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.