# α-time fractional Brownian motion: PDE connections and local times∗

Erkan Nane; Dongsheng Wu; Yimin Xiao

ESAIM: Probability and Statistics (2012)

- Volume: 16, page 1-24
- ISSN: 1292-8100

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topNane, Erkan, Wu, Dongsheng, and Xiao, Yimin. "α-time fractional Brownian motion: PDE connections and local times∗." ESAIM: Probability and Statistics 16 (2012): 1-24. <http://eudml.org/doc/222484>.

@article{Nane2012,

abstract = {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. },

author = {Nane, Erkan, Wu, Dongsheng, Xiao, Yimin},

journal = {ESAIM: Probability and Statistics},

keywords = {Fractional Brownian motion; strictlyα-stable Lévy process; α-time Brownian motion; α-time fractional Brownian motion; partial differential equation; local time; Hölder condition.; fractional Brownian motion; strictly -stable Lévy process; -time Brownian motion; -time fractional Brownian motion; PDE; Hölder condition},

language = {eng},

month = {3},

pages = {1-24},

publisher = {EDP Sciences},

title = {α-time fractional Brownian motion: PDE connections and local times∗},

url = {http://eudml.org/doc/222484},

volume = {16},

year = {2012},

}

TY - JOUR

AU - Nane, Erkan

AU - Wu, Dongsheng

AU - Xiao, Yimin

TI - α-time fractional Brownian motion: PDE connections and local times∗

JO - ESAIM: Probability and Statistics

DA - 2012/3//

PB - EDP Sciences

VL - 16

SP - 1

EP - 24

AB - 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.

LA - eng

KW - Fractional Brownian motion; strictlyα-stable Lévy process; α-time Brownian motion; α-time fractional Brownian motion; partial differential equation; local time; Hölder condition.; fractional Brownian motion; strictly -stable Lévy process; -time Brownian motion; -time fractional Brownian motion; PDE; Hölder condition

UR - http://eudml.org/doc/222484

ER -

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