# Calculation of characteristic functions of Levy processes

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?