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Queueing Systems

, Volume 89, Issue 1–2, pp 199–212 | Cite as

Discretization error for a two-sided reflected Lévy process

  • Søren Asmussen
  • Jevgenijs Ivanovs
Article

Abstract

An obvious way to simulate a Lévy process X is to sample its increments over time 1 / n, thus constructing an approximating random walk \(X^{(n)}\). This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resulting process Y and regulators LU at the lower and upper barriers at some fixed time. Under the weak assumption that \(X_\varepsilon /a_\varepsilon \) has a non-trivial weak limit for some scaling function \(a_\varepsilon \) as \(\varepsilon \downarrow 0\), it is proved in particular that \((Y_1-Y^{(n)}_n)/a_{1/n}\) converges weakly to \(\pm \, V\), where the sign depends on the last barrier visited. Here the limit V is the same as in the problem concerning approximation of the supremum as recently described by Ivanovs (Ann Appl Probab, 2018). Some further insight in the distribution of V is provided both theoretically and numerically.

Keywords

Brownian motion Conditioning Refraction Regular variation Regulator Scaling limits Self-similarity Skorokhod problem Stable process 

Mathematics Subject Classification

60G51 60G16 60G52 65C05 

Notes

Acknowledgements

This paper is dedicated to Ward Whitt in appreciation of his fundamental contributions to applied probability over several decades. The study links to Ward’s work in several directions. One is his work [1] (with Abate) on one-sided reflection; another is weak convergence as lucidly exposed in his book [18]. Also, the martingale technique developed by him and Kella in [14] was the key tool in our first study [7] of two-sided reflection for Lévy processes. Additionally, we are thankful to Krzysztof Bisewski for pointing out that the former version of (5.2) was incorrect.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNy MunkegadeAarhusDenmark

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