# Ito formulae for stochastic processes with finite cubic, quartic … n-tic variation

Many stochastic processes that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite 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"? 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 on "MathUnderflow" nearly two months ago, yet have not received any answers.

-
See ERRAMI M. and RUSSO F. (2003). n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes. Stochastic Process. Appl. 104 259–299. –  Did Mar 13 '13 at 13:37

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.

-
You are welcome. Always better to have an answer by the experts... Would you suggest that the OP reads Errami and Russo 2003 BEFORE these other, more recent, references, or not necessarily so? –  Did Mar 17 '13 at 10:09

Also Jason Swanson (University of Central Florida) has been working on processes with higher variations, in connection with fractional Brownian Motion (works with Chris Burdzy, David Nualart, as well as his PhD thesis and following papers)

-

Let $B$ denote a fractional Brownian motion with Hurst parameter $H=1/6$. This is a 6-variation process, so it has an infinite quadratic variation. Let $f$ be a smooth function, and let $X(t)=f(B(t))$. Then $X$ is also a 6-variation process. Let $g$ be another smooth function. In this case, the sequence of trapezoid-style Riemann sums, $\sum_{j=1}^{\lfloor nt\rfloor} \frac{g'(X((j-1)/n)) + g'(X(j/n))}2(X(j/n) - X((j-1)/n)),$ converges in distribution to a process, $t\mapsto \int_0^t g'(X(s))\\,{\bf d}X(s)$, which satisfies $g(X(t)) = g(X(0)) + \int_0^t g'(X(s))\,{\bf d}X(s) - \frac1{12}\int_0^t g'''(X(s))\,{\bf d}[\![X]\!]_s.$ This last integral is an ordinary Ito integral and $[\\![X]\\!]_t$ is a martingale which is the limit in distribution of the sums,

$\sum_{j=1}^{\lfloor nt\rfloor} (X(j/n) - X((j-1)/n))^3.$

This is a special case of a more general change-of-variable formula for integrals such as these. See http://arxiv.org/abs/1103.0341 and the references therein for further information. This paper is published in the Festschrift in Honor of David Nualart.

-