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Journal Articles ESAIM: Probability and Statistics Year : 2015

Gaussian and non-Gaussian processes of zero power variation

Frederi Viens
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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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Dates and versions

inria-00438532 , version 1 (03-12-2009)
inria-00438532 , version 2 (28-05-2012)



Francesco Russo, Frederi Viens. Gaussian and non-Gaussian processes of zero power variation. ESAIM: Probability and Statistics, 2015, 19 (9), pp.414-439. ⟨10.1051/ps/2014031⟩. ⟨inria-00438532v2⟩
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