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I'm writing a survey that involves Levy processes and wanted to mention the different forms of the Levy-Khintchine formula found in literature.

The most common version seems to give the Levy symbol as

$$\Psi(u) = i\langle b,u \rangle - \frac{1}{2} \langle u,\Sigma u\rangle + \int_{\mathbb{R}^d} {(} e^{i\langle u,y \rangle}-1 - i\langle u,y \rangle\mathbf{1}_{|y|\le1}{)}\, dK(y)$$

while in other versions it seems to be given as

$$\Psi(u) = i\langle b,u \rangle - \frac{1}{2} \langle u,\Sigma u\rangle + \int_{\mathbb{R}^d} {(} e^{i\langle u,y \rangle}-1 - \frac{ i\langle u,y \rangle}{1+|y|^2}{)} \, dK(y)$$

while at almostsure blog it is given as

$$\Psi(u) = i\langle b,u \rangle - \frac{1}{2} \langle u,\Sigma u\rangle + \int_{\mathbb{R}^d}{(} e^{i\langle u,y \rangle}-1 - \frac{ i\langle u,y \rangle}{1+|y|}{)} \, dK(y).$$

Are all of these correct and equivalent? If the last one is, does anyone know a published source I could cite that mentions it?

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    @ Dominic : I think the most general form I have seen described and explained in Jacod and Shyriaev' book Best regards2012-04-08

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