After searching the net, I couldn't find a suitable reference to some extension of the Itô calculus that involves wilder sources of randomness than mere brownian motion.
Motivation : Itô calculus is used extensively by option traders and quants, and the basic assumption is that the price of a stock follows roughly a geometric brownian motion. But
- financial markets are notoriously non brownian due to the fat tailedness of their increments as random processes.
- Itô calculus isn't suitable to something wilder than brownian motion (which is well enough for the vast majority of applications)
But in this particular case, does anyone knows of something equivalent but applicable to $\alpha$-stable Levi processes with $\alpha \neq 2$ (i.e. non- brownian increments).