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Let us say we have some Levy process $X_t$ and want to calculate its characteristic function, $E[e^{iuX_t}]$ for a certain value $u$. Is there a general procedure for this?

I can imagine a way of doing this for example for a Compound Poisson process with normally distributed jumps: We take the Levy-Khinchine formula $$ E[e^{iuX_t}] = exp \lbrace t [ ibu - \frac{u^2 c}{2} + \int_{\mathbb{R}}(e^{iux}-1-iux 1_{\vert x \vert < 1}) \nu (dx)] \rbrace $$ and use $\lambda F(dx)$ as the Levy measure, where $F(dx)$ is the normal density and $\lambda$ is the Poisson intensity. We can get pretty simple expressions for this. The last term in the integral for example, can be calculated as $$ iu \lambda\int^{1}_{-1}xf(x)dx $$ where $f$ is the normal density.

But how about more complicated Levy processes, for example an $\alpha$-stable process? Are there procedures to compute them numerically maybe?

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A general procedure is typically to try find the Laplace transform (or moment generating function) first, and then to use the concept of 'analytic continuation' to extend the result to the Fourier transform (i.e. the characteristic function). You will find some useful examples in the book linked to here, in the exercises to chapter 1 (follow the link underneath the subtitle 'A useful text'):

http://www.maths.bath.ac.uk/~ak257/pssMp.html

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