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Structural properties of reflected Lévy processes

By Lars Nørvang Andersen and Michel Mandjes
Thiele Research Reports
No. 15, December 2008

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 $\zeta(t):=\mathbb{E} V_t$ and $\xi(t):=\mathbb{V}{\rm{ar}} V_t$. We prove that for the one- and two-sided reflection we have $\zeta(t)$ is increasing and concave, whereas for the one-sided reflection we also show that $\xi(t)$ is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then 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 Lévy process.

Keywords Complete monotonicity, Lévy processes, One/Two-sided reflection, Mean function, Variance function, Stationary increments, concordance.

Mathematics Subject Classification (2000) Primary 60K25 Secondary 60F05 90B22

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