Gaussian and non-Gaussian processes of zero power variation

Details Gaussian and non-Gaussian processes of zero power variation Gaussian and non-Gaussian processes of zero power variation

2009-12-04

arxiv.org/abs/0912.0782v1

Mathematics

Probability

Scientific paper

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 It\^o's formula is proved to hold for all functions of class $C^{6}$.

Affiliated with

Also associated with

No associations

Rating

Gaussian and non-Gaussian processes of zero power variation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Gaussian and non-Gaussian processes of zero power variation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Gaussian and non-Gaussian processes of zero power variation will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-420239