A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

Abstract

We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger or equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H = 1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion.

A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index $H<1/4$

“…Other central limit theorems have been obtained for sums of integer powers of increments of one-dimensional fBm (see [2,17,14] for instance). The most closely related result is maybe that of I. Nourdin [13] which shows an Itô-type formula for a two-dimensional fBm with Hurst index H = 1/4 with a 'bracket-term' involving an independent Brownian motion, but the results are of a very different nature than ours (also, they hold precisely for H = 1/4, whereas our results concern the case H < 1/4). Note also the paper by Y. Hu and D. Nualart [7] which gives a Brownian scaling limit for the self-intersection local time for α large enough.…”

Gaussian and non-Gaussian processes of zero power variation

“…We first calculate the second mixed derivative ∂ 2δ2 /∂s∂t, whereδ is the canonical metric of Z, because we must show |µ| (OD) ≤ ε 2α , which is condition (12), and µ (dsdt) = ds dt ∂ 2δ2 /∂s∂t. We have, for 0 ≤ s ≤ t − ε, δ 2 (s, t) = We now prove (26). We can write…”

“…We now prove (26). We can write For A, we use the hypotheses of this proposition: for the last factor in A, we exploit the fact that g is decreasing in t while f is increasing in t; for the other factor in A, use the bound on ∂g/∂t; thus we have We separate the integral in u into two pieces, for u ∈ [0, s − ε] and u ∈ [s − ε, s].…”

“…The behavior at the threshold was worked out for H = 1/4, m = 2 in [28], boasting an exotic correction term with an independent Brownian motion, while the general open problem of Hermite variations with H = 1/ (2m) was settled in [27]. More questions arise, for instance, with a similar result in [26] for H = 1/4, but this time with bidimensional fBm, in which two independent Brownian motions are needed to characterize the exotic correction term. Compared to the above works, our work situates itself by…”

An Itô-type formula for the fractional Brownian motion in Brownian time

“…The formula (1.11) is related to a recent line of research in which, by means of Malliavin calculus, one aims to exhibit change-of-variable formulas in law with a correction term which is an Itô integral with respect to martingale independent of the underlying Gaussian processes. Papers dealing with this problem and which are prior to our work include [4,7,8,9,12,15,16]; however, it is worthwhile noting that all these mentioned references only deal with Gaussian processes, not with iterated processes (which are arguably more difficult to handle).…”