Integral representation of Skorokhod reflection

We present a very short derivation of the integral representation of the two-sided Skorokhod reflection Z of a continuous function X of bounded variation, which is a generalization of the integral representation of the one-sided map featured in Anantharam and Konstantopoulos (2011) and Konstantopoulos et al. (1996). We also show that Z satisfies a simpler integral representation when additional conditions are imposed on X.

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“…which is the integral representation found in [1], [3], and [7]. Our proof of Theorem 2.1 will make use of the following simple lemmas.…”

Section: Notation and Main Resultsmentioning

confidence: 99%

“…Subsequently, in [6] the representation was extended to the case where X is allowed to have discontinuity points. These integral representations are also briefly discussed in [3,Chapter 3], and an interesting result addressing the uniqueness of functions satisfying such representations can be found in [1].…”

Section: Introductionmentioning

confidence: 99%

A note on integral representations of the Skorokhod map

2016

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“…Generalized drift Skorokhod 17 In subsequent papers, the Skorokhod problem has been extended to multiple dimensions and also to include both smooth and nonsmooth domains (see, for example, [4], [6], [8], [13], and [17]), although we do not treat such cases in the present paper. There is a useful integral representation of the one-dimensional Skorokhod problem solution (see [2]). There is also an explicit solution to the (one-dimensional) Skorokhod problem when there is an upper boundary (see [9] and [10])) and to the (one-dimensional) Skorokhod problem in a time-dependent interval (see [3]).…”

Section: Introductionmentioning

confidence: 99%

On the Generalized Drift Skorokhod Problem in One Dimension

Reed

Ward

Zhan

2013

Journal of Applied Probability

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We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime.

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“…In subsequent papers, the Skorokhod problem has been extended to multiple dimensions and also to include both smooth and nonsmooth domains (see, for example, [4], [6], [8], [13], and [17]), although we do not treat such cases in the present paper. There is a useful integral representation of the one-dimensional Skorokhod problem solution (see [2]). There is also an explicit solution to the (one-dimensional) Skorokhod problem when there is an upper boundary (see [9] and [10])) and to the (one-dimensional) Skorokhod problem in a time-dependent interval (see [3]).…”

Section: Introductionmentioning

confidence: 99%

On the Generalized Drift Skorokhod Problem in One Dimension

Reed¹,

Ward²,

Zhan³

2013

J. Appl. Probab.

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We show how to write the solution to the generalized drift Skorokhod problem in onedimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein-Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime.

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Integral representation of Skorokhod reflection

Abstract: Abstract. We show that a certain integral representation of the one-sided Skorokhod reflection of a continuous bounded variation function characterizes the reflection in that it possesses a unique maximal solution which solves the Skorokhod reflection problem.

Cited by 4 publications

References 14 publications

A note on integral representations of the Skorokhod map

A note on integral representations of the Skorokhod map

On the Generalized Drift Skorokhod Problem in One Dimension

On the Generalized Drift Skorokhod Problem in One Dimension

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