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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.

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@ Grzenio : Why not applying the Lévy process version of Itô's lemma to the log of your process ? – The Bridge May 16 '12 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 '12 at 15:41
@ Grzenio : Do you have an explicit expression for $\nu$ ? – The Bridge May 16 '12 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: – Grzenio May 17 '12 at 7:26

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