HAL : inria-00438532, version 1

arXiv : 0912.0782

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Gaussian and non-Gaussian processes of zero power variation

Francesco Russo 1, 2, 3, Frederi Viens 4

(03/12/2009)

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 $C^{6}$.

1 :	Laboratoire d'Analyse, Géométrie et Applications (LAGA)
	CNRS : UMR7539 – Université Paris XIII - Paris Nord
2 :	Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
	Ecole des Ponts ParisTech
3 :	MATHFI (INRIA Rocquencourt)
	INRIA – Ecole des Ponts ParisTech – Université Paris XII - Paris Est Créteil Val-de-Marne
4 :	Statistics at Purdue
	Purdue University

Domaine

:

Mathématiques/Probabilités

Mots-clés : Power variation – martingale Volterra convolution – covariation – calculus via regularization – Gaussian processes – generalized Stratonovich integral – non-Gaussian processes

inria-00438532, version 1

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Contributeur : Francesco Russo <>

Soumis le : Jeudi 3 Décembre 2009, 20:38:51

Dernière modification le : Mardi 5 Janvier 2010, 15:01:35