Given a one dimensional Lévy process $X_t$ with characteristic exponent $\psi(\xi)$, so that \begin{align} \mathbf{E}[e^{i \xi X_t}]= e^{t \psi(\xi)} \end{align} for example we can find $\psi(\xi)$ from Lévy–Khintchine representation.
Now I'm looking for the object that has characteristic function $\frac{\psi(\xi)}{i \xi}$, do you know what could it be? For example in the case of subordinators we can find it easily from the Lévy measure, but for a generale one dimensional Lévy process? I think that it easy in the case such that \begin{align} \mathbf{E}[e^{i \xi X_t}]= e^{t \int_{-\infty}^\infty (e^{i \xi y} - 1) \nu(dy)} \quad (1) \end{align} for which Lévy processes (1) is true? Do you in general $\frac{\psi(\xi)}{i \xi}$ of what it can be the characteristic function?
Thank you!
If $X$ is a spectrally negative Lévy process (only negative jumps) with characteristic exponent $\xi \mapsto \psi(\xi)$, and $L$ is its local time at the infimum, then the process of infimum values $X(L^{-1}(t))$ is a negative subordinator of characteristic exponent $\xi \mapsto \dfrac {\psi(\xi)}{i \xi }$.
Also the time change $L^{-1}(t)$ itself is a subordinator of characteristic exponent $\dfrac {\xi}{\Phi(\xi)}$ where $\Phi$ is the functional inverse of $\psi$ (up to a $i$ somewhere, I'm tired and nobody uses the same definition for characteristic/laplace exponents.)
For instance, is $X$ is the Brownian motion ($\psi(\xi) = \xi^2/2$, the time change at local time $l$ is just the hitting time of negative level $-l/2$, which is well-known to be a 1/2-stable subordinator. Indeed it has Laplace exponent $\xi/ \sqrt(2\xi) = \sqrt(\xi/2)$.
The value of the process when it hits level $-l/2$ is obviously $-l/2$, and indeed the caracteristic exponent is $\xi^2/2/(i\xi) = -i \xi/2$ which is the caracteristic exponent of the deterministic process $l\mapsto -l/2$.
More generally, for $X$ a spectrally negative $\alpha$-stable process, $X\circ L^{-1}$ is a $\alpha-1$-stable negative subordinator, while $L^{-1}$ is a $1-1/\alpha$-stable subordinator. Pretty cool, huh!
This is the kind of things you find in books about fluctuation theory, like Kyprianou's : https://people.bath.ac.uk/ak257/book2e/book2e.pdf (see sections 6.4, 6.5)