This paper considers a number of structural properties of reflected L´evy processes, where both onesided reflection (at 0) and two-sided reflection (at both 0 and K > 0) are examined. With Vt being the position of the reflected process at time t, we focus on the analysis of (t) := EVt and (t) := VarVt. We prove that for the one- and two-sided reflection we have (t) is increasing and concave, whereas for the one-sided reflection we also show that (t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then we 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 L´evy process.
Additional Metadata
Keywords Complete monotonicity, Lévy processes, One/Two-sided reflection, Mean function, Variance function, Stationary increments, concordance
MSC Queueing theory (msc 60K25), Central limit and other weak theorems (msc 60F05), Queues and service (msc 90B22)
THEME Logistics (theme 3), Energy (theme 4)
Publisher CWI
Series CWI. Probability, Networks and Algorithms [PNA]
Citation
Andersen, L.N, & Mandjes, M.R.H. (2008). Structural properties of reflected Lévy processes. CWI. Probability, Networks and Algorithms [PNA]. CWI.