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

Abstract

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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.

How to cite

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Nane, 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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