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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 regards –  TheBridge Apr 8 '12 at 15:56

2 Answers 2

Here is part of Exercise 3.2.40 from Probability Theory: An Analytic View by Daniel W. Stroock. He refers to $$\int_{\{0<|y|<1\}} |y|\,K(dy)+K((-1,1)^c)<\infty\tag{3.2.2}$$

The difficulty of distinguishing between the drift and small jumps when (3.2.2) fails has a purely analytic antecedent which is reflected by an inherent arbitrariness in the Lévy-Khinchine formula. Namely, there is nothing sacrosanct, or even particularly compelling, about the way in which we corrected $e^{i\xi y}-1$ in order to accommodate $K$'s for which (3.2.2) fails. Indeed, show that we could have equally well taken any function of the form $$e^{i\xi y}-1-i\xi \psi(y),$$ where $\psi:\mathbb{R}\to\mathbb{R}\setminus\{0\}$ is any bounded measurable function with the property that $$\sup_{y\neq 0}\left| {\psi(y)-y\over y^2}\right|<\infty.$$

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(+1) Nice reference. :) –  cardinal Apr 8 '12 at 14:12

I consider this book: V. V. Petrov, Sums of independent random variables might be useful.

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