Many [stochastic processes](http://en.wikipedia.org/wiki/Stochastic_process) that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite [quadratic variation](http://en.wikipedia.org/wiki/Quadratic_variation).


My question revolves around stochastic processes that have infinite variation, infinite quadratic variation but finite cubic variation. Of course you can go on like this for all degrees (like finite quartic variation, finite quintic variation etc.) up to infinity.


**My question**  

Would you need one extra term with the respective degree for each step in new "[Ito formulae](http://en.wikipedia.org/wiki/Ito%27s_lemma)"? Could you give an example of an Ito formula for a stochastic process that has infinite (quadratic) variation but finite cubic variation - or even a generalized Ito formula (is it comparable to a normal taylor series with infinitely many terms?)


*Full disclosure*  

I originally asked this [question](https://math.stackexchange.com/questions/296468/ito-formulas-for-stochastic-processes-with-finite-cubic-quartic-n-tic-varia) on "MathUnderflow" nearly two months ago, yet have not received any answers.



Ito formulae for stochastic processes with finite cubic, quartic ... n-tic variation
Thank you Didier for giving references to my paper with Errami.


Other later related reference focusing on fractional Brownian motion are the following.


Gradinaru, Mihai; Nourdin, Ivan; Russo, Francesco; Vallois, Pierre $m$-order integrals and generalized Itô's formula: the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 781–806. 


Gradinaru, Mihai; Russo, Francesco; Vallois, Pierre Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index $H\ge\frac14$. Ann. Probab. 31 (2003), no. 4, 1772–1820. 


In spite of the title a relatively rich list of references is in
<http://uma.ensta-paristech.fr/publication.php?id=1251>
Francesco R.


