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Lévy Processes with Two-Sided Reflection

  • Chapter
Lévy Matters V

Part of the book series: Lecture Notes in Mathematics ((LEVY,volume 2149))

Abstract

Let X be a Lévy process and V the reflection at boundaries 0 and b > 0. A number of properties of V are studied, with particular emphasis on the behaviour at the upper boundary b. The process V can be represented as solution of a Skorokhod problem V (t) = V (0) + X(t) + L(t) − U(t) where L, U are the local times (regulators) at the lower and upper barrier. Explicit forms of V in terms of X are surveyed as well as more pragmatic approaches to the construction of V, and the stationary distribution π is characterised in terms of a two-barrier first passage problem. A key quantity in applications is the loss rate ℓ b at b, defined as \(\mathbb{E}_{\pi }U(1)\). Various forms of ℓ b and various derivations are presented, and the asymptotics as \(b \rightarrow \infty \) is exhibited in both the light-tailed and the heavy-tailed regime. The drift zero case \(\mathbb{E}X(1) = 0\) plays a particular role, with Brownian or stable functional limits being a key tool. Further topics include studies of the first hitting time of b, central limit theorems and large deviations results for U, and a number of explicit calculations for Lévy processes where the jump part is compound Poisson with phase-type jumps.

AMS Subject Classification 2000:Primary: 60G51, 60K25, 60K30; Secondary: 60G44, 60F99, 60J55, 34K50

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Notes

  1. 1.

    By this we mean that there exists a subexponential distribution G such that \(\overline{F}_{I}(x)\, =\, \overline{G}(x)\) for all large x. For background on heavy-tailed distributions, see e.g.[13, X.1], [62] and the start of Sect.  3 .

  2. 2.

    Strictly speaking, the argument requires \(\mathbb{E}_{\pi ^{\infty }}V (0) < \infty \) which amounts to a second moment assumption. For the general case, just use a truncation argument.

  3. 3.

    Note that the constants there should be replaced by their inverses.

  4. 4.

    The list of references contain a number of items not cited in the text, but judged to be relevant.

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Acknowledgements

We are very grateful to Victor Rivero for his extremely careful reading of the manuscript and for providing us with an extensive list of corrections and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Aarhus University, 8000, Aarhus C, Denmark

    Lars Nørvang Andersen & Søren Asmussen

  2. Department of Management Science and Engineering, Huang Engineering Center 357, Stanford, CA, 94305, USA

    Peter W. Glynn

  3. Faculty of Medicine, Lund University, Box 117, 221 00, Lund, Sweden

    Mats Pihlsgård

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  1. Lars Nørvang Andersen
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  2. Søren Asmussen
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  3. Peter W. Glynn
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  4. Mats Pihlsgård
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Correspondence to Lars Nørvang Andersen .

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Andersen, L.N., Asmussen, S., Glynn, P.W., Pihlsgård, M. (2015). Lévy Processes with Two-Sided Reflection. In: Lévy Matters V. Lecture Notes in Mathematics(), vol 2149. Springer, Cham. https://doi.org/10.1007/978-3-319-23138-9_2

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