Expected value of a logarithm of a Levy process

I have a strictly positive Levy process $(L_t)$ with no Brownian part, drift $\gamma$ and jump measure $\nu$. Is it possible to calculate the expected value of the logarithm of this process, so $\mathbf{E}[ln L_t]?$

I am trying to find an additive correction that would make $ln L_t$ a martingale. I am interested in the general case, but even the special case when $L_t$ has finite variation would help.

flag	asked May 16 2012 at 14:39 Grzenio 339●7

	@ Grzenio : Why not applying the Lévy process version of Itô's lemma to the log of your process ? – The Bridge May 16 2012 at 15:05
	@TheBridge, I tried but in the case of no Brownian part, for finite variation processes for simplicity, it seems to boil down to $d lnL_t = \Delta ln L_t$, which seems rather useless if I am not missing something... – Grzenio May 16 2012 at 15:41
	@ Grzenio : Do you have an explicit expression for $\nu$ ? – The Bridge May 16 2012 at 16:14
	@TheBridge, in general I don't know the expression for $\nu$, but I even failed to calculate the expectation for the gamma process: en.wikipedia.org/wiki/Gamma_process – Grzenio May 17 2012 at 7:26