Some applications of duality for Lévy processes in a half-line

Jean Bertoin (1, 2) , Mladen Savov (3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 Department of Statistics [Oxford]

The central result of this paper is an analytic duality relation for real-valued Lévy processes killed upon exiting a half-line. By Nagasawa's theorem, this yields a remarkable time-reversal identity involving the Lévy process conditioned to stay positive. As examples of applications, we construct a version of the Lévy process indexed by the entire real line and started from $-\infty$ which enjoys a natural spatial-stationarity property, and point out that the latter leads to a natural Lamperti-type representation for self-similar Markov processes in $(0,\infty)$ started from the entrance point $0+$.

Keywords: Lévy processes; time-reversal; fluctuation theory

Domains: Probability [math.PR]

Jean Bertoin, Mladen Savov. Some applications of duality for Lévy processes in a half-line. 2009. ⟨hal-00437716⟩