Gaussian and non-Gaussian processes of zero power variation and related stochastic calculus

Francesco Russo and Frederi Viens
septembre, 2015
Publication type:
Paper in peer-reviewed journals
Journal:
ESAIM P & S, vol. 19, pp. 414-439
arXiv:
images/icons/icon_arxiv.png 1407.4568
Keywords :
Power variation; martingale Volterra convolution; covariation; calculus via regularization; Gaussian processes. generalized Stratonovich integral, non-Gaussian processes.
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 $C^{6}$.
BibTeX:
@article{Rus-Vie-2015,
    author={Francesco Russo and Frederi Viens },
    title={Gaussian and non-Gaussian processes of zero power variation 
           and related stochastic calculus },
    doi={10.1051/ps/2014031 },
    journal={ESAIM P & S },
    year={2015 },
    month={9},
    volume={19 },
    pages={414--439},
}