Path properties of a class of locally asymptotically self similar processes

Abstract

Various paths properties of a stochastic process are obtained under mild conditions which allow for the integrability of the characteristic function of its increments and for the dependence among them. The main assumption is closely related to the notion of local asymptotic self-similarity. New results are obtained for the class of multifractional random processes .

Multistable Processes and Localizability

“…A well-known example is multifractional Brownian motion, where the Hurst exponent h of fractional Brownian motion is replaced by a functional parameter h(t), permitting the Hölder exponent to vary in a prescribed way, see [2,3,5,12] and references therein. Close to time t, the process behaves like index-h(t) fractional Brownian motion, but, nevertheless, this local form may vary considerably with time.…”

Brownian motion with variable drift can be space filling

“…The Hausdorff dimension of the level sets of B (H) (t) is, almost surely, bounded from below by 1−H; see Theorems 4 and 5 in Chapter 18 of [8] (actually, equality holds here, but we will need only the lower bound). The same arguments show that, for an independent Brownian motion B(t), the Hausdorff dimension of any level sets of B(t) − B (H) (t) is also bounded from below by 1 − H almost surely (see also Proposition 5.1 and Section 7.2 in [2]). Moreover, it is well known that, for a standard one-dimensional Brownian motion and any x ∈ R, any set of Hausdorff dimension strictly greater than 1/2 is intersected by the level set {t : B(t) = x} with positive probability.…”

“…• if a n = 3, let f a 1 ,...,a n (0) = f a 1 ,...,a n−1 (2), f a 1 ,...,a n (1) = f a 1 ,...,a n−1 (1).…”

“…. , a n is not 0-reverse, let f a 1 ,...,a n (0) = f a 1 ,...,a m (0); • if a n = 3, let f a 1 ,...,a n (0) = f a 1 ,...,a n−1 (2) and -if a 1 , . .…”

Multiparameter multifractional Brownian motion: Local nondeterminism and joint continuity of the local times

“…Ayache, Cohen and Lévy-Véhel [3] and Herbin [19] studied the covariance structure of mfBm with harmonisable representations. We refer to [14,29] for further information. We refer to [14,29] for further information.…”