On the local time of multifractional Brownian motion

On the local time of sub-fractional Brownian motion

“…We will proof only (3.13), the proof of (3.12) is similar. It follows from (25.7) in Geman and Horowitz [13](see also Boufoussi et al [9]) that for any x, y ∈ R, t, t + h ∈ [0, +∞[ and for every even integer m ≥ 2,…”

“…The joint continuity as well as Hölder conditions in both the space and the (time) set variable of the local time of locally nondeterministic (LND) Gaussian process and fields have been studied by Berman [4] and [6], Pitt [20], Kôno [15], Geman and Horowitz [13], and recently by Csörgo, Lin and Shao [11] and [23]. Recently, Boufoussi, Dozzi and Guerbaz [9] and Guerbaz [14] have studied respectively the local time of the multifractional Brownian motion (mBm) and the local time of the filtered white noises. Th multifractional Brownian motion extend the fBm in the sens that its Hurst parameter is not more constant, but a Hölder function of time.…”

Local time and related sample paths of filtered white noises

“…According to Remark 3.4 in [7], the LND property can be used on the whole interval [0, 1] instead of (0, 1). Lemma 3.5.…”

Local properties of a multifractional stable field

“…. , t n j follows from the local nondeterminism property for the multifractional Brownian motion (see [3]). Therefore, for all j = 1, .…”