Right inverses of Lévy processes

We call a right-continuous increasing process $K_x$ a partial right inverse (PRI) of a given Lévy process $X$ if $X_{K_x}=x$ for at least all $x$ in some random interval $[0,ζ)$ of positive length. In this paper, we give a necessary and sufficient condition for the existence of a PRI in terms of the Lévy triplet.

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