)-f(u,Yu,Au)]du1{τ≤τl*}}+E{∫0τ[f(u,Yu,l)-f(u,Yu,Au)]du1{τ>τl*}}=I11+I12.
Since on the set {τ≤τl*}, we have Au≤l, for all u∈[τ,τl*], almost surely. The monotonicity of f then yields that
(6.29)I11=E{(∫0τl*-∫ττl*)[f(u,Yu,l)-f(u,Yu,Au)]du1{τ≤τl*}}≥E{∫0τl*[f(u,Yu,l)-f(u,Yu,Au)]du1{τ≤τl*}}.
On the other hand, since A is an increasing process, thus Au≥l for all u≥τl*, In particular, on the set {τ>τl*}, it must hold that f(u,Yu,l)-f(u,Yu,Au)≥0 for all u∈[τl*,τ]. In other words, we have
(6.30)I12=E{(∫0τl*+∫τl*τ)[f(u,Yu,l)-f(u,Yu,Au)]du1{τ>τl*}}≥E{∫0τl*[f(u,Yu,l)-f(u,Yu,Au)]du1{τ>τl*}}≥0,
Combining (6.29) and (6.30) we obtain that
(6.31)I1≥E{∫0τl*[f(u,Yu,l)-f(u,Yu,Au)]du}.
We now analyze I2. First note that since X is the upper boundary, one must have
(6.32)I2=E{∫0τf(u,Yu,Au)du+Xτ}≥E{∫0τf(u,Yu,Au)du+Yτ}.
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 τλ*) that
(6.33)I2≥E{∫0τf(u,Yu,Au)du+Yτ}=EY0=E{∫0τl*f(u,Yu,Au)du+Yτl*}.
We now claim that,P-almost surely, τl* is either a point of increase of A or τl*=T. Indeed, for each fixed ω, let us assume without loss of generality that τl*(ω)<T. Then, we show that Aτl*-<Aτl*+ε for any ε>0 as long as τl*+ε≤T. To see this we first recall that by definition of τl*, and the fact that A is an increasing process we must have Au≥l for all u∈[τl*,T]. We are to show that for any given ε>0, there exists t0=t0(ε)∈[τl*,τl*+ε] such that At0>l. In fact, if not, then Au=l for all u∈[τl*,τl*+ε], and this will easily lead to a contradiction to the definition of τl*. It then follows that Aτl*+ε≥At0>l≤Aτl*-, proving the claim.
The direct consequence of the above claim is thatYτl*=Xτl*, thanks to the flat-off and the terminal conditions. We then derive from (6.33) that
(6.34)I2≥E{∫0τl*f(u,Yu,Au)du+Yτl*}=E{∫0τl*f(u,Yu,Au)du+Xτl*}.
This, together with (6.27) and (6.31), shows that
(6.35)E{∫0τf(u,Yu,l)du+Xτ}≥E{∫0τl*f(u,Yu,l)du+Xτl*}.
Namely, τl* it the optimal stopping time, proving the theorem.Theorem6.4 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.



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*Source: 854768-2009-06-18.xml*