Structural properties of reflected Lévy processes

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) := EVt and xi(t) := Var V-t. We prove that for the one- and two-sided reflection, zeta(t) is increasing and concave, whereas for the one-sided reflection, xi(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.