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

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.

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 ∫^∞₀𝐼_{{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 𝐷_𝑓 to denote the effective domain of 𝑓, that is, 𝐷_𝑓={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 every 𝑇>0, let ‖𝑓‖_𝑇=sup_{0≤𝑡≤𝑇}|𝑓(𝑡)| and ‖𝑓‖=sup_𝑡≥0|𝑓(𝑡)|. Let 𝑓_𝑛 be a sequence of functions in 𝐷⁺[0,∞) or in 𝐷⁻[0,∞). We say that 𝑓_𝑛 converges to 𝑓 uniformly on compact sets if for every 𝑇>0,lim_𝑛→∞‖𝑓_𝑛−𝑓‖_𝑇=0. Equivalently, we could say that 𝑓_𝑛 converges to 𝑓 uniformly on compact sets if 𝑓_𝑛=𝑓 on 𝐷^𝑐_𝑓 for large enough 𝑛 and 𝑓_𝑛 converges to 𝑓 uniformly on [0,𝑇]∩𝐷_𝑓 for every 𝑇>0.

Definition 1.1 (extended Skorokhod problem). Let 𝛼∈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 every 𝑡≥0,𝜙(𝑡)=𝜓(𝑡)+𝜂(𝑡)∈[𝛼(𝑡),𝛽(𝑡)]; (ii)for every 0≤𝑠≤𝑡,𝜂(𝑡)−𝜂(𝑠)≥0if𝜙(𝑟)<𝛽(𝑟)∀𝑟∈(𝑠,𝑡],𝜂(𝑡)−𝜂(𝑠)≤0if𝜙(𝑟)>𝛼(𝑟)∀𝑟∈(𝑠,𝑡]; (1.1)(iii)for every 𝑡≥0,𝜂(𝑡)−𝜂(𝑡−)≥0if𝜙(𝑡)<𝛽(𝑡),𝜂(𝑡)−𝜂(𝑡−)≤0if𝜙(𝑡)>𝛼(𝑡). (1.2)

Traditionally 𝜂(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). Let 𝛼,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 every 𝑡∈[0,∞),𝜙(𝑡)=𝜓(𝑡)+𝜂(𝑡)∈[𝛼(𝑡),𝛽(𝑡)];(ii)𝜂(0−)=0 and 𝜂 have the decomposition 𝜂=𝜂_𝑙−𝜂_𝑢, where 𝜂_𝑙,𝜂_𝑢∈𝐼[0,∞),^∞₀𝐼_{{𝜙(𝑠)>𝛼(𝑡)}}𝑑𝜂_𝑙(𝑠)=0,^∞₀𝐼_{{𝜙(𝑠)<𝛽(𝑠)}}𝑑𝜂_𝑢(𝑠)=0. (1.3)

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 Γ_𝛼,𝛽(𝜓)=𝜓−Ξ_𝛼,𝛽(𝜓), (1.4) where Ξ_𝛼,𝛽(𝜓)∶𝐷[0,∞)→𝐷[0,∞) is given by Ξ_𝛼,𝛽(𝜓)(𝑡)=max{[(𝜓(0)−𝛽(0))⁺∧inf_{0≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))],sup_{0≤𝑠≤𝑡}[(𝜓(𝑠)−𝛽(𝑠))∧inf_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))]}. (1.5) 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 inf_𝑡≥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. If inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0, then Γ_𝛼,𝛽=Γ_𝛼,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 times𝑇_𝛼=min{𝑡>0∣𝛼(𝑡)−𝜓(𝑡)≥0},𝑇^𝛽𝜏=min{𝑡>0∣𝜓(𝑡)−𝛽(𝑡)≥0},_𝛼=inf{𝑡>0∣𝛼(𝑡)−𝜓(𝑡)>0},𝜏^𝛽=inf{𝑡>0∣𝜓(𝑡)−𝛽(𝑡)>0}. (2.1)(2.2) Note that 𝑇_𝛼≤𝜏_𝛼,𝑇^𝛽≤𝜏^𝛽. (2.3) Also note that the four times depend on 𝜓. When necessary we will indicate it by using full notation such as 𝜏_𝛼(𝜓) or 𝜏^𝛽(𝜓).

Remark 2.1. Let 𝜓∈𝐷[0,∞),𝛼,𝛼_𝑛,∈𝐷⁻[0,∞),𝑛=1,2,3,…, and 𝛽,𝛽_𝑛∈𝐷⁺[0,∞), 𝑛=1,2,3,…. If 𝛼−−−−−_𝑛^𝑛→∞→𝛼 and 𝛽−−−−−_𝑛^𝑛→∞→𝛽 pointwise, then limsup_𝑛→∞𝜏^𝛽𝑛≤𝜏^𝛽,limsup_𝑛→∞𝜏_𝛼𝑛≤𝜏_𝛼. (2.4)

Proof. Let 𝜖>0. By (2.2) there is 𝜏^𝛽≤𝑡₀<𝜏^𝛽+𝜖 such that 𝜓(𝑡₀)−𝛽(𝑡₀)>0. Let 𝜓(𝑡₀)−𝛽(𝑡₀)=𝛿. There is 𝑛₀ such that |𝛽_𝑛(𝑡₀)−𝛽(𝑡₀)|<𝛿/2 for 𝑛≥𝑛₀. Hence 𝜓(𝑡₀)−𝛽_𝑛(𝑡₀)=𝜓(𝑡₀)−𝛽(𝑡₀)+𝛽(𝑡₀)−𝛽_𝑛(𝑡₀)≥𝛿₀−|𝛽(𝑡₀)−𝛽_𝑛(𝑡₀𝛿)|≥⁰2>0. (2.5) Thus 𝜏^𝛽𝑛≤𝑡₀<𝜏^𝛽+𝜖. Since 𝜖 is arbitrary, we conclude that limsup_𝑛→∞𝜏^𝛽𝑛≤𝜏^𝛽. By similar argument we can show that limsup_𝑛→∞𝜏_𝛼𝑛≤𝜏_𝛼.

The inequalities in Remark 2.1 distinguish 𝜏_𝛼 and 𝜏^𝛽 from 𝑇_𝛼 and 𝑇^𝛽 and will be essential for the proof of Theorem 2.11.

Remark 2.2. Let 𝛼∈𝐷⁻[0,∞) and 𝛽∈𝐷⁺[0,∞) be such that inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0. For any 𝜓∈𝐷[0,∞), there are three possibilities either𝛼(𝑡)<𝜓(𝑡)<𝛽(𝑡)forevery𝑡≥0,𝑇_𝛼<𝑇^𝛽or𝑇^𝛽<𝑇_𝛼. (2.6) Similarly, in terms of 𝜏_𝛼 and 𝜏^𝛽, the following three cases are possible: either𝛼(𝑡)≤𝜓(𝑡)≤𝛽(𝑡)forevery𝑡≥0,𝜏_𝛼<𝜏^𝛽or𝜏^𝛽<𝜏_𝛼. (2.7) Clearly, 𝛼(𝑡)<𝜓(𝑡)<𝛽(𝑡) for every 𝑡≥0 if and only if 𝑇_𝛼=𝑇^𝛽=∞, and 𝛼(𝑡)≤𝜓(𝑡)≤𝛽(𝑡) for every 𝑡≥0 if and only if 𝜏_𝛼=𝜏^𝛽=∞.

Remark 2.3. It follows from the definition of the ESM that for every 0≤𝑡≤𝜏_𝛼∧𝜏^𝛽,𝛼(𝑡)≤𝜓(𝑡)≤𝛽(𝑡),Γ_𝛼,𝛽(𝜓)(𝑡)=𝜓(𝑡) and Ξ_𝛼,𝛽(𝜓)(𝑡)=0. Similarly, we obtain that for every 0≤𝑡<𝑇_𝛼∧𝑇^𝛽,𝛼(𝑡)<𝜓(𝑡)<𝛽(𝑡) and Γ_𝛼,𝛽(𝜓)(𝑡)=𝜓(𝑡).

Under the assumption that inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0, we define two increasing sequences of times {𝑇_𝑘∣𝑘=0,1,2,…} and {𝑆_𝑘∣𝑘=0,1,2,…} similar to the sequences 𝜏_𝑘,𝜎_𝑘 used in [6]. If 𝑇^𝛽<𝑇_𝛼, we set 𝑇₀=0 and 𝑆₀=𝑇^𝛽; for 𝑘≥1, we set𝑇_𝑘=min{𝑡>𝑆_𝑘−1∣sup_{𝑆𝑘−1≤𝑠≤𝑡}𝑆[𝜓(𝑠)−𝛽(𝑠)]≥𝜓(𝑡)−𝛼(𝑡)},_𝑘=min{𝑡>𝑇_𝑘∣𝜓(𝑡)−𝛽(𝑡)≥inf_{𝑇𝑘≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))}. (2.8)(2.9) If 𝑇_𝛼<𝑇^𝛽, we set 𝑇₀=𝑇_𝛼, we define 𝑆_𝑘 for all 𝑘 by (2.9), and we define 𝑇_𝑘 for 𝑘≥1 by (2.8).

It is easy to see that 0≤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 inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0, then lim_𝑘→∞𝑇_𝑘=∞,lim_𝑘→∞𝑆_𝑘=∞. (2.10) The following observations follow immediately from the definition of 𝑆_𝑘.

Remark 2.4. If 𝑘>1 or if 𝑘=0 and 𝑇_𝛼<𝑇^𝛽, then 𝜓(𝑆_𝑘)−𝛽(𝑆_𝑘)≥inf_{𝑇𝑘≤𝑟≤𝑆𝑘}(𝜓(𝑟)−𝛼(𝑟)) (2.11) and for every 𝑠∈[𝑇_𝑘,𝑆_𝑘)𝜓(𝑠)−𝛽(𝑠)<inf_{𝑇𝑘≤𝑟≤𝑠}(𝜓(𝑟)−𝛼(𝑟)). (2.12) If 𝑇^𝛽<𝑇_𝛼, then 𝜓(𝑆₀)−𝛽(𝑆₀)≥0, (2.13) and for every 𝑠∈[𝑇₀,𝑆₀), 𝜓(𝑠)−𝛽(𝑠)<0. (2.14)

Similarly, by definition of 𝑇_𝑘, we make the following conclusions.

Remark 2.5. If 𝑘>1, then 𝜓(𝑇_𝑘)−𝛼(𝑇_𝑘)≤sup_{𝑆𝑘−1≤𝑠≤𝑇𝑘}[𝜓(𝑠)−𝛽(𝑠)], (2.15) and for every 𝑡∈[𝑆_𝑘−1,𝑇_𝑘)sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)]<𝜓(𝑡)−𝛼(𝑡). (2.16) If 𝑇_𝛼<𝑇^𝛽, then 𝛼(𝑇₀)−𝜓(𝑇₀)≥0, (2.17) and for every 𝑠∈[0,𝑇₀), 𝛼(𝑠)−𝜓(𝑠)<0. (2.18)

It follows from (2.16) that [𝜓(𝑠)−𝛽(𝑠)]<[𝜓(𝑡)−𝛼(𝑡)] whenever 𝑆_𝑘−1≤𝑠≤𝑡<𝑇_𝑘,𝑘≥1. Therefore[𝜓(𝑠)−𝛽(𝑠)]≤inf_{𝑠≤𝑟<𝑇𝑘}[𝜓(𝑟)−𝛼(𝑟)]whenever𝑆_𝑘−1≤𝑠<𝑇_𝑘. (2.19) Also note that by (2.11),inf_{𝑇𝑘−1≤𝑟≤𝑆𝑘−1}(𝜓(𝑟)−𝛼(𝑟))≤sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)]. (2.20)

The following result establishes a straight-forward representation for the constraining term Ξ_𝛼,𝛽 of the ESM similar to the representation of Theorem  2.2 of [6].

Theorem 2.6. Let 𝛼∈𝐷⁻[0,∞),𝛽∈𝐷⁺[0,∞) be such that inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0, let 𝜓∈𝐷[0,∞) and let Ξ be defined by (1.5).
If 𝑇^𝛽<𝑇_𝛼, then Ξ_𝛼,𝛽(𝜓)(𝑡)=0if0≤𝑡<𝑆₀,sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)]if𝑆_𝑘−1≤𝑡<𝑇_𝑘,𝑘≥1,inf_{𝑇𝑘≤𝑟≤𝑡}[𝜓(𝑟)−𝛼(𝑟)]if𝑇_𝑘≤𝑡<𝑆_𝑘,𝑘≥1. (2.21) If 𝑇_𝛼<𝑇^𝛽, then Ξ_𝛼,𝛽(𝜓)(𝑡)=0if0≤𝑡<𝑇₀,inf_{𝑇𝑘≤𝑟≤𝑡}[𝜓(𝑟)−𝛼(𝑟)]if𝑇_𝑘≤𝑡<𝑆_𝑘,𝑘≥0,sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)]if𝑆_𝑘−1≤𝑡<𝑇_𝑘,𝑘≥1. (2.22)

We precede the proof with two technical lemmas. The first one examines Ξ_𝛼,𝛽(𝜓) on [𝑆_𝑘−1,𝑇_𝑘).

Lemma 2.7. Under the assumptions of Theorem 2.6, for every 𝑘≥1 and for every 𝑡∈[𝑆_𝑘−1,𝑇_𝑘), Ξ_𝛼,𝛽(𝜓)(𝑡)=sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)]. (2.23)

Proof. Let 𝑡∈[𝑆_𝑘−1,𝑇_𝑘). Then Ξ_𝛼,𝛽(𝜓)(𝑡)=𝑋₁(𝑡)∨𝑋₂^𝑘−1(𝑡)∨𝑋₃^𝑘−1(𝑡)∨𝑋₄^𝑘−1(𝑡), where 𝑋₁(𝑡)=(𝜓(0)−𝛽(0))⁺∧inf_{0≤𝑟≤𝑡}𝑋(𝜓(𝑟)−𝛼(𝑟)),_2𝑘−1(𝑡)=sup_{0≤𝑠≤𝑇𝑘−1}[(𝜓(𝑠)−𝛽(𝑠))∧inf_{𝑠≤𝑟≤𝑡}𝑋(𝜓(𝑟)−𝛼(𝑟))],_3𝑘−1(𝑡)=sup_{𝑇𝑘−1≤𝑠≤𝑆𝑘−1}[(𝜓(𝑠)−𝛽(𝑠))∧inf_{𝑠≤𝑟≤𝑡}𝑋(𝜓(𝑟)−𝛼(𝑟))],_4𝑘−1(𝑡)=sup_{𝑆𝑘−1≤𝑠≤𝑡}[(𝜓(𝑠)−𝛽(𝑠))∧inf_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))]. (2.24)(2.25)(2.26)(2.27) By (2.20), 𝑋₁(𝑡)≤inf_{0≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≤inf_{𝑇𝑘−1≤𝑟≤𝑆𝑘−1}(𝜓(𝑟)−𝛼(𝑟))≤sup_{𝑆𝑘−1≤𝑠≤𝑡}𝑋[𝜓(𝑠)−𝛽(𝑠)],_2𝑘−1(𝑡)≤sup_{0≤𝑠≤𝑇𝑘−1}[inf_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))]≤inf_{𝑇𝑘−1≤𝑟≤𝑆𝑘−1}(𝜓(𝑟)−𝛼(𝑟))≤sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)]. (2.28)(2.29) By (2.12) and (2.20), 𝑋₃^𝑘−1(𝑡)≤sup_{𝑇𝑘−1≤𝑠≤𝑆𝑘−1}[inf_{𝑇𝑘−1≤𝑟≤𝑠}(𝜓(𝑟)−𝛼(𝑟))∧inf_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))]≤inf_{𝑇𝑘−1≤𝑟≤𝑆𝑘−1}(𝜓(𝑟)−𝛼(𝑟))≤sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)]. (2.30) Finally, by (2.19), 𝑋₄^𝑘−1(𝑡)=sup_{𝑆𝑘−1≤𝑠≤𝑡}[𝜓(𝑠)−𝛽(𝑠)] which completes the proof.

The next lamma examines Ξ_𝛼,𝛽(𝜓) on [𝑇_𝑘,𝑆_𝑘).

Lemma 2.8. Under the assumptions of Theorem 2.6, if 𝑘≥1 or if 𝑇_𝛼<𝑇^𝛽 and 𝑘=0, then for every 𝑡∈[𝑇_𝑘,𝑆_𝑘), Ξ_𝛼,𝛽(𝜓)(𝑡)=inf_{𝑇𝑘≤𝑠≤𝑡}[𝜓(𝑠)−𝛼(𝑠)]. (2.31)

Proof. Let 𝑘 be a nonnegative integer, and let 𝑡∈[𝑇_𝑘,𝑆_𝑘). We can write Ξ_𝛼,𝛽(𝜓)(𝑡)=𝑋₁(𝑡)∨𝑋^𝑘₂(𝑡)∨𝑌^𝑘(𝑡), where 𝑋₁ and 𝑋^𝑘₂ are defined by (2.24) and (2.25), respectively, and 𝑌^𝑘(𝑡)=sup_{𝑇𝑘≤𝑠≤𝑡}[(𝜓(𝑠)−𝛽(𝑠))∧inf_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))]. (2.32) We first show the upper bound, 𝑋₁(𝑡)≤inf_{0≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≤inf_{𝑇𝑘≤𝑟≤𝑡}𝑋(𝜓(𝑟)−𝛼(𝑟)),_𝑘2(𝑡)≤sup_{0≤𝑠≤𝑇𝑘}inf_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≤inf_{𝑇𝑘≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟)). (2.33)(2.34) By (2.12) 𝑌^𝑘(𝑡)≤sup_{𝑇𝑘≤𝑠≤𝑡}[inf_{𝑇𝑘≤𝑟≤𝑠}(𝜓(𝑟)−𝛼(𝑟))∧inf_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))]=inf_{𝑇𝑘≤𝑠≤𝑡}(𝜓(𝑠)−𝛼(𝑠)). (2.35) Thus we have shown that Ξ_𝛼,𝛽(𝜓)(𝑡)≤inf_{𝑇𝑘≤𝑠≤𝑡}(𝜓(𝑠)−𝛼(𝑠)). To show the opposite inequality for 𝑘≥1, it suffices to show that 𝑋^𝑘₂(𝑡)≥inf_{𝑇𝑘≤𝑠≤𝑡}(𝜓(𝑠)−𝛼(𝑠)). (2.36) Let 𝜖>0 be arbitrary and 𝑘≥1, and let 𝜌∈[𝑆_𝑘−1,𝑇_𝑘] be such that sup_{𝑆𝑘−1≤𝑠≤𝑇𝑘}[𝜓(𝑠)−𝛽(𝑠)]≤𝜓(𝜌)−𝛽(𝜌)+𝜖. (2.37) Then, by (2.15), 𝜓(𝜌)−𝛽(𝜌)≥𝜓(𝑇_𝑘)−𝛼(𝑇_𝑘)−𝜖. (2.38) Therefore, by (2.19) and (2.38), 𝑋^𝑘₂(𝑡)≥(𝜓(𝜌)−𝛽(𝜌))∧inf_{𝜌≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≥(𝜓(𝜌)−𝛽(𝜌))∧inf_{𝜌≤𝑟<𝑇𝑘}(𝜓(𝑟)−𝛼(𝑟))∧inf_{𝑇𝑘≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≥(𝜓(𝜌)−𝛽(𝜌))∧inf_{𝑇𝑘≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≥(𝜓(𝑇_𝑘)−𝛼(𝑇_𝑘)−𝜖)∧inf_{𝑇𝑘≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≥inf_{𝑇𝑘≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))−𝜖. (2.39) 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 𝑇_𝛼<𝑇^𝛽, it suffices to show that 𝑋₁(𝑡)=inf_{𝑇0≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟)),forevery𝑡∈[𝑇₀,𝑆₀). (2.40) In this case 𝑇^𝛽>0, and so 𝜓(0)−𝛽(0)<0. Also inf_{0≤𝑟<𝑇0}(𝜓(𝑟)−𝛼(𝑟))≥0 since 𝑇₀=𝑇_𝛼 and inf_{𝑇0≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))≤𝜓(𝑇₀)−𝛼(𝑇₀)≤0. Therefore 𝑋₁(𝑡)=0∧inf_{0≤𝑟<𝑇0}(𝜓(𝑟)−𝛼(𝑟))∧inf_{𝑇0≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟))=inf_{𝑇0≤𝑟≤𝑡}(𝜓(𝑟)−𝛼(𝑟)), (2.41) which ends the proof.

Proof of Theorem 2.6. Let 𝑇^𝛽<𝑇_𝛼. If 𝑡∈[0,𝑆₀), then Ξ_𝛼,𝛽(𝜓)(𝑡)=0 by Remark 2.3. If 𝑡∈[𝑆_𝑘−1,𝑇_𝑘) for some 𝑘≥1, then (2.21) holds by Lemma 2.7, and if 𝑡∈[𝑇_𝑘,𝑆_𝑘) for some 𝑘≥1, then (2.21) holds by Lemma 2.8.
Similarly, when 𝑇_𝛼<𝑇^𝛽, then 𝑇₀=𝑇_𝛼 and (2.22) holds for 𝑡∈[0,𝑇₀) by Remark 2.3, for 𝑡∈[𝑆_𝑘−1,𝑇_𝑘), 𝑘≥1 by Lemma 2.7, and for 𝑡∈[𝑇_𝑘,𝑆_𝑘), 𝑘≥1 by Lemma 2.8.

We introduce two functions𝐻_𝛼,𝛽(𝜓)(𝑡)=sup_{0≤𝑠≤𝑡}[(𝜓(𝑠)−𝛽(𝑠))∧inf_{𝑠≤𝑟≤𝑡}𝐿(𝜓(𝑟)−𝛼(𝑟))],_𝛼,𝛽(𝜓)(𝑡)=inf_{0≤𝑠≤𝑡}[(𝜓(𝑠)−𝛼(𝑠))∨sup_{𝑠≤𝑟≤𝑡}(𝜓(𝑟)−𝛽(𝑟))]. (2.42)(2.43) It is easy to verify that the following relationship holds:𝐿_𝛼,𝛽(𝜓)(𝑡)=−𝐻_{−𝛽,−𝛼}(−𝜓)(𝑡). (2.44)

Corollary 2.9. Let 𝛼∈𝐷⁻[0,∞), 𝛽∈𝐷⁺[0,∞) be such that inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0. Then for every 𝜓∈𝐷[0,∞), Ξ_𝛼,𝛽(𝜓)(𝑡)=𝐼_{𝑇^𝛽_<𝑇𝛼}𝐼_[𝑇^𝛽_,∞)(𝑡)𝐻_𝛼,𝛽(𝜓)(𝑡)+𝐼_{{𝑇𝛼<𝑇}^𝛽_}𝐼_{[𝑇𝛼,∞)}(𝑡)𝐿_𝛼,𝛽(𝜓)(𝑡). (2.45)

Proof. If 𝑇_𝛼=∞ and 𝑇^𝛽=∞, then both sides of (2.45) are zero.
Suppose that 𝑇^𝛽<𝑇_𝛼. Then 𝑆₀=𝑇^𝛽. If 𝑡∈[0,𝑇^𝛽), then Ξ_𝛼,𝛽(𝜓)(𝑡)=0 by Remark 2.3.
Let 𝑡∈[𝑆_𝑘−1,𝑇_𝑘) for some 𝑘≥1. We have shown in the proof of Lemma 2.7 that 𝐻_𝛼,𝛽(𝜓)(𝑡)=𝑋₂^𝑘−1(𝑡)∨𝑋₃^𝑘−1(𝑡)∨𝑋₄^𝑘−1(𝑡)=𝑋₄^𝑘−1(𝑡). (2.46) Therefore (2.45) holds by (2.19) and (2.21).
Consider now 𝑡∈[𝑇_𝑘,𝑆_𝑘) for some 𝑘≥1. We have shown in the proof of Lemma 2.8 that 𝐻_𝛼,𝛽(𝜓)(𝑡)=𝑋^𝑘₂(𝑡)∨𝑌^𝑘(𝑡)=inf_{𝑇𝑘≤𝑠≤𝑡}(𝜓(𝑠)−𝛼(𝑠)). (2.47) Hence, again, we conclude (2.45) from (2.21).
Suppose now that 𝑇_𝛼<𝑇^𝛽 and set 𝜓^′=−𝜓,𝛼^′=−𝛽, and 𝛽^′=−𝛼. Then 𝑇^𝛽′(𝜓^′)=𝑇_𝛼(𝜓)<𝑇^𝛽(𝜓)=𝑇_𝛼′(𝜓^′) and so we can apply the already proven case of (2.45) to 𝜓^′,𝛼^′ and 𝛽^′. We obtain, by (2.44), Ξ_{𝛼′,𝛽′}(𝜓^′)(𝑡)=𝐼_{[𝑇𝛽′,∞)}(𝑡)𝐻_{𝛼′,𝛽′}(𝜓^′)(𝑡)=−𝐼_{[𝑇𝛼,∞)}(𝑡)𝐿_𝛼,𝛽(𝜓)(𝑡). (2.48) By Remark 1.3 and Remark 2.5 in [6], Ξ_𝛼,𝛽(𝜓)(𝑡)=−Ξ_{𝛼′,𝛽′}(𝜓^′)(𝑡), and so the proof of (2.45) is complete.

We are going to show next that the times 𝑇_𝛼 and 𝑇^𝛽 in (2.45) can be replaced by 𝜏_𝛼 and 𝜏^𝛽. Their properties described in Remark 2.1 will be essential in expanding the representation to a general ESM.

Corollary 2.10. Let 𝛼∈𝐷⁻[0,∞), 𝛽∈𝐷⁺[0,∞) be such that inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0. Then for every 𝜓∈𝐷[0,∞), Ξ_𝛼,𝛽(𝜓)(𝑡)=𝐼_{{𝜏𝛽<𝜏𝛼}}𝐼_{[𝜏𝛽,∞)}(𝑡)𝐻_𝛼,𝛽(𝜓)(𝑡)+𝐼_{{𝜏𝛼<𝜏𝛽}}𝐼_{[𝜏𝛼,∞)}(𝑡)𝐿_𝛼,𝛽(𝜓)(𝑡). (2.49)

Proof. By Remark 2.2, there are three possible cases. If 𝜏_𝛼=𝜏^𝛽=∞, then Ξ_𝛼,𝛽(𝜓)(𝑡)=0, and (2.49) holds trivially.
Consider the case when 𝜏^𝛽<𝜏_𝛼. If 𝑡≤𝜏^𝛽, then, as in Remark 2.2, Ξ_𝛼,𝛽(𝜓)(𝑡)=0, and (2.49) holds.
If 𝑡>𝜏^𝛽, then by (2.3), 𝑡>𝑇^𝛽, and so, by (2.10), there is 𝑘≥1 such that 𝑡∈[𝑆_𝑘−1,𝑇_𝑘) or 𝑡∈[𝑇_𝑘,𝑆_𝑘). If 𝑡∈[𝑆_𝑘−1,𝑇_𝑘), then, as in the proof of Lemma 2.7, Ξ_𝛼,𝛽(𝜓)(𝑡)=𝑋₄^𝑘−1(𝑡), and therefore, by (2.46), Ξ_𝛼,𝛽(𝜓)(𝑡)=𝐻_𝛼,𝛽(𝜓)(𝑡).
If 𝑡∈[𝑇_𝑘,𝑆_𝑘), then by (2.31) and (2.47), we have Ξ_𝛼,𝛽(𝜓)(𝑡)=𝐻_𝛼,𝛽(𝜓)(𝑡) again.
In the case of 𝜏^𝛼<𝜏_𝛽 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. Let 𝛼∈𝐷⁻[0,∞), 𝛽∈𝐷⁺[0,∞) be such that 𝛼≤𝛽. Then for every 𝜓∈𝐷[0,∞) and every 𝑡≥0, Ξ_𝛼,𝛽(𝜓)(𝑡)=𝐼_{{𝜏𝛽≤𝜏𝛼}}𝐼_{[𝜏𝛽,∞)}(𝑡)𝐻_𝛼,𝛽(𝜓)(𝑡)+𝐼_{{𝜏𝛼<𝜏𝛽}}𝐼_{[𝜏𝛼,∞)}(𝑡)𝐿_𝛼,𝛽(𝜓)(𝑡). (2.50)

Proof. Let 𝛼(𝑡)≤𝛽(𝑡) for every 𝑡≥0, and define 𝛼_𝑛(𝑡)=𝛼(𝑡)∧(𝛽(𝑡)−𝑛⁻¹) and 𝛽_𝑛(𝑡)=(𝛼(𝑡)+𝑛⁻¹)∨𝛽(𝑡). Then 𝛼_𝑛∈𝐷⁻[0,∞),𝛽_𝑛∈𝐷⁺[0,∞),𝛼_𝑛↑𝛼,𝛽_𝑛↓𝛽,−𝑛⁻¹≤𝛼_𝑛−𝛼≤0≤𝛽_𝑛−𝛽≤𝑛⁻¹, and inf_𝑡≥0(𝛽_𝑛(𝑡)−𝛼_𝑚(𝑡))≥𝑛⁻¹∨𝑚⁻¹. By Corollary 2.10, for every 𝑛,𝑚 and every 𝑡≥0, Ξ_{𝛼𝑚,𝛽𝑛}(𝜓)(𝑡)=𝐼_{𝜏^𝛽_{𝑛<𝜏𝛼𝑚}}𝐼_[𝜏^𝛽_𝑛,∞)(𝑡)𝐻_{𝛼𝑚,𝛽𝑛}(𝜓)(𝑡)+𝐼_{{𝜏𝛼𝑚<𝜏}^𝛽_𝑛}𝐼_{[𝜏𝛼𝑚,∞)}(𝑡)𝐿_{𝛼𝑚,𝛽𝑛}(𝜓)(𝑡). (2.51)
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 lim_𝑛→∞Ξ_{𝛼𝑛,𝛽𝑛}(𝜓)=Ξ_𝛼,𝛽(𝜓). In fact we will show the uniform convergence of the right-hand sides.
It is easy to see that for every 𝑛𝜏_𝛼≤𝜏_𝛼𝑛,𝜏^𝛽≤𝜏^𝛽𝑛. (2.52) Hence, by Remark 2.1, lim_𝑛→∞𝜏_𝛼𝑛=𝜏_𝛼,lim_𝑛→∞𝜏^𝛽𝑛=𝜏^𝛽. (2.53)
To show the convergence of 𝐻_{𝛼𝑛,𝛽𝑛}(𝜓) we consider a mapping 𝑅∶𝐷⁺[0,∞)×𝐷⁻[0,∞)→𝐷[0,∞) defined by 𝑅(𝑓,𝑔)(𝑡)=sup_{0≤𝑠≤𝑡}[𝑓(𝑠)∧inf_{𝑠≤𝑟≤𝑡}𝑔(𝑟)]. (2.54) It is easy to see that 𝑅 is continuous in the uniform metric. In fact, it can be shown that ‖𝑅(𝑓₁,𝑔₁)−𝑅(𝑓₂,𝑔₂)‖≤‖𝑓₁−𝑓₂‖∨‖𝑔₁−𝑔₂‖. (2.55) Since 𝐻_𝛼,𝛽(𝜓)=𝑅(𝛽−𝜓,𝜓−𝛼), we get that lim_𝑛→∞𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)=𝐻_𝛼,𝛽(𝜓) uniformly. Similarly, lim_𝑛→∞𝐿_{𝛼𝑛,𝛽𝑛}(𝜓)=𝐿_𝛼,𝛽(𝜓) uniformly.
We consider four possible cases: 𝜏_𝛼=𝜏^𝛽=∞,𝜏_𝛼<𝜏^𝛽,𝜏^𝛽<𝜏_𝛼, and 𝜏_𝛼=𝜏^𝛽<∞. If 𝜏_𝛼=𝜏^𝛽=∞, then by (2.52), 𝜏_𝛼𝑛=𝜏^𝛽𝑛=∞ for every 𝑛, and so the right-hand sides of both (2.51) and (2.50) are zero.
Suppose that 𝜏^𝛽<𝜏_𝛼, then also 𝜏^𝛽𝑛<𝜏_𝛼𝑛 for almost all 𝑛. We show first that 𝐼_[𝜏^𝛽_𝑛,∞)𝐻_{𝛼𝑛,𝛽𝑛}(𝜓) converges to 𝐼_{[𝜏𝛽,∞)}𝐻_𝛼,𝛽(𝜓) uniformly. If 𝛽−𝜓 has a jump at 𝜏^𝛽, then 𝜏^𝛽𝑛=𝜏^𝛽 and so also 𝐼_{[𝜏𝛽𝑛,∞)}=𝐼_{[𝜏𝛽,∞)} for sufficiently large 𝑛. Thus for large enough 𝑛 we have that 𝐼_[𝜏^𝛽_𝑛,∞)𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)=𝐼_{[𝜏𝛽,∞)}𝐻_{𝛼𝑛,𝛽𝑛}(𝜓) converges uniformly to 𝐼_{[𝜏𝛽,∞)}𝐻_𝛼,𝛽(𝜓). If 𝛽−𝜓 is continuous at 𝜏^𝛽, then 𝛽(𝜏^𝛽)=𝜓(𝜏^𝛽) and so 𝐻_𝛼,𝛽(𝜓)(𝜏^𝛽)=0. Because 𝐻_𝛼,𝛽(𝜓) is right-continuous, lim_𝑛→∞sup_{𝜏𝛽≤𝑡≤𝜏𝛽𝑛}𝐻_𝛼,𝛽(𝜓)(𝑡)=0. Hence lim_𝑛→∞‖𝐼_{[𝜏𝛽𝑛,∞)}𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)−𝐼_{[𝜏𝛽,∞)}𝐻_𝛼,𝛽(𝜓)‖≤lim_𝑛→∞‖𝐼_{[𝜏𝛽𝑛,∞)}𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)−𝐼_{[𝜏𝛽𝑛,∞)}𝐻_𝛼,𝛽(𝜓)‖+lim_𝑛→∞‖𝐼_{[𝜏𝛽𝑛,∞)}𝐻_𝛼,𝛽(𝜓)−𝐼_{[𝜏𝛽,∞)}𝐻_𝛼,𝛽(𝜓)‖≤lim_𝑛→∞‖𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)−𝐻_𝛼,𝛽(𝜓)‖+lim_𝑛→∞sup_{𝜏𝛽≤𝑡≤𝜏𝛽𝑛}|𝐻_𝛼,𝛽(𝜓)(𝑡)|=0. (2.56) Thus we have established that lim_𝑛→∞𝐼_{[𝜏𝛽𝑛,∞)}𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)=𝐼_{[𝜏𝛽,∞)}𝐻_𝛼,𝛽(𝜓), (2.57) and therefore lim_𝑛→∞[𝐼_{{𝜏𝛽𝑛<𝜏𝛼𝑛}}𝐼_{[𝜏𝛽𝑛,∞)}𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)−𝐼_{{𝜏𝛼𝑛<𝜏𝛽𝑛}}𝐼_{[𝜏𝛼𝑛,∞)}𝐿_{𝛼𝑛,𝛽𝑛}(𝜓)]=lim_𝑛→∞𝐼_{[𝜏𝛽𝑛,∞)}𝐻_{𝛼𝑛,𝛽𝑛}(𝜓)=𝐼_{[𝜏𝛽,∞)}𝐻_𝛼,𝛽(𝜓)=𝐼_{{𝜏𝛽≤𝜏𝛼}}𝐼_{[𝜏𝛽,∞)}𝐻_𝛼,𝛽(𝜓)−𝐼_{{𝜏𝛼<𝜏𝛽}}𝐼_{[𝜏𝛼,∞)}𝐿_𝛼,𝛽(𝜓), (2.58) where the convergence is uniform.
By a similar argument we show convergence in the case of 𝜏_𝛼<𝜏^𝛽. We consider now the final case of 𝜏_𝛼=𝜏^𝛽<∞. Because inf_𝑡≥0[𝛽_𝑛(𝑡)−𝛼(𝑡)]≥𝑛⁻¹, it follows by (2.7), that 𝜏^𝛽𝑛≠𝜏_𝛼. Analogously we can show that 𝜏_𝛼𝑛≠𝜏^𝛽. Hence, for every 𝑛, 𝜏^𝛽<𝜏^𝛽𝑛+1≤𝜏^𝛽𝑛,𝜏_𝛼<𝜏_𝛼𝑛+1≤𝜏_𝛼𝑛. (2.59) By (2.53) and (2.59), we can find an increasing sequence of positive integers {𝑛_𝑘∣𝑘=1,2,…} such that 𝜏^𝛽𝑛1>𝜏_𝛼𝑛2>𝜏^𝛽𝑛3>𝜏_𝛼𝑛4>𝜏^𝛽𝑛5>⋯. (2.60) Let 𝛼^′_2𝑘+1=𝛼_𝑛2𝑘+2for𝑘≥0,𝛼^′_2𝑘=𝛼_𝑛2𝑘𝛽for𝑘≥1,_′2𝑘+1=𝛽_𝑛2𝑘+1for𝑘≥1,𝛽^′_2𝑘=𝛽_𝑛2𝑘+1for𝑘≥1. (2.61)(2.62) Then it follows from (2.60) that 𝜏_{𝛼′2𝑘+1}<𝜏^{𝛽′2𝑘+1}forevery𝑘≥0,𝜏_𝛼′2𝑘>𝜏^𝛽′2𝑘forevery𝑘≥1. (2.63) Therefore, as in (2.58), lim_𝑘→∞[𝐼_{{𝜏𝛽′2𝑘+1<𝜏𝛼′2𝑘+1}}𝐼_{[𝜏𝛽′2𝑘+1,∞)}𝐻_{𝛼′2𝑘+1,𝛽′2𝑘+1}(𝜓)+𝐼_{{𝜏𝛼′2𝑘+1<𝜏𝛽′2𝑘+1}}𝐼_{[𝜏𝛼′2𝑘+1,∞)}𝐿_{𝛼′2𝑘+1,𝛽′2𝑘+1}(𝜓)]=lim_𝑘→∞𝐼_{[𝜏𝛼′2𝑘+1,∞)}𝐿_{𝛼′2𝑘+1,𝛽′2𝑘+1}(𝜓)=𝐼_{[𝜏𝛼,∞)}𝐿_𝛼,𝛽(𝜓). (2.64) Similarly, lim_𝑘→∞[𝐼_{{𝜏𝛽′2𝑘<𝜏𝛼′2𝑘}}𝐼_{[𝜏𝛽′2𝑘,∞)}𝐻_{𝛼′2𝑘,𝛽2𝑘}(𝜓)+𝐼_{{𝜏𝛼′2𝑘<𝜏𝛽′2𝑘}}𝐼_{[𝜏𝛼′2𝑘,∞)}𝐿_{𝛼′2𝑘,𝛽′2𝑘}(𝜓)]=lim_𝑘→∞𝐼_{[𝜏𝛽′2𝑘,∞)}(𝑡)𝐻_{𝛼′2𝑘,𝛽′2𝑘}(𝜓)=𝐼_{[𝜏𝛽,∞)}(𝑡)𝐻_𝛼,𝛽(𝜓). (2.65) Both limits, in (2.64) and in (2.65), are in fact the uniform limits. Since, by Proposition  2.5 in [5], lim_𝑘→∞Ξ_{𝛼′2𝑘,𝛽′2𝑘}(𝜓)=Ξ_𝛼,𝛽(𝜓) and lim_𝑘→∞Ξ_{𝛼′2𝑘+1,𝛽′2𝑘+1}(𝜓)=Ξ_𝛼,𝛽(𝜓), the limits in (2.64) and (2.65) must be the same, that is, Ξ_𝛼,𝛽(𝜓)=𝐼_{[𝜏𝛼,∞)}(𝑡)𝐿_𝛼,𝛽(𝜓)(𝑡)=𝐼_{[𝜏𝛽,∞)}(𝑡)𝐻_𝛼,𝛽(𝜓)(𝑡). (2.66) Therefore lim_𝑛→∞Ξ_{𝛼𝑛,𝛽𝑛}(𝜓)=Ξ_𝛼,𝛽(𝜓) uniformly, and so (2.50) holds again.

3. Continuity Properties of the ESM in Metrics on 𝐷[0,∞)

In [6] we have established a number of continuity properties of the Skorokhod map under the assumptions that 𝛼,𝛽∈𝐷[0,∞) and inf_𝑡≥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 inf_𝑡≥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 Definions 1.2 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. Let 𝛼∈𝐷⁻[0,∞) and 𝛽∈𝐷⁺[0,∞) be such that inf_𝑡≥0[𝛽(𝑡)−𝛼(𝑡)]>0. For any 𝜓∈𝐷[0,∞) there is a nonincreasing sequence {𝛼_𝑛∣𝑛≥1}⊂𝐷[0,∞) and a nondecreasing sequence {𝛽_𝑛∣𝑛≥1}⊂𝐷[0,∞) such that the following conditions hold:(i)𝛼≤𝛼_𝑛≤Γ_𝛼,𝛽(𝜓)≤𝛽_𝑛≤𝛽for every 𝑛≥1;(ii)inf_𝑡≥0[𝛽_𝑛(𝑡)−𝛼_𝑛(𝑡)]=inf_𝑡≥0[𝛽(𝑡)−𝛼(𝑡)]for every 𝑛≥1; (iii)for every 𝑇>0, there is 𝑁_𝑇 such that for all 𝑛≥𝑁_𝑇 and all 𝑡∈[0,𝑇],𝛼_𝑛(𝑡)=𝛼(𝑡)if𝑡∈𝐷_𝛼,−𝑛if𝑡∉𝐷_𝛼,𝛽_𝑛(𝑡)=𝛽(𝑡)if𝑡∈𝐷_𝛽,𝑛if𝑡∉𝐷_𝛽; (3.1)(iv)Γ_{𝛼𝑛,𝛽𝑛}(𝜓)=Γ_𝛼,𝛽(𝜓) for every 𝑛≥1.

Proof. Let 𝛼∈𝐷⁻[0,∞),𝛽∈𝐷⁺[0,∞) with 𝑑=inf_𝑡≥0[𝛽(𝑡)−𝛼(𝑡)]>0, let 𝜓∈𝐷[0,∞), and let 𝜙=Γ_𝛼,𝛽(𝜓). For every 𝑛≥1, we define 𝛼_𝑛=𝛼∨[(−𝑛)∧𝜙∧0.5(𝛼+𝛽−𝑑)𝐼_{{𝛼>−∞,𝛽<∞}}]=(𝛼∨(−𝑛))∧𝜙∧0.5(𝛼+𝛽−𝑑)𝐼_{{𝛼>−∞,𝛽<∞}},𝛽_𝑛=𝛽∧[𝑛∨𝜙∨0.5(𝛼+𝛽+𝑑)𝐼_{{𝛼>−∞,𝛽<∞}}]=(𝛽∧𝑛)∨𝜙∨0.5(𝛼+𝛽+𝑑)𝐼_{{𝛼>−∞,𝛽<∞}}. (3.2) It is easy to verify that for every 𝑛≥1, 𝛼≤𝛼_𝑛+1≤𝛼_𝑛≤𝜙≤𝛽_𝑛≤𝛽_𝑛+1𝛼≤𝛽,_𝑛≤𝛼+𝛽−𝑑2≤𝛼+𝛽+𝑑2≤𝛽_𝑛on𝐷_𝛼∩𝐷_𝛽. (3.3)(3.4) Inequalities in (3.3) show the monotonic properties of 𝛼_𝑛 and 𝛽_𝑛 and prove statement (i) as well.
To prove (ii) we note that, by (3.4), for 𝑡∈𝐷_𝛼∩𝐷_𝛽𝛽_𝑛(𝑡)−𝛼_𝑛(𝑡)≥𝛼+𝛽+𝑑2−𝛼+𝛽−𝑑2=𝑑 (3.5) and so (ii) immediately follows.
For 𝑇>0, define 𝑁^𝛼_𝑇=sup_{𝑡∈𝐷𝛼∩[0,𝑇]}|𝛼(𝑡)|,𝑁^𝛽_𝑇=sup_{𝑡∈𝐷𝛽∩[0,𝑇]}|𝛽(𝑡)|,𝑁^𝜙_𝑇=sup_{𝑡∈[0,𝑇]}|𝜙(𝑡)|, (3.6) and set 𝑁_𝑇=𝑁^𝛼_𝑇∨𝑁^𝛽_𝑇∨𝑁^𝜙_𝑇. Let 𝑡∈[0,𝑇] and let 𝑛≥𝑁_𝑇. If 𝛽(𝑡)<∞, then 𝛽_𝑛(𝑡)=(𝛽(𝑡)∧𝑛)∨𝜙(𝑡)∨0.5(𝛼(𝑡)+𝛽(𝑡)+𝑑)𝐼_{{𝛼(𝑡)>−∞}}=𝛽(𝑡). If 𝛽(𝑡)=∞, then 𝛽_𝑛(𝑡)=(𝛽(𝑡)∧𝑛)∨𝜙(𝑡)=𝑛. Similarly, we show that 𝛼_𝑛(𝑡)=𝛼(𝑡) if 𝛼(𝑡)>−∞ and 𝛼_𝑛(𝑡)=−𝑛 if 𝛼(𝑡)=−∞, which completes the proof of (iii).
Finally, (iv) follows from (i) and Remark 3.1.

The limiting process used in the proof of Theorem 2.11 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 inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0 by a weaker assumption of 𝛽≥𝛼.

Definition 3.3. Consider 𝒞, a family of mappings 𝑇_𝛼,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 𝑇_{𝛼𝑛,𝛽𝑛} in 𝒞 if 𝛼−−−−−_𝑛^𝑛→∞→𝛼,𝛽−−−−−_𝑛^𝑛→∞→𝛽, and 𝑇_{𝛼𝑛,𝛽𝑛}(𝜓)−−−−−^𝑛→∞→𝑇_𝛼,𝛽(𝜓) for every 𝜓∈𝐷[0,∞) uniformly, then 𝑇_𝛼,𝛽∈𝒞.

Proposition 3.4. Let 𝒞 be a family of mappings 𝑇_𝛼,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 Γ_𝛼,𝛽, such that inf_𝑡≥0(𝛽(𝑡)−𝛼(𝑡))>0, then 𝒞 contains all extended Skorokhod maps Γ_𝛼,𝛽, where 𝛼≤𝛽.

Proof. Suppose that 𝒞 is closed and contains all Skorokhod maps Γ_𝛼,𝛽 with inf_𝑡≥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 {𝛼_𝑛∣𝑛≥1}⊂𝐷⁻[0,∞) and {𝛽_𝑛∣𝑛≥1}⊂𝐷⁺[0,∞) such that 𝛼_𝑛↑𝛼,𝛽_𝑛↓𝛽, and inf_𝑡≥0(𝛽_𝑛(𝑡)−𝛼_𝑛(𝑡))≥1/𝑛, so that, by Remark 1.3, Γ_{𝛼𝑛,𝛽𝑛} is an SM for every 𝑛≥1, and Γ_{𝛼𝑛,𝛽𝑛}(𝜓) converges to Γ_𝛼,𝛽(𝜓) uniformly for every 𝜓∈𝐷[0,∞). Since Γ_{𝛼𝑛,𝛽𝑛}∈𝒞, for every 𝑛≥1 we get Γ_𝛼,𝛽∈𝒞.

Theorem 3.5. For any 𝜓₁,𝜓₂∈𝐷[0,∞),𝛼₁,𝛼₂∈𝐷⁻[0,∞),𝛽₁,𝛽₂∈𝐷⁺[0,∞) such that 𝛼₁≤𝛽₁ and 𝛼₂≤𝛽₂‖Γ_𝛼1,𝛽1(𝜓₁)−Γ_𝛼2,𝛽2(𝜓₂)‖≤4‖𝜓₁−𝜓₂‖+3[‖𝛼₁−𝛼₂‖∨‖𝛽₁−𝛽₂‖]. (3.7)

Proof. Assume first that 𝛼₁,𝛼₂,𝛽₁,𝛽₂∈𝐷[0,∞), and that the condition inf_𝑡≥0(𝛽_𝑖(𝑡)−𝛼_𝑖(𝑡))>0 holds for 𝑖=1,2. By Proposition  4.1 and Remark  4.3 in [6], for any 𝜓₁,𝜓₂∈𝐷[0,∞),‖Γ_𝛼1,𝛽1(𝜓₁)−Γ_𝛼2,𝛽2(𝜓₂)‖≤4‖𝜓₁−𝜓₂‖+2‖𝛼₁−𝛼₂‖+[‖𝛼₁−𝛼₂‖∨‖𝛽₁−𝛽₂‖]. (3.8) By Remark  2.5 in [6], Γ_𝛼,𝛽(𝜓)=−Γ_{−𝛽,−𝛼}(−𝜓); therefore applying (3.8) to Γ_{−𝛽,−𝛼}(−𝜓), we get ‖Γ_𝛼1,𝛽1(𝜓₁)−Γ_𝛼2,𝛽2(𝜓₂)‖≤4‖𝜓₁−𝜓₂‖+2‖𝛽₁−𝛽₂‖+[‖𝛼₁−𝛼₂‖∨‖𝛽₁−𝛽₂‖]. (3.9) 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 that 𝛼_𝑖∈𝐷⁻[0,∞) and 𝛽_𝑖∈𝐷⁺[0,∞) for 𝑖=1,2. We can find sequences {𝛼^𝑛_𝑖∣𝑛≥1},{𝛽^𝑛_𝑖∣𝑛≥1}⊂𝐷[0,∞),𝑖=1,2, satisfying (i)–(iv) of Lemma 3.2 and we already know that for every 𝑛≥1‖Γ_𝛼^𝑛_1,𝛽^𝑛₁(𝜓₁)−Γ_𝛼^𝑛_2,𝛽^𝑛₂(𝜓₂)‖≤4‖𝜓₁−𝜓₂‖+3[‖𝛼^𝑛₁−𝛼^𝑛₂‖∨‖𝛽^𝑛₁−𝛽^𝑛₂‖]. (3.10) If 𝐷_𝛼1≠𝐷_𝛼2 or 𝐷_𝛽1≠𝐷_𝛽2 then ‖𝛼₁−𝛼₂‖=∞, or ‖𝛽₁−𝛽₂‖=∞ and so (3.7) holds trivially. We can assume therefore that 𝐷_𝛼1=𝐷_𝛼2 and 𝐷_𝛽1=𝐷_𝛽2. By part (iii) of Lemma 3.2, ‖𝛼^𝑛₁−𝛼^𝑛₂‖=‖𝛼₁−𝛼₂‖ and ‖𝛽^𝑛₁−𝛽^𝑛₂‖=‖𝛽₁−𝛽₂‖ for large enough 𝑛. On the other hand, by part (iv) of Lemma 3.2, ‖Γ_𝛼^𝑛_1,𝛽^𝑛₁(𝜓₁)−Γ_𝛼^𝑛_2,𝛽^𝑛₂(𝜓₂)‖=‖Γ_𝛼1,𝛽1(𝜓₁)−Γ_𝛼2,𝛽2(𝜓₂)‖. Thus, for large 𝑛, we can replace 𝛼^𝑛₁,𝛽^𝑛₁,𝛼^𝑛₂,and𝛽^𝑛₂ 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 that 𝛼_𝑖,𝛽_𝑖 are separated for 𝑖=1,2. For a fixed 𝛼₁∈𝐷⁻[0,∞) and 𝛽₁∈𝐷⁺[0,∞) satisfying inf_𝑡≥0[𝛽₁(𝑡)−𝛼₁(𝑡)]>0, consider a family 𝒞_𝛼1,𝛽1 of mappings 𝑇_𝛼2,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,∞)‖Γ_𝛼1,𝛽1(𝜓₁)−𝑇_𝛼2,𝛽2(𝜓₂)‖≤4‖𝜓₁−𝜓₂‖+3[‖𝛼₁−𝛼₂‖∨‖𝛽₁−𝛽₂‖]. (3.11) We have already established that Γ_𝛼2,𝛽2∈𝒞_𝛼1,𝛽1 whenever inf_𝑡≥0[𝛽₂(𝑡)−𝛼₂(𝑡)]>0. It is also easy to verify that 𝒞_𝛼1,𝛽1 is closed and so, by Proposition 3.4, Γ_𝛼2,𝛽2∈𝒞_𝛼1,𝛽1 for any 𝛼₂≤𝛽₂. Thus we have shown so far that (3.7) holds whenever inf_𝑡≥0[𝛽₁(𝑡)−𝛼₁(𝑡)]>0.
Finally, for a fixed 𝛼₂∈𝐷⁻[0,∞) and 𝛽₂∈𝐷⁺[0,∞) such that 𝛼₂≤𝛽₂, let 𝒞_𝛼2,𝛽2 be a family of mappings 𝑇_𝛼1,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,∞)‖𝑇_𝛼1,𝛽1(𝜓₁)−Γ_𝛼2,𝛽2(𝜓₂)‖≤4‖𝜓₁−𝜓₂‖+3[‖𝛼₁−𝛼₂‖∨‖𝛽₁−𝛽₂‖]. (3.12) Then 𝒞_𝛼2,𝛽2 contains all Γ_𝛼1,𝛽1 with inf_𝑡≥0[𝛽₁(𝑡)−𝛼₁(𝑡)]>0, and it is closed. Applying again Proposition 3.4, we obtain that Γ_𝛼1,𝛽1∈𝒞_𝛼2,𝛽2 for any 𝛼₁≤𝛽₁, and so the proof of (3.7) is complete.

Applying Theorem 3.5 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). Let 𝛼∈𝐷⁻[0,∞),𝛽∈𝐷⁺[0,∞) be such that 𝛼≤𝛽. Then for any 𝜓₁,𝜓₂∈𝐷[0,∞)‖Γ_𝛼,𝛽(𝜓₁)−Γ_𝛼,𝛽(𝜓₂)‖≤2‖𝜓₁−𝜓₂‖. (3.13)

Proof. By Proposition  4.6 in [6], (3.13) holds for any 𝛼,𝛽∈𝐷[0,∞) such that inf_𝑡≥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 {𝛼_𝑛∣𝑛≥1},{𝛽_𝑛∣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, ‖Γ_𝛼,𝛽(𝜓₁)−Γ_𝛼,𝛽(𝜓₂)‖=‖Γ_{𝛼𝑛,𝛽𝑛}(𝜓₁)−Γ_{𝛼𝑛,𝛽𝑛}(𝜓₂)‖≤2‖𝜓₁−𝜓₂‖. (3.14)
Finally, to relax the assumption of inf_𝑡≥0[𝛽₁(𝑡)−𝛼₁(𝑡)]>0, we apply Proposition 3.4 to the family 𝒞 of mappings 𝑇_𝛼,𝛽∶𝐷[0,∞)→𝐷[0,∞) indexed by pairs (𝛼,𝛽), where 𝛼∈𝐷⁻[0,∞) and 𝛽∈𝐷⁺[0,∞) and satisfying ‖𝑇_𝛼,𝛽(𝜓₁)−𝑇_𝛼,𝛽(𝜓₂)‖≤2‖𝜓₁−𝜓₂‖. (3.15) Since 𝒞 contains all Γ_𝛼,𝛽 with inf_𝑡≥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 Skorokhod 𝐽₁ metric 𝑑₀. The Skorokhod metric 𝑑₀ on 𝐷[0,∞) is defined by 𝑑₀(𝑓,𝑔)=inf_𝜆(‖𝜆−𝐼‖∨‖𝑓−𝑔∘𝜆‖), (3.16) 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. Let 𝜓∈𝐷[0,∞),𝛼∈𝐷⁻[0,∞),and𝛽∈𝐷⁺[0,∞) be such that 𝛼≤𝛽. For any strictly increasing continuous bijection 𝜆 of [0,∞)Γ_𝛼,𝛽(𝜓)∘𝜆=Γ_{𝛼∘𝜆,𝛽∘𝜆}(𝜓∘𝜆). (3.17)

Proof. Using formula (2.42) we can verify that 𝐻_𝛼,𝛽(𝜓)∘𝜆=𝐻_{𝛼∘𝜆,𝛽∘𝜆}(𝜓∘𝜆). Then from (2.44) we obtain 𝐿_𝛼,𝛽(𝜓)∘𝜆=𝐿_{𝛼∘𝜆,𝛽∘𝜆}(𝜓∘𝜆). Using (2.2), we can verify that 𝜆⁻¹(𝜏_𝛼(𝜓))=𝜏_𝛼∘𝜆(𝜓∘𝜆) and 𝜆⁻¹(𝜏^𝛽(𝜓))=𝜏^𝛽∘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 Ξ_𝛼,𝛽(𝜓)∘𝜆=Ξ_{𝛼∘𝜆,𝛽∘𝜆}(𝜓∘𝜆), and so, applying (1.4), we conclude (3.17).

Theorem 3.8. For any 𝜓₁,𝜓₂∈𝐷[0,∞),𝛼∈𝐷⁻[0,∞),𝛽∈𝐷⁺[0,∞) such that 𝛼≤𝛽𝑑₀(Γ_𝛼,𝛽(𝜓₁),Γ_𝛼,𝛽(𝜓₂))≤4𝑑₀(𝜓₁,𝜓₂)+3sup_𝑟,𝑠>0|𝛼(𝑟)−𝛼(𝑠)|∨sup_𝑟,𝑠>0|𝛽(𝑟)−𝛽(𝑠)|. (3.18)

Proof. Let 𝜆 be any strictly increasing continuous bijection of [0,∞). By (3.17) and (3.13), 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(sup_𝑟,𝑠>0|𝛼(𝑟)−𝛼(𝑠)|∨sup_𝑟,𝑠>0|𝛽(𝑟)−𝛽(𝑠)|). (3.19) Taking inf_𝜆, we obtain (3.18).

Note that in cases when 𝐷_𝛼 or 𝐷_𝛽 is a proper subset of [0,∞) the oscillation terms sup_𝑟,𝑠>0|𝛼(𝑟)−𝛼(𝑠)| or sup_𝑟,𝑠>0|𝛽(𝑟)−𝛽(𝑠)| become infinite thus rendering the upper bound of (3.18) useless.

Remark 3.9. In general Γ_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(Γ_𝛼,𝛽(𝜓₁),Γ_𝛼,𝛽(𝜓₂)) is arbitrarily large. Thus Γ_𝛼,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 general Γ_𝛼,𝛽 is not continuous in the Skorokhod 𝑀₁ metric as indicated in Example  4.2 of [6].