Calculation of characteristic functions of Levy processes

Question

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?

Answer 1

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

Answer 2

You need to solve the integrals or calculate them numerically or asymptotically if there's no closed-form solution. For the alpha-stable process, you will get the known formula for the characteristic exponent that satisfies $[^{_}] = ^{\Psi_{X_t}(u)}$ $$\Psi_{X_t}(u) = - t (\sigma)^\alpha |u|^\alpha \left(1 - i \beta \mathrm{sign}(u)\tan\frac{\pi \alpha}{2} \right) + i \mu u $$ For $\alpha \neq 1$ and $$\Psi_{X_t}(u) = - t \sigma |u| \left(1 - i \beta \frac{2}{\pi} \mathrm{sign}(u)\log |t| \right) + i \mu u $$ For the case $\alpha = 1$.

This is one of the possible parametrizations. The integrals involved in the stable process are related to the gamma function, and there is another parametrization that makes this more clear. For simplicity, for the case $\alpha \neq 1$:

$$\Psi_{X_t}(u) = t c \Gamma(\alpha) \cos\left(\frac{\pi \alpha}{2}\right) |u|^\alpha \left(1 - i \beta \mathrm{sign}(u)\tan\frac{\pi \alpha}{2} \right) + i \mu u $$

Which corresponds to the Lévy measure of the alpha stable process:

$$ \nu(dx) = c \frac{1+\beta}{2}x^{-(\alpha + 1)}\mathrm{I}_{x\geq0 }dx + c \frac{1-\beta}{2}|x|^{-(\alpha + 1)}\mathrm{I}_{x<0 }dx$$

Which I believe makes it more clear as being the result of calculating the integrals in the Lévy-Kintchine formula.

You can find more information on the alpha stable process on the books "One-dimensional Stable Distributions" by Zolotarev and "Stable Non-Gaussian Random Processes Stochastic Models with Infinite Variance" by Samoradnitsky and Taqqu.