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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

  1. financial markets are notoriously non brownian due to the fat tailedness of their increments as random processes.
  2. 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).

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2 Answers

David Applebaums' "Lévy processes and stochastic calculus"

It looks like integrating with respect to an $\alpha$-stable process is addressed directly on page 212 with the Ito formula to follow starting on page 218. Sadly the Ito formula section is not included in the Google preview.

I have not read the new version of this book but I did read the first edition in graduate school and I found it quite readable.

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Thank you very much, and what about differential stochastic calculus ? –  Samuel Vidal Feb 26 '12 at 19:46
    
Then you might look at chapter 6 of Applebaum and references therein. Unfortunately Google preview doesn't have much of that chapter. –  BSteinhurst Feb 26 '12 at 23:24
    
Ok thank you again ;-) –  Samuel Vidal Mar 4 '12 at 23:22
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You might want to look at the stochastic calculus of semi-martingales. The Brownian motion is a special case of a semi-martingale. There are quite a few books on stochastic calculus, for example Protter's "Stochastic Integration and Differential Equations".

You mention $\alpha$-stable Levy processes in connection with finance. The following link might therefore be of interest to you: "http://www.crm.es/Publications/quaderns/Quadern24.pdf". It was the first link Google gave me when I searched for "Ito Levy process".

I hope this helps.

Best regards, Emil Hedevang

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Tank you very much. –  Samuel Vidal Feb 26 '12 at 19:46
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