# Gaussian and non-Gaussian processes of zero power variation

1 MATHFI - Financial mathematics
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech, UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
Abstract : This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r)$ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)})$. In the case of a Gaussian process with homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on $\delta$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô's formula is proved to hold for all functions of class $C^{6}$.
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Preprints, Working Papers, ...
Domain :

https://hal.inria.fr/inria-00438532
Contributor : Francesco Russo Connect in order to contact the contributor
Submitted on : Monday, May 28, 2012 - 9:57:14 PM
Last modification on : Tuesday, October 19, 2021 - 4:06:38 PM
Long-term archiving on: : Wednesday, August 29, 2012 - 2:19:49 AM

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### Identifiers

• HAL Id : inria-00438532, version 2
• ARXIV : 0912.0782

### Citation

Francesco Russo, Frederi Viens. Gaussian and non-Gaussian processes of zero power variation. 2012. ⟨inria-00438532v2⟩

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