For 0 < α ≤ 2 and 0 < H < 1, an α-time fractional Brownian motion is an iterated process Z = {Z(t) = W(Y(t)), t ≥ 0} obtained by taking a fractional Brownian motion {W(t), t ∈ ℝ} with Hurst index 0 < H < 1 and replacing the time parameter with a strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when Y is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a d-dimensional α-time fractional Brownian motion X = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ..., Xd(t)), where t ≥ 0 and X1, ..., Xd are independent copies of Z, are investigated. Our methods rely on the strong local nondeterminism of fractional Brownian motion.
Keywords: fractional brownian motion, strictlyα-stable Lévy process, α-time brownian motion, α-time fractional brownian motion, partial differential equation, local time, Hölder condition
@article{PS_2012__16__1_0,
author = {Nane, Erkan and Wu, Dongsheng and Xiao, Yimin},
title = {$\alpha $-time fractional brownian motion: {PDE} connections and local times},
journal = {ESAIM: Probability and Statistics},
pages = {1--24},
year = {2012},
publisher = {EDP Sciences},
volume = {16},
doi = {10.1051/ps/2011103},
mrnumber = {2900521},
zbl = {1278.60074},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2011103/}
}
TY - JOUR AU - Nane, Erkan AU - Wu, Dongsheng AU - Xiao, Yimin TI - $\alpha $-time fractional brownian motion: PDE connections and local times JO - ESAIM: Probability and Statistics PY - 2012 SP - 1 EP - 24 VL - 16 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2011103/ DO - 10.1051/ps/2011103 LA - en ID - PS_2012__16__1_0 ER -
%0 Journal Article %A Nane, Erkan %A Wu, Dongsheng %A Xiao, Yimin %T $\alpha $-time fractional brownian motion: PDE connections and local times %J ESAIM: Probability and Statistics %D 2012 %P 1-24 %V 16 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ps/2011103/ %R 10.1051/ps/2011103 %G en %F PS_2012__16__1_0
Nane, Erkan; Wu, Dongsheng; Xiao, Yimin. $\alpha $-time fractional brownian motion: PDE connections and local times. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 1-24. doi: 10.1051/ps/2011103
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