Abstract
This paper considers a number of structural properties of reflected Lévy processes, where both one-sided reflection (at 0) and two-sided reflection (at both 0 and K>0) are examined. With V t being the position of the reflected process at time t, we focus on the analysis of \(\zeta(t):=\mathbb{E}V_{t}\) and \(\xi(t):=\mathbb{V}\mathrm{ar}V_{t}\) . We prove that for the one- and two-sided reflection, ζ(t) is increasing and concave, whereas for the one-sided reflection, ξ(t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then use a limiting argument. A key step in our proofs for the two-sided reflection is a new representation of the position of the reflected process in terms of the driving Lévy process.
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Part of this work was done while both authors were at Stanford University, Stanford, CA 94305, USA.
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Andersen, L.N., Mandjes, M. Structural properties of reflected Lévy processes. Queueing Syst 63, 301 (2009). https://doi.org/10.1007/s11134-009-9116-y
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DOI: https://doi.org/10.1007/s11134-009-9116-y
Keywords
- Complete monotonicity
- Lévy processes
- One/Two-sided reflection
- Mean function
- Variance function
- Stationary increments
- Concordance