Structural properties of reflected Lévy processes
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.
|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)|
|Series||CWI. Probability, Networks and Algorithms [PNA]|
Andersen, L.N, & Mandjes, M.R.H. (2008). Structural properties of reflected Lévy processes. CWI. Probability, Networks and Algorithms [PNA]. CWI.