# An Explicit Representation of the Extended Skorokhod Map with Two Time-Dependent Boundaries

Authors: Marek Slaby
Journal: Journal of Probability and Statistics
 (2010)

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## Abstract

We consider the extended Skorokhod problem for real-valued càdlàg functions with the constraining interval[α,β], where α and β change in time as values of two càdlàg functions. We find an explicit
form of the solution and discuss its continuity properties with respect to the uniform, J1 and M1, metrics on the space of càdlàg functions. We develop a useful technique of extending known results
for the Skorokhod maps onto the larger class of extended Skorokhod maps.


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## Body

## 1. Introduction

The Skorokhod problem (SP) was introduced originally in [1] as a tool for solving stochastic differential equations in a domain with one fixed reflective boundary. Given a function ψ, a solution of the Skorokhod problem on [0,∞) is a pair of functions (ϕ,η) such that ϕ≥0,η is nondecreasing, and ∫0∞I{ϕ(s)>0}dη(s)=0. The mapping Γ0(ψ)=ϕ is called a Skorokhod map (SM) on [0,∞) and is well defined for all càdlàg functions ψ.The Skorokhod problem has been studied since [1] in more general settings. Chaleyat-Maural et al. studied in [2] a Skorokhod map constraining functions to [0,a]. A multidimensional version of the SP was introduced by Tanaka in [3]. Over the years, numerous applications were found for the SM particularly in queueing theory. In [4], Kruk et al. provided an explicit formula and studied the properties of the two-sided Skorokhod map constraining the function to remain in the interval [0,a], where a is a positive constant. More recently attempts were made towards relaxing the rigidity of the constrains. Burdzy et al. in [5] found an explicit representation for the so called extended Skorokhod map (ESM), which is a relaxed version of the SM. [The constraining interval in that paper varies with time.] Another explicit representation for the SM with two time-dependent boundaries, different from the representation in [5] and based on the approach in [4], was developed by the author in [6]. In addition, a number of properties of the SM were studied in [6] including its continuity and Lipschitz conditions.In this paper we obtain an alternative form of the explicit formula for the ESM with two time-dependent boundaries developed in [5] that is simpler to understand and potentially more useful for applications and generalizations to higher dimensions. We develop methods of extending certain properties of the SM onto the ESM and use them to analyze continuity properties of the ESM.Throughout the paper,D[0,∞) will denote real-valued càdlàg functions on [0,∞),D-[0,∞), and D+[0,∞) will denote càdlàg functions on [0,∞) taking values in ℝ∪{-∞} and in ℝ∪{∞}, respectively. A function is càdlàg if it is right-continuous and has finite left limits at every t≥0. Similarly, we will use I[0,∞) and BV[0,∞) to denote subspaces of D[0,∞) consisting of nondecreasing functions and functions with bounded variation on every finite interval, respectively. We will use Df to denote the effective domain of f, that is, Df={t≥0∣∞<f(t)<∞}.On the space of the càdlàg functions we will consider the topology of the uniform convergence and the topology of the uniform convergence on compact sets. For everyT>0, let ∥f∥T=sup0≤t≤T|f(t)| and ∥f∥=supt≥0|f(t)|. Let fn be a sequence of functions in D+[0,∞) or in D-[0,∞). We say that fn converges to f uniformly on compact sets if for every T>0,limn→∞∥fn-f∥T=0. Equivalently, we could say that fn converges to f uniformly on compact sets if fn=f on Dfc for large enough n and fn converges to f uniformly on [0,T]∩Df for every T>0.Definition 1.1 (extended Skorokhod problem).
Letα∈D-[0,∞),β∈D+[0,∞) be such that α≤β, and let ψ∈D[0,∞). A pair of real-valued càdlàg functions, (ϕ,η), is said to be a solution of the extended Skorokhod problem (ESP) on [α,β] for ψ if the following three properties are satisfied:(i)
for everyt≥0,ϕ(t)=ψ(t)+η(t)∈[α(t),β(t)];(ii)
for every0≤s≤t,(1.1)η(t)-η(s)≥0ifϕ(r)<β(r)∀r∈(s,t],η(t)-η(s)≤0ifϕ(r)>α(r)∀r∈(s,t];(iii)
for everyt≥0,(1.2)η(t)-η¯(t-)≥0ifϕ(t)<β(t),η(t)-η(t-)≤0ifϕ(t)>α(t).Traditionallyη(0-) in (iii) is defined as zero so that η has a jump at 0, whenever η(0)>0. The map Γ¯α,β:D[0,∞)→D[0,∞) defined by Γ¯α,β(ψ)=ϕ is called the extended Skorokhod map (ESM) on [α,β]. In the traditional Skorokhod problem, conditions (ii) and (iii) are replaced by a stronger condition.Definition 1.2 (Skorokhod problem).
Letα,β∈D[0,∞) be such that α≤β. Given ψ∈D[0,∞), a pair of functions (ϕ,η)∈D[0,∞)×BV[0,∞) is said to be a solution of the Skorokhod problem on [α,β] for ψ if the following two properties are satisfied:(i)
for everyt∈[0,∞),ϕ(t)=ψ(t)+η(t)∈[α(t),β(t)];(ii)
η(0-)=0 and η have the decomposition η=ηl-ηu, where ηl,ηu∈I[0,∞),(1.3)∫0∞I{ϕ(s)>α(t)}dηl(s)=0,∫0∞I{ϕ(s)<β(s)}dηu(s)=0.Burdzy et al. have shown in Theorem  2.6 of [5] that for any α∈D-[0,∞) and β∈D+[0,∞) such that α≤β, there is a well-defined ESM Γ¯α,β and it is represented by
(1.4)Γ¯α,β(ψ)=ψ-Ξα,β(ψ),
where Ξα,β(ψ):D[0,∞)→D[0,∞) is given by
(1.5)Ξα,β(ψ)(t)=max{sup0≤s≤t[(ψ(0)-β(0))+∧inf0≤r≤t(ψ(r)-α(r))],sup0≤s≤t[(ψ(s)-β(s))∧infs≤r≤t(ψ(r)-α(r))]}.
They obtained their result first for simple functions and then extended it by the limiting process. In Section 2 we will develop an alternative version of this formula.It is easy to see that if(ϕ,η) is a solution of the SP on [α,β] for ψ, then it is also a solution of the ESP on [α,β] for ψ. Conversely, it is shown in Proposition  2.3 of [5] that a solution (ϕ,η) of the ESP solves also the corresponding SP whenever η∈BV[0,∞). Furthermore, Corollary  2.4 of [5] shows that if inft≥0(β(t)-α(t))>0 and (ϕ,η) is a solution of the ESP on [α,β] for ψ, then η∈BV[0,∞). Therefore we can identify the ESM with the SM in this special case.Remark 1.3.
Ifinft≥0(β(t)-α(t))>0, then Γ¯α,β=Γα,β.


## 2. Alternative Explicit Formula for the Two-Sided Extended Skorokhod Map with Time-Dependent Boundaries

We will make the explicit formula (1.4) more user friendly by developing a new expression for the constraining term Ξ that is easier to understand and shows more promise for possible extensions to higher dimensions than (1.5). Given α∈D-[0,∞) and β∈D+[0,∞) such that α≤β, we introduce two pairs of times(2.1)Tα=min{t>0∣α(t)-ψ(t)≥0},Tβ=min{t>0∣ψ(t)-β(t)≥0},(2.2)τα=inf{t>0∣α(t)-ψ(t)>0},τβ=inf{t>0∣ψ(t)-β(t)>0}.
Note that
(2.3)Tα≤τα,Tβ≤τβ.
Also note that the four times depend on ψ. When necessary we will indicate it by using full notation such as τα(ψ) or τβ(ψ).Remark 2.1.
Letψ∈D[0,∞),α,αn,∈D-[0,∞),n=1,2,3,…, and β,βn∈D+[0,∞), n=1,2,3,…. If αn→n→∞α and βn→n→∞β pointwise, then
(2.4)lim supn→∞τβn≤τβ,lim supn→∞ταn≤τα.Proof.
Letϵ>0. By (2.2) there is τβ≤t0<τβ+ϵ such that ψ(t0)-β(t0)>0. Let ψ(t0)-β(t0)=δ. There is n0 such that |βn(t0)-β(t0)|<δ/2 for n≥n0. Hence
(2.5)ψ(t0)-βn(t0)=ψ(t0)-β(t0)+β(t0)-βn(t0)≥δ0-|β(t0)-βn(t0)|≥δ02>0.
Thus τβn≤t0<τβ+ϵ. Since ϵ is arbitrary, we conclude that lim supn→∞τβn≤τβ. By similar argument we can show that lim supn→∞ταn≤τα.The inequalities in Remark2.1 distinguish τα and τβ from Tα and Tβ and will be essential for the proof of Theorem 2.11.Remark 2.2.
Letα∈D-[0,∞) and β∈D+[0,∞) be such that inft≥0(β(t)-α(t))>0. For any ψ∈D[0,∞), there are three possibilities
(2.6)eitherα(t)<ψ(t)<β(t)foreveryt≥0,Tα<TβorTβ<Tα.
Similarly, in terms of τα and τβ, the following three cases are possible:
(2.7)eitherα(t)≤ψ(t)≤β(t)foreveryt≥0,τα<τβorτβ<τα.
Clearly, α(t)<ψ(t)<β(t) for every t≥0 if and only if Tα=Tβ=∞, and α(t)≤ψ(t)≤β(t) for every t≥0 if and only if τα=τβ=∞.Remark 2.3.
It follows from the definition of the ESM that for every0≤t≤τα∧τβ,α(t)≤ψ(t)≤β(t),Γ¯α,β(ψ)(t)=ψ(t) and Ξα,β(ψ)(t)=0. Similarly, we obtain that for every 0≤t<Tα∧Tβ,α(t)<ψ(t)<β(t) and Γ¯α,β(ψ)(t)=ψ(t).Under the assumption thatinft≥0(β(t)-α(t))>0, we define two increasing sequences of times {Tk∣k=0,1,2,…} and {Sk∣k=0,1,2,…} similar to the sequences τk,σk used in [6]. If Tβ<Tα, we set T0=0 and S0=Tβ; for k≥1, we set(2.8)Tk=min{t>Sk-1∣supSk-1≤s≤t[ψ(s)-β(s)]≥ψ(t)-α(t)},(2.9)Sk=min{t>Tk∣ψ(t)-β(t)≥infTk≤r≤t(ψ(r)-α(r))}.
If Tα<Tβ, we set T0=Tα, we define Sk for all k by (2.9), and we define Tk for k≥1 by (2.8).It is easy to see that0≤T0≤S0<T1<S1<T2<S2<⋯ unless one of the times equals ∞, at which point all the following times are also ∞. Also note that the time sequences depend on ψ, α and β. Finally, as in Proposition  2.1 of [6], if inft≥0(β(t)-α(t))>0, then
(2.10)limk→∞Tk=∞,limk→∞Sk=∞.
The following observations follow immediately from the definition of Sk.Remark 2.4.
Ifk>1 or if k=0 and Tα<Tβ, then
(2.11)ψ(Sk)-β(Sk)≥infTk≤r≤Sk(ψ(r)-α(r))
and for every s∈[Tk,Sk)(2.12)ψ(s)-β(s)<infTk≤r≤s(ψ(r)-α(r)).
If Tβ<Tα, then
(2.13)ψ(S0)-β(S0)≥0,
and for every s∈[T0,S0),
(2.14)ψ(s)-β(s)<0.Similarly, by definition ofTk, we make the following conclusions.Remark 2.5.
Ifk>1, then
(2.15)ψ(Tk)-α(Tk)≤supSk-1≤s≤Tk[ψ(s)-β(s)],
and for every t∈[Sk-1,Tk)(2.16)supSk-1≤s≤t[ψ(s)-β(s)]<ψ(t)-α(t).
If Tα<Tβ, then
(2.17)α(T0)-ψ(T0)≥0,
and for every s∈[0,T0),
(2.18)α(s)-ψ(s)<0.It follows from (2.16) that [ψ(s)-β(s)]<[ψ(t)-α(t)] whenever Sk-1≤s≤t<Tk,k≥1. Therefore(2.19)[ψ(s)-β(s)]≤infs≤r<Tk[ψ(r)-α(r)]wheneverSk-1≤s<Tk.
Also note that by (2.11),(2.20)infTk-1≤r≤Sk-1(ψ(r)-α(r))≤supSk-1≤s≤t[ψ(s)-β(s)].The following result establishes a straight-forward representation for the constraining termΞα,β of the ESM similar to the representation of Theorem  2.2 of [6].Theorem 2.6.
Letα∈D-[0,∞),β∈D+[0,∞) be such that inft≥0(β(t)-α(t))>0, let ψ∈D[0,∞) and let Ξ be defined by (1.5).
IfTβ<Tα, then
(2.21)Ξα,β(ψ)(t)={0if0≤t<S0,supSk-1≤s≤t[ψ(s)-β(s)]ifSk-1≤t<Tk,k≥1,infTk≤r≤t[ψ(r)-α(r)]ifTk≤t<Sk,k≥1.
If Tα<Tβ, then
(2.22)Ξα,β(ψ)(t)={0if0≤t<T0,infTk≤r≤t[ψ(r)-α(r)]ifTk≤t<Sk,k≥0,supSk-1≤s≤t[ψ(s)-β(s)]ifSk-1≤t<Tk,k≥1.We precede the proof with two technical lemmas. The first one examinesΞα,β(ψ) on [Sk-1,Tk).Lemma 2.7.
Under the assumptions of Theorem2.6, for every k≥1 and for every t∈[Sk-1,Tk),
(2.23)Ξα,β(ψ)(t)=supSk-1≤s≤t[ψ(s)-β(s)].Proof.
Lett∈[Sk-1,Tk). Then Ξα,β(ψ)(t)=X1(t)∨X2k-1(t)∨X3k-1(t)∨X4k-1(t), where
(2.24)X1(t)=(ψ(0)-β(0))+∧inf0≤r≤t(ψ(r)-α(r)),(2.25)X2k-1(t)=sup0≤s≤Tk-1[(ψ(s)-β(s))∧infs≤r≤t(ψ(r)-α(r))],(2.26)X3k-1(t)=supTk-1≤s≤Sk-1[(ψ(s)-β(s))∧infs≤r≤t(ψ(r)-α(r))],(2.27)X4k-1(t)=supSk-1≤s≤t[(ψ(s)-β(s))∧infs≤r≤t(ψ(r)-α(r))].
By (2.20),
(2.28)X1(t)≤inf0≤r≤t(ψ(r)-α(r))≤infTk-1≤r≤Sk-1(ψ(r)-α(r))≤supSk-1≤s≤t[ψ(s)-β(s)],(2.29)X2k-1(t)≤sup0≤s≤Tk-1[infs≤r≤t(ψ(r)-α(r))]≤infTk-1≤r≤Sk-1(ψ(r)-α(r))≤supSk-1≤s≤t[ψ(s)-β(s)].
By (2.12) and (2.20),
(2.30)X3k-1(t)≤supTk-1≤s≤Sk-1[infTk-1≤r≤s(ψ(r)-α(r))∧infs≤r≤t(ψ(r)-α(r))]≤infTk-1≤r≤Sk-1(ψ(r)-α(r))≤supSk-1≤s≤t[ψ(s)-β(s)].
Finally, by (2.19), X4k-1(t)=supSk-1≤s≤t[ψ(s)-β(s)] which completes the proof.The next lamma examinesΞα,β(ψ) on [Tk,Sk).Lemma 2.8.
Under the assumptions of Theorem2.6, if k≥1 or if Tα<Tβ and k=0, then for every t∈[Tk,Sk),
(2.31)Ξα,β(ψ)(t)=infTk≤s≤t[ψ(s)-α(s)].Proof.
Letk be a nonnegative integer, and let t∈[Tk,Sk). We can write Ξα,β(ψ)(t)=X1(t)∨X2k(t)∨Yk(t), where X1 and X2k are defined by (2.24) and (2.25), respectively, and
(2.32)Yk(t)=supTk≤s≤t[(ψ(s)-β(s))∧infs≤r≤t(ψ(r)-α(r))].
We first show the upper bound,
(2.33)X1(t)≤inf0≤r≤t(ψ(r)-α(r))≤infTk≤r≤t(ψ(r)-α(r)),(2.34)X2k(t)≤sup0≤s≤Tkinfs≤r≤t(ψ(r)-α(r))≤infTk≤r≤t(ψ(r)-α(r)).
By (2.12)
(2.35)Yk(t)≤supTk≤s≤t[infTk≤r≤s(ψ(r)-α(r))∧infs≤r≤t(ψ(r)-α(r))]=infTk≤s≤t(ψ(s)-α(s)).
Thus we have shown that Ξα,β(ψ)(t)≤infTk≤s≤t(ψ(s)-α(s)). To show the opposite inequality for k≥1, it suffices to show that
(2.36)X2k(t)≥infTk≤s≤t(ψ(s)-α(s)).
Let ϵ>0 be arbitrary and k≥1, and let ρ∈[Sk-1,Tk] be such that
(2.37)supSk-1≤s≤Tk[ψ(s)-β(s)]≤ψ(ρ)-β(ρ)+ϵ.
Then, by (2.15),
(2.38)ψ(ρ)-β(ρ)≥ψ(Tk)-α(Tk)-ϵ.
Therefore, by (2.19) and (2.38),
(2.39)X2k(t)≥(ψ(ρ)-β(ρ))∧infρ≤r≤t(ψ(r)-α(r))≥(ψ(ρ)-β(ρ))∧infρ≤r<Tk(ψ(r)-α(r))∧infTk≤r≤t(ψ(r)-α(r))≥(ψ(ρ)-β(ρ))∧infTk≤r≤t(ψ(r)-α(r))≥(ψ(Tk)-α(Tk)-ϵ)∧infTk≤r≤t(ψ(r)-α(r))≥infTk≤r≤t(ψ(r)-α(r))-ϵ.
Since (2.39) holds for every ϵ>0, the proof of (2.36) is complete for k≥1. To complete the proof for k=0 in the case of Tα<Tβ, it suffices to show that
(2.40)X1(t)=infT0≤r≤t(ψ(r)-α(r)),foreveryt∈[T0,S0).
In this case Tβ>0, and so ψ(0)-β(0)<0. Also inf0≤r<T0(ψ(r)-α(r))≥0 since T0=Tα and infT0≤r≤t(ψ(r)-α(r))≤ψ(T0)-α(T0)≤0. Therefore
(2.41)X1(t)=0∧inf0≤r<T0(ψ(r)-α(r))∧infT0≤r≤t(ψ(r)-α(r))=infT0≤r≤t(ψ(r)-α(r)),
which ends the proof.Proof of Theorem2.6.
LetTβ<Tα. If t∈[0,S0), then Ξα,β(ψ)(t)=0 by Remark 2.3. If t∈[Sk-1,Tk) for some k≥1, then (2.21) holds by Lemma 2.7, and if t∈[Tk,Sk) for some k≥1, then (2.21) holds by Lemma 2.8.
Similarly, whenTα<Tβ, then T0=Tα and (2.22) holds for t∈[0,T0) by Remark 2.3, for t∈[Sk-1,Tk), k≥1 by Lemma 2.7, and for t∈[Tk,Sk), k≥1 by Lemma 2.8.We introduce two functions(2.42)Hα,β(ψ)(t)=sup0≤s≤t[(ψ(s)-β(s))∧infs≤r≤t(ψ(r)-α(r))],(2.43)Lα,β(ψ)(t)=inf0≤s≤t[(ψ(s)-α(s))∨sups≤r≤t(ψ(r)-β(r))].
It is easy to verify that the following relationship holds:(2.44)Lα,β(ψ)(t)=-H-β,-α(-ψ)(t).Corollary 2.9.
Letα∈D-[0,∞), β∈D+[0,∞) be such that inft≥0(β(t)-α(t))>0. Then for every ψ∈D[0,∞),
(2.45)Ξα,β(ψ)(t)=I{Tβ<Tα}I[Tβ,∞)(t)Hα,β(ψ)(t)+I{Tα<Tβ}I[Tα,∞)(t)Lα,β(ψ)(t).Proof.
IfTα=∞ and Tβ=∞, then both sides of (2.45) are zero.
Suppose thatTβ<Tα. Then S0=Tβ. If t∈[0,Tβ), then Ξα,β(ψ)(t)=0 by Remark 2.3.
Lett∈[Sk-1,Tk) for some k≥1. We have shown in the proof of Lemma 2.7 that
(2.46)Hα,β(ψ)(t)=X2k-1(t)∨X3k-1(t)∨X4k-1(t)=X4k-1(t).
Therefore (2.45) holds by (2.19) and (2.21).
Consider nowt∈[Tk,Sk) for some k≥1. We have shown in the proof of Lemma 2.8 that
(2.47)Hα,β(ψ)(t)=X2k(t)∨Yk(t)=infTk≤s≤t(ψ(s)-α(s)).
Hence, again, we conclude (2.45) from (2.21).
Suppose now thatTα<Tβ and set ψ′=-ψ,α′=-β, and β′=-α. Then Tβ′(ψ′)=Tα(ψ)<Tβ(ψ)=Tα′(ψ′) and so we can apply the already proven case of (2.45) to ψ′,α′ and β′. We obtain, by (2.44),
(2.48)Ξα′,β′(ψ′)(t)=I[Tβ′,∞)(t)Hα′,β′(ψ′)(t)=-I[Tα,∞)(t)Lα,β(ψ)(t).
By Remark 1.3 and Remark 2.5 in [6], Ξα,β(ψ)(t)=-Ξα′,β′(ψ′)(t), and so the proof of (2.45) is complete.We are going to show next that the timesTα and Tβ in (2.45) can be replaced by τα and τβ. Their properties described in Remark 2.1 will be essential in expanding the representation to a general ESM.Corollary 2.10.
Letα∈D-[0,∞), β∈D+[0,∞) be such that inft≥0(β(t)-α(t))>0. Then for every ψ∈D[0,∞),
(2.49)Ξα,β(ψ)(t)=I{τβ<τα}I[τβ,∞)(t)Hα,β(ψ)(t)+I{τα<τβ}I[τα,∞)(t)Lα,β(ψ)(t).Proof.
By Remark2.2, there are three possible cases. If τα=τβ=∞, then Ξα,β(ψ)(t)=0, and (2.49) holds trivially.
Consider the case whenτβ<τα. If t≤τβ, then, as in Remark 2.2, Ξα,β(ψ)(t)=0, and (2.49) holds.
Ift>τβ, then by (2.3), t>Tβ, and so, by (2.10), there is k≥1 such that t∈[Sk-1,Tk) or t∈[Tk,Sk). If t∈[Sk-1,Tk), then, as in the proof of Lemma 2.7, Ξα,β(ψ)(t)=X4k-1(t), and therefore, by (2.46), Ξα,β(ψ)(t)=Hα,β(ψ)(t).
Ift∈[Tk,Sk), then by (2.31) and (2.47), we have Ξα,β(ψ)(t)=Hα,β(ψ)(t) again.
In the case ofτα<τβ we can apply the already proven case of (2.49) to -ψ, -β, and -α, as in the proof of Corollary 2.9, to complete the proof.In the following final result, we extend the representation of (2.49) to a general case, thus producing an alternative representation formula for the ESM with two time-dependent reflective boundaries.Theorem 2.11.
Letα∈D-[0,∞), β∈D+[0,∞) be such that α≤β. Then for every ψ∈D[0,∞) and every t≥0,
(2.50)Ξα,β(ψ)(t)=I{τβ≤τα}I[τβ,∞)(t)Hα,β(ψ)(t)+I{τα<τβ}I[τα,∞)(t)Lα,β(ψ)(t).Proof.
Letα(t)≤β(t) for every t≥0, and define αn(t)=α(t)∧(β(t)-n-1) and βn(t)=(α(t)+n-1)∨β(t). Then αn∈D-[0,∞),βn∈D+[0,∞),αn↑α,βn↓β,-n-1≤αn-α≤0≤βn-β≤n-1, and inft≥0(βn(t)-αm(t))≥n-1∨m-1. By Corollary 2.10, for every n,m and every t≥0,
(2.51)Ξαm,βn(ψ)(t)=I{τβn<ταm}I[τβn,∞)(t)Hαm,βn(ψ)(t)+I{ταm<τβn}I[ταm,∞)(t)Lαm,βn(ψ)(t).
To complete the proof of (2.50), it suffices to show that the right-hand side of (2.51) converges to the right-hand side of (2.50) uniformly on compact sets. By Proposition  2.5 in [5], we could then conclude that limn→∞Ξαn,βn(ψ)=Ξα,β(ψ). In fact we will show the uniform convergence of the right-hand sides.
It is easy to see that for everyn(2.52)τα≤ταn,τβ≤τβn.
Hence, by Remark 2.1,
(2.53)limn→∞ταn=τα,limn→∞τβn=τβ.
To show the convergence ofHαn,βn(ψ) we consider a mapping R:D+[0,∞)×D-[0,∞)→D[0,∞) defined by
(2.54)R(f,g)(t)=sup0≤s≤t[f(s)∧infs≤r≤tg(r)].
It is easy to see that R is continuous in the uniform metric. In fact, it can be shown that
(2.55)∥R(f1,g1)-R(f2,g2)∥≤∥f1-f2∥∨∥g1-g2∥.
Since Hα,β(ψ)=R(β-ψ,ψ-α), we get that limn→∞Hαn,βn(ψ)=Hα,β(ψ) uniformly. Similarly, limn→∞Lαn,βn(ψ)=Lα,β(ψ) uniformly.
We consider four possible cases:τα=τβ=∞,τα<τβ,τβ<τα, and τα=τβ<∞. If τα=τβ=∞, then by (2.52), ταn=τβn=∞ for every n, and so the right-hand sides of both (2.51) and (2.50) are zero.
Suppose thatτβ<τα, then also τβn<ταn for almost all n. We show first that I[τβn,∞)Hαn,βn(ψ) converges to I[τβ,∞)Hα,β(ψ) uniformly. If β-ψ has a jump at τβ, then τβn=τβ and so also I[τβn,∞)=I[τβ,∞) for sufficiently large n. Thus for large enough n we have that I[τβn,∞)Hαn,βn(ψ)=I[τβ,∞)Hαn,βn(ψ) converges uniformly to I[τβ,∞)Hα,β(ψ). If β-ψ is continuous at τβ, then β(τβ)=ψ(τβ) and so Hα,β(ψ)(τβ)=0. Because Hα,β(ψ) is right-continuous, limn→∞supτβ≤t≤τβnHα,β(ψ)(t)=0. Hence
(2.56)limn→∞∥I[τβn,∞)Hαn,βn(ψ)-I[τβ,∞)Hα,β(ψ)∥≤limn→∞∥I[τβn,∞)Hαn,βn(ψ)-I[τβn,∞)Hα,β(ψ)∥+limn→∞∥I[τβn,∞)Hα,β(ψ)-I[τβ,∞)Hα,β(ψ)∥≤limn→∞∥Hαn,βn(ψ)-Hα,β(ψ)∥+limn→∞supτβ≤t≤τβn|Hα,β(ψ)(t)|=0.
Thus we have established that
(2.57)limn→∞I[τβn,∞)Hαn,βn(ψ)=I[τβ,∞)Hα,β(ψ),
and therefore
(2.58)limn→∞[I{τβn<ταn}I[τβn,∞)Hαn,βn(ψ)-I{ταn<τβn}I[ταn,∞)Lαn,βn(ψ)]=limn→∞I[τβn,∞)Hαn,βn(ψ)=I[τβ,∞)Hα,β(ψ)=I{τβ≤τα}I[τβ,∞)Hα,β(ψ)-I{τα<τβ}I[τα,∞)Lα,β(ψ),
where the convergence is uniform.
By a similar argument we show convergence in the case ofτα<τβ. We consider now the final case of τα=τβ<∞. Because inft≥0[βn(t)-α(t)]≥n-1, it follows by (2.7), that τβn≠τα. Analogously we can show that ταn≠τβ. Hence, for every n,
(2.59)τβ<τβn+1≤τβn,τα<ταn+1≤ταn.
By (2.53) and (2.59), we can find an increasing sequence of positive integers {nk∣k=1,2,…} such that
(2.60)τβn1>ταn2>τβn3>ταn4>τβn5>⋯.
Let
(2.61)α2k+1′=αn2k+2fork≥0,α2k′=αn2kfork≥1,(2.62)β2k+1′=βn2k+1fork≥1,β2k′=βn2k+1fork≥1.
Then it follows from (2.60) that
(2.63)τα2k+1′<τβ2k+1′foreveryk≥0,τα2k′>τβ2k′foreveryk≥1.
Therefore, as in (2.58),
(2.64)limk→∞[I{τβ2k+1′<τα2k+1′}I[τβ2k+1′,∞)Hα2k+1′,β2k+1′(ψ)+I{τα2k+1′<τβ2k+1′}I[τα2k+1′,∞)Lα2k+1′,β2k+1′(ψ)]=limk→∞I[τα2k+1′,∞)Lα2k+1′,β2k+1′(ψ)=I[τα,∞)Lα,β(ψ).
Similarly,
(2.65)limk→∞[I{τβ2k′<τα2k′}I[τβ2k′,∞)Hα2k′,β2k(ψ)+I{τα2k′<τβ2k′}I[τα2k′,∞)Lα2k′,β2k′(ψ)]=limk→∞I[τβ2k′,∞)(t)Hα2k′,β2k′(ψ)=I[τβ,∞)(t)Hα,β(ψ).
Both limits, in (2.64) and in (2.65), are in fact the uniform limits. Since, by Proposition  2.5 in [5], limk→∞Ξα2k′,β2k′(ψ)=Ξα,β(ψ) and limk→∞Ξα2k+1′,β2k+1′(ψ)=Ξα,β(ψ), the limits in (2.64) and (2.65) must be the same, that is,
(2.66)Ξα,β(ψ)=I[τα,∞)(t)Lα,β(ψ)(t)=I[τβ,∞)(t)Hα,β(ψ)(t).
Therefore limn→∞Ξαn,βn(ψ)=Ξα,β(ψ) uniformly, and so (2.50) holds again.


## 3. Continuity Properties of the ESM in Metrics onD[0,∞)

In [6] we have established a number of continuity properties of the Skorokhod map under the assumptions that α,β∈D[0,∞) and inft≥0[β-α]>0. We are going to extend some of these properties onto the ESM. This will be done in two steps. First we will allow α∈D-[0,∞), β∈D+[0,∞), and secondly we will let inft≥0[β(t)-α(t)]≥0.We begin by observing the following nesting property of the SP and the ESP constrains. It can be readily verified by checking the conditions of Definions1.2 and 1.1.Remark 3.1.
Let(ϕ,η) be the solution of the SP [resp., ESP] for ψ on [α1,β1] for some α1∈D-[0,∞) and β1∈D+[0,∞). Consider another pair of constrains α2∈D-[0,∞) and β2∈D+[0,∞). If α1≤α2≤ϕ≤β2≤β1, then (ϕ,η) is also the solution of the SP [resp., ESP] for ψ on [α2,β2].Instead of checking that the proofs of all the continuity properties in [6] are valid when the constraining functions α,β are allowed to take infinite values, we develop in the next lemma a convenient tool for expanding such properties to this more general case.Lemma 3.2.
Letα∈D-[0,∞) and β∈D+[0,∞) be such that inft≥0[β(t)-α(t)]>0. For any ψ∈D[0,∞) there is a nonincreasing sequence {αn∣n≥1}⊂D[0,∞) and a nondecreasing sequence {βn∣n≥1}⊂D[0,∞) such that the following conditions hold:(i)
α≤αn≤Γα,β(ψ)≤βn≤βfor every n≥1;(ii)
inft≥0[βn(t)-αn(t)]=inft≥0[β(t)-α(t)]for every n≥1;(iii)
for everyT>0, there is NT such that for all n≥NT and all t∈[0,T],(3.1)αn(t)={α(t)ift∈Dα,-nift∉Dα,βn(t)={β(t)ift∈Dβ,nift∉Dβ;(iv)
Γαn,βn(ψ)=Γα,β(ψ) for every n≥1.Proof.
Letα∈D-[0,∞),β∈D+[0,∞) with d=inft≥0[β(t)-α(t)]>0, let ψ∈D[0,∞), and let ϕ=Γα,β(ψ). For every n≥1, we define
(3.2)αn=α∨[(-n)∧ϕ∧0.5(α+β-d)I{α>-∞,β<∞}]=(α∨(-n))∧ϕ∧0.5(α+β-d)I{α>-∞,β<∞},βn=β∧[n∨ϕ∨0.5(α+β+d)I{α>-∞,β<∞}]=(β∧n)∨ϕ∨0.5(α+β+d)I{α>-∞,β<∞}.
It is easy to verify that for every n≥1,
(3.3)α≤αn+1≤αn≤ϕ≤βn≤βn+1≤β,(3.4)αn≤α+β-d2≤α+β+d2≤βnonDα∩Dβ.
Inequalities in (3.3) show the monotonic properties of αn and βn and prove statement (i) as well.
To prove (ii) we note that, by (3.4), for t∈Dα∩Dβ(3.5)βn(t)-αn(t)≥α+β+d2-α+β-d2=d
and so (ii) immediately follows.
ForT>0, define
(3.6)NTα=supt∈Dα∩[0,T]|α(t)|,NTβ=supt∈Dβ∩[0,T]|β(t)|,NTϕ=supt∈[0,T]|ϕ(t)|,
and set NT=NTα∨NTβ∨NTϕ. Let t∈[0,T] and let n≥NT. If β(t)<∞, then βn(t)=(β(t)∧n)∨ϕ(t)∨0.5(α(t)+β(t)+d)I{α(t)>-∞}=β(t). If β(t)=∞, then βn(t)=(β(t)∧n)∨ϕ(t)=n. Similarly, we show that αn(t)=α(t) if α(t)>-∞ and αn(t)=-n if α(t)=-∞, which completes the proof of (iii).
Finally, (iv) follows from (i) and Remark3.1.The limiting process used in the proof of Theorem2.11 provides a useful technique of extending properties of the Skorokhod map with separated constraining boundaries onto the general ESM. In the following proposition we set a formal method that will allow us to replace the assumption of inft≥0(β(t)-α(t))>0 by a weaker assumption of β≥α.Definition 3.3.
Consider𝒞, a family of mappings Tα,β from D[0,∞) to D[0,∞) indexed by a set of pairs (α,β), where α∈D-[0,∞),β∈D+[0,∞). We will say that 𝒞 is closed if the following condition is satisfied: for any sequence Tαn,βn in 𝒞 if αn→n→∞α,βn→n→∞β, and Tαn,βn(ψ)→n→∞Tα,β(ψ) for every ψ∈D[0,∞) uniformly, then Tα,β∈𝒞.Proposition 3.4.
Let𝒞 be a family of mappings Tα,β from D[0,∞) to D[0,∞) indexed by a set of pairs (α,β), where α∈D-[0,∞),β∈D+[0,∞). If 𝒞 is closed and contains all Skorokhod maps Γα,β, such that inft≥0(β(t)-α(t))>0, then 𝒞 contains all extended Skorokhod maps Γ¯α,β, where α≤β.Proof.
Suppose that𝒞 is closed and contains all Skorokhod maps Γα,β with inft≥0(β(t)-α(t))>0. Let Γ¯α,β be an ESM, where α∈D-[0,∞),β∈D+[0,∞), and α≤β. As in the proof of Theorem 2.11 we can construct sequences {αn∣n≥1}⊂D-[0,∞) and {βn∣n≥1}⊂D+[0,∞) such that αn↑α,βn↓β, and inft≥0(βn(t)-αn(t))≥1/n, so that, by Remark 1.3, Γαn,βn is an SM for every n≥1, and Γαn,βn(ψ) converges to Γ¯α,β(ψ) uniformly for every ψ∈D[0,∞). Since Γαn,βn∈𝒞, for every n≥1 we get Γ¯α,β∈𝒞.Theorem 3.5.
For anyψ1,ψ2∈D[0,∞),α1,α2∈D-[0,∞),β1,β2∈D+[0,∞) such that α1≤β1 and α2≤β2(3.7)∥Γ¯α1,β1(ψ1)-Γ¯α2,β2(ψ2)∥≤4∥ψ1-ψ2∥+3[∥α1-α2∥∨∥β1-β2∥].Proof.
Assume first thatα1,α2,β1,β2∈D[0,∞), and that the condition inft≥0(βi(t)-αi(t))>0 holds for i=1,2. By Proposition  4.1 and Remark  4.3 in [6], for any ψ1,ψ2∈D[0,∞),(3.8)∥Γα1,β1(ψ1)-Γα2,β2(ψ2)∥≤4∥ψ1-ψ2∥+2∥α1-α2∥+[∥α1-α2∥∨∥β1-β2∥].
By Remark  2.5 in [6], Γα,β(ψ)=-Γ-β,-α(-ψ); therefore applying (3.8) to Γ-β,-α(-ψ), we get
(3.9)∥Γα1,β1(ψ1)-Γα2,β2(ψ2)∥≤4∥ψ1-ψ2∥+2∥β1-β2∥+[∥α1-α2∥∨∥β1-β2∥].
Combining (3.8) and (3.9), we get that (3.7) holds for Skorokhod maps when the constraining boundaries take finite values and are separated.
Suppose now thatαi∈D-[0,∞) and βi∈D+[0,∞) for i=1,2. We can find sequences {αin∣n≥1},{βin∣n≥1}⊂D[0,∞),i=1,2, satisfying (i)–(iv) of Lemma 3.2 and we already know that for every n≥1(3.10)∥Γα1n,β1n(ψ1)-Γα2n,β2n(ψ2)∥≤4∥ψ1-ψ2∥+3[∥α1n-α2n∥∨∥β1n-β2n∥].
If Dα1≠Dα2 or Dβ1≠Dβ2 then ∥α1-α2∥=∞, or ∥β1-β2∥=∞ and so (3.7) holds trivially. We can assume therefore that Dα1=Dα2 and Dβ1=Dβ2. By part (iii) of Lemma 3.2, ∥α1n-α2n∥=∥α1-α2∥ and ∥β1n-β2n∥=∥β1-β2∥ for large enough n. On the other hand, by part (iv) of Lemma 3.2, ∥Γα1n,β1n(ψ1)-Γα2n,β2n(ψ2)∥=∥Γα1,β1(ψ1)-Γα2,β2(ψ2)∥. Thus, for large n, we can replace α1n,β1n,α2n,andβ2n in (3.10) by α1,β1,α2,andβ2, and so (3.9) holds for Skorokhod maps with separated constraining boundaries.
Next, we are going to relax the assumption thatαi,βi are separated for i=1,2. For a fixed α1∈D-[0,∞) and β1∈D+[0,∞) satisfying inft≥0[β1(t)-α1(t)]>0, consider a family 𝒞α1,β1 of mappings Tα2,β2:D[0,∞)→D[0,∞) indexed by pairs (α2,β2), where α2∈D-[0,∞) and β2∈D+[0,∞) and satisfying for every ψ1,ψ2∈D[0,∞)(3.11)∥Γα1,β1(ψ1)-Tα2,β2(ψ2)∥≤4∥ψ1-ψ2∥+3[∥α1-α2∥∨∥β1-β2∥].
We have already established that Γα2,β2∈𝒞α1,β1 whenever inft≥0[β2(t)-α2(t)]>0. It is also easy to verify that 𝒞α1,β1 is closed and so, by Proposition 3.4, Γ¯α2,β2∈𝒞α1,β1 for any α2≤β2. Thus we have shown so far that (3.7) holds whenever inft≥0[β1(t)-α1(t)]>0.
Finally, for a fixedα2∈D-[0,∞) and β2∈D+[0,∞) such that α2≤β2, let 𝒞α2,β2 be a family of mappings Tα1,β1:D[0,∞)→D[0,∞) indexed by pairs (α1,β1), where α1∈D-[0,∞) and β1∈D+[0,∞) and satisfying for every ψ1,ψ2∈D[0,∞)(3.12)∥Tα1,β1(ψ1)-Γ¯α2,β2(ψ2)∥≤4∥ψ1-ψ2∥+3[∥α1-α2∥∨∥β1-β2∥].
Then 𝒞α2,β2 contains all Γα1,β1 with inft≥0[β1(t)-α1(t)]>0, and it is closed. Applying again Proposition 3.4, we obtain that Γ¯α1,β1∈𝒞α2,β2 for any α1≤β1, and so the proof of (3.7) is complete.Applying Theorem3.5 in the special case of α1=α2 and β1=β2, we can get the Lipschitz continuity of the ESM with the Lipschitz constant 4. However, reapplying our techniques based on Lemma 3.2 and Proposition 3.4, we will obtain the following stronger result, which is a generalization of Proposition  4.6 in [6].Theorem 3.6 (Lipschitz continuity).
Letα∈D-[0,∞),β∈D+[0,∞) be such that α≤β. Then for any ψ1,ψ2∈D[0,∞)(3.13)∥Γ¯α,β(ψ1)-Γ¯α,β(ψ2)∥≤2∥ψ1-ψ2∥.Proof.
By Proposition  4.6 in [6], (3.13) holds for any α,β∈D[0,∞) such that inft≥0(β(t)-α(t))>0. If α∈D-[0,∞) and β∈D+[0,∞) then, as in the proof of Theorem 3.5, we find sequences {αn∣n≥1},{βn∣n≥1}⊂D[0,∞), i=1,2, satisfying (i)–(iv) of Lemma 3.2. By part (iv) of Lemma 3.2, for any positive integer n,
(3.14)∥Γα,β(ψ1)-Γα,β(ψ2)∥=∥Γαn,βn(ψ1)-Γαn,βn(ψ2)∥≤2∥ψ1-ψ2∥.
Finally, to relax the assumption ofinft≥0[β1(t)-α1(t)]>0, we apply Proposition 3.4 to the family 𝒞 of mappings Tα,β:D[0,∞)→D[0,∞) indexed by pairs (α,β), where α∈D-[0,∞) and β∈D+[0,∞) and satisfying
(3.15)∥Tα,β(ψ1)-Tα,β(ψ2)∥≤2∥ψ1-ψ2∥.
Since 𝒞 contains all Γα,β with inft≥0[β(t)-α(t)]>0 and it is closed, we conclude, by Proposition 3.4, that 𝒞 must contain all Γ¯α,β with β≥α. Thus, (3.13) holds for every ESM.We next examine the continuity of the ESM under the SkorokhodJ1 metric d0. The Skorokhod metric d0 on D[0,∞) is defined by
(3.16)d0(f,g)=infλ(∥λ-I∥∨∥f-g∘λ∥),
where the infimum is over all strictly increasing continuous bijections of [0,∞).We are going to need the following scaling property of the ESM.Remark 3.7.
Letψ∈D[0,∞),α∈D-[0,∞),andβ∈D+[0,∞) be such that α≤β. For any strictly increasing continuous bijection λ of [0,∞)(3.17)Γ¯α,β(ψ)∘λ=Γ¯α∘λ,β∘λ(ψ∘λ).Proof.
Using formula (2.42) we can verify that Hα,β(ψ)∘λ=Hα∘λ,β∘λ(ψ∘λ). Then from (2.44) we obtain Lα,β(ψ)∘λ=Lα∘λ,β∘λ(ψ∘λ). Using (2.2), we can verify that λ-1(τα(ψ))=τα∘λ(ψ∘λ) and λ-1(τβ(ψ))=τβ∘λ(ψ∘λ), which in turn implies that I[τα(ψ),∞)∘λ=I[τα∘λ(ψ∘λ),∞) and I[τβ(ψ),∞)∘λ=I[τβ∘λ(ψ∘λ),∞). Then by (2.50) we obtain Ξα,β(ψ)∘λ=Ξα∘λ,β∘λ(ψ∘λ), and so, applying (1.4), we conclude (3.17).Theorem 3.8.
For anyψ1,ψ2∈D[0,∞),α∈D-[0,∞),β∈D+[0,∞) such that α≤β(3.18)d0(Γ¯α,β(ψ1),Γ¯α,β(ψ2))≤4d0(ψ1,ψ2)+3supr,s>0|α(r)-α(s)|∨supr,s>0|β(r)-β(s)|.Proof.
Letλ be any strictly increasing continuous bijection of [0,∞). By (3.17) and (3.13),
(3.19)d0(Γ¯α,β(ψ1),Γ¯α,β(ψ2))≤∥λ-I∥∨∥Γ¯α,β(ψ1)-Γ¯α,β(ψ2)∘λ∥=∥λ-I∥∨∥Γ¯α,β(ψ1)-Γ¯α∘λ,β∘λ(ψ2∘λ)∥≤∥λ-I∥∨[4∥ψ1-ψ2∘λ∥+3(∥α-α∘λ∥∨∥β-β∘λ∥)]≤∥λ-I∥∨4∥ψ1-ψ2∘λ∥+3(∥α-α∘λ∥∨∥β-β∘λ∥)≤4(∥λ-I∥∨∥ψ1-ψ2∘λ∥)+3(supr,s>0|α(r)-α(s)|∨supr,s>0|β(r)-β(s)|).
Taking infλ, we obtain (3.18).Note that in cases whenDα or Dβ is a proper subset of [0,∞) the oscillation terms supr,s>0|α(r)-α(s)| or supr,s>0|β(r)-β(s)| become infinite thus rendering the upper bound of (3.18) useless.Remark 3.9.
In generalΓ¯α,β is not continuous in d0 metric on D[0,∞].Proof.
Example  4.1 in [6] shows how to construct α, β, ψ1, and ψ2 so that d0(ψ1,ψ2) is arbitrarily small while d0(Γα,β(ψ1),Γα,β(ψ2)) is arbitrarily large. Thus Γα,β is not continuous in d0 metric and therefore neither is Γ¯α,β.In fact the same example can be used to demonstrate that in generalΓ¯α,β is not continuous in the Skorokhod M1 metric as indicated in Example  4.2 of [6].



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