Structural properties of reflected Lévy processes

  • Authors:

  • Lars Nørvang Andersen;Michel Mandjes

  • Affiliations:

  • Department of Mathematical Sciences, Aarhus University, Aarhus C, Denmark 8000;Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands 1018 TV and CWI, Amsterdam, the Netherlands and Eurandom, Eindhoven, the Netherlands

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 2009

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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 K0) 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, 驴(t) is increasing and concave, whereas for the one-sided reflection, 驴(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.